Hugging Face's logo

Datasets:
WINGNUS
/
ACL-OCL

Tasks:
Token Classification
Languages:
English
Size:
10K<n<100K
Tags:
research papers
acl
License:
Dataset card Files Community
5
ACL-OCL / Base_JSON /prefixN /json /N12 /N12-1027.json
Benjamin Aw
Add updated pkl file v3
6fa4bc9
raw
history blame contribute delete
130 kB
 {
"paper_id": "N12-1027",
"header": {
"generated_with": "S2ORC 1.0.0",
"date_generated": "2023-01-19T14:04:33.952578Z"
},
"title": "Every sensible extended top-down tree transducer is a multi bottom-up tree transducer",
"authors": [
{
"first": "Andreas",
"middle": [],
"last": "Maletti",
"suffix": "",
"affiliation": {
"laboratory": "",
"institution": "Universit\u00e4t Stuttgart",
"location": {
"addrLine": "Pfaffenwaldring 5b",
"postCode": "70569",
"settlement": "Stuttgart",
"country": "Germany"
}
},
"email": "andreas.maletti@ims.uni-stuttgart.de"
}
],
"year": "",
"venue": null,
"identifiers": {},
"abstract": "A tree transformation is sensible if the size of each output tree is uniformly bounded by a linear function in the size of the corresponding input tree. Every sensible tree transformation computed by an arbitrary weighted extended top-down tree transducer can also be computed by a weighted multi bottom-up tree transducer. This further motivates weighted multi bottom-up tree transducers as suitable translation models for syntax-based machine translation.",
"pdf_parse": {
"paper_id": "N12-1027",
"_pdf_hash": "",
"abstract": [
{
"text": "A tree transformation is sensible if the size of each output tree is uniformly bounded by a linear function in the size of the corresponding input tree. Every sensible tree transformation computed by an arbitrary weighted extended top-down tree transducer can also be computed by a weighted multi bottom-up tree transducer. This further motivates weighted multi bottom-up tree transducers as suitable translation models for syntax-based machine translation.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Abstract",
"sec_num": null
}
],
"body_text": [
{
"text": "Several different translation models are used in syntax-based statistical machine translation. Koehn (2010) presents an introduction to statistical machine translation, and Knight (2007) presents an overview of syntax-based statistical machine translation. The oldest and best-studied tree transformation device is the top-down tree transducer of Rounds (1970) and Thatcher (1970) . G\u00e9cseg and Steinby (1984) and F\u00fcl\u00f6p and Vogler (2009) present the existing results on the unweighted and weighted model, respectively. Knight (2007) promotes the use of weighted extended top-down tree transducers (XTOP), which have also been implemented in the toolkit TIBURON by May and Knight (2006) [more detail is reported by May (2010) ]. In the context of bimorphisms, Dauchet (1976) investigated XTOP, and Lilin (1978) and Arnold and Dauchet (1982) investigated multi bottom-up tree transducers (MBOT) [as k-morphisms]. Recently, weighted XTOP and MBOT, which are the central devices in this contribution, were investigated by Maletti (2011a) in the context of statistical machine translation.",
"cite_spans": [
{
"start": 95,
"end": 107,
"text": "Koehn (2010)",
"ref_id": "BIBREF20"
},
{
"start": 173,
"end": 186,
"text": "Knight (2007)",
"ref_id": "BIBREF19"
},
{
"start": 347,
"end": 360,
"text": "Rounds (1970)",
"ref_id": "BIBREF32"
},
{
"start": 365,
"end": 380,
"text": "Thatcher (1970)",
"ref_id": "BIBREF36"
},
{
"start": 383,
"end": 408,
"text": "G\u00e9cseg and Steinby (1984)",
"ref_id": "BIBREF13"
},
{
"start": 413,
"end": 436,
"text": "F\u00fcl\u00f6p and Vogler (2009)",
"ref_id": "BIBREF11"
},
{
"start": 518,
"end": 531,
"text": "Knight (2007)",
"ref_id": "BIBREF19"
},
{
"start": 663,
"end": 684,
"text": "May and Knight (2006)",
"ref_id": "BIBREF28"
},
{
"start": 713,
"end": 723,
"text": "May (2010)",
"ref_id": "BIBREF30"
},
{
"start": 758,
"end": 795,
"text": "Dauchet (1976) investigated XTOP, and",
"ref_id": null
},
{
"start": 796,
"end": 808,
"text": "Lilin (1978)",
"ref_id": "BIBREF21"
},
{
"start": 813,
"end": 838,
"text": "Arnold and Dauchet (1982)",
"ref_id": "BIBREF2"
},
{
"start": 1017,
"end": 1032,
"text": "Maletti (2011a)",
"ref_id": "BIBREF26"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Several tree transformation devices are used as translation models in statistical machine translation. Chiang (2007) uses synchronous context-free grammars, which force translations to be very similar as observed by Eisner (2003) and Shieber (2004) . This deficiency is overcome by synchronous tree substitution grammars, which are state-less linear and nondeleting XTOP. Recently, Maletti (2010b) proposed MBOT, and Zhang et al. (2008b) and Sun et al. (2009) proposed the even more powerful synchronous tree-sequence substitution grammars. Those two models allow certain translation discontinuities, and the former device also offers computational benefits over linear and nondeleting XTOP as argued by Maletti (2010b) .",
"cite_spans": [
{
"start": 103,
"end": 116,
"text": "Chiang (2007)",
"ref_id": "BIBREF3"
},
{
"start": 216,
"end": 229,
"text": "Eisner (2003)",
"ref_id": "BIBREF5"
},
{
"start": 234,
"end": 248,
"text": "Shieber (2004)",
"ref_id": "BIBREF34"
},
{
"start": 382,
"end": 397,
"text": "Maletti (2010b)",
"ref_id": "BIBREF25"
},
{
"start": 407,
"end": 437,
"text": "MBOT, and Zhang et al. (2008b)",
"ref_id": null
},
{
"start": 442,
"end": 459,
"text": "Sun et al. (2009)",
"ref_id": "BIBREF35"
},
{
"start": 704,
"end": 719,
"text": "Maletti (2010b)",
"ref_id": "BIBREF25"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "The simplicity of XTOP makes them very appealing as translation models. In 2010 the ATANLP participants [workshop at ACL] identified 'copying' as the most exciting and promising feature of XTOP, but unrestricted copying can lead to an undesirable explosion of the size of the translation. According to Engelfriet and Maneth (2003) a tree transformation has linear size-increase if the size of each output tree is linearly bounded by the size of its corresponding input tree. The author believes that this is a very sensible restriction that intuitively makes sense and at the same time suitably limits the copying power of XTOP.",
"cite_spans": [
{
"start": 302,
"end": 330,
"text": "Engelfriet and Maneth (2003)",
"ref_id": "BIBREF6"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "We show that every sensible tree transformation that can be computed by an XTOP can also be computed by an MBOT. For example, linear XTOP (i.e., no copying) compute only sensible tree transformations, and Maletti (2008) shows that for each linear XTOP there exists an equivalent MBOT. Here, we do not make any restrictions on the XTOP besides some sanity conditions (see Section 3). In particular, we consider copying XTOP. If we accept the restriction to linear size-increase tree transformation, then our main result further motivates MBOT as a suitable translation model for syntax-based machine translation because MBOT can implement each reasonable (even copying) XTOP. In addition, our result allows us to show that each reasonable XTOP preserves regularity under backward application. As demonstrated by backward application is the standard application of XTOP in the machine translation pipeline, and preservation of regularity is the essential property for several of the evaluation algorithms of .",
"cite_spans": [
{
"start": 205,
"end": 219,
"text": "Maletti (2008)",
"ref_id": "BIBREF23"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "We start by introducing our notation for trees, whose nodes are labeled by elements of an alphabet \u03a3 and a set V . However, only leaves can be labeled by elements of V . For every set T , we let",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "\u03a3(T ) = {\u03c3(t 1 , . . . , t k ) | \u03c3 \u2208 \u03a3, t 1 , . . . , t k \u2208 T } ,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "which contains all trees with a \u03a3-labeled root and direct successors in T . The set T \u03a3 (V ) of \u03a3-trees with V -leaves is the smallest set T such that V \u222a \u03a3(T ) \u2286 T . We use X = {x 1 , x 2 , . . . } as a set of formal variables. Each node of the tree t \u2208 T \u03a3 (V ) is identified by a position p \u2208 N + , which is a sequence of positive integers. The root is at position \u03b5 (the empty string), and the position ip with i \u2208 N + and p \u2208 N * + is the position p in the i-th direct subtree. The set pos(t) contains all positions of t, and the size of t is |t| = |pos(t)|. For each p \u2208 pos(t), the label of t at p is t(p). Given a set L \u2286 \u03a3 \u222a V of labels, we let pos L (t) = {p \u2208 pos(t) | t(p) \u2208 L} be the positions with L-labels. We write pos l (t) for pos {l} (t) for each l \u2208 L. Finally, we write t[u] p for the tree obtained from t by replacing the subtree at position p by the tree u \u2208 T \u03a3 (V ).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "The following notions refer to the variables X.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "The tree t \u2208 T \u03a3 (V ) [potentially V \u2229 X = \u2205] is S \u03b5 NP 1 PP 11 x 111 2 VP 2 VBD 21 ran 211 RB 22",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "away 221 Figure 1 : The tree t (with positions indicated as superscripts) is linear and var(t) = {x 2 }. The tree t [He] 111 is the same tree with x 2 replaced by 'He'.",
