diff --git "a/CdFQT4oBgHgl3EQf_DcV/content/tmp_files/load_file.txt" "b/CdFQT4oBgHgl3EQf_DcV/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/CdFQT4oBgHgl3EQf_DcV/content/tmp_files/load_file.txt" @@ -0,0 +1,1325 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf,len=1324 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='13456v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='FL] 31 Jan 2023 ONE DETERMINISTIC-COUNTER AUTOMATA PRINCE MATHEW, VINCENT PENELLE, PRAKASH SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We introduce one deterministic-counter automata (odca), which are one-counter automata where all runs labelled by a given word have the same counter effect, a property we call counter-determinacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' odcas are an extension of visibly one-counter automata - one-counter automata (oca) where the input alphabet determines the actions on the counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' They are a natural way to introduce non-determinism/weights to ocas while maintaining the decidabil- ity of crucial problems, that are undecidable on general ocas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For example, the equivalence problem is decidable for deterministic ocas whereas it is un- decidable for non-deterministic ocas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We consider both non-deterministic and weighted odcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This work shows that the equivalence problem is decidable in polynomial time for weighted odcas over a field and polynomial space for non-deterministic odcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' As a corollary, we get that the regularity problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=', the problem of checking whether an input weighted odca is equivalent to some weighted automaton, is also in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Furthermore, we show that the covering and coverable equivalence problems for uninitialised weighted odcas are decidable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We also introduce a few reachability problems that are of independent interest and show that they are in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' These reachability problems later help in solving the equivalence problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Introduction Visibly pushdown automata (vpda) was introduced by Alur and Madhusudan in 2004 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' They have received a lot of attention as they are a strict subclass of pushdown automata, suitable for program analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' vpdas enjoy tractable decid- able properties, which are undecidable in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The visibly restriction, in essence, is that the stack operations are input-driven, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=', only depends on the letter read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In this paper, we investigate a relaxation in the visibly constraint on one-counter automata (oca): the counter actions are no longer input-driven, but are determin- istic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We could summarise this as: “any run on a given word has a fixed counter effect”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We give a model satisfying this new restriction, which includes all visibly oca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Syntactically, one deterministic-counter automata (odca) contain two parts: (1) Counter structure: This is a deterministic oca without epsilon transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The transitions are deterministic, and the state transitions depend only on the current state, the alphabet, and whether the counter is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' (2) Finite state machine: This machine has finite states and no counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' It can be deterministic, non-deterministic, or weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The transitions of Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' One counter automata, Equivalence, Reachability, Weighted automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 1 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH this machine depend on its current state, current counter structure state, input alphabet, and whether the counter value is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' An odca will be called deterministic, non-deterministic, or weighted depending on the type of the finite state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' One can observe that the class of deter- ministic oca and the class of visibly oca are specific cases of odcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In a visibly oca, the input alphabet determines the counter structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' An odca represents a function that maps words (over a finite alphabet) to a weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The run of a word over an odca determines its accepting weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In the case of weighted odca, the weights come from a field, and in the case of deterministic and non-deterministic odca, the weights come from the boolean semiring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Hence a deterministic or non-deterministic odca represents a language, which is the set of all words whose weight is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A non-deterministic odca can have a succinct representation compared to the deterministic odca recognising the same language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For example, for any k ∈ N, let Lk denote the language {an(b + c)mb(b + c)k | m, n ∈ N and m > n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The non- deterministic odca recognising the language Lk guesses whether a b encountered after reading the string an(b + c)n+1 for some n ∈ N is at the kth position from the end of the string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' An example of a non-deterministic odca that recognises L2 is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The deterministic odca that recognises the same language will have to check whether every b encountered after reading the string an(b + c)n+1 is at the kth position from the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This will require an additional 2k states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Two odcas are equivalent if the functions they represent are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Observe that deterministic real-time ocas are deterministic odcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We also note that deterministic odcas are deterministic real-time ocas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' B¨ohm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' [5] proved that the equivalence of deterministic oca is in non-deterministic log space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We show that a non-deterministic odca is equivalent to an exponentially sized deter- ministic odca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, unlike non-deterministic ocas, the equivalence of two non-deterministic odcas is decidable and can be determined by a PSPACE machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This paper also presents a polynomial time algorithm for deciding the equivalence problem of two weighted odcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If the two odcas are non-equivalent, we output a word (whose length is polynomial in the size of the two odcas) that the two weighted odcas accept with different weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We dedicate Section 4 to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' There exists a polynomial time algorithm that decides if two weighted odcas are equivalent and outputs a word that distinguishes them, if they are non- equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' To solve the equivalence problem for weighted odca, we introduce a few reacha- bility problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' These problems are also of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The complement to vector space (co-VS) reachability problem takes a weighted odca, a vector space, and an initial configuration as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' It asks whether it is possible, starting from the initial configuration, to reach a configuration in the complement of the given vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We develop novel ideas to show that the unary (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' binary) co-VS reachability problem is in P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' NP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let us call a word a witness if the run of the word ‘reaches’ a configuration desired by the reachability problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Through a series of lemmas, we identify two interesting properties of witnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' (1) The witnesses satisfy a small model property - a witness that is longer than a polynomial can be ‘cut’ to get a shorter witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We remove parts of a ONE DETERMINISTIC-COUNTER AUTOMATA 3 q0 q1 q2 q3 q4 q5 ({a}, {0, 1}) ({b, c}, {0}) ({b, c}, {1}) ({b, c}, {0}) ({b}, {0}) ({b, c}, {1}) ({b, c}, {0}) ({b, c}, {0}) ({b, c}, {0}) Finite state machine Counter structure p0 p1 ({b, c}, {1}, −1) ({a}, {0, 1}, +1) ({b}, {0}, 0) ({b, c}, {0, 1}, 0) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The figure shows a non-deterministic odca recognising the language L = {an(b + c)mb(b + c)2 | m, n ∈ N and m > n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let S ⊆ Σ, R ⊆ {0, 1} and T ∈ {−1, 0, +1} are non-empty sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For i, j ∈ N, if a transition from qi to qj is labelled (S, R) and (s, r) ∈ S × R, then there is a transition from qi to qj on reading the symbol s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The current counter value should be 0 if r = 0 and greater than 0 if r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Similarly, if a transition from pi to pj is labelled (S, R, T ) and (s, r, t) ∈ S × R × {T }, then there is a transition from pi to pj on reading the symbol s with counter action t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The current counter value should be 0 if r = 0 and greater than 0 if r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' long run and join the remaining portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The challenge is identifying cuts that preserve the counter actions during the run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' (2) The lexicographically smallest word that witnesses the reachability is of the form uyr1 1 vyr2 2 w where u, v, w, y1 and y2 are ‘small’ words and r1, r2 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The reachability problems, along with the ideas developed in the context of real- time oca by B¨ohm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' [3] (also see [4] [6]), and Valiant, Paterson [21] help us solve the equivalence problem for odcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Next, we consider the regularity problem - the problem of deciding whether a weighted odca is equivalent to some weighted automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In Theorem 36, we show that regularity of odca is decidable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is done by showing the existence of infinitely many equivalence classes by “pumping up” some parts of a run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Next, we look at uninitialised odcas - an odca without initial finite state dis- tribution and initial counter state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We show that the “equivalence” problem for unitialised odcas are in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH Related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Extensive studies have been conducted on weighted automata with weights from semirings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Tzeng [20] gave a polynomial time algorithm to decide the equivalence of two probabilistic automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The result has been extended to weighted automata with weights over a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' On the other hand, the problem is undecidable if the weights are over the semiring (N, min, +) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Unlike the exten- sive literature on weighted automata, the study on weighted versions of pushdown or one-counter machines is limited [11] [12] [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' One of the major bottlenecks is the undecidability of many interesting problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Probabilistic Pushdown Automata (pPDA) is equivalent to probabilistic recur- sive state machines (RSMs) or recursive Markov chains [8] [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' These models have been studied extensively for the analysis and model checking of procedural programs [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' pPDAs can model probabilistic sequential programs with recursive procedure calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' They are also a generalisation of stochastic context-free grammars [1] used in natural language processing, molecular biology, and many variants of one-dimensional random walks [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Kucera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' [16] have looked at model-checking of probabilistic pushdown systems and Br´azdil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' [8] studied temporal proper- ties of probabilistic pushdown automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The equivalence problem of pPDA was examined by Forejt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' [10] and they showed that it is equivalent to the multiplic- ity equivalence of context-free grammars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The decidability of the latter problem is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Kiefer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' [13] show that the equivalence of probabilistic vpda is logspace equivalent to polynomial identity testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The later problem in known to be in coRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The bisimilarity problem of probabilistic vpda (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' probabilistic oca) was shown to be EXPTIME-complete (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PSPACE-complete) by Forejt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' They also proved the decidability of the bisimilarity problem of pPDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Etessami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' [9] show that probabilistic oca and Quasi-Birth-Death processes are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Moving on to the non-weighted models, for non-deterministic pushdown au- tomata the equivalence problem is known to be undecidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' On the other hand, from the seminal result by S´enizergues [17], we know that the equivalence prob- lem for deterministic pushdown automata is decidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The lower bound, though, is primitive recursive [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The equivalence problem for deterministic one-counter automata (with and without ǫ transitions) is decidable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In fact, similar to that of deterministic finite automata, the problem is NL-complete [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The rest of this paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Section 1 contains the basic definitions and some lemmas from linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We also give a formal definition of odca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In Section 2, we look at the special cases of non- deterministic and deterministic odcas and show the decidability of their equiva- lence problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Section 3 analyses a few reachability problems of weighted odcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In Section 4, we prove Theorem 1 and show that the equivalence of weighted odcas is in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Section 5, gives a polynomial time algorithm for the regular- ity problem of weighted odca, and in Section 6, we prove that the covering problem for weighted odca is in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Section 7 gives a short conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Preliminaries 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Basic notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' An alphabet is a non-empty finite set of letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In this paper, we denote the alphabet by Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We use Σ∗ to denote the set of finite length words over Σ, and for all l ∈ N, we use Σ≤l (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Σl) to denote the set of words over Σ having