{ "paper_id": "P94-1024", "header": { "generated_with": "S2ORC 1.0.0", "date_generated": "2023-01-19T09:18:58.493387Z" }, "title": "A MARKOV LANGUAGE LEARNING MODEL FOR FINITE PARAMETER SPACES", "authors": [ { "first": "Partha", "middle": [], "last": "Niyogi", "suffix": "", "affiliation": { "laboratory": "", "institution": "Massachusetts Institute of Technology", "location": { "postCode": "E25-201, 02139", "settlement": "Cambridge", "region": "MA", "country": "USA" } }, "email": "" }, { "first": "Robert", "middle": [ "C" ], "last": "Berwick", "suffix": "", "affiliation": { "laboratory": "", "institution": "Massachusetts Institute of Technology", "location": { "postCode": "E25-201, 02139", "settlement": "Cambridge", "region": "MA", "country": "USA" } }, "email": "berwick@ai.nfit.edu" } ], "year": "", "venue": null, "identifiers": {}, "abstract": "This paper shows how to formally characterize language learning in a finite parameter space as a Markov structure, hnportant new language learning results follow directly: explicitly calculated sample complexity learning times under different input distribution assumptions (including CHILDES database language input) and learning regimes. We also briefly describe a new way to formally model (rapid) diachronic syntax change.", "pdf_parse": { "paper_id": "P94-1024", "_pdf_hash": "", "abstract": [ { "text": "This paper shows how to formally characterize language learning in a finite parameter space as a Markov structure, hnportant new language learning results follow directly: explicitly calculated sample complexity learning times under different input distribution assumptions (including CHILDES database language input) and learning regimes. We also briefly describe a new way to formally model (rapid) diachronic syntax change.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Abstract", "sec_num": null } ], "body_text": [ { "text": "Recently, several researchers, including Gibson and Wexler (1994) , henceforth GW, Dresher and Kaye (1990) ; and Clark and Roberts (1993) have modeled language learning in a (finite) space whose grammars are characterized by a finite number of parameters or nlength Boolean-valued vectors. Many current linguistic theories now employ such parametric models explicitly or in spirit, including Lexical-Functional Grammar and versions of HPSG, besides GB variants. With all such models, key questions about sample complexity, convergence time, and alternative modeling assumptions are difficult to assess without a precise mathematical formalization. Previous research has usually addressed only the question of convergence in the limit without probing the equally important question of sample complexity: it is of not much use that a learner can acquire a language if sample complexity is extraordinarily high, hence psychologically implausible. This remains a relatively undeveloped area of language learning theory. The current paper aims to fill that gap. We choose as a starting point the GW Triggering Learning Algorithm (TLA). Our central result is that the performance of this algorithm and others like it is completely modeled by a Markov chain. We explore the basic computational consequences of this, including some surprising results about sample complexity and convergence time, the dominance of random walk over gradient ascent, and the applicability of these results to actual child language acquisition and possibly language change.", "cite_spans": [ { "start": 41, "end": 65, "text": "Gibson and Wexler (1994)", "ref_id": "BIBREF3" }, { "start": 83, "end": 106, "text": "Dresher and Kaye (1990)", "ref_id": "BIBREF2" }, { "start": 113, "end": 137, "text": "Clark and Roberts (1993)", "ref_id": "BIBREF0" } ], "ref_spans": [], "eq_spans": [], "section": "BACKGROUND MOTIVATION: TRIGGERS AND LANGUAGE ACQUISITION", "sec_num": null }, { "text": "Background. Following Gold (1967) the basic framework is that of identification in the limit. We assume some familiarity with Gold's assumptions. The learner receives an (infinite) sequence of (positive) example sentences from some target language. After each, the learner either (i) stays in the same state; or (ii) moves to a new state (change its parameter settings). If after some finite number of examples the learner converges to the correct target language and never changes its guess, then it has correctly identified the target language in the limit; otherwise, it fails.", "cite_spans": [ { "start": 22, "end": 33, "text": "Gold (1967)", "ref_id": "BIBREF4" } ], "ref_spans": [], "eq_spans": [], "section": "BACKGROUND MOTIVATION: TRIGGERS AND LANGUAGE ACQUISITION", "sec_num": null }, { "text": "In the GW model (and others) the learner obeys two additional fundamental constraints: (1) the single.value constraint--the learner can change only 1 parameter value each step; and (2) the greediness constraint--if the learner is given a positive example it cannot recognize and changes one parameter value, finding that it can accept the example, then the learner retains that new value. The TLA essentially simulates this; see Gibson and Wexler (1994) for details.", "cite_spans": [ { "start": 429, "end": 453, "text": "Gibson and Wexler (1994)", "ref_id": "BIBREF3" } ], "ref_spans": [], "eq_spans": [], "section": "BACKGROUND MOTIVATION: TRIGGERS AND LANGUAGE ACQUISITION", "sec_num": null }, { "text": "Previous parameter models leave open key questions addressable by a more precise formalization as a Markov chain. The correspondence is direct. Each point i in the Markov space is a possible parameter setting. Transitions between states stand for probabilities b that the learner will move from hypothesis state i to state j. As we show below, given a distribution over L(G), we can calculate the actual b's themselves. Thus, we can picture the TLA learning space as a directed, labeled graph V with 2 n vertices. See figure 1 for an example in a 3-parameter system. 1 We can now use Markov theory to describe TLA parameter spaces, as in lsaacson and 1GW construct an identical transition diagram in the description of their computer program for calculating local maxima. However, this diagram is not explicitly presented as a Markov structure and does not include transition probabilities.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "THE MARKOV FORMULATION", "sec_num": null }, { "text": "Madsen (1976) . By the single value hypothesis, the system can only move 1 Hamming bit at a time, either toward the target language or 1 bit away. Surface strings can force the learner from one hypothesis state to another. For instance, if state i corresponds to a grammar that generates a language that is a proper subset of another grammar hypothesis j, there can never be a transition from j to i, and there must be one from i to j. Once we reach the target grammar there is nothing that can move the learner from this state, since all remaining positive evidence will not cause the learner to change its hypothesis: an Absorbing State (AS) in the Markov literature. Clearly, one can conclude at once the following important learnability result:", "cite_spans": [ { "start": 7, "end": 13, "text": "(1976)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "THE MARKOV FORMULATION", "sec_num": null }, { "text": "Theorem 1 Given a Markov chain C corresponding to a GW TLA learner, 3 exactly 1 AS (corresponding to the target grammar/language) iff C is learnable.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "THE MARKOV FORMULATION", "sec_num": null }, { "text": "Proof. \u00a2::. By assumption, C is learnable. Now assume for sake of contradiction that there is not exactly one AS. Then there must be either 0 AS or > 1 AS. In the first case, by the definition of an absorbing state, there is no hypothesis in which the learner will remain forever. Therefore C is not learnable, a contradiction. In the second case, without loss of generality, assume there are exactly two absorbing states, the first S corresponding to the target parameter setting, and the second S ~ corresponding to some other setting. By the definition of an absorbing state, in the limit C will with some nonzero probability enter S I, and never exit S I. Then C is not learnable, a contradiction. Hence our assumption that there is not exactly 1 AS must be false.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "THE MARKOV FORMULATION", "sec_num": null }, { "text": "=\u00a2.. Assume that there exists exactly 1 AS i in the Markov chain M. Then, by the definition of an absorbing state, after some number of steps n, no matter what the starting state, M will end up in state i, corresponding to the target grammar. | Corollary 0.1 Given a Markov chain corresponding to a (finite) family of grammars in a G W learning system, if there exist 2 or more AS, then that family is not learnable.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "THE MARKOV FORMULATION", "sec_num": null }, { "text": "We now derive the transition probabilities for the Markov TLA structure, the key to establishing sample complexity results. Let the target language L~ be L~ = {sl, s2, s3, ...} and P a probability distribution on these strings. Suppose the learner is in a state corresponding to language Ls. With probability P(sj), it receives a string sj. There are two cases given current parameter settings.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "DERIVATION OF TRANSITION PROBABILITIES FOR THE MARKOV TLA STRUCTURE", "sec_num": null }, { "text": "Case I. The learner can syntactically analyze the received string sj. Then parameter values are unchanged. This is so only when sj \u2022 L~. The probability of remaining in the state s is P(sj).