"cite_spans": [
{
"start": 116,
"end": 120,
"text": "[He]",
"ref_id": null
}
],
"ref_spans": [
{
"start": 9,
"end": 17,
"text": "Figure 1",
"ref_id": null
}
],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "linear if every x \u2208 X occurs at most once in t (i.e., |pos",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "x (t)| \u2264 1). Moreover, var(t) = {x \u2208 X | pos x (t) = \u2205}",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "contains the variables that occur in t. A substitution \u03b8 is a mapping \u03b8 : X \u2192 T \u03a3 (V ). When applied to t, it returns the tree t\u03b8, which is obtained from t by replacing all occurrences of x \u2208 X in t by \u03b8(x). Our notions for trees are illustrated in Figure 1 .",
"cite_spans": [],
"ref_spans": [
{
"start": 249,
"end": 257,
"text": "Figure 1",
"ref_id": null
}
],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "Finally, we present weighted tree grammars (WTG) as defined by F\u00fcl\u00f6p and Vogler (2009) , who defined it for arbitrary semirings as weight structures. In contrast, our weights are always nonnegative reals, which form the semiring (R + , +, \u2022, 0, 1) and are used in probabilistic grammars. For each weight assignment f :",
"cite_spans": [
{
"start": 63,
"end": 86,
"text": "F\u00fcl\u00f6p and Vogler (2009)",
"ref_id": "BIBREF11"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "T \u2192 R + , we let supp(f ) = {t \u2208 T | f (t) = 0} .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "WTG offer an efficient representation of weighted forests (i.e., set of weighted trees), which is even more efficient than the packed forests of Mi et al. (2008) because they can be minimized efficiently using an algorithm of Maletti and Quernheim (2011) . In particular, WTG can share more than equivalent subtrees and can even represent infinite sets of trees. A WTG is a system G = (Q, \u03a3, q 0 , P, wt) with",
"cite_spans": [
{
"start": 145,
"end": 161,
"text": "Mi et al. (2008)",
"ref_id": "BIBREF31"
},
{
"start": 226,
"end": 254,
"text": "Maletti and Quernheim (2011)",
"ref_id": "BIBREF22"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "\u2022 a finite set Q of states (nonterminals),",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "\u2022 an alphabet \u03a3 of symbols, \u2022 a starting state q 0 \u2208 Q, \u2022 a finite set P of productions q \u2192 r, where q \u2208 Q and r \u2208 T \u03a3 (Q) \\ Q, and \u2022 a mapping wt : P \u2192 R + that assigns production weights. Without loss of generality, we assume that we can distinguish states and symbols (i.e., Q \u2229 \u03a3 = \u2205). For all \u03be, \u03b6 \u2208 T \u03a3 (Q) and a production \u03c1 we write \u03be \u21d2 \u03c1 G \u03b6 if \u03be = \u03be[q] p and \u03b6 = \u03be[r] p , where p is the lexicographically least element of pos Q (\u03be). The WTG G generates the weighted tree language L G :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "= q \u2192 r, S t 1 VP t 2 t 3 \u2192 S t 2 t 1 t 3",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "T \u03a3 \u2192 R + such that L G (t) = n\u2208N,\u03c1 1 ,...,\u03c1n\u2208P q 0 \u21d2 \u03c1 1 G \u2022\u2022\u2022\u21d2 \u03c1n G t wt(\u03c1 1 ) \u2022 . . . \u2022 wt(\u03c1 n )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "for every t \u2208 T \u03a3 . Each such language is regular, and Reg(\u03a3) contains all those languages over the alphabet \u03a3. A thorough introduction to tree languages is presented by G\u00e9cseg and Steinby (1984) and G\u00e9cseg and Steinby (1997) for the unweighted case and by F\u00fcl\u00f6p and Vogler (2009) for the weighted case.",
"cite_spans": [
{
"start": 170,
"end": 195,
"text": "G\u00e9cseg and Steinby (1984)",
"ref_id": "BIBREF13"
},
{
"start": 200,
"end": 225,
"text": "G\u00e9cseg and Steinby (1997)",
"ref_id": "BIBREF14"
},
{
"start": 257,
"end": 280,
"text": "F\u00fcl\u00f6p and Vogler (2009)",
"ref_id": "BIBREF11"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Notation",
"sec_num": "2"
},
{
"text": "We start by introducing the main model of this contribution. Extended top-down tree transducers (XTOP) are a generalization of the top-down tree transducers (TOP) of Rounds (1970) and Thatcher (1970) . XTOP allow rules with several (non-state and non-variable) symbols in the left-hand side (as in the rule of Figure 3 ), whereas a TOP rule contains exactly one symbol in the left-hand side. Shieber (2004) and Knight (2007) identified that this extension is essential for many NLP applications because without it linear (i.e., non-copying) cannot compute rotations (see Figure 2 ). In the form of bimorphisms XTOP were investigated by Arnold and Dauchet (1976) and Arnold and Dauchet (1982) in the 1970s, and Knight (2007) invigorated research. As demonstrated by Graehl et al. (2009) the most general XTOP model includes copying, deletion, and regular look-ahead in the spirit of Engelfriet (1977) . More powerful models (such as synchronous tree-sequence substitution grammars and multi bottom-up tree transducers) can handle translation discontinuities naturally as evidenced by Zhang et al. (2008a) and Maletti (2011b) , but",
"cite_spans": [
{
"start": 166,
"end": 179,
"text": "Rounds (1970)",
"ref_id": "BIBREF32"
},
{
"start": 184,
"end": 199,
"text": "Thatcher (1970)",
"ref_id": "BIBREF36"
},
{
"start": 400,
"end": 406,
"text": "(2004)",
"ref_id": null
},
{
"start": 411,
"end": 424,
"text": "Knight (2007)",
"ref_id": "BIBREF19"
},
{
"start": 636,
"end": 661,
"text": "Arnold and Dauchet (1976)",
"ref_id": "BIBREF1"
},
{
"start": 666,
"end": 691,
"text": "Arnold and Dauchet (1982)",
"ref_id": "BIBREF2"
},
{
"start": 710,
"end": 723,
"text": "Knight (2007)",
"ref_id": "BIBREF19"
},
{
"start": 765,
"end": 785,
"text": "Graehl et al. (2009)",
"ref_id": "BIBREF17"
},
{
"start": 882,
"end": 899,
"text": "Engelfriet (1977)",
"ref_id": "BIBREF10"
},
{
"start": 1083,
"end": 1103,
"text": "Zhang et al. (2008a)",
"ref_id": "BIBREF37"
},
{
"start": 1108,
"end": 1123,
"text": "Maletti (2011b)",
"ref_id": "BIBREF27"
}
],
"ref_spans": [
{
"start": 310,
"end": 318,
"text": "Figure 3",
"ref_id": null
},
{
"start": 571,
"end": 579,
"text": "Figure 2",
"ref_id": "FIGREF0"
}
],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "q 0 S x 1 VP x 2 x 3 \u2192 S q VB x 2 q NP x 1 q NP x 3",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "Figure 3: Example XTOP rule by Graehl et al. (2008) .",
"cite_spans": [
{
"start": 31,
"end": 51,
"text": "Graehl et al. (2008)",
"ref_id": "BIBREF16"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "XTOP need copying and deletion to handle them. Copying essentially allows an XTOP to translate certain parts of the input several times and was identified by the ATANLP 2010 participants as one of the most interesting and promising features of XTOP. Currently, the look-ahead feature is not used in machine translation, but we need it later on in the theoretical development.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "Given an alphabet Q and a set T , we let",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "Q[T ] = {q(t) | q \u2208 Q, t \u2208 T },",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "in which the root always has exactly one successor from T in contrast to Q(T ). We treat elements of",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "Q[T \u03a3 (V )] as special trees of T \u03a3\u222aQ (V ).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "Moreover, we let 1 \u03a3 (t) = 1 for every t \u2208 T \u03a3 . XTOP with regular look-ahead (XTOP R ) were also studied by Knight and Graehl (2005) and Graehl et al. (2008) . Formally, an XTOP R is a system",
"cite_spans": [
{
"start": 109,
"end": 133,
"text": "Knight and Graehl (2005)",
"ref_id": "BIBREF19"
},
{
"start": 138,
"end": 158,
"text": "Graehl et al. (2008)",