length less than or equal to l (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' exactly equal to l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Given a ONE DETERMINISTIC-COUNTER AUTOMATA 5 word w ∈ Σ∗, we use |w| to denote the length of the word w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We use the notation [i, j] to denote the interval {i, i + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' , j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We say that a word u = a1 · · · ak is a subword of a word w, if w = u0a1u1a2 · · · akuk, where ai ∈ Σ, uj ∈ Σ∗ for all i ∈ [1, k] and j ∈ [0, k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We call u a proper subword of w if u ̸= w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We say that a word u is a prefix of a word w if there exists v ∈ Σ∗ such that w = uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Given a word w = a0 · · · an, we write w[i · · · j] to denote the factor ai · · · aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Given d ∈ N, sign(d) = 0 if d = 0 and is 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A field F = (S, +, ·, 0, 1) is a set S with operations + and and distinguished elements 0 and 1 such that (S, +, 0) and (S, ·, 1) are groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In this paper, we use x, y, z to denote row vectors over a field F, s, t, r to denote elements in a field F and A, B, M to denote matrices over a field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We use U, V to denote vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We recall the following facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 2 ([19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The following are true for a field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' (1) For any set X of n vectors in Fm with n > m, there exists a vector x ∈ X that is a linear combination of the other vectors in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' (2) Given a set B of n vectors in Fm and a vector x ∈ Fm, we can check if x is a linear combination of vectors in B in time polynomial in m and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' (3) Let k, r ∈ N and M ∈ Fk×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The matrix Mr can be computed in time polynomial in k and log(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ The following properties of vector spaces are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let V be a vector space, k ∈ N and for all r ∈ [0, k] zr ∈ Fk and Mr ∈ Fk×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Then, there exists an i ∈ [1, k] such that the following conditions are true: (1) zi is a linear combination of z0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' zi−1, and (2) if ziMi /∈ V, then there exists j < i such that zjMi /∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let k ∈ N, r ∈ [0, k], zr ∈ Fk, Mr ∈ Fk×k be matrices over F and V be a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the set {z0, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' , zk} of k +1 vectors of dimension k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' It follows from Lemma 2 that there are at most k independent vectors of dimension k, and hence not all elements of the set can be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let i ∈ [1, k] be such that zi is a linear combination of z0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' zi−1 and ziMi /∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let us assume for contradiction that zjMi ∈ V for all j ∈ [0, i − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since zi is a linear combination on z0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' zi−1, there exists s0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' si−1 ∈ F such that zi = s0 · z0 + s1 · z1 + · · · + si−1 · zi−1 Since ziMi = �i−1 j=0 sj · zjMi and V is closed under linear combinations, we get that ziMi ∈ V contradicting our initial assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let V be a vector space, k ∈ N and for all r ∈ [0, k2] Ar, Mr, Br ∈ Fk×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Then, there exists an i ∈ [1, k2] such that for all x ∈ Fk the following conditions are true: (1) Mi is a linear combination of M0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' , Mi−1, and (2) if xAiMiBi /∈ V, then there exists a j < i such that xAiMjBi /∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 6 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let Ar, Mr, Br ∈ Fk×k for r ∈ [0, k2], be matrices over F and V a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the set {M0, M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' , Mk2} of k2 + 1 matrices of dimension k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' It follows from Lemma 2 that there are at most k2 independent vectors of dimension k2, and hence not all elements of this set can be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let i ∈ [1, k2] be such that Mi is a linear combination of M0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' , Mi−1 and xAiMiBi /∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since Mi is dependent on M0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' , Mi−1, we prove that there exists j < i such that xAiMjBi /∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let us assume for contradiction that this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since Mi is a linear combination on M0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Mi−1, there exists s0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' si−1 ∈ F such that Mi = s0 · M0 + s1 · M1 + · · · + si−1 · Mi−1 Since xAiMjBi ∈ V for all j ∈ [0, i−1] we get that xAiMiBi = �i−1 j=0 sj ·xAiMjBi ∈ V, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let k ∈ N, A ∈ Fk×k and V ⊆ Fk be a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Then the following set is a vector space, U = {y ∈ Fk | yA ∈ V}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' To prove that U is a vector space, it suffices to show that it is closed under vector addition and scalar multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' First, we prove that U is closed under vector addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z1, z2 ∈ U be two vectors, since z1A, z2A ∈ V, (z1 + z2)A = z1A+z2A ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, z1 +z2 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now we prove that U is closed under scalar multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For any vector z1 ∈ U, we know that z1A ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since V is a vector space, for any scalar r ∈ F, (r · z1)A ∈ V, and therefore r · z1 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ In particular, the above lemma holds for the vector space {0 ∈ Fk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' One deterministic-counter automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A one deterministic-counter au- tomata (odca) consists of two parts, a finite state machine, which is a weighted automaton over a semiring, and a counter structure, which is a deterministic oca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' An odca is defined as follows: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A one deterministic-counter automata ( odca), A over an alphabet Σ and a semiring S is as defined below: A = (C, δ, p0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q, λ, ∆, η) C is a non-empty finite set of counter states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' δ : C ×Σ×{0, 1} → C ×{−1, 0, +1} is the deterministic counter transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' p0 ∈ C is the start state for counter transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q is a non-empty finite set of states of the finite state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We assume |C| = |Q| and use K to denote |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' λ ∈ SK is the initial distribution where the ith component of λ indicates the initial weight on state qi ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' ∆ : C × Σ × {0, 1} → SK×K gives the transition matrix for all p ∈ C, a ∈ Σ and d ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The component in the ith row and jth column of ∆(p, a, d) denotes the weight on the transition from state qi ∈ Q to state qj ∈ Q on reading symbol a from counter state p and counter value n with sign(n) = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' ONE DETERMINISTIC-COUNTER AUTOMATA 7 η ∈ SK is the final distribution, where the ith component of η indicates the output weight on state qi ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A configuration c of an odca is of the form (xc, pc, nc) ∈ SK × C × N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The configuration (λ, p0, 0) is the initial configuration of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A transition is a tuple τ = (ιτ, dτ, aτ, ceτ, Aτ, θτ) where ιτ, θτ ∈ C are counter states, dτ ∈ {0, 1} is to denote whether the current counter value is zero or not, aτ ∈ Σ, ceτ ∈ {−1, 0, 1} is the counter-effect, Aτ ∈ SK×K such that ∆(ιτ, aτ, dτ) = Aτ, and δ(ciτ, aτ, dτ) = θτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Given a transition τ and a configuration c, we denote the application of τ to c as τ(c) = (xcAτ , θτ, nc + ceτ) if pc = ιτ and dτ = 0 if and only if nc = 0, and is undefined otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that the counter values always stay positive, implying that we cannot perform a decrement operation on the counter from a configuration with a counter value of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Given a sequence of transitions T = τ0 · · · τℓ−1, we denote word(T ) = aτ0 · · · aτℓ−1 the word labelling it, we(T ) = Aτ0 · · · Aτℓ−1 its weight-effect matrix, and ce(T ) = ceτ0 + · · · + ceτℓ−1 its counter-effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For all 0 ≤ i < j ≤ |ℓ − 1|, we use Ti···j to denote the sequence of transitions τi · · · τj and |T | to denote ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We call a sequence of transition T = τ0 · · · τℓ floating if for all i ∈ [0, ℓ−1] dτi = 1 and non-floating otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We denote mince(T ) = mini(ce(τ0 · · · τi)) the minimal effect of its prefixes and call it its decrease and maxce(T ) = maxi(ce(τ0 · · · τi)) is the maximal effect of its prefixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We say that the sequence of transitions T is valid if for every i ∈ [0, ℓ − 2], θτi = ιτi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We will only consider valid sequences of transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A run π is an alternate sequence of configurations and transitions denoted as π = c0τ0c1 · · · τℓ−1cℓ such that for every i, ci+1 = τi(ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Given a sequence of transition T and a configuration c, we denote T (c) the run obtained by applying T to c sequentially (if it is defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The word labelling it, its length, weight effect, and counter-effect are those of its underlying sequence of transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Observe that, for a valid floating sequence of transitions, T (c) is defined if and only if nc > − mince(T ), and for a valid non-floating sequence of transitions, T (c) is defined if and only if nc = − mince(T ) and for every i, dτi = 0 if and only if ce(τ0 · · · τi−1) = mince(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In particular, observe that if a valid floating sequence of transition T is applicable to a configuration (xc, pc, nc), then for every n′ ≥ nc and vector x′ ∈ SK, it is applicable to (x′, pc, n′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For any word w, there is at most one run labelled by w starting from a given configuration c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We denote this run π(w, c0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A run π(w, c0) = c0τ0c1 · · · τℓ−1cℓ is also represented as c0 w −→ cℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We use the notation c0 →∗ cℓ to denote the existence of some word w such that c0 w −→ cℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The counter effect of a word w on a floating run c0 w −→ cℓ is ncℓ − nc0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The weight with which a word w is accepted by A along the run c0 w −→ cℓ is denoted by fA(w, c0) = λwe(π(w, c0))η⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We use the notation fA(w) to denote fA(w, (λ, p0, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let A and B be two odcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the configurations c of A and d of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We say that c ≡l d if and only if for all w ∈ Σ≤l, fA(w, c) = fB(w, d) otherwise c ̸≡l d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We say that the configurations c and d are equivalent if and only if c ≡l d for all l ∈ N and we denote this by c ≡ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We say that A and B are equivalent if for all w ∈ Σ∗, fA(w) = fB(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If the odca is defined over a semiring which is also a field, then we call the model a weighted odca, and if it is the boolean semiring then we call it a non- deterministic/deterministic odca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that the equivalence problem of odca 8 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH defined over an arbitrary semiring is undecidable because of the undecidability of equivalence of weighted automata over semirings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The class of weighted odcas includes deterministic oca, visibly weighted oca, and deterministic weighted oca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We have to bring in known examples from literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Have others considered weighted oca?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Does our model capture it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Also, note that the δ need not be a function and A is always a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In that case, the control states are non- deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Like in the previous section, we can determinise it (with an expo- nential blow-up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The question is, Is equivalence checking in this model (as well as the weighted case) in PTIME?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Given a weighted odca A over the alphabet Σ and a field F, we define it’s M-unfolding weighted automata AM as a finite state weighted automaton that recognises the same function as A for all runs where the counter value does not exceed M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A formal definition in given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Definition 7 (M-unfolding weighted automata).