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "DERIVATION OF TRANSITION PROBABILITIES FOR THE MARKOV TLA STRUCTURE", "sec_num": null }, { "text": "Case II. The learner cannot syntactically analyze the string. Then sj ~ Ls; the learner is in state s, and has n neighboring states (Hamming distance of 1). The learner picks one of these uniformly at random. If nj of these neighboring states correspond to languages which contain sj and the learner picks any one of them (with probability nj/n), it stays in that state. If the learner picks any of the other states (with probability ( n -nj)/n) then it remains in state s. Note that nj could take values between 0 and n. Thus the probability that the learner remains in state s is P(sj)(( n -nj )/n).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "DERIVATION OF TRANSITION PROBABILITIES FOR THE MARKOV TLA STRUCTURE", "sec_num": null }, { "text": "The probability of moving to each of the other nj states is P(sj)(nj/n).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "DERIVATION OF TRANSITION PROBABILITIES FOR THE MARKOV TLA STRUCTURE", "sec_num": null }, { "text": "The probability that the learner will remain in its original state s is the sum of the probabilities of these two cases: ~,jEL, P(sj) + E,jCL,(1 -nj/n)P(sj).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "DERIVATION OF TRANSITION PROBABILITIES FOR THE MARKOV TLA STRUCTURE", "sec_num": null }, { "text": "To compute the transition probability from s to k, note that this transition will occur with probability 1/n for all the strings sj E Lk but not in L~. These strings occur with probability P(sj) each and so the transition probability is: , jeL, , , j\u00a2L, , , jeLk (1/n) ", "cite_spans": [ { "start": 238, "end": 239, "text": ",", "ref_id": null }, { "start": 240, "end": 244, "text": "jeL,", "ref_id": null }, { "start": 245, "end": 246, "text": ",", "ref_id": null }, { "start": 247, "end": 248, "text": ",", "ref_id": null }, { "start": 249, "end": 253, "text": "j\u00a2L,", "ref_id": null }, { "start": 254, "end": 255, "text": ",", "ref_id": null }, { "start": 256, "end": 257, "text": ",", "ref_id": null }, { "start": 258, "end": 268, "text": "jeLk (1/n)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "DERIVATION OF TRANSITION PROBABILITIES FOR THE MARKOV TLA STRUCTURE", "sec_num": null }, { "text": "P[s ~ k] = ~", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "DERIVATION OF TRANSITION PROBABILITIES FOR THE MARKOV TLA STRUCTURE", "sec_num": null }, { "text": "P(si) \u2022 Summing over all strings sj E ( Lt N Lk ) \\ L, (set dif- ference) it is easy to see that sj \u2022 ( Lt N Lk ) \\ Ls \u00a2~ sj \u2022 (L, N nk) \\ (L, n Ls). Rewriting, we have P[s ---* k] = ~,je(L,nLk)\\(L,nL.)(1/n)P(sj)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "DERIVATION OF TRANSITION PROBABILITIES FOR THE MARKOV TLA STRUCTURE", "sec_num": null }, { "text": ". Now we can compute the transition probabilities between any two states. Thus the self-transition probability can be given as,", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "DERIVATION OF TRANSITION PROBABILITIES FOR THE MARKOV TLA STRUCTURE", "sec_num": null }, { "text": "P[s --, s] = 1-~-'~ k is a neighboring state of, P[s ---, k].", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "DERIVATION OF TRANSITION PROBABILITIES FOR THE MARKOV TLA STRUCTURE", "sec_num": null }, { "text": "Consider the 3-parameter natural language system described by Gibson and Wexler (1994) , designed to cover basic word orders (X-bar structures) plus the verbsecond phenomena of Germanic languages, lts binary parameters are: (1) Spec(ifier) initial (0) or final (1);", "cite_spans": [ { "start": 62, "end": 86, "text": "Gibson and Wexler (1994)", "ref_id": "BIBREF3" } ], "ref_spans": [], "eq_spans": [], "section": "Example.", "sec_num": null }, { "text": "(2) Compl(ement) initial (0) or final (1); and Verb Second (V2) does not exist (0) or does exist (l). Possible \"words\" in this language include S(ubject), V(erb), O(bject), D(irect) O(bject), Adv(erb) phrase, and so forth. Given these alternatives, Gibson and Wexler (1994) show that there are 12 possible surface strings for each (-V2) grammar and 18 possible surface strings for each (+V2) grammar, restricted to unembedded or \"degree-0\" examples for reasons of psychological plausibility (see Gibson and Wexler for discussion). For instance, the parameter setting [0 1 0]= Specifier initial, Complement final, and -V2, works out to the possible basic English surface phrase order of Subject-Verb-Object (SVO).", "cite_spans": [ { "start": 249, "end": 273, "text": "Gibson and Wexler (1994)", "ref_id": "BIBREF3" } ], "ref_spans": [], "eq_spans": [], "section": "Example.", "sec_num": null }, { "text": "As in figure 1 below, suppose the SVO (\"English\", setting #5=[0 1 0]) is the target grammar. The figure's shaded rings represent increasing Hamming distances from the target. Each labeled circle is a Markov state. Surrounding the bulls-eye target are the 3 other parameter arrays that differ from [0 1 0] by one binary digit: e.g., [0, 0, 0], or Spec-first, Comp-first, -V2, basic order SOV or \"Japanese\". Around it are the three settings that differ from the target by exactly one binary digit; surrounding those are the 3 hypotheses two binary digits away from the target; the third ring out contains the single hypothesis that differs from the target by 3 binary digits.