"ref_id": "BIBREF16"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "M = (Q, \u03a3, \u2206, q 0 , R, c, wt) with \u2022 a finite set Q of states,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "\u2022 alphabets \u03a3 and \u2206 of input and output symbols,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "\u2022 a starting state q 0 \u2208 Q, \u2022 a finite set R of rules of the form \u2192 r with linear \u2208 Q[T \u03a3 (X)] and r \u2208 T \u2206 (Q[var( )]), \u2022 c : R \u00d7 X \u2192 Reg(\u03a3) assigns a regular look- ahead to each deleted variable of a rule [i.e., c( \u2192 r, x) = 1 \u03a3 for all \u2192 r \u2208 R and x \u2208 X \\ (var( ) \\ var(r))]",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": ", and",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "\u2022 wt : R \u2192 R + assigns rule weights. The XTOP R M is linear [respectively, nondeleting]",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "if r is linear [respectively, var( ) = var(r)] for every rule \u2192 r \u2208 R. It has no look-ahead (XTOP) if c(\u03c1, x) = 1 \u03a3 for all \u03c1 \u2208 R and x \u2208 X. Figure 3 shows a rule of a linear and nondeleting XTOP.",
"cite_spans": [],
"ref_spans": [
{
"start": 141,
"end": 149,
"text": "Figure 3",
"ref_id": null
}
],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "The look-ahead can be used to restrict rule applications. It can inspect subtrees that are deleted by a rule application, so for each rule \u03c1 = \u2192 r, we let del(\u03c1) = var( ) \\ var(r) be the set of deleted variables in \u03c1. If we suppose that a variable x \u2208 del(\u03c1) matches to an input subtree t, then the weight of the look-ahead c(\u03c1, x)(t), which we also write c \u03c1,x (t), is applied to the derivation. If it is 0, then this lookahead essentially prohibits the application of \u03c1. It is important that the look-ahead is regular (i.e., there exists a WTG accepting it). The toolkit TIBURON by May and Knight (2006) implements XTOP together with a number of essential operations. Lookahead is not implemented in TIBURON, but it can be simulated using a composition of two XTOP, in which the first XTOP performs the look-ahead and marks the results, so that the second XTOP can access the look-ahead information. As for WTG the semantics for the XTOP R M = (Q, \u03a3, \u2206, I, R, c, wt) is presented using rewriting. Without loss of generality, we again suppose that Q \u2229 (\u03a3 \u222a \u2206) = \u2205. Let \u03be, \u03b6 \u2208 T \u2206 (Q[T \u03a3 ]), w \u2208 R + , and \u03c1 = \u2192 r be a rule of R. We write \u03be \u21d2 \u03c1,w M \u03b6 if there exists a substitution \u03b8 :",
"cite_spans": [
{
"start": 584,
"end": 605,
"text": "May and Knight (2006)",
"ref_id": "BIBREF28"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "u q 0 S t 1 VP t 2 t 3 \u21d2 \u03c1,.5 M u S q VB t 2 q NP t 1 q NP t 3",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "X \u2192 T \u03a3 such that \u2022 \u03be = \u03be[ \u03b8] p , \u2022 \u03b6 = \u03be[r\u03b8] p , and \u2022 w = wt(\u03c1) \u2022 x\u2208del(\u03c1) c \u03c1,x (x\u03b8)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": ", where p \u2208 pos Q (\u03be) is the lexicographically least Q-labeled position in \u03be. Figure 4 illustrates a derivation step.",
"cite_spans": [],
"ref_spans": [
{
"start": 78,
"end": 86,
"text": "Figure 4",
"ref_id": "FIGREF1"
}
],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "The XTOP R M computes a weighted tree transformation by applying rewrite steps to the tree q 0 (t), where t \u2208 T \u03a3 is the input tree, until an output tree u \u2208 T \u2206 has been produced. The weight of a particular derivation is obtained by multiplying the weights of the rewrite steps. The weight of the transformation from t to u is obtained by summing all weights of the derivations from q 0 (t) to u. Formally 1 , the weighted tree transformation computed by M in state q \u2208 Q is",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "\u03c4 q M (t, u) = n\u2208N,\u03c1 1 ,...,\u03c1n\u2208R q(t)\u21d2 \u03c1 1 ,w 1 M \u2022\u2022\u2022\u21d2 \u03c1n,wn M u w 1 \u2022 . . . \u2022 w n (1)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "for every t \u2208 T \u03a3 and u \u2208 T \u2206 . The XTOP R M computes the weighted tree transformation",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "\u03c4 q 0 M . Two XTOP R M and N are equivalent, if \u03c4 M = \u03c4 N .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "The sum (1) can be infinite, which we avoid by simply requiring that all our XTOP R are producing, which means that r / \u2208 Q[X] for every rule \u2192 r \u2208 R. 2 In a producing XTOP R each rule application produces at least one output symbol, which limits the number n of rule applications to the size of the output tree u. A detailed exposition to XTOP R is presented by Arnold and Dauchet (1982) and Graehl et al. (2009) for the unweighted case and by F\u00fcl\u00f6p and Vogler (2009) for the weighted case.",
"cite_spans": [
{
"start": 363,
"end": 388,
"text": "Arnold and Dauchet (1982)",
"ref_id": "BIBREF2"
},
{
"start": 393,
"end": 413,
"text": "Graehl et al. (2009)",
"ref_id": "BIBREF17"
},
{
"start": 445,
"end": 468,
"text": "F\u00fcl\u00f6p and Vogler (2009)",
"ref_id": "BIBREF11"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "Example 1. Let M ex = (Q, \u03a3, \u03a3, q, R, c, wt) be the nondeleting XTOP with \u2022 Q = {q}, \u2022 \u03a3 = {\u03c3, \u03b3, \u03b1},",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "\u2022 the two rules",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "q(\u03b1) \u2192 \u03b1 (\u03c1) q(\u03b3(x 1 )) \u2192 \u03c3(q(x 1 ), q(x 1 )) (\u03c1 )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "\u2022 trivial look-ahead (i.e., c(\u03c1, x) = 1 \u03a3 ), and \u2022 wt(\u03c1) = 2 and wt(\u03c1 ) = 1. The XTOP R M ex computes the tree transformation that turns the input tree \u03b3 n (\u03b1) into the fully balanced binary tree u of the same height with weight 2 (2 n ) . An example derivation is presented in Figure 5 .",
"cite_spans": [],
"ref_spans": [
{
"start": 278,
"end": 286,
"text": "Figure 5",
"ref_id": "FIGREF2"
}
],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "Unrestricted copying (as in Example 1) yields very undesirable phenomena and is most likely not needed in the machine translation task. In fact, it is almost universally agreed that a translation model should be \"linear-size increase\", which means that the size of each output tree should be linearly bounded in the size of the corresponding input tree according to Aho and Ullman (1971) and Engelfriet and Maneth (2003) .",
"cite_spans": [
{
"start": 366,
"end": 387,
"text": "Aho and Ullman (1971)",
"ref_id": "BIBREF0"
},
{
"start": 392,
"end": 420,
"text": "Engelfriet and Maneth (2003)",
"ref_id": "BIBREF6"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "Definition 2. A mapping \u03c4 :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "T \u03a3 \u00d7 T \u2206 \u2192 R + is linear-size increase if there exists an integer n \u2208 N such that |u| \u2264 n \u2022 |t| for all (t, u) \u2208 supp(\u03c4 ). An XTOP R M is sensible if \u03c4 M is linear-size in- crease.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "'Sensible' is not a syntactic property of an XTOP R as it does not depend on the actual rules, but only on its computed weighted tree transformation. The XTOP M ex of Example 1 is not sensible because |u| = 2 |t| \u2212 1 for every (t, u) \u2208 \u03c4 Mex . Intuitively, the number of times that M ex can use the copying rule \u03c1 is not uniformly bounded.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "We need an auxiliary result in the main part. Let \u03c4 : T \u03a3 \u00d7 T \u2206 \u2192 R + be a weighted tree transformation. We need the weighted tree language \u03c4 \u22121 (u) : T \u03a3 \u2192 R + of input trees weighted by their translation weight to a given output tree u \u2208 T \u2206 . Formally, \u03c4 \u22121 (u) (t) = \u03c4 (t, u) for every t \u2208 T \u03a3 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "Theorem 3. For every producing XTOP R M and output tree u \u2208 T \u2206 , the weighted tree language \u03c4 \u22121 M (u ) is regular.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "Proof sketch. We use some properties that are only defined in the next sections (for proof economy). It is recommended to skip this proof on the first reading and revisit it later. Maletti (2010a) shows that we can construct an XTOP R M such that",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "\u03c4 M (t, u) = \u03c4 M (t, u) if u = u 0 otherwise",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "for every t \u2208 T \u03a3 and u \u2208 T \u2206 . This operation is called 'output product' by Maletti (2010a) . The obtained XTOP R M is also producing, so we know that M can take at most |u | rewrite steps to derive u . Since M can only produce the output tree u , this also limits the total number of rule applications in any successful derivation. Consequently, M can only apply a copying rule at most |u | times, which shows that M is finitely copying (see Definition 8). By Theorem 11 we can implement M by an equivalent MBOT M (i.e., \u03c4 M = \u03c4 M ; see Section 5), for which we know by Theorem 14 of Maletti (2011a) ",