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let A = (C, δ, p0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q, λ, ∆, η) be a weighted odca over the alphabet Σ and a field F, let K = |Q| = |C|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For a given M ∈ N, we define an M-unfolding weighted automata AM of A as follows, AM = (C′, δ′, p′ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q′, λ′, ∆′, η′ F ) where, C′ = C × [0, M] is the finite set of counter states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' δ′ : C′×Σ → C′ is the deterministic counter transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let p, q ∈ C, m ∈ N, a ∈ Σ and d ∈ {−1, 0, +1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' δ′((p, m), a) = (q, m + d), if δ(p, a, sign(m)) = (q, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' p′ 0 = (p0, 0) is the initial counter state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q′ = Q × [0, M] is the finite set of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' λ′ ∈ F|Q′| is the initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' λ′[i] = � λ[i], if i < K 0, otherwise ∆′ : C′ × Σ → F|Q|′×|Q′| gives the transition matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For i, j ∈ |Q′|, p ∈ C, m ∈ N and a ∈ Σ, ∆′((p, m), a)[i][j] = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∆(p, a, 0)[i][j], if i, j < K ∆(p, a, 1)[i mod K][j mod K], if i K = j K 0, otherwise η′ F ∈ F|Q′| is the final distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' η′ F [i] = η[i mod K] An uninitialised weighted odca A is a weighted odca without an initial counter state and initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Formally, A = (C, δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q, ∆, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Given an uninitialised weighted odca A and an initial configuration c0 = (x, p, 0), we define the weighted odca A⟨c0⟩ = (C, δ, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q, x, ∆, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Weighted automata (WA) is a restricted form of an odca where the counter value is fixed at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The above notions of transitions, runs, acceptance, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' are used for WA also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We also use the classical notion and represent weighted automata as A = (Q, λ, ∆, η), without counter states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' ONE DETERMINISTIC-COUNTER AUTOMATA 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Nondeterministic / deterministic odca A deterministic/non-deterministic odca A is an odca over the boolean semiring S = ({0, 1}, ∨, ∧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The language recognised by A is given by L(A) = {w | fA(w) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We say an odca A = (C, δ, p0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q, λ, ∆, η) is deterministic if for every transition sequence T = τ0 · · · τℓ−1, the vector λwe(T ) contains exactly one 1 and non-deterministic otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The following theorem is ‘analogous’ to the case of finite automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The idea is a simple subset construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For every language recognised by a non-deterministic odca, there is a deterministic odca of at most exponential size that recognise it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let A = (C, δ, p0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q, λ, ∆, η) be a non-deterministic odca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Given a vector x ∈ Sk for some k ∈ N, we define the function IsDet: Sk → {true, false} as follows: IsDet(x) = � true, if ∃i < k s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='t x[i] = 1 and ∀j ̸= i, x[i] = 0 false, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Given a transition matrix A corresponding to the states Q, we define its determin- isation det(A) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' There are rows and columns corresponding to each set in 2Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For any qi ∈ Q, let M(qi, A) = {qj | A[i][j] = 1} be the set of all states in the row of qi whose entries are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' With the notation that det(A)[s][s′] corresponds to the entry of the cell corresponding to the sets s, s′ ∈ 2Q, we let det(A)[s][s′] = 1 if and only if s′ = � q∈s M(qi, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We claim that Adet = (C, δ, p0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q, λ, ∆′, η′), with η′ such that for any S ∈ 2Q,η′[S] = � s∈S η[s] and for all p ∈ C, a ∈ Σ and d ∈ {0, 1}, ∆′(p, a, d) = det(∆(p, a, d)) is such that it is deterministic and L(A) = L(Adet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For this, for any sequence of operations T = τ0 · · · τℓ−1, let vT , v′ T be the vectors corresponding to λwe(T ) in A and Adet respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Then we have IsDet(v′ T ) = 1 and for any S ∈ 2Q, v′ T [S] = 1 if and only if for all qi ∈ S, vT [i] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ The equivalence of deterministic odcas can be decided in non-deterministic log space [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From the above theorem, it follows that a PSPACE machine can decide on the equivalence of non-deterministic odcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Equivalence of non-deterministic odca is in PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' As a corollary, we get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The emptiness and the universality problems of non-deterministic odca are in PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Reachability problems in weighted odca In this section, we examine two reachability problems of weighted odcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In the subsequent section, we develop the techniques that play a key role in proving the equivalence of weighted odca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A weighted odca A = (C, δ, p0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q, λ, ∆, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Without loss of generality, assume |C| = |Q| and denote |Q| by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We use V ⊆ FK to denote a vector space and V = FK \\ V to denote the set complement of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let S ⊆ C be a subset of the set of counter states, X ⊆ N a set of counter values and w ∈ Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The notation c w −→ V × S × X denotes the run c w −→ d where d ∈ V × S × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We call z ∈ Σ∗ a reachability witness for (c, V, S, X) if c z−→ V × S × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Moreover, we say z is a 10 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH minimal reachability witness for (c, V, S, X) if c z−→ V × S × X and for all u ∈ Σ∗ with c u−→ V × S × X, |u| ≥ |z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We use c ∗−→ V × S × X to denote that there exists a word u ∈ Σ∗ such that c u−→ V × S × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We assume that the vector space V ⊆ FK will be provided by giving a suitable basis for V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' (1) co-VS reachability problem: Input: a weighted odca A, an initial configuration c, a vector space V, set of counter states S and counter value m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Output: Yes, if there exists a run c ∗−→ V × S × {m} in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' No, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' (2) co-VS coverability problem: Input: a weighted odca A, an initial configuration c, a vector space V, and set of counter states S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Output: Yes, if there exists a run c ∗−→ V × S × N in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' No, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that in the second problem, the counter value of the final configuration is not part of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We consider the cases where the counter values of the initial configuration and the final counter value, if part of the input, are given in unary or in binary notation separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that the size of the unary representation is exponentially larger than the binary representation for the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' First, we look at the particular case of co-VS reachability problem for weighted automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that for weighted automata, the counter value is always zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Given a weighted automata B, with k states, an initial configuration ¯c, a vector space U ⊆ Fk and a set of counter states S, the co-VS reachability problem asks whether there exists a run ¯c ∗−→ U × S × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' There is a polynomial time algorithm that decides the co-VS reach- ability problem for weighted automata and outputs a minimal reachability witness if it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Tzeng [20] gives a polynomial time algorithm for the equivalence of two probabilistic automata by reducing the problem to the co-VS reachability problem where V = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The same algorithm can be modified to solve the general co-VS reachability problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ The following lemma will help us break down both the reachability problems into smaller sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z ∈ Σ∗ be a minimal reachability witness for (c, V, S, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider arbitrary z1z2 ∈ Σ∗ such that z = z1z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let d, e be configurations such that c z1 −→ d z2 −→ e and A ∈ FK×K be such that xdA = xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Then z1 is a minimal reachability witness for (c, U, {pd}, {nd}), where U = {y ∈ FK | yA ∈ V}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z ∈ Σ∗ be a minimal reachability witness for (c, V, S, X), d, e be config- urations such that c z1 −→ d z2 −→ e where z1, z2 ∈ Σ∗ with z = z1z2 and A ∈ FK×K be such that xdA = xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let U = {y ∈ FK | yA ∈ V}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Assume for contradic- tion that there exists z′ 1 ∈ Σ∗ smaller than z1 and c z′ 1 −→ f for some configuration f ∈ U × {pd} × {nd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that for all y ∈ U, the vector yA ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since nf = nd and pf = pd, the run c z′ 1 −→ f z2 −→ V × {pe} × {ne} is a valid run and the word z′ 1z2 contradicts the minimality of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ ONE DETERMINISTIC-COUNTER AUTOMATA 11 The following subsection shows that the unary version of co-VS reachability and coverability are in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In the subsection after, we show that binary version of both problems are in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Unary reachability in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In this subsection, we show that both the reacha- bility problems of weighted odca are solvable in polynomial time when the counter values are given in unary representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Unary co-VS reachability and co-VS coverability problems are de- cidable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The theorem is proved by showing a small model property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=', the length of a minimal witness of reachability is bounded by a polynomial in the number of states K and the input counter value(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is proved by showing that the maximum and minimum counter values encountered during the run of a minimal reachability witness do not exceed some bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Assume this is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In this case, there are two sub-runs of the run which satisfy the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In the first part, the counter values increases and reaches a maximum counter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In the second part, the counter values decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We show that in such a scenario, we can cut parts from both the sub-runs by maintaining the reachability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is proved in Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now, we prove that if the number of distinct counter values encountered during the run of a minimal reachability witness is polynomially bounded, then we can bound the length of that witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z ∈ Σ∗ be a minimal reachability witness for (c, V, S, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If the number of distinct counter values encountered during the run c z−→ V × S × X is t, then |z| ≤ K2 · t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let c = c1 and T (c1) = c1τ1c2 · · · τh−1ch be the run on word z from c1 and T the corresponding sequence of transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let t be the number of distinct counter values encountered during this run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now assume for contradiction that h > K2 · t, then by Pigeon-hole principle, there are K + 1 many configurations ci0, ci1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' , ciK with the same counter state and counter value during this run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let Aj denote the matrix such that xcij Aj = xch for all j ∈ [0, K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 4 we get that there exists r ≤ K, and t ∈ [0, r − 1] such that xcit Ar ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the sequence of transitions T ′ = τ1···itτr···ℓ−1 and v = word(T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The run π(v, c1) = T ′(c1) is a valid run since nct = ncr and pct = pcr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is a shorter run than π(z, c1) and c1 v−→ V × S × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is a contradiction since z was assumed to be minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ It now suffices to show that the number of distinct counter values encountered during the run of a minimal witness is polynomially bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We first show that if the run of a minimal reachability witness of (c, V, S, {m}) is a floating run, then the maximum and minimum counter values encountered during this run are bounded by a polynomial in K and the initial and final counter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z ∈ Σ∗ be a minimal reachability witness for (c, V, S, {m}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If c z−→ V × S × {m} is a floating run, then the maximum counter value during this run is less than max(nc, m) + K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z ∈ Σ∗ be a reachability witness for (c, V, S, {m}) and c z−→ V ×S ×{m} be a floating run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let f ∈ V × S × {m}, such that c z−→ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We prove that the 12 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH vj d c e gi ui g′ i wi gj g′ j word length nc1 + M nc1 ngi ngj counter value Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The figure shows the floating run from a configuration c to e such that xe ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The configurations gi and gj (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' g′ i and g′ j) are where the counter values ngi and ngj are encountered for the last (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' first) time before (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' after) reaching counter value nc + M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Also, pgi = pg′ i = pgj = pg′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The dashed line denotes the part of the run which can be removed to get a shorter reachability witness for (c, U, {pe}, {ne}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' maximum counter value encountered during this run are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let us assume that max(nc, m) = nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The case where max(nc, m) = m can be proven analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Assume for contradiction that the maximum counter value encountered during this run is greater than nc + K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' There exists z1, z2, z3 ∈ Σ∗ such that z = z1z2z3 and configurations d, e such that the run on z from c can be written as follows: c z1 −→ d z2 −→ e z3 −→ f where ne = nc and nd = nc + maxce(π(z, c)) (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let M ∈ FK×K such that xf = xeM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 5 we know that the set U = {y ∈ FK | yM ∈ V} is a vector space and hence the vector xe ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 12 we know that z1z2 is a minimal reachability witness for (c, U, {pe}, {ne}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We contradict the minimality of z1z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let c1 = c and T (c1) = c1τ1c2 · · · τℓ−1cℓ denote the run on word z1z2 from the configuration c1 and T the corresponding sequence of transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let M = maxce(π(z, c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that M = nd − nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For any i ∈ [0, M], we denote by li and di the indices such that the counter value nc1 + i is encountered for the last (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' first) time before (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' after) reaching counter value nc1 + M in T (c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' That is, ce(T1···li−1) = ce(T1···di−1) = i, and for any j ∈ [li, di − 2], ceT1···j > i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We call gi = T1···li−1(c1) and g′ i = T1···di−1(c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let r = K2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since M > K4, by Pigeonhole principle, there exists set of indices X = {i1, i2, · · · , ir} ⊆ [0, M] such that for any k < r, we have ik < ir and for all h, j ∈ X pgh = pg′ h = pgj = pg′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For all j ∈ X, let uj, vj, wj be words such that c1 uj −→ gj vj −→ g′ j wj −−→ g′ 1 as depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For all j ∈ X, let matrix Aj and Bj be such that xg′ j = xgjAj and xg′ 1 = xg′ jBj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We know that for all j ∈ X, xgjAjBj ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now we list the matrices in the following sequence ONE DETERMINISTIC-COUNTER AUTOMATA 13 Air, Air−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' , Ai1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 4, it follows that, there exists h, j ∈ X with h < j such that xghAjBh ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the sequence of transitions T ′ = τ1···lh−1τlj···dj−1τdh···ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The word uhvjwh = word(T ′) is a proper subword of z1z2 and the run π(uhvjwh, c1) = T ′(c1) is a valid floating run shorter than π(w, c1) and c1 uhvjwh −−−−−→ e′ such that e′ ∈ U × {pe} × {ne}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This contradicts the minimality of z1z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ Now, we prove that for any run (need not necessarily be a floating run) of a mini- mal reachability witness z for (c, V, S, m), the maximum counter value encountered during the run c z−→ V ×S×{m} is bounded by a polynomial in the number of states of the machine and the initial and final counter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is achieved by applying Lemma 15 multiple times on the run of the minimal witness (refer Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' e1 e2 e3 c e4 d word length counter value Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The figure shows a run from configuration c to d such that xd ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Configurations e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' , e4 denote the configurations where counter value zero is encountered during the run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The dashed lines denote the portions that can be removed to get a shorter reachability witness for (c, V, {pd}, {nd}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If z ∈ Σ∗ is a minimal reachability witness for (c, V, S, {m}) then the maximum counter value encountered during the run c z−→ V × S × {m} is less than max(nc, m) + K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z ∈ Σ∗ be a minimal reachability witness for (c, V, S, {m}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the run of word z from c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let d ∈ V × S × {m} such that c z−→ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Assume for contradiction that the maximum counter value encountered during the run c z−→ d is greater than max(nc, m) + K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let e1, e2, · · · , et be all the configurations in this run such that nei = 0 for all i ∈ [1, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' There exists words u1, u2, · · · , ut+1 ∈ Σ∗ such that z = u1u2 · · · ut+1 and c u1 −→ e1 u2 −→ e2 u3 −→ · · · ut −→ et ut+1 −−−→ d Note that c u1 −→ e1, et ut+1 −−−→ d and ei ui+1 −−−→ ei+1 for all i ∈ [1, t − 1] are floating runs (refer Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 14 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH We show that the counter values are bounded during each of these floating runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' First, we consider the floating run c u1 −→ e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let A ∈ FK×K be such that xd = xe1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 5 we know that the set U = {y ∈ FK | yA ∈ V} is a vector space and hence the vector xe1 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 12, we know that u1 is a minimal reachability witness for (c, U, {pe1}, {0}) and therefore by Lemma 15 we know that the maximum counter value encountered during the run π(u1, c) is less than nc+K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Similarly for the floating run et ut+1 −−−→ d, the maximum counter value is bounded by nd + K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now consider the floating runs ei ui+1 −−−→ ei+1 for all i ∈ [1, t − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Again by applying Lemma 15 we get that the maximum counter value encountered during each of these sub-runs is less than K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, the maximum counter value encountered during the run c z−→ V × S × {m} is less than max(nc, m) + K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ We have shown that the counter values are polynomially bounded during the run of a minimal reachability witness for the co-VS reachability problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Our next objective is to prove an analogous result for the co-VS coverability problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The problem is similar to co-VS reachability, except that now we are not given a final counter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A crucial ingredient in proving this is Lemma 17 where we prove that if the run of a minimal reachability witness z for (c, V, S, N) is a floating run, then the number of distinct counter values encountered during the run c z−→ V × S × N is polynomially bounded in K and nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Using this and the ideas presented earlier for co-VS reachability, we can prove the existence of a polynomial length witness for the co-VS coverability problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If z ∈ Σ∗ is a minimal reachability witness for (c, V, S, N) and c z−→ V × S × {m} is a floating run then |m − nc| ≤ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' word length counter value c1 ci0 ci1 cil cik ciK cℓ Ad Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The figure shows a run from c1 to cℓ such that xcℓ ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The configurations cil and cik are where the counter values ncil and ncik are encountered for the last time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Also pcil = pcik .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The dashed lines denote a part that can be removed to get a shorter reachability witness for (c, V, {pcℓ}, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z ∈ Σ∗ be a minimal reachability witness for (c, V, S, N) and c z−→ V × S × {m} is a floating run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Assume for contradiction that m > nc + K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The ONE DETERMINISTIC-COUNTER AUTOMATA 15 case where nc > m + K2 can be proven analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let c1 = c and π(z, c1) = c1τ1c2 · · · τℓ−1cℓ be such that m = ncℓ > nc1 + K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the sequence of transitions T = τ0τ1 · · · τℓ−1 in π(z, c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since there are only K counter states, by Pigeon-hole principle, there exists a strictly increasing sequence I = 0 < i0 < i1 < · · < iK ≤ ℓ such that for all j, j′ ∈ I (Condition 1) pcj = pcj′ and (Condition 2) if j < j′ then ncj < ncj′ and for all d ∈ [j + 1, j′ − 1], ncj < ncd < ncj′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the set of configurations ci0, ci1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' , ciK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For any j ∈ [0, K], let Aj denote the matrix such that xcij Aj = xcℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since xcidAd ∈ V for all d ∈ [0, K], from Lemma 3 we get that there exists l, k ∈ [0, K] with l < k such that xcilAk ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider a configuration e = (x, p, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If π(u, e) is a valid floating run with mince(π(u, e)) > 0, then for all m ∈ N and y ∈ FK, π(u, (y, p, m)) is a valid run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the sequence of transitions T ′ = τik···ℓ−1 and let u = word(T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Because of Condition 2, mince(π(u, cik)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore the run T ′′(c1) where T ′′ = τ1···il−1τik···ℓ−1 is a valid run shorter than π(z, c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This contradicts the minimality of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ Now we show that for any run (need not be floating) of a minimal reachability witness z for (c, V, S, N), the maximum counter value encountered during the run c z−→ V × S × N is bounded by a polynomial in K and the initial counter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If z ∈ Σ∗ is a minimal reachability witness for (c, V, S, N) then the maximum counter value encountered during the run c z−→ V × S × N is less than max(nc, K2) + K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z ∈ Σ∗ be a minimal reachability witness for (c, V, S, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the run of word z from c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let d ∈ V × S × N such that c z−→ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If c z−→ d is a floating run, then by Lemma 17 the maximum counter value encountered during this run will be less than nc + K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now if c z−→ d is not a floating run, then there exists u1, u2 ∈ Σ∗ such that z = u1u2 and c u1 −→ e u2 −→ d where, nei = 0 and e u2 −→ d is a floating run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let A ∈ FK×K be such that xd = xeA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 5, we know that the the set U = {y ∈ FK | yA ∈ V} is a vector space and hence the vector xe ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that for all y ∈ U, the vector yA ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 12, we know that u1 is a minimal reachability witness for (c, U, {pe}, {0}) and therefore by Lemma 16, we know that the maximum counter value encountered during the run π(u1, c) is less than nc + K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now since e u2 −→ d is a floating run and u2 is the minimal such word, from Lemma 17, we get that nd ≤ K2, and by Lemma 15, we know that the maximum counter value encountered during this run is less than K2+K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, we get that the maximum counter value encountered during the run c z−→ d is less than max(nc, K2) + K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ Proof of Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For solving the co-VS reachability problem when the weighted odca A = (C, δ, p0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q, λ, ∆, η) with K = |Q| = |C| states, initial configura- tion c, vector space V, set of counter states S and counter value m are given as inputs, we first consider the max(nc, m) + K4-unfolding weighted automata Amax(nc,m)+K4 = (C′, δ′, p′ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q′, λ′, ∆′, η′ F ) of A as described in Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 16, we know that the maximum counter value encountered during the run of the minimal reachability witness z for (c, V, S, {m}) is less than max(nc, m)+K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We define a vector space U ⊆ F|Q′| as follows: A vector x ∈ F|Q′| is in U if there 16 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH exists y ∈ V such that for all i ∈ [0, K − 1], x[K · m + i] = y[i] and for all n ̸= m and i ∈ [0, K − 1], x[K · n + i] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Given a configuration c of a weighted odca, we define the vector zc ∈ F|Q′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' zc[i] = � xc[i mod K], if i K = nc 0, otherwise Now, consider the configuration ¯c = (zc, (pc, nc)) of Amax(nc,m)+K4 and check whether ¯c ∗−→ U × S × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is a co-VS reachability problem of weighted automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Using Theorem 11, this can be solved in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For solving the co-VS coverability problem when the weighted odca A with K states, an initial configuration c, a vector space V and a set of counter states S are given as inputs, we consider the max(nc, K2) + K4-unfolding weighted automata Amax(nc,K2)+K4 = (C′, δ′, p′ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q′, λ′, ∆′, η′ F ) of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 18, we know that the maximum counter value encountered during the run of a minimal reachability witness z for (c, V, S, N) is less than max(nc, K2) + K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We define a vector space U ⊆ F|Q′| as follows: A vector x ∈ F|Q′| is in U if there exists y ∈ V and m ∈ N such that for all i ∈ [0, K−1], x[K·m+i] = y[i] and for all n ̸= m and i ∈ [0, K−1], x[K·n+i] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now, consider the configuration ¯c = (zc, (pc, nc)) of Amax(nc,K2)+K4 and check whether ¯c ∗−→ U × S × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is a co-VS reachability problem of a weighted automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Theorem 11, we know that this can be solved in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Binary reachability in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the case where the counter values are specified in binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Theorem 13 can still be applied to get an algorithm whose running time is polynomial in the input counter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since the counter values are represented in binary, their values can be exponentially large compared to their size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, we only get an exponential time algorithm for reachability from Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This section shows that co-VS reachability can be tested in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The technically challenging part of the proof is proved in Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' It shows that the “lexicographically minimal” reachability witness z is of the form uyr1 1 vyr2 2 w, where the length of the words u, y1, y2, v and w are polynomially bounded in K and r1, r2 are polynomial values dependent on K and the input counter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is a polynomial sized representation of the witness (r1, r2 in binary) whose reachability can be verified in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A non-deterministic machine guesses the words u, y1, y2, v, and w and verifies reachability in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Theorem 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Binary co-VS reachability and co-VS coverability problems are in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We aim to show that there is an “encoding” of a minimal reachability witness of polynomial size with respect to the input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The following lemma shows that the length of a minimal reachability witness is bounded by a polynomial in the input counter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that this can be exponential in size with respect to the input size when the counter values are represented in binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' (1) If z is a minimal reachability witness for (c, V, S, {m}) then |z| ≤ K2 · (max(nc, m) + K4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' (2) If z is a minimal reachability witness for (c, V, S, N) then |z| ≤ K2·(max(nc, K2)+ K4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' ONE DETERMINISTIC-COUNTER AUTOMATA 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z ∈ Σ∗ be a minimal reachability witness for (c, V, S, {m}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 16, we know that the maximum counter value encountered during the run c z−→ V × S × {m} is less than max(nc, m) + K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, there are at most max(nc, m) + K4 many distinct counter values encountered during this run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now from Lemma 14 we get that |z| ≤ K2 · (max(nc, m) + K4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z ∈ Σ∗ be a minimal reachability witness for (c, V, S, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 18, we know that the maximum counter value encountered during the run c z−→ V×S×N is less than max(nc, K2)+K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, there are at most max(nc, K2)+K4 many distinct counter values encountered during this run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now from Lemma 14 we get that |z| ≤ K2 · (max(nc, K2) + K4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ We define the counter effect of a word w with respect to a counter state q ∈ C as ce(π(w, c)) where c is any configuration with nc = |w| and pc = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that for any two configuration c, d, ce(π(u, c)) = ce(π(u, d) if nc = nd = |w| and pc = pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' First, we consider the case of the run of a minimal reachability witness from c, which is a floating run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The following lemma is required for the special case where |nc − m| is bounded by a polynomial in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If z ∈ Σ∗ is a minimal reachability witness for (c, V, S, {m}) and c z−→ V × S × {m} is a floating run, then (1) the minimum counter value during this run is greater than min(nc, m)−K4, and (2) |z| ≤ K2 · (|nc − m| + 2K4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z ∈ Σ∗ be a minimal reachability witness for (c, V, S, {m}) and c z−→ V × S × {m} is a floating run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We prove the claims in the lemma one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This case is symmetric to that of Lemma 15 and can be proven analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 15 and Point 1, we get that the counter values encountered during the run c z−→ V × S × {m} lies between max(nc, m) + K4 and min(nc, m) − K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let t = |nc − m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' There are at most t + 2 · K4 distinct counter values during this run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now from Lemma 14 we get that |z| ≤ K2 · (t + 2 · K4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ We assume a total order on the symbols in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Given two words u, v ∈ Σ∗, we say that u precedes v in the lexicographical ordering if |u| < |v| or if |u| = |v| and there exists an i ∈ [0, |u| − 1] such that u[0, i − 1] = v[0, i − 1] and u[i] precedes v[i] in the total ordering assumed on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A word z ∈ Σ∗ is called the lexicographically minimal reachability witness for (c, V, S, {m}), if c z−→ V × S × X and for all u ∈ Σ∗ with c u−→ V × S × X, z precedes u in the lexicographical ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We show that the lexicographically minimal reachability witness z for (c, V, S, {m}) has a special form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' First, we consider the case of floating runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If z ∈ Σ∗ is the lexicographically minimal reachability witness for (c, V, S, {m}) and c z−→ V × S × {m} is a floating run, then there exists u, y, w ∈ Σ∗ and r ∈ N such that z = uyrw and for all configurations dk, k ∈ [0, r], where c uyk −−→ dk the following conditions hold: (1) |u|, |y| ≤ 3K7 and |w| < 6K7, (2) either ndi > ndj for all i, j such that 0 ≤ i < j ≤ r or ndi < ndj for all i, j such that 0 ≤ i < j ≤ r, (3) for all i, j ∈ [0, r], pdi = pdj, and 18 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH (4) r ∈ [0, K2 · |nc − m| + K6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z be the lexicographically minimal reachability witness for (c, V, S, {m}) such that c z−→ V × S × {m} is a floating run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let t = |nc − m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If t ≤ K4, then from Lemma 21, Item 2, we get that |z| ≤ 3K6 and the claim is trivially true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the case where nc > m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The case where m > nc can be proven analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let us assume t > K4 and let d ∈ Z be such that d = −t + K4 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let c = c1 and T (c1) = c1τ1c2 · · · τℓ−1cℓ denote the run on word z from the configuration c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For any i ∈ [0, K4], we denote by li the index such that the counter value nc1 −i is encountered for the first time and ri the index such that the counter value nc1 − i + d is encountered for the last time in T (c1) (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since t > K4, there are at least K4 + 1 pairs of positions (li, ri), i ∈ [0, K4] such that for all i ∈ [0, K4] the factor z[li, ri] has counter effect d with respect to counter state pcli .