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Example.", "sec_num": null }, { "text": "Plainly there are exactly 2 absorbing states in this Markov chain. One is the target grammar (by definition); the other is state 2. State 4 is also a sink that leads only to state 4 or state 2. GW call these two nontarget states local maxima because local gradient ascent will converge to these without reaching the desired target. Hence this system is not learnable. More importantly though, in addition to these local maxima, we show (see below) that there are other states (not detected in GW or described by Clark) from which the learner will never reach the target with (high) positive probability. Example: we show that if the learner starts at hypothesis VOS-V2, then with probability 0.33 in the limit, the learner will never converge to the SVO target. Crucially, we must use set differences to build the Markov figure straightforwardly, as indicated in the next section. In short, while it is possible to reach \"English\"from some source languages like \"Japanese,\" this is not possible for other starting points (exactly 4 other initial states).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Example.", "sec_num": null }, { "text": "It is easy to imagine alternatives to the TLA that avoid the local maxima problem. As it stands the learner only changes a parameter setting if that change allows the learner to analyze the sentence it could not analyze before. If we relax this condition so that under unanalyzability the learner picks a random parameter to change, then the problem with local maxima disappears, because there can be only 1 Absorbing State, the target grammar. All other states have exit arcs. Thus, by our main theorem, such a system is learnable. We discuss other alternatives below.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Example.", "sec_num": null }, { "text": "Perhaps the most significant advantage of the Markov chain formulation is that one can calculate the number of examples needed to acquire a language. Recall it is not enough to demonstrate convergence in the limit; learning must also be feasible. This is particularly true in the case of finite parameter spaces where convergence might not be as much of a problem as feasibility. Fortunately, given the transition matrix of a Markov chain, the problem of how long it takes to converge has been well studied.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "CONVERGENCE TIMES FOR THE MARKOV CHAIN MODEL", "sec_num": null }, { "text": "Consider the example in the previous section. The target grammar is SVO-V2 (grammar ~5 in GW). For simplicity, assume a uniform distribution on L5. Then the probability of a particular string sj in L5 is 1/12 because there are 12 (degree-0) strings in L~. We directly compute the transition matrix (0 entries elsewhere): States 2 and 5 are absorbing; thus this chain contains local maxima. Also, state 4 exits only to either itself or to state 2, hence is also a local maximum. If T is the transition probability matrix of a chain, then the corresponding i, j element of T m is the probability that the learner moves from state i to state j in m steps. For learnability to hold irrespective starting state, the probability of reaching state 5 should approach 1 as m goes to infinity, i.e., column 5 of T m should contain all l's, and O's elsewhere. Direct computation shows this to be false:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "SOME TRANSITION MATRICES AND THEIR CONVERGENCE CURVES", "sec_num": null }, { "text": "L1 L2", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "SOME TRANSITION MATRICES AND THEIR CONVERGENCE CURVES", "sec_num": null }, { "text": "L1 L2 L3 L4 Ls L6 L7 Ls L1 L2 L3 L4 L5 L6 L7 Ls ! 3 1 1 3 1", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "SOME TRANSITION MATRICES AND THEIR CONVERGENCE CURVES", "sec_num": null }, { "text": "We see that if the learner starts out in states 2 or 4, it will certainly end up in state 2 in the limit. These two states correspond to local maxima grammars in the GW framework. We also see that if the learner starts in states 5 through 8, it will certainly converge in the limit to the target grammar.