"cite_spans": [
{
"start": 77,
"end": 92,
"text": "Maletti (2010a)",
"ref_id": "BIBREF24"
},
{
"start": 586,
"end": 601,
"text": "Maletti (2011a)",
"ref_id": "BIBREF26"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "that \u03c4 \u22121 M (u) = \u03c4 \u22121 M (u) is regu- lar.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "Finally, let us illustrate the overall structure of our arguments to show that every sensible XTOP R can be implemented by an equivalent MBOT. We first normalize the given XTOP R such that the semantic property 'sensible' yields a syntactic property called 'finitely copying' (see Section 4). In a second step, we show that each finitely copying XTOP R can be implemented by an equivalent MBOT (see Section 5). Figure 6 illustrates these steps towards our main result. In the final section, we derive some consequences from our main result (see Section 6).",
"cite_spans": [],
"ref_spans": [
{
"start": 411,
"end": 419,
"text": "Figure 6",
"ref_id": "FIGREF3"
}
],
"eq_spans": [],
"section": "Extended top-down tree transducers",
"sec_num": "3"
},
{
"text": "First, we adjust a normal form of Engelfriet and Maneth (2003) to our needs. This section borrows heavily from Aho and Ullman (1971) and Engelfriet and Maneth (2003) , where \"sensible\" (unweighted) deterministic macro tree transducers (MAC) [see Engelfriet and Vogler (1985) ] are considered. Our setting is simpler on the one hand because XTOP R do not have context parameters as MAC, but more difficult on the other hand because we consider nondeterministic and weighted transducers.",
"cite_spans": [
{
"start": 34,
"end": 62,
"text": "Engelfriet and Maneth (2003)",
"ref_id": "BIBREF6"
},
{
"start": 111,
"end": 132,
"text": "Aho and Ullman (1971)",
"ref_id": "BIBREF0"
},
{
"start": 137,
"end": 165,
"text": "Engelfriet and Maneth (2003)",
"ref_id": "BIBREF6"
},
{
"start": 246,
"end": 274,
"text": "Engelfriet and Vogler (1985)",
"ref_id": "BIBREF7"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "From sensible to finite copying",
"sec_num": "4"
},
{
"text": "Intuitively, a sensible XTOP R cannot copy a lot since the size of each output tree is linearly bounded in the size of the corresponding input tree. However, the actual presentation of the XTOP R M might con- tain rules that allow unbounded copying. This unbounded copying might not manifest due to the lookahead restrictions or due to the fact that those rules cannot be used in a successful derivation. The purpose of the normal form is the elimination of those artifacts. To this end, we eliminate all states (except the initial state) that can only produce finitely many outputs. Such a state can simply be replaced by one of the output trees that it can produce and an additional look-ahead that checks whether the current input tree indeed allows that translation (and inserts the correct translation weight).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From sensible to finite copying",
"sec_num": "4"
},
{
"text": "Normalized XTOP R are called 'proper', and we define this property next. For the rest of this section, let M = (Q, \u03a3, \u2206, q 0 , R, c, wt) be the considered sensible XTOP R . Without loss of generality, we assume that the state q 0 does not occur in the righthand sides of rules. Moreover, we write \u03be \u21d2 * M \u03b6 if there exist nonzero weights w 1 , . . . , w n \u2208 R + \\ {0} and rules \u03c1 1 , . . . , \u03c1 n \u2208 R with",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From sensible to finite copying",
"sec_num": "4"
},
{
"text": "\u03be \u21d2 \u03c1 1 ,w 1 M \u2022 \u2022 \u2022 \u21d2 \u03c1n,wn M \u03b6 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From sensible to finite copying",
"sec_num": "4"
},
{
"text": "In essence, \u03be \u21d2 * M \u03b6 means that M can transform \u03be into \u03b6 (in the unweighted setting). Definition 4. A state q \u2208 Q is proper if there are infinitely many u \u2208 T \u2206 such that there exists a derivation",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From sensible to finite copying",
"sec_num": "4"
},
{
"text": "q 0 (t) \u21d2 * M \u03be[q(s)] p \u21d2 * M u[u ] p where s, t \u2208 T \u03a3 are input trees, \u03be \u2208 T \u2206 (Q[T \u03a3 ])",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From sensible to finite copying",
"sec_num": "4"
},
{
"text": ", p \u2208 pos(\u03be), and u \u2208 T \u2206 is an output tree. Figure 7 . In other words, a proper state is reachable from the initial state and can transform infinitely many input trees into infinitely many output trees. The latter is an immediate consequence of Definition 4 since each input tree can be transformed into only finitely many output trees due to sensibility. The restriction includes the look-ahead (because we require that the rewrite step weights are nonzero), which might further restrict the input trees.",
"cite_spans": [],
"ref_spans": [
{
"start": 45,
"end": 53,
"text": "Figure 7",
"ref_id": "FIGREF4"
}
],
"eq_spans": [],
"section": "From sensible to finite copying",
"sec_num": "4"
},
{
"text": "Example 5. The state q of the XTOP M ex is proper because we already demonstrated that it can transform infinitely many input trees into infinitely many output trees.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "The derivation in Definition 4 is illustrated in",
"sec_num": null
},
{
"text": "The XTOP R M is proper if all its states except the initial state q 0 are proper. Next, we show that each XTOP R can be transformed into an equivalent proper XTOP R using a simplified version of the construction of Lemma 5.4 by Engelfriet and Maneth (2003) . Mind that we generally assume that all considered XTOP R are producing.",
"cite_spans": [
{
"start": 228,
"end": 256,
"text": "Engelfriet and Maneth (2003)",
"ref_id": "BIBREF6"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "The derivation in Definition 4 is illustrated in",
"sec_num": null
},
{
"text": "Theorem 6. For every XTOP R there exists an equivalent proper XTOP R .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "The derivation in Definition 4 is illustrated in",
"sec_num": null
},
{
"text": "Proof sketch. The construction is iterative. Suppose that M is not yet proper. Then there exists a state q \u2208 Q, which can produce only finitely many outputs U . It can be decided whether a state is proper using Theorem 4.5 of Drewes and Engelfriet (1998) , and in case it is proper, the set U can also be computed effectively. The cited theorem applies to unweighted XTOP R , but it can be applied also in our setting because \u21d2 * M in Definition 4 disregards weights. Now we consider each u \u2208 U individually. Clearly, (\u03c4 q M ) \u22121 (u) is regular by Theorem 3. For each u and each occurrence of q in the right-hand side of a rule \u03c1 \u2208 R of M , we create a copy \u03c1 of \u03c1, in which the selected occurrence of q(x) is replaced by u and the new lookahead is c(\u03c1 , x) = c(\u03c1, x) \u2022 (\u03c4 q M ) \u22121 (u), which restricts the input tree appropriately and includes the adjustment of the weights. Since regular weighted tree languages are closed under HADAMARD products [see F\u00fcl\u00f6p and Vogler (2009) ], the look-ahead c(\u03c1, x) \u2022 (\u03c4 q M ) \u22121 (u) is again regular. Essentially, we precompute the action of q as much as possible, and immediately output one of the finitely many output trees, check that the input tree has the required shape using the look-ahead, and charge the weight for the precomputed transformation again using the look-ahead. This process is done for each occurrence, so if a rule contains two occurrences of q, then the process must be done twice to this rule. In this way, we eventually purge all occurrences of q from the right-hand sides of rules of M without changing the computed transformation. Since q = q 0 and q is now unreachable, it is useless and can be deleted, which removes one non-proper state. This process is repeated until all states except the initial state q 0 are proper.",
"cite_spans": [
{
"start": 226,
"end": 254,
"text": "Drewes and Engelfriet (1998)",
"ref_id": "BIBREF4"
},
{
"start": 954,
"end": 977,
"text": "F\u00fcl\u00f6p and Vogler (2009)",
"ref_id": "BIBREF11"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "The derivation in Definition 4 is illustrated in",
"sec_num": null
},
{
"text": "q 0 t \u21d2 * M . . . . . . q s \u21d2 * M . . . . . . u",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "The derivation in Definition 4 is illustrated in",
"sec_num": null
},
{
"text": "Clearly, the construction of Theorem 6 applied to a sensible XTOP R M yields a sensible proper XTOP R M since the property 'sensible' refers to the computed transformation and \u03c4 M = \u03c4 M . Let us illustrate the construction on a small example.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "The derivation in Definition 4 is illustrated in",