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that these factors need not be all distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let X = {(li, ri)}i∈[0,K4] be the set containing these pairs of positions and W = {z[l, r] | (l, r) ∈ X} be the set containing the corresponding factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that |X| > K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' z[li, ri] c g ei fi word length nc m nc − i + d nc − i counter value Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The figure shows the floating run from a configuration c to g such that xg ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The configurations cli and cri are where the counter values nc − i and nc − i + d are encountered for the first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' last) time during this run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The dashed line denotes the part of the run due to the factor z[li, ri] and has a counter effect d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' |W| ≤ K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof: Assume for contradiction that |W| > K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let g ∈ V × S × {m} be such that c z−→ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since number of counter states is K, by Pigeon-hole principle there exists Y ⊆ X with |Y | = K2 + 1 such that for all (l, r), (l′, r′) ∈ Y , pcl = pcl′, pcr = pcr′ , and z[l, r] ̸= z[l′, r′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We say (l, r) < (l′, r′) if z[l, r] precedes z[l′, r′] in the lexicographical order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, the elements in Y have an ordering as follows: (l0, r0) < (l1, r1) < · · · < (lK2, rK2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For all i ∈ [0, K2], let ui = z[1, li], xi = z[li, ri], wi = z[ri, ℓ], configurations ei, fi be such that c ui −→ ei xi −→ fi wi −→ g and matrices Ai, Mi, Bi be such that xei = xcAi , xfi = xeiMi , xg = xfiBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We know that for all i ∈ [0, K2], xcAiMiBi ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the sequence of matrices M0, M1, · · · , MK2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 4 we know that there exists an i ∈ [0, K2− ONE DETERMINISTIC-COUNTER AUTOMATA 19 1] and j < i such that xcAiMjBi ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that the word uixjwi precedes z in the lexicographical ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore the run c uixjwi −−−−→ V × S × {m} contradicts minimality of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □Claim:1 Since |W| ≤ K4 and |X| > K4, there exists i, j ∈ [0, K4], with i < j and x ∈ Σ∗ such that (li, ri) ∈ X, (lj, rj) ∈ X and x = w[li, ri] = w[lj, rj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let u1, w1, u2, w2 ∈ Σ∗ such that z = u1xw1 = u2xw2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since u1 ̸= u2, either u1 is a prefix of u2 or u2 a prefix of u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Without loss of generality, let us assume u1 is prefix of u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, there exists v ∈ Σ∗ such that u2 = u1v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let e be a configuration such that c u1 −→ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' |u1|, |v|, |w1| ≤ 3K6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof: Consider the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For any i, j ∈ [0, K4], with i < j, ncli − nclj ≤ K4 + 1 and ncrj − ncri ≤ K4 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore the counter-effect of u2 and w2 can be at most K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' So the counter-effect of v with respect to counter state pe can be at most K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since it is a minimal floating run from Lemma 21, we get that |v| ≤ 3K6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' By similar arguments, the counter-effect of u1 and w1 can be at most K4, and again by Lemma 21, we get that their lengths are at most 3K6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □Claim:2 u1 x w1 u2 x w2 v v v v v v v′ v v v v v v v′ Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The figure shows the factorisation of a word z = u1xw1 = u2xw2, where x is an overlapping factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The factor v is a prefix of x such that u2 = u1v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The word z can be written as u1viv′w2 for some i ∈ N and v′ prefix of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' There exists v′ ∈ Σ∗ and r ∈ [0, K2 · |nc − m| + K6] such that x = vrv′ with |v′| ≤ |v|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof: Let r ∈ N be the largest number such that x is of the form vrv′ for some v′ ∈ Σ∗ (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We know that z = u2xw2 and u2 = u1v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, z = u1vxw2 = u1vvrv′w2 = u1vrvv′w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Furthermore, z = u1xw1 = u1vrv′w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now since u1vrvv′w2 = u1vrv′w1, we get that vv′w2 = v′w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Hence, if |v′| ≥ |v|, then v is a prefix of v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is a contradiction since r was chosen to be the largest number such that x is of the form vrv′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In order to show the bound on the value r, we observe the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We know that the counter effect of the run π(x, e) is d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore from Lemma 21 Point 2, we get that |x| ≤ K2 · (|d| + 2K4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, the value of r is less than or equal to K2 · (|d| + 2K4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □Claim:3 From Claim 3 and Claim 2, we get that |u1v′w1| ≤ 9K6 and z = u1vrv′w1 for some r ∈ [0, K2 · (|d| + 2K4)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that the factor v might start and end in different counter states during the run and, therefore need not always have a negative counter effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' However, we also know that the word vr has a negative counter effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For i ∈ [1, 2K], let gi be the configuration such that e vi −→ gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' By 20 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH Pigeon-hole principle there exists j, k ∈ [1, 2K] with j < K and k − j ≤ K such that pgj = pgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Also, note that the word y = vk−j has a negative counter-effect from the counter state pgj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let r′ = r−j k−j and j′ = (r − j) (mod (k − j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now consider the word z = u1vjyr′vj′v′w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since |u1|, |w1|, |v| ≤ 3K6, j < K and k − j ≤ K, we get that |u1vj| ≤ 3K7, |vj′v′w1| < 6K7, |y| ≤ 3K7 and r′ ∈ [0, K2 · (|d| + 2K4)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If z ∈ Σ∗ is the lexicographically minimal reachability witness for (c, V, S, {m}), then there exists u, y1, v1, v2, v3, y2, w ∈ Σ∗ and r1, r2 ∈ N such that (1) z = uyr1 1 v1v2v3yr2 2 w, (2) |uy1v1v2v3y2w| ≤ 25K7, (3) r1, r2 ≤ max{m, nc} · K2 + K6, (4) π(uyr1 1 v1, c) and π(v3yr2 2 w, d) are floating runs for configuration d where c uyr1 1 v1v2 −−−−−−→ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' , and (5) ce(π(uyr1 1 v1, c)) = ce(π(uyr1 1 v1v2, c)) = −nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z ∈ Σ∗ be the lexicographically minimal reachability witness for (c, V, S, {m}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the run of word z from c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let d ∈ V × S × {m} such that c z−→ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let c = c1 and T (c1) = c1τ1c2 · · · τℓ−1cℓ denote the run on word z from the configuration c1 and T the corresponding sequence of transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let e1 be the first configuration with counter value zero and e2 be the last configuration with counter value zero during this run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z1, z2, z3 ∈ Σ∗ be such that c z1 −→ e1 z2 −→ e2 z3 −→ cℓ and z = z1z2z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Observe that c z1 −→ e1 and e2 z3 −→ cℓ are floating runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 22, we know that there exists u1, u3, v1, v3, y1, y3 ∈ Σ∗ and r1, r3 ∈ N such that z1 = u1yr1 1 v1, z3 = u3yr3 3 v3, |u1|, |u3| ≤ 3K7, |v1|, |v3| ≤ 6K7, |y1|, |y3| ≤ 3K7, r1 ∈ [0, nc · K2 + K6] and r3 ∈ [0, m · K2 + K6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Also, from Lemma 20 we get that |z2| ≤ K6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ We now prove that the binary co-VS reachability and co-VS coverability prob- lems are in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 23 we observe there is a polynomial-size encoding of the lexicographically minimal word (where r1 and r2 are in binary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A non- deterministic machine can guess this encoding and verify the reachability in poly- nomial time since Mr can be computed in log(r) time (see Lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A detailed proof is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof of Theorem 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let us first look at the binary co-VS reachability problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z ∈ Σ∗ be the lexicographically minimal reachability witness for (c, V, S, {m}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the run of word z from c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let d ∈ V×S×{m} such that c z−→ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let c = c1 and T (c1) = c1τ1c2 · · · τℓ−1cℓ denote the run on word z from the configuration c1 and T the corresponding sequence of transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let e1 be the first configuration with counter value zero and e2 be the last configuration with counter value zero during this run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z1, z2, z3 ∈ Σ∗ be such that c z1 −→ e1 z2 −→ e2 z3 −→ cℓ and z = z1z2z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Observe that c z1 −→ e1 and e2 z3 −→ cℓ are floating runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 22, we know that there exists u1, u3, v1, v3, y1, y3 ∈ Σ∗ and r1, r3 ∈ N such that z1 = u1yr1 1 v1, z3 = u3yr3 3 v3, |u1|, |u3| ≤ 3K7, |v1|, |v3| ≤ 6K7, |y1|, |y3| ≤ 3K7, r1 ∈ [0, nc · K2 + K6] and r3 ∈ [0, m · K2 + K6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Also, from Lemma 20 we get that |z2| ≤ K6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Our NP algorithm starts by guessing the words u1, y1, v1, z2, u3, y3, v3, the values r1, r2, and the configurations e1 and e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We first show how to verify if c u1yr1 1 v1 −−−−−→ e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' ONE DETERMINISTIC-COUNTER AUTOMATA 21 The algorithm computes configuration f0 such that c u1 −→ f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now it constructs the matrix My1 and computes the configuration f1 such that f0 y1 −→ f1 and xf1 = xf0My1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 2, we know that (My1)r1 can be computed by repeated powering in time polynomial in log(r1) and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let fr1 be a configuration such that f0 yr1 −−→ fr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 22, we know that pf0 = pfr1 and nfr1 = pf0 − r1 · (nf0 − nf1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since xfr1 = xf0(My1)r1, we can construct the configuration fr1 in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We now verify in polynomial time whether fr1 v1 −→ e1 or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We can verify if e2 u3yr3 3 v3 −−−−−→ d in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The algorithm can also check whether e1 z2 −→ e2 in polynomial time since |z2| ≤ K6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' It finally checks whether d ∈ V × S × {m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Hence the binary co-VS reachability problem is decidable in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' As for the binary co-VS coverability problem, either the run of a minimal witness is a floating run or is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In the former case where the run is floating, from Lemma 17, we know that the difference between the final and initial counter values is at most K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In the latter case where the run is grounded, by Lemma 17, we get that the final value is at most K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In both the cases, the algorithm guesses the final counter value, and the problem is reduced to the co-VS reachability problem, which is in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Hence the binary co-VS coverability problem is decidable in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Equivalence of weighted odca In this section, we give a polynomial time algorithm to check the equivalence of two weighted odcas (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The algorithm returns a minimal distinguishing word if the odcas are non-equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We use the reachability results presented in the previous section to show that the length of a minimal distinguishing word is short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The idea here is to prove that the maximum counter value encountered during the run of a minimal witness is polynomially bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We use this to reduce the equivalence problem to that of weighted automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In the remainder of this section, we fix two weighted odcas A1 and A2 over an alphabet Σ and a field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For i ∈ {1, 2}, Ai = (Ci, δi, p0i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Qi, λi, ∆i, ηi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Without loss of generality assume K = |C1| = |Q1| = |C2| = |Q2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We will reason on the synchronised runs on pairs of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Given two weighted odcas, A1 and A2 and i ∈ N, we denote a configuration pair as hi = ⟨ci, di⟩ where ci is a configuration of A1 and di is a configuration of A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We similarly consider transition pairs of A1 and A2, and consider synchronised runs as the application of a sequence of transition pairs to a configuration pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We fix a minimal word z (also called witness) that distinguishes A1 and A2 and ℓ = |z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Henceforth we will denote by Π = h0τ0h1 · · · τℓ−1hℓ the run pair of z from the initial configuration pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We denote by T = τ0 · · · τℓ−1 the sequence of transition pairs of this run pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' To prove Theorem 1, we use the following lemma, which states that the counter values in Π are bounded by a polynomial poly0(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' There is a polynomial poly0 : N → N such that if two weighted odcas A1 and A2 are not equivalent, then there exists a witness z such that the counter values encountered during the run of z are less than poly0(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 22 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH We use Lemma 24 to show that the length of the witness z is bounded by a polynomial poly1(K) = 2K5poly0(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' There is a polynomial poly1 : N → N such that if two weighted odcas A1 and A2 are not equivalent, then there exists a witness z such that |z| is less than or equal to poly1(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Assume for contradiction that the length of a minimal witness z is greater than poly1(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 24, we know that the counter values encountered during the run Π in less than poly0(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since |z| > poly1(K), by the Pigeonhole principle, we get that there exist indices 0 ≤ i0 < i2 < · · · < i2K ≤ ℓ such that for all configuration pairs hij, j ∈ [1, 2K], ncij = ncij−1 , ndij = ndij−1 , pcij = pcij−1 and pdij = ndij−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For all j ∈ [0, 2K] we define the vector xj ∈ F2K such that xj[r] = xcij [r], if r < K and xdij [r − K], otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We also define the vector η ∈ F2K such that η[r] = η1[r], if r < K and η2[r − K], otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For all j ∈ [0, 2K], let Aj denote the matrix such that xjAj = xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since z is a minimal witness, we know that for all j ∈ [0, 2K], xjAjη⊤ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 4, we get that there exists r, r′ ∈ [0, 2K], with r′ < r such that xr′Arη⊤ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The sequence of transitions τir+1 · · · τℓ can be taken from hi′r since the counter values and counter states are the same for both configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the sequence of transitions T ′ = τ0 · · · τi′ rτir+1 · · · τℓ and let w = word(T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The word w is a shorter witness than z and contradicts its minimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ Lemma 25 helps us to reduce the equivalence problem of weighted odca to that of weighted automata by “simulating” the runs of weighted odcas up to length poly1(K) by two weighted automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The naive algorithm will only give us a PSPACE procedure, but there is a polynomial time procedure to do this, and the proof is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We consider the two weighted odcas A1 and A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 25, we know that the length of the minimal witness z is less than poly1(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let M = poly1(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We construct the M-unfolding weighted automata AM 1 and AM 2 as described in Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' It follows that, A1 is non-equivalent to A2 if and only if there exists a word w ∈ Σ≤M such that fAM 1 (w) ̸= fAM 2 (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Tzeng [20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='4] gives a polynomial time algorithm to output a minimal word that distinguishes two probabilistic automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We conclude the proof by noting that the algorithm can be extended to the case of weighted automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ The rest of this section is dedicated to proving Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We adapt techniques developed by B¨ohm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' [3] for ocas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We start by labelling some configuration pairs as background points (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the case where there is no background point in Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' By reducing the problem to co-VS reachability/coverability we show that the counter values in Π are polynomially bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now consider the case where there is a background point hj in Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We show that the counter values encountered during the run of Π till hj is polynomially bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is shown by Lemma 29 and Lemma 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We conclude by arguing that the length of the run from ONE DETERMINISTIC-COUNTER AUTOMATA 23 hj is polynomially bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' counters poly-bounded � �� � h0τ0h1τ1h2 · · · hj−1τj−1 hj = ⟨cj, dj⟩ � �� � 1st configuration pair in background space counters poly-bounded � �� � τj · · · τℓ−1hℓ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Configuration Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Each pair of configuration h = ⟨c, d⟩ is mapped to a point in the space N × N × (C1 × C2) × FK × FK, henceforth referred to as the configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Here, the first two dimensions represent the two counter val- ues, the third dimension C1 × C2 corresponds to the pair of counter states, and the remaining dimensions represent the weight vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The projection of the configu- ration space onto the first two dimensions is depicted in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We partition the configuration space into three: initial space, belt space, and background space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The size of the initial space and, thickness and number of belts will be polynomially bounded in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This partition is indexed on two polynomials, poly2 : N → N and poly3 : N → N chosen so that all belts are disjoint outside the initial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We use some properties of these partitions to show that the length of a minimal witness is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We assume poly2(K) = 516K21 and poly3(K) = 42K14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The precise polynomials are required in the proofs of Lemma 26 and Lemma 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' initial space: All configuration pairs ⟨c, d⟩ such that nc, nd < poly2(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' belt space: Let α, β ∈ [1, 3K7] be co-prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A belt of slope α β consists of those configuration pairs ⟨c, d⟩ outside the initial space that satisfies |α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='nc − β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='nd| ≤ poly3(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The belt space contains all configuration pairs ⟨c, d⟩ that is inside belts with slope α β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' background space: All remaining configuration pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' N N initial space background space belt space belt space belt space poly2(K) poly3(K) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Projection of configuration space The proof of the following lemma is similar to that of the non-weighted case presented in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If ⟨c, d⟩ and ⟨e, f⟩ are configuration pairs inside two distinct belts and lie outside the initial space, then there is no a ∈ Σ such that ⟨c, d⟩ a−→ ⟨e, f⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 24 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Recall poly2(K) = 516K21 and poly3(K) = 42K14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let B and B′ be two distinct belts with µ being the slope of the belt B and µ′ the slope of the belt B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Hence µ ̸= µ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Without loss of generality, let us assume that µ′ > µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' It suffices to show that for all x > poly2(K), we have µx + poly3(K) + 1 < µ′x − poly3(K) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We know that µ′ − µ ≥ 1 3K7 and x > 516K21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, 516K21 6K7 < (µ′ − µ) · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' =⇒ µx + 86K14 2 < µ′x − 86K14 2 =⇒ µx + 42K14 + K14 < µ′x − 42K14 − K14 =⇒ µx + 42K14 + 1 < µ′x − 42K14 − 1 □ Lemma 26 ensures that the belts are disjoint outside the initial space and that no run can go from one belt to another without passing through the initial space or background space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Belt space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We look at two scenarios in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' First, we show that if a sub-run of the run of a minimal witness enters and exists a belt from the initial space, then the counter values encountered during this sub-run are polynomially bounded in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Secondly, we show that if the run of a minimal witness never enters the background space, then the counter values encountered during this run are polynomially bounded in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is shown by reducing to co-VS reachability of an odca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let Πb = hiτihi+1 · · · τj−1hj be a sub-run of the run of z inside a belt with slope α β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Similar to the technique mentioned in [5], each configuration pair hr, where r ∈ [i, j] can alternatively be presented as ((xcr, xdr), pcr, pdr, lr) where lr denotes a line with slope α β inside the given belt that contains the point (ncr, ndr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let L be the set of all lines with slope α β inside the given belt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that |L| = poly3(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The run Πb is similar to the run of a weighted odca D that has the tuple (pcr, pdr, lr) as the state of the finite state machine and xr ∈ F2K as its weight vector where xr[i] = xcr[i], if i < K and xr[i] = xdr[i − K], otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A formal definition of the odca D is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Definition 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let Ai = (Ci, δi, p0i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Qi, λi, ∆i, ηi) for i ∈ {1, 2}, be the two odcas given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let L be the set of all lines with slope α β inside the given belt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We define the odca D = (C, δ, p0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q, λ, ∆, η), where the initial state p0 and the initial distribution λ are arbitrarily chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' C = C1 × C2 × L is a non-empty finite set of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' δ : C × Σ → C × {−1, 0, +1} is the deterministic counter transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let p1, q1 ∈ C1, p2, q2 ∈ C2, a ∈ Σ and d1, d2 ∈ {−1, 0, +1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let l1, l2 ∈ L and m1, m2 ∈ N, such that the point (m1, m2) lies on the line l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' δ((p1, p2, l1), a) = ((q1, q2, l2), d1), if δ1(p1, a, 1) = (q1, d1) and δ2(p2, a, 1) = (q2, d2) and the point (m1 + d1, m2 + d2) lies on the line l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' It is undefined otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q = Q1 ∪ Q2 is a non-empty finite set of states of the finite state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' ONE DETERMINISTIC-COUNTER AUTOMATA 25 ∆ : C × Σ × {0, 1} → F2K×2K gives the transition matrix for all p ∈ C, a ∈ Σ and d ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For p1 ∈ C1, p2 ∈ C2, l ∈ L, m ∈ N and a ∈ Σ, ∆((p1, p2, l), a)[i][j] = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∆(p1, a, 1)[i][j], if i, j < K ∆(p2, a, 1)[i − K][j − K], if i, j > K 0, otherwise η ∈ F2K is the final distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' η[i] = � η1[i], if i < K η2[i − K], otherwise The sub-run Πb can now be seen as a floating run of a weighted odca D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If the run Π ends inside a belt, then Πb = hiτi · · · τℓ−1hℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In this case, we show that the difference between the counter values of the first and last configuration pairs is smaller than a polynomial in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' There is a polynomial poly : N → N, such that if Πb = hiτi · · · τℓ−1hℓ lies inside a belt, then |ncℓ − nci| ≤ poly(K) and |ndℓ − ndi| ≤ poly(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let Πb = hiτihi+1 · · · τℓ−1hℓ be a sub-run of the run of a minimal wit- ness inside a belt and ends in the belt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' As mentioned in Definition 27, we con- sider this as the run of the weighted odca D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since it is the run of a witness, xjη⊤ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the vector space U = {y ∈ F2K | yη⊤ = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Our problem now reduces to the co-VS coverability problem in machine D and asks whether (xi, (pci, pdi, li), nci) ∗−→ U × {(pcℓ, pdℓ, lℓ)} × N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 20, we know that the length of a minimal reachability witness for ((xi, (pci, pdi, li), nci), U, (pcℓ, pdℓ, lℓ), N) is polynomially bounded in nci and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Hence proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ In the following lemma, we show that if Πb = hiτihi+1 · · · τj−1hj is a sub-run of Π inside a belt and either nci = ncj or ndi = ndj, then the counter values in Πb cannot increase more than a polynomial in K from nci and ndi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' There is a polynomial poly : N → N such that, if Πb = hiτihi+1 · · · τj−1hj is a run inside a belt with nci = ncj or ndi = ndj, then the counter effect of any sub-run of Πb is less than or equal to poly(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let Πb = hiτihi+1 · · · τj−1hj be a sub-run of the run of a minimal witness inside a belt such that nci = ncj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We consider this as the run of the weighted odca D as mentioned in Definition 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since it is the run of a witness, we know that there exists A ∈ F2K×2K such that xjAη⊤ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the vector space U = {y ∈ F2K | yAη⊤ = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Our problem now reduces to the co-VS reachability problem in machine D and asks whether (xi, (pci, pdi, li), nci) ∗−→ U × {(pcj, pdj, lj)} × {nci}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 20, length of a minimal reachability witness for ((xi, (pci, pdi, li), nci), U, (pcj, pdj, lj), {nci}) is bounded by a polynomial in nci and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Hence proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ We have now shown that if the run of a minimal witness does not enter the background space, then the counter values in this run are polynomially bounded in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now we look at the case where the run enters the background space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 26 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Background space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In this subsection, we consider the case where the run of a minimal witness enters the background space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We show that during the run of the minimal witness the counter values of the first configuration pair in the background space and the remaining length of the run is polynomially bounded in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Floating runs of a weighted ODCA are isomorphic to runs of a weighted au- tomaton obtained by ignoring counter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In order to bound the length of the run of a minimal witness in the background space, we introduce the notion of an underlying uninitialised weighted automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Definition 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For l ∈ {1, 2}, the underlying uninitialised weighted automaton of A is the uninitialised weighted automaton U(Al) = (Q′ l, ∆′ l, η′ l), where Q′ l = Cl ×Ql and η′ l ∈ FK2 is the final distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For i < K2, η′ l[i] = ηl[i mod K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The transition matrix is given by ∆′ l : Σ → FK2×K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let a ∈ Σ, d ∈ {−1, 0, +1}, i, j < K2, ∆′ l(a)[i][j] = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ∆l(p i K , a, 1)[i mod K][j mod K], if δl(p i K , a, 1) = (p j K , d) 0 otherwise Note that a configuration of U(Al) is a vector of dimension K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A configuration c of a weighted odca A is said to be k-equivalent to a configuration ¯c of an uninitialised weighted automata B, denoted c ∼k ¯c, if for all w ∈ Σ≤k, fA(w, c) = fB(w, ¯c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We say that c is not k-equivalent to ¯c otherwise and denote this as c ̸∼k ¯c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' As we need to test the equivalence of configurations from A1 and A2, we con- sider the uninitialised weighted automata B, which is a disjoint union of U(A1) and U(A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This gives us a single automaton with which we can compare their configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let i ∈ {1, 2} and c be a configuration of Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For all p ∈ Ci and m < 2K2, we define the sets Wp,m i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The set Wp,m i contains vectors x ∈ FK such that the configuration (x, p, m) is 2K2-equivalent to some configuration of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The set W p,m i is the set FK \\ Wp,m i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Formally, Wp,m i = {x ∈ FK|∃¯c ∈ F2K2, c = (x, p, m) ∼2K2 ¯c} Lemma 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For any i ∈ {1, 2}, p ∈ Ci and m < 2K2, the set Wp,m i is a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' To prove this, it suffices to show that it is closed under vector addition and scalar multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We fix a set Wp,m i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' First, we prove that it is closed under scalar multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For any vector z1 ∈ Wp,m i , we know that there exists a configuration c = (z1, p, m) and ¯c ∈ F2K2 such that c ∼2K2 ¯c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now, for any scalar r ∈ F, the configuration (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='z1, p, m) ∼2K2 r · ¯c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore r · z1 ∈ Wp,m i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now, we show that it is closed under vector addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let z1, z2 ∈ Wp,m i be two vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, there exists configurations c1 = (z1, p, m), c2 = (z2, p, m), ¯c1 ∈ F2K2 and ¯c2 ∈ F2K2, such that c1 ∼2K2 ¯c1 and c2 ∼2K2 ¯c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the configuration c3 = (z1 + z2, p, m), c3 ∼2K2 ¯c1 + ¯c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, z1 + z2 ∈ Wp,m i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ The distance of a configuration c of Ai is the length of a minimal word that takes you from c to a configuration (x, p, m) for some m < 2K2 and p ∈ Ci such that x ∈ W p,m i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We define distAi(c) as: min{|w| | c w −→ (x, p, m) ∃p ∈ Ci, m < 2K2, x ∈ W p,m i } ONE DETERMINISTIC-COUNTER AUTOMATA 27 The notion of distance play a key role in determining which parts of the run of a witness can be pumped out if it is not minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The following lemma is a crucial component in proving the equivalence of weighted odca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' By Lemma 22, the lexicographically minimal reachability witness has a special form and this plays a crucial role in proving Lemma 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let c be a configuration of weighted odca A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If distA(c) < ∞ then, distA(c) = a b nc + d where a, b ∈ [0, 3K7] and |d| < 42K14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Without loss of generality, let us consider the weighted odca A1 and a configuration c of A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let us assume that distA1(c) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This means that c →∗ d with xd ∈ W p,m 1 for some p ∈ C1 and m < 2K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since nd = m, by Lemma 22, we know that there is a word u = u1ur 2u3 (with r ≥ 0) such that that c u−→ d where |u| = distA1(c), |u1u3| ≤ 9K7, |u2| ≤ 3K7 and u2 has a negative counter effect ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let g be the combined counter effect of u1, u3 and α = |u2| ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since |u1u3| ≤ 9K7, we have |g| ≤ 9K7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' distA1(c) = nc − nd − g ℓ |u2| + |u1u3| = αnc − α(nd + g) + |u1u3| � �� � d Since 1 ≤ α ≤ 3K7 it follows that −42K14 < d < 42K14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Hence proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ The polynomials poly1 and poly2 were picked so that the configuration pairs with equal distance always lie in the belt space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, the background space points either have unequal or infinite distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For any configuration pair ⟨c, d⟩, in the background space, either distA1(c) ̸= distA2(d) or distA1(c) = distA2(d) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Assume for contradiction that there is a configuration pair ⟨c, d⟩, in the back- ground space such that distA1(c) = distA2(d) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since distA1(c) = distA2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 32, there exists a1, b1, a2, b2 ∈ [0, 3K7] and d1, d2 < 42K14 such that a1 b1 nc + d1 = distA1(c) = distA1(d) = a2 b2 nd + d2 Therefore | a1 b1 nc − a2 b2 nd| ≤ |d2 −d1| < 42K14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This satisfies the belt condition and is a configuration pair in the belt space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This contradicts our initial assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ The following lemma shows that the length of the run Π in the background space is polynomially bounded in K and the counter values of the first background point in Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The proof is similar to that in [3] and is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If hj = ⟨cj, dj⟩ is the first configuration pair in the background space during Π, then ℓ − j is bounded by a polynomial in ncj, ndj and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let hj = ⟨cj, dj⟩ be the first configuration pair in the background space during the run Π, then from Lemma 33, either distA1(cj) = distA2(dj) = ∞ or distA1(cj) ̸= distA2(dj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We separately consider the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Case-1, distA1(cj) = distA2(dj) = ∞: then we prove that the remaining length of the witness from ⟨cj, dj⟩ is bounded by 2K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Assume for contradiction that this is not the case and cj ∼2K2 dj but cj ̸≡ dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let v ∈ Σ>2K2 be the word which 28 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH N N ⟨c, d⟩ ⟨e, f⟩ Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The figure depicts an α-β repetition inside a belt with slope α β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The configuration pairs ⟨c, d⟩ and ⟨e, f⟩ are α-β-related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=', they line on a line with slope α β , pc = pe and pd = pf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' distinguishes c and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, there exists a prefix of v, u ∈ Σ|v|−2K2, and i = ℓ − 2K2 such that ⟨cj, dj⟩ u−→ ⟨ci, di⟩ and ci ̸≡2K2 di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since v is a minimal witness ci ≡2K2−1 di and ci ̸≡2K2 di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since distA1(cj) = distA2(dj) = ∞, there exists configurations ¯ci and ¯di in the underlying automa- ton B such that ci ∼2K2 ¯ci and di ∼2K2 ¯di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since ci ≡2K2−1 di, it follows that ¯ci ∼2K2−1 ¯di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From the equivalence result of weighted automata, we know that if two configurations of a weighted automata with k states are non-equivalent, then there is a word of length less than k which distinguishes them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, this is sufficient to prove that the underlying weighted automata with ¯ci and ¯di as initial distributions are equivalent, and thus ¯ci ∼2K2 ¯di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This allows us to deduce that ci ≡2K2 di, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, the remaining length of the witness from ⟨cj, dj⟩ is bounded by 2K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Case-2, distA1(cj) ̸= distA2(dj): Without loss of generality, we suppose distA1(cj) > distA2(dj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' By definition of distA2, there exists u ∈ ΣdistA2 (dj), i > j and a configuration ¯c of the underlying automaton B such that cj u−→ ci, dj u−→ di, ci ∼2K2 ¯ci and di ̸∼2K2 ¯ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore ci ̸≡2K2 di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' By definition, there exists v ∈ Σ≤2K2 such that fA1(v, ci) ̸= fA2(v, di) and hence fA1(uv, cj) ̸= fA2(uv, dj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' As uv ∈ ΣdistA2(dj)+2K2, we get that cj ̸≡distA2 (dj)+2K2 dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, there is w ∈ Σ≤min{distA1 (cj),distA2 (dj)}+2K2 that distinguishes cj and dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ Let α, β ∈ [1, 3K7] be co-prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We say configuration pairs ⟨c, d⟩ and ⟨e, f⟩ are α-β related if pc = pe, pd = pf and α·nc −β ·nd = α·ne −β ·nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Roughly speaking, two configuration pairs are α-β related if they have the same state pairs and lie on a line with slope α β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' An α-β repetition is a run ¯π1 = ciτici+1τi+1 · · · τj−1cj that lies inside a belt with slope α β such that ci and cj are α-β related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The following lemma bounds the counter values of the first configuration in the background space, if it exists, during the run Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If hj is the first background point in Π then, counter values of hj are less than K5 · 42K14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let hj be the first point in the background space during the run Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Assume for contradiction that ncj is greater than K5·42K14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let Π = h0τ0 · · · hj−1τj−1hj · · · hℓ be a run of a minimal witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since hj is the first point in the background space ONE DETERMINISTIC-COUNTER AUTOMATA 29 N N Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The figure shows the run of a word that enters the background space from the belt such that the counter values of the first configuration pair in the background space exceed a poly- nomial bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' in this run and ncj > K5 · 42K14, there exists 0 < i < j such that the sub-run Πb = hiτihi+1 · · · τj−2hj−1 lies inside a belt B with slope α β for some α, β ∈ [1, 3K7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since we are looking at the run of a minimal witness, from Lemma 33 either cj ̸≡2K2 dj or dist(cj) ̸= dist(dj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We separately consider the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Case-1: distA1(cj) ̸= distA2(dj): Without loss of generality, let us assume distA1(cj) < distA2(dj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore there exists t ∈ N with j < t ≤ ℓ and con- figuration pair ht such that m = nct < 2K2, p = pct and xct ∈ W p,m 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We show that we can pump some portion out from Πb to reach a configuration in the background space with unequal distance and smaller counter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since ncj > K5 · 42K14, by Pigeonhole principle, there exists indices i0 < i1 < i2 < · · · , iK2 < i′ 0 < i′ 1 < i′ 2, · · · , < i′ K2 such that for all r ∈ [1, K2], (1) hir−1 and hir are α-β related and lie in belt B, (2) ncir−1 < ncir = nci′r , (3) pci′r = pci′ r−1 , (4) for all t ∈ N with ir < t < j, nct > ncir , and (5) for all t ∈ N with j < t < i′ r, nct < nci′r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For r ∈ [0, K2] let Ar ∈ FK×K denote the matrix such that xcir Ar = xci′r and Br ∈ FK×K denote the matrix such that xci′r Br = xct ∈ W p,m 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore for all r ∈ [0, K2], we have xcir ArBr ∈ W p,m 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 4, we have that there exits s, r ∈ [0, K2] with s < r such that xcisArBs ∈ W p,m 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the sequence of transitions T ′ = τ0, · · · , τis−1τir, · · · , τj−1 and let w = word(T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let hj′ be the configuration such that h0 w −→ hj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since we have removed an α-β repetitions inside the belt, the configuration pair hj′ is a point in the background space (see Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Moreover, ncj′ < ncj and distA1(cj′) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since it is a point in the background space, from Lemma 33, we get that distA1(cj′) ̸= distA2(dj′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, there is a shorter path to a configuration in background space with smaller counter values and unequal distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Case-2: cj ̸≡2K2 dj: Consider the sub-run Πb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since it is a run inside a belt, we can consider this as the run of the weighted odca D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since ncj > K4 · 42K14, by Pigeon-hole principle, there exists indices i0, i1, i2, · · · , iK2 such that for all r ∈ 30 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH [1, K2], hir−1 and hir are α-β related with ncir−1 < ncir and for all t ∈ N with ir < t < j, nct > ncir .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since it is the run of a minimal witness, we know that there exists A ∈ F2K×2K such that xj−1Aη⊤ F ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the vector space U = {y ∈ F2K | yAη⊤ F = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For r ∈ [0, K2], let Ar denote the matrices such that xirAr = xj ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since xirAr ∈ U for all r ∈ [0, K2], from Lemma 4, we get that there exists r′ ∈ [0, r − 1] such that xci′r Ar ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The sequence of transitions τir+1 · · · τℓ can be taken from hi′r since the counter values always stay positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the sequence of transitions T ′ = τ0 · · · τi′rτir+1 · · · τℓ and let w = word(T ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The word w is a shorter witness than z and contradicts its minimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ Finally, we prove that the counter values encountered during the run Π are polynomially bounded in K using above lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof of Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the run Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 28, Lemma 29 and Lemma 35, we get that the counter values of configuration pairs inside the belt space during this run in polynomially bounded in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore, if it exists, the first background point in Π has polynomially bounded counter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 34, the length of Π after the first background point is polynomially bounded in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since initial space is already bounded by a polynomial in K, the maximum counter value in Π is polynomially bounded in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Regularity of ODCA is in P We say that a weighted odca A = (C, δ, p0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q, λ, ∆, η) is regular if there is a weighted automaton B that is equivalent to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We look at the regularity problem the problem of deciding whether a weighted odca is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We fix a weighted odca A = (C, δ, p0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q, λ, ∆, η) and use N to denote |C| · |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The proof technique is adapted from the ideas developed by B¨ohm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' [6] in the context of real-time oca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The crucial idea in proving regularity is to check for the existence of infinitely many equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The