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "SOME TRANSITION MATRICES AND THEIR CONVERGENCE CURVES", "sec_num": null }, { "text": "States 1 and 3 are much more interesting, and constitute new results about this parameterization. If the learner starts in either of these states, it reaches the target grammar with probability 2/3 and state 2 with probability 1/3. Thus, local maxima are not the only problem for parameter space learnability. To our knowledge, GW and other researchers have focused exclusively on local maxima. However, while it is true that states 2 and 4 will, with probability l, not converge to the target grammar, it is also true that states l and 3 will not converge to the target, with probability 1/3. Thus, the number of \"bad\" initial hypotheses is significantly larger than realized generally (in fact, 12 out of 56 of the possible source-target grammar pairs in the 3parameter system). This difference is again due to the new probabilistic framework introduced in the current paper. The quantity p(m) is easy to interpret. Thus p(m) = 0.95 rneans that for every initial state of the learner the probability that it is in the target state after m examples is at least 0.95. Further there is one initial state (the worst initial state with respect to the target, which in our example is Ls) for which this probability is exactly 0.95. We find on looking at the curve that the learner converges with high probability within 100 to 200 (degree-0) example sentences, a psychologically plausible number.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "SOME TRANSITION MATRICES AND THEIR CONVERGENCE CURVES", "sec_num": null }, { "text": "We can now compare the convergence time of TLA to other algorithms. Perhaps the simplest is random walk: start the learner at a random point in the 3-parameter space, and then, if an input sentence cannot be analyzed, move 1-bit randomly from state to state. Note that this regime cannot suffer from the local maxima problem, since there is always some finite probability of exiting a non-target state.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "SOME TRANSITION MATRICES AND THEIR CONVERGENCE CURVES", "sec_num": null }, { "text": "Computing the convergence curves for a random walk algorithm (RWA) on the 8 state space, we find that the convergence times are actually faster than for the TLA; see figure 2. Since the RWA is also superior in that it does not suffer from the same local maxima problem as TLA, the conceptual support for the TLA is by no means clear. Of course, it may be that the TLA has empirical support, in the sense of independent evidence that children do use this procedure (given by the pattern of their errors, etc.), but this evidence is lacking, as far as we know.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "SOME TRANSITION MATRICES AND THEIR CONVERGENCE CURVES", "sec_num": null }, { "text": "In the earlier section we assumed that the data was uniformly distributed. We computed the transition matrix for a particular target language and showed that convergence times were of the order of 100-200 samples. In this section we show that the convergence times depend crucially upon the distribution. In particular we can choose a distribution which will make the convergence time as large as we want. Thus the distribution-free convergence time for the 3-parameter system is infinite.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "DISTRIBUTIONAL ASSUMPTIONS: PART I", "sec_num": null }, { "text": "As before, we consider the situation where the target language is L1. There are no local maxima problems for this choice. We begin by letting the distribution be parametrized by the variables a, b, c, d where", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "DISTRIBUTIONAL ASSUMPTIONS: PART I", "sec_num": null }, { "text": "a = P(A = {Adv(erb)Phrase V S}) b = P(B = {Adv V O S, Adv Aux V S}) c = P(C={AdvV O1 O2S, AdvAuxVOS, Adv Aux V O1 02 S}) d = P(D={VS})", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "DISTRIBUTIONAL ASSUMPTIONS: PART I", "sec_num": null }, { "text": "Thus each of the sets A, B, C and D contain different degree-O sentences of L1. Clearly the probability of the set L, \\{AUBUCUD} is 1-(a+b+c+d). The elements of each defined subset of La are equally likely with respect to each other. Setting positive values for a, b, c, d such that a + b + c + d < 1 now defines a unique probability for each degree(O) sentence in L1. For example, the probability of AdvVOS is b/2, the probability of AdvAuxVOS is c/3, that of VOS is (1-(a+b+c+d))/6 and so on; see figure 3. We can now obtain the transition matrix corresponding to this distribution. If we compare this matrix with that obtained with a uniform distribution on the sentences of La in the earlier section. This matrix has non-zero elements (transition probabilities) exactly where the earlier matrix had non-zero elements. However, the value of each transition probability now depends upon a,b, c, and d. In particular if we choose a = 1/12, b = 2/12, c = 3/12, d = 1/12 (this is equivalent to assuming a uniform distribution) we obtain the appropriate transition matrix as before. Looking more closely at the general transition matrix, we see that the transition probability from state 2 to state 1 is (1-(a+b+c))/3.