"sec_num": null
},
{
"text": "Example 7. Let \u03c1 be the rule displayed in Figure 3 , and let us assume that the state q VB is not proper. Moreover, suppose that q VB can yield the output tree u and that we already computed the translation options that yield u. Let t 1 , . . . , t n \u2208 T \u03a3 be those translation options. Then we create the copy \u03c1",
"cite_spans": [],
"ref_spans": [
{
"start": 42,
"end": 50,
"text": "Figure 3",
"ref_id": null
}
],
"eq_spans": [],
"section": "The derivation in Definition 4 is illustrated in",
"sec_num": null
},
{
"text": "q 0 (S(x 1 , VP(x 2 , x 3 ))) \u2192 S(u, q NP (x 1 ), q NP (x 3 )) of the rule \u03c1 with look-ahead c (\u03c1 , x) such that c \u03c1 ,x (t) = c \u03c1,x (t) if x = x 2 \u03c4 q VB M (t, u) if x = x 2 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "The derivation in Definition 4 is illustrated in",
"sec_num": null
},
{
"text": "In general, there can be infinitely many input trees t i that translate to a selected output tree u, so we cannot simply replace the variable in the lefthand side by all the options for the input tree. This is the reason why we use the look-ahead because the set \u03c4 \u22121 M (u) is a regular weighted tree language. From now on, we assume that the XTOP R M is proper. Next, we want to invoke Theorem 7.1 of Engelfriet and Maneth (2003) to show that a proper sensible XTOP R is finitely copying. Engelfriet and Maneth (2003) present a formal definition of finite copying, but we only present a high-level description of it.",
"cite_spans": [
{
"start": 402,
"end": 430,
"text": "Engelfriet and Maneth (2003)",
"ref_id": "BIBREF6"
},
{
"start": 490,
"end": 518,
"text": "Engelfriet and Maneth (2003)",
"ref_id": "BIBREF6"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "The derivation in Definition 4 is illustrated in",
"sec_num": null
},
{
"text": "Definition 8. The XTOP R M is finitely copying if there is a copying bound n \u2208 N such that no input subtree is copied more than n times in any derivation q(t) \u21d2 * M u with q \u2208 Q, t \u2208 T \u03a3 , and u \u2208 T \u2206 . Example 9. The XTOP of Example 1 is not finitely copying as the input subtree \u03b1 is copied 2 n times if the input tree is \u03b3 n (\u03b1). Clearly, this shows that there is no uniform bound on the number of copies.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "The derivation in Definition 4 is illustrated in",
"sec_num": null
},
{
"text": "It is worth noting that the properties 'sensible' and 'finitely copying' are essentially unweighted properties. They largely disregard the weights and a weighted XTOP R does have one of those properties if and only if its associated unweighted XTOP R has it. We now use this tight connection to lift Theorem 7.1 of Engelfriet and Maneth (2003) from the unweighted (and deterministic) case to the weighted (and nondeterministic) case.",
"cite_spans": [
{
"start": 315,
"end": 343,
"text": "Engelfriet and Maneth (2003)",
"ref_id": "BIBREF6"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "The derivation in Definition 4 is illustrated in",
"sec_num": null
},
{
"text": "Theorem 10. If a proper XTOP R is sensible, then it is finitely copying.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "The derivation in Definition 4 is illustrated in",
"sec_num": null
},
{
"text": "Proof. Let M be the input XTOP R . Since M is sensible, its associated unweighted XTOP R N , which is obtained by setting all weights to 1 and computing in the BOOLEAN semiring, is sensible. Consequently, N is finitely copying by Theorem 7.1 of Engelfriet and Maneth (2003) . Thus, also M is finitely copying, which concludes the proof. We remark that Theorem 7.1 of Engelfriet and Maneth (2003) only applies to deterministic XTOP R , but the essential pumping argument, which is Lemma 6.2 of Engelfriet and Maneth (2003) also works for nondeterministic XTOP R . Essentially, the pumping argument shows the contraposition. If M is not finitely copying, then M can copy a certain subtree an arbitrarily often. Due to the properness of M , all these copies have an impact on the output tree, which yields that its size grows beyond any uniform linear bound, which in turn demonstrates that M is not sensible.",
"cite_spans": [
{
"start": 245,
"end": 273,
"text": "Engelfriet and Maneth (2003)",
"ref_id": "BIBREF6"
},
{
"start": 367,
"end": 395,
"text": "Engelfriet and Maneth (2003)",
"ref_id": "BIBREF6"
},
{
"start": 493,
"end": 521,
"text": "Engelfriet and Maneth (2003)",
"ref_id": "BIBREF6"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "The derivation in Definition 4 is illustrated in",
"sec_num": null
},
{
"text": "We showed that each sensible XTOP R can be implemented by a finitely copying XTOP R via the construction of the proper normal form. This approach actually yields a characterization because finitely copying XTOP R are trivially sensible by Theorem 4.19 of Engelfriet and Maneth (2003) .",
"cite_spans": [
{
"start": 255,
"end": 283,
"text": "Engelfriet and Maneth (2003)",
"ref_id": "BIBREF6"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "The derivation in Definition 4 is illustrated in",
"sec_num": null
},
{
"text": "We complete the argument by showing how to implement a finitely copying XTOP R by a weighted multi bottom-up tree transducer (MBOT). First, we recall the MBOT, which was introduced by Arnold and Dauchet (1982) and Lilin (1978) in the unweighted case. Engelfriet et al. (2009) give an English presentation. We present the linear and nondeleting MBOT of Engelfriet et al. (2009) .",
"cite_spans": [
{
"start": 184,
"end": 209,
"text": "Arnold and Dauchet (1982)",
"ref_id": "BIBREF2"
},
{
"start": 214,
"end": 226,
"text": "Lilin (1978)",
"ref_id": "BIBREF21"
},
{
"start": 251,
"end": 275,
"text": "Engelfriet et al. (2009)",
"ref_id": "BIBREF8"
},
{
"start": 352,
"end": 376,
"text": "Engelfriet et al. (2009)",
"ref_id": "BIBREF8"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "A weighted multi bottom-up tree transducer is a system M = (Q, \u03a3, \u2206, F, R, wt) with",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "\u2022 an alphabet Q of states,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "\u2022 alphabets \u03a3 and \u2206 of input and output symbols,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "\u2022 a set F \u2286 Q of final states,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "\u2022 a finite set R of rules of the form \u2192 r where \u2208 T \u03a3 (Q(X)) and r \u2208 Q(T \u2206 (X)) are linear and var( ) = var(r), and \u2022 wt : R \u2192 R + assigning rule weights. We now use T \u03a3 (Q(X)) and Q(T \u2206 (X)) instead of T \u03a3 (Q[X]) and Q[T \u2206 (X)], which highlights the difference between XTOP R and MBOT. First, MBOT are a bottom-up device, which yields that \u03a3 and \u2206 as well as and r exchange their place. More importantly, MBOT can use states with more than 1 successor (e.g, Q(X) instead of Q [X] ). An example rule is displayed in Figure 8 .",
"cite_spans": [
{
"start": 477,
"end": 480,
"text": "[X]",
"ref_id": null
}
],
"ref_spans": [
{
"start": 516,
"end": 524,
"text": "Figure 8",
"ref_id": "FIGREF5"
}
],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "Let M = (Q, \u03a3, \u2206, F, R, wt) be an MBOT such that Q\u2229(\u03a3\u222a\u2206) = \u2205. 3 We require that r / \u2208 Q(X) for each rule \u2192 r \u2208 R to guarantee finite derivations and thus a well-defined semantics. 4 As before, we present a rewrite semantics. Let \u03be, \u03b6 \u2208 T \u03a3 (Q(T \u2206 )), and let \u03c1 = \u2192 r be a rule. We write \u03be \u21d2 \u03c1 M \u03b6 if there exists a substitution \u03b8 : X \u2192 T \u2206 such that \u03be = \u03be[ \u03b8] p and \u03b6 = \u03be[r\u03b8] p , where p \u2208 pos(\u03be) be is the lexicographically least reducible position in \u03be. A rewrite step is illustrated in Figure 8 .",
"cite_spans": [
{
"start": 62,
"end": 63,
"text": "3",
"ref_id": null
},
{
"start": 180,
"end": 181,
"text": "4",
"ref_id": null
}
],
"ref_spans": [
{
"start": 489,
"end": 497,
"text": "Figure 8",
"ref_id": "FIGREF5"
}
],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "The weighted tree transformation computed by M in state q \u2208 Q is",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "\u03c4 q M (t, u 1 \u2022 \u2022 \u2022 u k ) = n\u2208N,\u03c1 1 ,...,\u03c1n\u2208R t\u21d2 \u03c1 1 M \u2022\u2022\u2022\u21d2 \u03c1n M q(u 1 ,...,u k ) wt(\u03c1 1 ) \u2022 . . . \u2022 wt(\u03c1 n )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "for all t \u2208 T \u03a3 and u 1 , . . . , u k \u2208 T \u2206 . The semantics of M is \u03c4 M (t, u) = q\u2208F \u03c4 q M (t, u) for all t \u2208 T \u03a3 and u \u2208 T \u2206 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "We move to the last step for our main result, in which we show how to implement each finitely copying XTOP R by an MBOT using a weighted version of the construction in Lemma 15 of Maletti (2008) . The computational benefits (binarization, composition, efficient parsing, etc.) of MBOT over XTOP R are described by Maletti (2011a) .",