proof relies on the notion of distance of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Distance of a configuration is the length of a minimal word to be read to reach a configuration that does not have an N equivalent configuration in the underlying automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The challenge is to find infinitely many configurations reachable from the initial configuration, so that no two of them have same distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Our main contribution is in designing a “pumping” like argument to show this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Theorem 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' There is a polynomial time algorithm to decide whether a weighted odca is equivalent to some weighted automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Recall the definition of U(A) from Definition 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We use c to denote some configuration of A and ¯c to denote some configuration of U(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For a p ∈ C and m ∈ N, we define Wp,m = {x ∈ F|Q||∃¯c ∈ FN, c = (x, p, m) ∼N ¯c} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The set W p,m is F|Q|\\Wp,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The distance of a configuration c (denoted by dist(c)) is min{|w| | c w −→ (x, p, m) ∃p ∈ C, m < N, and x ∈ W p,m} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The following lemma shows when A is not regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Clean up, move to appendix, proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' ONE DETERMINISTIC-COUNTER AUTOMATA 31 Lemma 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let c be an initial configuration of an odca A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Then the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' (1) A is not regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' (2) for all t ∈ N, there exists configurations d, e s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' ne < N,c ∗−→ d ∗−→ e, xe ∈ W pe,ne and t < dist(d) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' (3) there exists configurations d, e and a run c ∗−→ d ∗−→ e s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' N2 + N ≤ nd ≤ 2N2 + N, xe ∈ W pe,ne with ne < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 3 → 2: Consider an arbitrary q ∈ C, m < N and vector space V = Wq,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let us assume for contradiction the complement of Point 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' That is, there exists a t ∈ N such that for all configurations d′ where c ∗−→ d′ ∗−→ V ×{q}×{m}, dist(d′) ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Note that for all d′ where nd′ > N, dist(d′) ≥ nd′ −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Hence there exists an M ∈ N such that for all d′ where c ∗−→ d′ ∗−→ V × {q} × {m}, nd′ ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider a configuration d where nd > N2+N and a run c ∗−→ d ∗−→ V ×{q}×{m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Point 3 shows the existence of such a run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For contradiction, it suffices to show there exists a d′ such that c ∗−→ d′ ∗−→ V × {q} × {m} and nd′ > nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let m = |Q|2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since nd > N2 + N, by Pigeonhole principle, there exists set of indices X = {i1, i2, · · · , im} such that for k, l ∈ [1, m] with k < l, we have ik < il, and for all h, j ∈ X pch = pc′ h = pcj = pc′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let uj, vj, wj be words such that for all j ∈ X, c uj −→ cj vj −→ c′ j wj −−→ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For all j ∈ X, let matrix Aj and Bj be such that xc′ j = xcjAj and xe = xc′ jBj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We know that for all j ∈ X, xcjAjBj ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' List the matrices Ai1, Ai2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' , Aim in sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' From Lemma 4, it follows that, there exists i, j ∈ X with i < j such that xcjAiBj ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Consider the run π(ujviwj, c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' It contains a configuration d′ where nd′ > nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 2 → 1: Assume for contradiction that for all t ∈ N, there exists configurations d, e such that c ∗−→ d ∗−→ e, xe ∈ W pe,ne, ne < N and t < dist(d) < ∞ but A is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let B be the weighted automaton equivalent to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We use |B| to represent the number of states of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' t|B|+1 ∈ N such that for i ∈ [1, |B|], ti < ti+1, and dti be such that c ∗−→ dti ∗−→ (xi, pe, ne), xi ∈ W pe,ne and ti < dist(dti) < ti+1 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Clearly dti ̸≡ dtj for i ̸= j and hence corresponds to two different states of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Since we can find more than |B| pairwise non-equivalent configurations, it contradicts the assumption that B is equivalent to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 1 → 3: We prove the contrapositive of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let us assume that there is no configurations d, e and a run c ∗−→ d ∗−→ e such that N2 + N ≤ nd ≤ 2N2 + N, xe ∈ W pe,ne with ne < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This implies that there does not exists a configuration d′ such that nd′ > 2N2, c ∗−→ d′ ∗−→ (y, pe, ne) for some y ∈ W pe,ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Assume for contradiction that there is such a run, then there exists a portion in this run that can be “pumped down” to get a run c ∗−→ d′′ ∗−→ (y′, pe, ne) for some configuration d′′ such that N2 + N ≤ nd′′ ≤ 2N2 + N and y′ ∈ W pe,ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Therefore all runs starting from configuration with counter value greater than or equal to N2 + N “looks” similar to a run on a weighted automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This allows us to simulate the runs of A using a weighted automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □We now prove that the regularity problem for weighted odca is decidable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof of Theorem 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let A be a weighted odca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Lemma 37 shows that if A is not regular, then there are words u, v ∈ Σ∗ and configurations d, e such that there 32 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH is a run of the form c u−→ d v−→ e such that N2 + N ≤ nd ≤ 2N2 + N, xe ∈ W pe,ne with ne < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The existence of such words u and v can be decided in polynomial time since the minimal length of such a path if it exists, is polynomially bounded in the number of states of the weighted odca by Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Covering Let A1 and A2 be two uninitialised weighted odcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We say A2 covers A1 if for all initial configurations c0 of A1 there exists an initial configuration d0 of A2 such that A1⟨c0⟩ and A2⟨d0⟩ are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We say A1 and A2 are coverable equivalent if A1 covers A2 and A2 covers A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We show that the covering and coverable equivalence problems for uninitialised weighted odcas are decidable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The proof relies on the algorithm to check the equivalence of two weighted odcas and is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Theorem 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Covering and coverable equivalence problems of uninitialised weighted odcas are in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We fix two uninitialised weighted odcas A1 = (C1, δ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q1, ∆1, η1) and A2 = (C2, δ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Q2, ∆2, η2) for this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Without loss of generality, assume K = |C1| = |Q1| = |C2| = |Q2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For i ∈ [1, K] we define the vector ei ∈ FK as follows: ei[j] = � 1, if i = j 0, otherwise For j ∈ [1, K], q ∈ C1, we use hj,q to denote the configuration (ej, q, 0) of A1 and for i ∈ [1, K], p ∈ C2, we use gi,p to denote the configuration (ei, p, 0) of A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' There is a polynomial time algorithm to decide whether A2 covers A1⟨hj,q⟩ for any j ∈ [1, K] and q ∈ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Proof: First, we check, in polynomial time (equivalence with a zero machine), whether A1⟨hj,q⟩ accepts all words with weight 0 ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If that is the case, then A1⟨hj,q⟩ and A2⟨g0⟩ are equivalent for the configuration g0 = ({0}K, p, 0), for any p ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Otherwise, there is some word w0 accepted by A1⟨hj,q⟩ with non-zero weight s (returned by the previous equivalence check).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Without loss of generality, we consider the smallest one, whose size is polynomial in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We pick a p ∈ C2 and check whether there exists an initial distribution from counter state p that makes the two machines equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Assume that such an initial distribution exists and for all i ∈ [1, K], let αi denote the initial weight on state qi ∈ Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We use α to denote the resultant initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We initialise an empty set B to store a system of linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The following steps will be repeated at most K times to check the existence of an initial distribution with initial state p ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let w be the counter-example returned by the equivalence query in the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For all i ∈ [1, K], we compute fA2⟨gi,p⟩(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We add the linear equation �K i=1 αi · fA2⟨gi,p⟩(w) = fA1⟨hj,q⟩(w) to B and compute values for αi, i ∈ [1, K], such that it satisfies the system of linear equations in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We check whether A1⟨hj,q⟩ and A2⟨(α, p, 0)⟩ are equivalent or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If they are not equivalent, we get a new counter example that distinguishes them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now we repeat the procedure to compute a new initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' ONE DETERMINISTIC-COUNTER AUTOMATA 33 Note that the above procedure is executed at most K times to find an initial dis- tribution if it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is because we can find only K many linearly independent linear equations in K variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Suppose the above procedure fails to find an initial distribution for which the machines are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In that case, there is an initial distribution of A1 with initial counter state q, for which A2 with initial counter state p does not have an equivalent initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We now pick a different counter state of C2 and repeat the process until we find a p ∈ C2 for which the algorithm finds an equivalent initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If for all p ∈ C2, the algorithm returns false, then A2 does not cover A1⟨hj,q⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □Claim:1 First, we show the existence of a polynomial time procedure to check whether A2 covers A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For all j ∈ [1, K], we check whether there exists an initial state p ∈ C2 such that A2 with initial counter state p has an initial distribution that makes it equivalent to A1⟨hj,q⟩ using Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If we fail to find such a state in C2 then we return false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We repeat this procedure for all q ∈ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' If for all q ∈ C1 there exists a p ∈ C2 such that A2 with initial counter state p has an initial distribution that makes it equivalent to A1⟨hj,q⟩ for all j ∈ [1, K], then we say that A2 covers A1 otherwise we say that A2 does not cover A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Let us see why this is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' For simplifying the arguments we ��x a q ∈ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Assume that for all j ∈ [1, K], there exits p ∈ C2 such that A1⟨hj,q⟩ is equivalent to the configuration (xj,q, p, 0) for some xj,q ∈ FK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Now, any initial configuration (λ, q, 0) of A1 is equivalent to the configuration (�K j=1 λ[j] · xj,q, p, 0) of A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The coverable equivalence problem can now be solved by checking whether A1 covers A2 and A2 covers A1, which can be done in time polynomial in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Conclusion We introduced a new model called odca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The equivalence problem for non- deterministic odcas is in PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' This is in contrast to non-deterministic oca, where the equivalence problem is undecidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We observe that undecidability is a consequence of non-determinism occurring in counter actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In the case of weighted odcas, we show that the reachability, equivalence, regularity, and covering problems are in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' The natural way to extend the work is to consider epsilon transitions in the odca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We conjecture that the equivalence, regularity, and covering problems will be polynomial time decidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Another possible direction is to look at pushdown systems partitioned into a deterministic stack and a finite state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' In this case, a non-deterministic model can be determinized (similar to Theorem 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Even though all our algorithms are in polynomial time, they may not be ‘practical’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Con- siderable effort is required to find faster algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' We also leave open questions on learning and approximate equivalence of odcas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Acknowledgment The authors would like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Rahul C S, School of Mathematics and Computer Science, IIT Goa, for his valuable and intuitive suggestions which helped us in solving the binary Co-VS reachability problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Sreejith would like to thank DST Matrics grant for the project “Probabilistic Pushdown Automata”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' 34 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' MATHEW, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' PENELLE, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SAIVASAN, AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SREEJITH References [1] Steven P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Abney, David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' McAllester, and Fernando Pereira.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Relating probabilistic gram- mars and automata.' metadata={'source': 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and its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Thomson, Brooks/Cole, Belmont, CA, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' [20] Wen-Guey Tzeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' A polynomial-time algorithm for the equivalence of probabilistic automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Comput, 21(2):216–227, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' [21] Valiant and Paterson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Deterministic one-counter automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' JCSS: Journal of Computer and System Sciences, 10, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content=' Indian Institute of Technology Goa Email address: prince@iitgoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='in Universit´e de Bordeaux Email address: vincent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='penelle@u-bordeaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='fr The Institute of Mathematical Sciences, HBNI Email address: prakashs@imsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='in Indian Institute of Technology Goa Email address: sreejithav@iitgoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'} +page_content='in' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQf_DcV/content/2301.13456v1.pdf'}