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "DISTRIBUTIONAL ASSUMPTIONS: PART I", "sec_num": null }, { "text": "Clearly if we make a arbitrarily close to 1, then this transition probability is arbitrarily close to 0 so that the number of samples needed to converge can be made arbitrarily large. Thus choosing large values for a and small values for b will result in large convergence times.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "DISTRIBUTIONAL ASSUMPTIONS: PART I", "sec_num": null }, { "text": "This means that the sample complexity cannot be bounded in a distribution-free sense, because by choosing a highly unfavorable distribution the sample complexity can be made as high as possible. For example, we now give the convergence curves calculated for different choices of a, b,c, d. We see that for a uniform distribution the convergence occurs within 200 samples. By choosing a distribution with a = 0.9999 and b = c = d = 0.000001, the convergence time can be pushed up to as much as 50 million samples. (Of course, this distribution is presumably not psychologically realistic.) For a = 0.99, b = c = d = 0.0001, the sample complexity is on the order of 100,000 positive examples. Remark. The preceding calculation provides a worstcase convergence time. We can also calculate average convergence times using standard results from Markov chain theory (see Isaacson and Madsen, 1976) , as in table 2. These support our previous results.", "cite_spans": [ { "start": 865, "end": 891, "text": "Isaacson and Madsen, 1976)", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "DISTRIBUTIONAL ASSUMPTIONS: PART I", "sec_num": null }, { "text": "There are also well-known convergence theorems derived from a consideration of the eigenvalues of the transition matrix. We state without proof a convergence result for transition matrices stated in terms of its eigenvalues. Table 1 : Complete list of problem states, i.e., all combinations of starting grammar and target grammar which result in non-learnability of the target. The items marked with an asterisk are those listed in the original paper by Gibson and Wexler (1994) . ~l , . . . , .~n. Let x0 (an ndimensional vector) represent the starting probability of being in each state of the chain and r be the limiting probability of being in each state. Then after k transitions, the probability of being in each state x0T k can be described by", "cite_spans": [ { "start": 454, "end": 478, "text": "Gibson and Wexler (1994)", "ref_id": "BIBREF3" }, { "start": 481, "end": 485, "text": "~l ,", "ref_id": null }, { "start": 486, "end": 493, "text": ". . . ,", "ref_id": null }, { "start": 494, "end": 530, "text": ".~n. Let x0 (an ndimensional vector)", "ref_id": null } ], "ref_spans": [ { "start": 225, "end": 232, "text": "Table 1", "ref_id": null } ], "eq_spans": [], "section": "DISTRIBUTIONAL ASSUMPTIONS: PART I", "sec_num": null }, { "text": "Initial Grammar Target Grammar (svo-v2) (svo+v2)* (soy-v2) (SOV+V2)* (VOS-V2) (VOS+V2)* (OVS-V2) (ovs+v2)* (vos-v2) (VOS+V2)* (OVS-V2) (OVS+V2)* (OVS-V2) (ovs-v2) (ovs-v2) (ovs-v2) (svo-v2) (svo-v2) (svo-v2) (svo-v2) (sov-v2) (soy-v2) (soy-v2) (sov-v2)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "DISTRIBUTIONAL ASSUMPTIONS: PART I", "sec_num": null }, { "text": "n I1 x0T k-~ I1=11 ~ mfx0y~x, I1~< max I~,lk ~ II x0y,x, II 2State of Initial GrammarProbability of Not(Markov Structure)Converging to TargetNot Sink0.5Sink1.0Not Sink0.15Sink1.0Not Sink0.33Sink1.0Not Sink0.33Not Sink1.0Not Sink0.33Sink1.0Not Sink0.08Sink1.0~m~o-@ ;\u00b01 616o260 Number of examples (m}360460", "html": null, "num": null }, "TABREF2": { "type_str": "table", "text": "Mean and standard deviation convergence times to target 5 (English) given different distributions over the target language, and a uniform distribution over initial states. The first distribution is uniform over the target", "content": "
language; the other distributionsalter the value of a as discussed in the main text.
LearningMean abs.Std. Dev.
scenariotimeof abs. time
TEA (uniform)34.822.3
TLA (a = 0.99)4500033000
TLA (a = 0.9999)4.5 \u00d7 1063.3 \u00d7 l06
RW9.610.1
", "html": null, "num": null }, "TABREF4": { "type_str": "table", "text": "Convergence rates derived from eigenvalue calculations.", "content": "
Rate of Convergence
0(0.94 ~)
", "html": null, "num": null } } } }