"cite_spans": [
{
"start": 180,
"end": 194,
"text": "Maletti (2008)",
"ref_id": "BIBREF23"
},
{
"start": 314,
"end": 329,
"text": "Maletti (2011a)",
"ref_id": "BIBREF26"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "Theorem 11. Every finitely copying XTOP R can be implemented by an MBOT.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "Proof sketch. We plan to utilize Theorem 18 of Engelfriet et al. (2009) , which proves the same statement in the unweighted and deterministic case. Again, the weights are not problematic, but we need to remove the nondeterminism before we can apply it. This is achieved by a decomposition into two XTOP R . The first XTOP R annotates the input tree with the rules that the second XTOP R is supposed to use. Thus, the first XTOP R remains nondeterministic, but the second XTOP R , which simply executes the annotated rules, is now deterministic. This standard approach due to Engelfriet (1975) is used in many similar constructions.",
"cite_spans": [
{
"start": 47,
"end": 71,
"text": "Engelfriet et al. (2009)",
"ref_id": "BIBREF8"
},
{
"start": 575,
"end": 592,
"text": "Engelfriet (1975)",
"ref_id": "BIBREF9"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "Suppose that n is a copying bound for the input XTOP R M , which means that no more than n rules are applied to each input symbol. The first XTOP R is actually a nondeterministic linear and nondeleting XTOP that annotates each input tree symbol with exactly n rules of M that are consistent with the state behavior of M . Moreover, the annotation also prescribes with which of n rules the processing should continue at each subtree. Since we know all the rules that will potentially be applied for a certain symbol, we can make the assignment such that no annotated rule is used twice in the same derivation. The details for this construction can be found in Lemma 15 of Maletti (2008) .",
"cite_spans": [
{
"start": 671,
"end": 685,
"text": "Maletti (2008)",
"ref_id": "BIBREF23"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "In this way, we obtain a weighted linear and nondeleting XTOP M 1 , which includes the look-ahead, and an unweighted deterministic XTOP M 2 . Only the weight and look-ahead of rules that are actually executed are applied (e.g., although we annotate n rules at the root symbol, we only execute the first rule and thus only apply its weight and lookahead). The look-ahead of different rules is either resolved (i.e., pushed to the next rules) or multiplied using the HADAMARD product [see F\u00fcl\u00f6p and Vogler (2009) ], which preserves regularity. This process is also used by Seemann et al. (2012) . Now we can use Theorem 4 of Maletti (2011a) to obtain an MBOT N 1 that is equivalent to M 1 . Similarly, we can use Theorem 18 of Engelfriet et al. (2009) to obtain an MBOT N 2 that is equivalent to M 2 . Since MBOT are closed under composition by Theorem 23 of Engelfriet et al. (2009) , we can compose N 1 and N 2 to obtain a single MBOT N that is equivalent to M .",
"cite_spans": [
{
"start": 487,
"end": 510,
"text": "F\u00fcl\u00f6p and Vogler (2009)",
"ref_id": "BIBREF11"
},
{
"start": 571,
"end": 592,
"text": "Seemann et al. (2012)",
"ref_id": "BIBREF33"
},
{
"start": 623,
"end": 638,
"text": "Maletti (2011a)",
"ref_id": "BIBREF26"
},
{
"start": 725,
"end": 749,
"text": "Engelfriet et al. (2009)",
"ref_id": "BIBREF8"
},
{
"start": 857,
"end": 881,
"text": "Engelfriet et al. (2009)",
"ref_id": "BIBREF8"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "S q NP x 1 VP q VB x 2 q NP x 3 x 4 \u2192 q S S x 2 x 1 x 3 x 4 t S q NP u 1 VP q VB u 2 q NP u 3 u 4 \u21d2 \u03c1 M t q S S u 2 u 1 u 3 u 4",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "Corollary 12. For every sensible producing XTOP R there exists an equivalent MBOT.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "Proof. Theorem 6 shows that there exists an equivalent proper XTOP R , which must be finitely copying by Theorem 10. This last fact allows us to construct an equivalent MBOT by Theorem 11.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "From finite copying to an MBOT",
"sec_num": "5"
},
{
"text": "Finally, we present an application of Corollary 12 to solve an open problem. The translation model is often used in a backwards manner in a machine translation system as demonstrated, for example, by , which means that an output tree is supplied and the corresponding input trees are sought.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preservation of regularity",
"sec_num": "6"
},
{
"text": "This starting output tree is typically the best parse of the string that we want to translate. However, instead of a single tree, we want to use all parses of this sentence together with their parse scores. Those parses form a regular weighted tree language, and applying them backwards to the translation model yields another weighted tree language L of corresponding input trees. For an efficient representation and efficient modification algorithms (such a k-best extraction) we would like L to be regular. However, F\u00fcl\u00f6p et al. (2011) demonstrate that the backward application of a regular weighted tree language to an XTOP R is not necessarily regular. The counterexample uses a variant of the XTOP of Example 1 and is thus not sensible. Theorem 14 of Maletti (2011a) shows that MBOT preserve regularity under backward application.",
"cite_spans": [
{
"start": 519,
"end": 538,
"text": "F\u00fcl\u00f6p et al. (2011)",
"ref_id": "BIBREF12"
},
{
"start": 757,
"end": 772,
"text": "Maletti (2011a)",
"ref_id": "BIBREF26"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Preservation of regularity",
"sec_num": "6"
},
{
"text": "Corollary 13. Sensible XTOP R preserve regularity under backward application.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preservation of regularity",
"sec_num": "6"
},
{
"text": "We demonstrated that each sensible XTOP R can be implemented by an MBOT. The latter formalism offers many computational advantages, so that the author believes that MBOT should be used instead of XTOP. We used real number weights, but the author believes that our results carry over to at least all zerosum and zero-divisor free semirings [see Hebisch and Weinert (1998) and Golan (1999) ], which are semirings such that (i) a + b = 0 implies a = 0 and (ii) a \u2022 b = 0 implies 0 \u2208 {a, b}. Whether our results hold in other semirings (such as the semiring of all reals where \u22121 + 1 = 0) remains an open question.",
"cite_spans": [
{
"start": 344,
"end": 370,
"text": "Hebisch and Weinert (1998)",
"ref_id": "BIBREF18"
},
{
"start": 375,
"end": 387,
"text": "Golan (1999)",
"ref_id": "BIBREF15"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Conclusion",
"sec_num": null
},
{
"text": "There is an additional restriction that is discussed in the next paragraph.2 This is a convenience requirement. We can use other conditions on the XTOP R or the used weight structures to guarantee a well-defined semantics.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
},
{
"text": "This restriction can always be achieved by renaming the states.4 Again this could have been achieved with the help of other conditions on the MBOT or the used weight structure.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
}
],
"back_matter": [],
"bib_entries": {
"BIBREF0": {
"ref_id": "b0",
"title": "Translations on a context-free grammar",
"authors": [
{
"first": "Alfred",
"middle": [
"V"
],
"last": "Aho",
"suffix": ""
},
{
"first": "Jeffrey",
"middle": [
"D"
],
"last": "Ullman",
"suffix": ""
}
],
"year": 1971,
"venue": "Inform. and Control",
"volume": "19",
"issue": "5",
"pages": "439--475",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Alfred V. Aho and Jeffrey D. Ullman. 1971. Transla- tions on a context-free grammar. Inform. and Control, 19(5):439-475.",
"links": null
},
"BIBREF1": {
"ref_id": "b1",
"title": "Bi-transductions de for\u00eats",
"authors": [
{
"first": "Andr\u00e9",
"middle": [],
"last": "Arnold",
"suffix": ""
},
{
"first": "Max",
"middle": [],
"last": "Dauchet",
"suffix": ""
}
],
"year": 1976,
"venue": "Proc. 3th Int. Coll. Automata, Languages and Programming",
"volume": "",
"issue": "",
"pages": "74--86",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Andr\u00e9 Arnold and Max Dauchet. 1976. Bi-transductions de for\u00eats. In Proc. 3th Int. Coll. Automata, Languages and Programming, pages 74-86. University of Edin- burgh.",
"links": null
},
"BIBREF2": {
"ref_id": "b2",
"title": "Morphismes et bimorphismes d'arbres",
"authors": [
{
"first": "Andr\u00e9",
"middle": [],
"last": "Arnold",
"suffix": ""
},
{
"first": "Max",
"middle": [],
"last": "Dauchet",
"suffix": ""
}
],
"year": 1982,
"venue": "Theoret. Comput. Sci",
"volume": "20",
"issue": "1",
"pages": "33--93",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Andr\u00e9 Arnold and Max Dauchet. 1982. Morphismes et bimorphismes d'arbres. Theoret. Comput. Sci., 20(1):33-93.",
"links": null
},
"BIBREF3": {
"ref_id": "b3",
"title": "Hierarchical phrase-based translation",
"authors": [
{
"first": "David",
"middle": [],
"last": "Chiang",
"suffix": ""
}
],
"year": 2007,
"venue": "Comput. Linguist",
"volume": "33",
"issue": "2",
"pages": "201--228",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "David Chiang. 2007. Hierarchical phrase-based transla- tion. Comput. Linguist., 33(2):201-228.",
"links": null
},
"BIBREF4": {
"ref_id": "b4",
"title": "Decidability of the finiteness of ranges of tree transductions",
"authors": [
{
"first": "Frank",
"middle": [],
"last": "Drewes",
"suffix": ""
},
{
"first": "Joost",
"middle": [],
"last": "Engelfriet",
"suffix": ""
}
],
"year": 1998,
"venue": "Inform. and Comput",
"volume": "145",
"issue": "1",
"pages": "1--50",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Frank Drewes and Joost Engelfriet. 1998. Decidability of the finiteness of ranges of tree transductions. In- form. and Comput., 145(1):1-50.",
"links": null
},
"BIBREF5": {
"ref_id": "b5",
"title": "Learning non-isomorphic tree mappings for machine translation",
"authors": [
{
"first": "Jason",
"middle": [],
"last": "Eisner",
"suffix": ""
}
],
"year": 2003,
"venue": "Proc. 41st Ann. Meeting Association for Computational Linguistics",
"volume": "",
"issue": "",
"pages": "205--208",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Jason Eisner. 2003. Learning non-isomorphic tree map- pings for machine translation. In Proc. 41st Ann. Meeting Association for Computational Linguistics, pages 205-208. Association for Computational Lin- guistics.",
"links": null
},
"BIBREF6": {
"ref_id": "b6",
"title": "Macro tree translations of linear size increase are MSO definable",
"authors": [
{
"first": "Joost",
"middle": [],
"last": "Engelfriet",
"suffix": ""
},
{
"first": "Sebastian",
"middle": [],
"last": "Maneth",
"suffix": ""
}
],
"year": 2003,
"venue": "SIAM J. Comput",
"volume": "32",
"issue": "4",
"pages": "950--1006",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Joost Engelfriet and Sebastian Maneth. 2003. Macro tree translations of linear size increase are MSO definable. SIAM J. Comput., 32(4):950-1006.",
"links": null
},
"BIBREF7": {
"ref_id": "b7",
"title": "Macro tree transducers",
"authors": [
{
"first": "Joost",
"middle": [],
"last": "Engelfriet",
"suffix": ""
},
{
"first": "Heiko",
"middle": [],
"last": "Vogler",
"suffix": ""
}
],
"year": 1985,
"venue": "J. Comput. System Sci",
"volume": "31",
"issue": "1",
"pages": "71--146",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Joost Engelfriet and Heiko Vogler. 1985. Macro tree transducers. J. Comput. System Sci., 31(1):71-146.",
"links": null
},
"BIBREF8": {
"ref_id": "b8",
"title": "Extended multi bottom-up tree transducers -composition and decomposition",
"authors": [
{
"first": "Joost",
"middle": [],
"last": "Engelfriet",
"suffix": ""
},
{
"first": "Eric",
"middle": [],
"last": "Lilin",
"suffix": ""
},
{
"first": "Andreas",
"middle": [],
"last": "Maletti",
"suffix": ""
}
],
"year": 2009,
"venue": "Acta Inform",
"volume": "46",
"issue": "8",
"pages": "561--590",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Joost Engelfriet, Eric Lilin, and Andreas Maletti. 2009. Extended multi bottom-up tree transducers -compo- sition and decomposition. Acta Inform., 46(8):561- 590.",
"links": null
},
"BIBREF9": {
"ref_id": "b9",
"title": "Bottom-up and top-down tree transformations -a comparison",
"authors": [
{
"first": "Joost",
"middle": [],
"last": "Engelfriet",
"suffix": ""
}
],
"year": 1975,
"venue": "Math. Systems Theory",
"volume": "9",
"issue": "",
"pages": "198--231",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Joost Engelfriet. 1975. Bottom-up and top-down tree transformations -a comparison. Math. Systems The- ory, 9(3):198-231.",
"links": null
},
"BIBREF10": {
"ref_id": "b10",
"title": "Top-down tree transducers with regular look-ahead",
"authors": [
{
"first": "Joost",
"middle": [],
"last": "Engelfriet",
"suffix": ""
}
],
"year": 1977,
"venue": "Math. Systems Theory",
"volume": "10",
"issue": "1",
"pages": "289--303",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Joost Engelfriet. 1977. Top-down tree transducers with regular look-ahead. Math. Systems Theory, 10(1):289-303.",
"links": null
},
"BIBREF11": {
"ref_id": "b11",
"title": "Weighted tree automata and tree transducers",
"authors": [
{
"first": "Zolt\u00e1n",
"middle": [],
"last": "F\u00fcl\u00f6p",
"suffix": ""
},
{
"first": "Heiko",
"middle": [],
"last": "Vogler",
"suffix": ""
}
],
"year": 2009,
"venue": "Handbook of Weighted Automata",
"volume": "9",
"issue": "",
"pages": "313--403",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Zolt\u00e1n F\u00fcl\u00f6p and Heiko Vogler. 2009. Weighted tree automata and tree transducers. In Manfred Droste, Werner Kuich, and Heiko Vogler, editors, Handbook of Weighted Automata, EATCS Monographs on Theo- ret. Comput. Sci., chapter 9, pages 313-403. Springer.",
"links": null
},
"BIBREF12": {
"ref_id": "b12",
"title": "Weighted extended tree transducers",
"authors": [
{
"first": "Zolt\u00e1n",
"middle": [],
"last": "F\u00fcl\u00f6p",
"suffix": ""
},
{
"first": "Andreas",
"middle": [],
"last": "Maletti",
"suffix": ""
},
{
"first": "Heiko",
"middle": [],
"last": "Vogler",
"suffix": ""
}
],
"year": 2011,
"venue": "Fundam. Inform",
"volume": "111",
"issue": "2",
"pages": "163--202",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Zolt\u00e1n F\u00fcl\u00f6p, Andreas Maletti, and Heiko Vogler. 2011. Weighted extended tree transducers. Fundam. Inform., 111(2):163-202.",
"links": null
},
"BIBREF13": {
"ref_id": "b13",
"title": "Tree Automata",
"authors": [
{
"first": "Ferenc",
"middle": [],
"last": "G\u00e9cseg",
"suffix": ""
},
{
"first": "Magnus",
"middle": [],
"last": "Steinby",
"suffix": ""
}
],
"year": 1984,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Ferenc G\u00e9cseg and Magnus Steinby. 1984. Tree Au- tomata. Akad\u00e9miai Kiad\u00f3, Budapest.",
"links": null
},
"BIBREF14": {
"ref_id": "b14",
"title": "Tree languages",
"authors": [
{
"first": "Ferenc",
"middle": [],
"last": "G\u00e9cseg",
"suffix": ""
},
{
"first": "Magnus",
"middle": [],
"last": "Steinby",
"suffix": ""
}
],
"year": 1997,
"venue": "Grzegorz Rozenberg and Arto Salomaa",
"volume": "3",
"issue": "",
"pages": "1--68",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Ferenc G\u00e9cseg and Magnus Steinby. 1997. Tree lan- guages. In Grzegorz Rozenberg and Arto Salomaa, editors, Handbook of Formal Languages, volume 3, chapter 1, pages 1-68. Springer.",
"links": null
},
"BIBREF15": {
"ref_id": "b15",
"title": "Semirings and their Applications",
"authors": [
{
"first": "Jonathan",
"middle": [
"S"
],
"last": "Golan",
"suffix": ""
}
],
"year": 1999,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Jonathan S. Golan. 1999. Semirings and their Applica- tions. Kluwer Academic, Dordrecht.",
"links": null
},
"BIBREF16": {
"ref_id": "b16",
"title": "Training tree transducers",
"authors": [
{
"first": "Jonathan",
"middle": [],
"last": "Graehl",
"suffix": ""
},
{
"first": "Kevin",
"middle": [],
"last": "Knight",
"suffix": ""
},
{
"first": "Jonathan",
"middle": [],
"last": "",
"suffix": ""
}
],
"year": 2008,
"venue": "Comput. Linguist",
"volume": "34",
"issue": "3",
"pages": "391--427",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Jonathan Graehl, Kevin Knight, and Jonathan May. 2008. Training tree transducers. Comput. Linguist., 34(3):391-427.",
"links": null
},
"BIBREF17": {
"ref_id": "b17",
"title": "The power of extended top-down tree transducers",
"authors": [
{
"first": "Jonathan",
"middle": [],
"last": "Graehl",
"suffix": ""
},
{
"first": "Mark",
"middle": [],
"last": "Hopkins",
"suffix": ""
},
{
"first": "Kevin",
"middle": [],
"last": "Knight",
"suffix": ""
},
{
"first": "Andreas",
"middle": [],
"last": "Maletti",
"suffix": ""
}
],
"year": 2009,
"venue": "SIAM J. Comput",
"volume": "39",
"issue": "2",
"pages": "410--430",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Jonathan Graehl, Mark Hopkins, Kevin Knight, and An- dreas Maletti. 2009. The power of extended top-down tree transducers. SIAM J. Comput., 39(2):410-430.",
"links": null
},
"BIBREF18": {
"ref_id": "b18",
"title": "Semirings-Algebraic Theory and Applications in Computer Science",
"authors": [
{
"first": "Udo",
"middle": [],
"last": "Hebisch",
"suffix": ""
},
{
"first": "J",
"middle": [],
"last": "Hanns",
"suffix": ""
},
{
"first": "",
"middle": [],
"last": "Weinert",
"suffix": ""
}
],
"year": 1998,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Udo Hebisch and Hanns J. Weinert. 1998. Semirings- Algebraic Theory and Applications in Computer Sci- ence. World Scientific.",
"links": null
},
"BIBREF19": {
"ref_id": "b19",
"title": "An overview of probabilistic tree transducers for natural language processing",
"authors": [
{
"first": "Kevin",
"middle": [],
"last": "Knight",
"suffix": ""
},
{
"first": "Jonathan",
"middle": [],
"last": "Graehl",
"suffix": ""
}
],
"year": 2005,
"venue": "Proc. 6th Int. Conf. Computational Linguistics and Intelligent Text Processing",
"volume": "3406",
"issue": "",
"pages": "121--133",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Kevin Knight and Jonathan Graehl. 2005. An overview of probabilistic tree transducers for natural language processing. In Proc. 6th Int. Conf. Computational Lin- guistics and Intelligent Text Processing, volume 3406 of LNCS, pages 1-24. Springer. Kevin Knight. 2007. Capturing practical natu- ral language transformations. Machine Translation, 21(2):121-133.",
"links": null
},
"BIBREF20": {
"ref_id": "b20",
"title": "Statistical Machine Translation",
"authors": [
{
"first": "Philipp",
"middle": [],
"last": "Koehn",
"suffix": ""
}
],
"year": 2010,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Philipp Koehn. 2010. Statistical Machine Translation. Cambridge University Press.",
"links": null
},
"BIBREF21": {
"ref_id": "b21",
"title": "Une g\u00e9n\u00e9ralisation des transducteurs d'\u00e9tats finis d'arbres: les S-transducteurs",
"authors": [
{
"first": "Eric",
"middle": [],
"last": "Lilin",
"suffix": ""
}
],
"year": 1978,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Eric Lilin. 1978. Une g\u00e9n\u00e9ralisation des transduc- teurs d'\u00e9tats finis d'arbres: les S-transducteurs. Th\u00e8se 3\u00e8me cycle, Universit\u00e9 de Lille.",
"links": null
},
"BIBREF22": {
"ref_id": "b22",
"title": "Pushing for weighted tree automata",
"authors": [
{
"first": "Andreas",
"middle": [],
"last": "Maletti",
"suffix": ""
},
{
"first": "Daniel",
"middle": [],
"last": "Quernheim",
"suffix": ""
}
],
"year": 2011,
"venue": "Proc. 36th Int. Symp. Mathematical Foundations of Computer Science",
"volume": "",
"issue": "",
"pages": "460--471",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Andreas Maletti and Daniel Quernheim. 2011. Pushing for weighted tree automata. In Proc. 36th Int. Symp. Mathematical Foundations of Computer Science, vol- ume 6907 of LNCS, pages 460-471. Springer.",
"links": null
},
"BIBREF23": {
"ref_id": "b23",
"title": "Compositions of extended topdown tree transducers",
"authors": [
{
"first": "Andreas",
"middle": [],
"last": "Maletti",
"suffix": ""
}
],
"year": 2008,
"venue": "Inform. and Comput",
"volume": "206",
"issue": "9",
"pages": "1187--1196",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Andreas Maletti. 2008. Compositions of extended top- down tree transducers. Inform. and Comput., 206(9- 10):1187-1196.",
"links": null
},
"BIBREF24": {
"ref_id": "b24",
"title": "Input and output products for weighted extended top-down tree transducers",
"authors": [
{
"first": "Andreas",
"middle": [],
"last": "Maletti",
"suffix": ""
}
],
"year": 2010,
"venue": "Proc. 14th Int. Conf. Developments in Language Theory",
"volume": "6224",
"issue": "",
"pages": "316--327",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Andreas Maletti. 2010a. Input and output products for weighted extended top-down tree transducers. In Proc. 14th Int. Conf. Developments in Language The- ory, volume 6224 of LNCS, pages 316-327. Springer.",
"links": null
},
"BIBREF25": {
"ref_id": "b25",
"title": "Why synchronous tree substitution grammars?",
"authors": [
{
"first": "Andreas",
"middle": [],
"last": "Maletti",
"suffix": ""
}
],
"year": 2010,
"venue": "Proc. Human Language Technologies: Conf. North American Chapter of the ACL",
"volume": "",
"issue": "",
"pages": "876--884",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Andreas Maletti. 2010b. Why synchronous tree substitu- tion grammars? In Proc. Human Language Technolo- gies: Conf. North American Chapter of the ACL, pages 876-884. Association for Computational Linguistics.",
"links": null
},
"BIBREF26": {
"ref_id": "b26",
"title": "An alternative to synchronous tree substitution grammars",
"authors": [
{
"first": "Andreas",
"middle": [],
"last": "Maletti",
"suffix": ""
}
],
"year": 2011,
"venue": "J. Natur. Lang. Engrg",
"volume": "17",
"issue": "2",
"pages": "221--242",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Andreas Maletti. 2011a. An alternative to synchronous tree substitution grammars. J. Natur. Lang. Engrg., 17(2):221-242.",
"links": null
},
"BIBREF27": {
"ref_id": "b27",
"title": "How to train your multi bottomup tree transducer",
"authors": [
{
"first": "Andreas",
"middle": [],
"last": "Maletti",
"suffix": ""
}
],
"year": 2011,
"venue": "Proc. 49th Ann. Meeting Association for Computational Linguistics: Human Language Technologies",
"volume": "",
"issue": "",
"pages": "825--834",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Andreas Maletti. 2011b. How to train your multi bottom- up tree transducer. In Proc. 49th Ann. Meeting Asso- ciation for Computational Linguistics: Human Lan- guage Technologies, pages 825-834. Association for Computational Linguistics.",
"links": null
},
"BIBREF28": {
"ref_id": "b28",
"title": "Tiburon: A weighted tree automata toolkit",
"authors": [
{
"first": "Jonathan",
"middle": [],
"last": "May",
"suffix": ""
},
{
"first": "Kevin",
"middle": [],
"last": "Knight",
"suffix": ""
}
],
"year": 2006,
"venue": "Proc. 11th Int. Conf. Implementation and Application of Automata",
"volume": "4094",
"issue": "",
"pages": "102--113",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Jonathan May and Kevin Knight. 2006. Tiburon: A weighted tree automata toolkit. In Proc. 11th Int. Conf. Implementation and Application of Automata, volume 4094 of LNCS, pages 102-113. Springer.",
"links": null
},
"BIBREF29": {
"ref_id": "b29",
"title": "Efficient inference through cascades of weighted tree transducers",
"authors": [
{
"first": "Jonathan",
"middle": [],
"last": "May",
"suffix": ""
},
{
"first": "Kevin",
"middle": [],
"last": "Knight",
"suffix": ""
},
{
"first": "Heiko",
"middle": [],
"last": "Vogler",
"suffix": ""
}
],
"year": 2010,
"venue": "Proc. 48th Ann. Meeting Association for Computational Linguistics",
"volume": "",
"issue": "",
"pages": "1058--1066",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Jonathan May, Kevin Knight, and Heiko Vogler. 2010. Efficient inference through cascades of weighted tree transducers. In Proc. 48th Ann. Meeting Association for Computational Linguistics, pages 1058-1066. As- sociation for Computational Linguistics.",
"links": null
},
"BIBREF30": {
"ref_id": "b30",
"title": "Weighted Tree Automata and Transducers for Syntactic Natural Language Processing",
"authors": [
{
"first": "Jonathan",
"middle": [
"May"
],
"last": "",
"suffix": ""
}
],
"year": 2010,
"venue": "",
"volume": "",
"issue": "",
"pages": "",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Jonathan May. 2010. Weighted Tree Automata and Transducers for Syntactic Natural Language Process- ing. Ph.D. thesis, University of Southern California, Los Angeles.",
"links": null
},
"BIBREF31": {
"ref_id": "b31",
"title": "Forestbased translation",
"authors": [
{
"first": "Haitao",
"middle": [],
"last": "Mi",
"suffix": ""
},
{
"first": "Liang",
"middle": [],
"last": "Huang",
"suffix": ""
},
{
"first": "Qun",
"middle": [],
"last": "Liu",
"suffix": ""
}
],
"year": 2008,
"venue": "Proc. 46th Ann. Meeting Association for Computational Linguistics",
"volume": "",
"issue": "",
"pages": "192--199",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Haitao Mi, Liang Huang, and Qun Liu. 2008. Forest- based translation. In Proc. 46th Ann. Meeting Associ- ation for Computational Linguistics, pages 192-199. Association for Computational Linguistics.",
"links": null
},
"BIBREF32": {
"ref_id": "b32",
"title": "Mappings and grammars on trees",
"authors": [
{
"first": "C",
"middle": [],
"last": "William",
"suffix": ""
},
{
"first": "",
"middle": [],
"last": "Rounds",
"suffix": ""
}
],
"year": 1970,
"venue": "Math. Systems Theory",
"volume": "4",
"issue": "3",
"pages": "257--287",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "William C. Rounds. 1970. Mappings and grammars on trees. Math. Systems Theory, 4(3):257-287.",
"links": null
},
"BIBREF33": {
"ref_id": "b33",
"title": "Preservation of recognizability for weighted linear extended top-down tree transducers",
"authors": [
{
"first": "Nina",
"middle": [],
"last": "Seemann",
"suffix": ""
},
{
"first": "Daniel",
"middle": [],
"last": "Quernheim",
"suffix": ""
},
{
"first": "Fabienne",
"middle": [],
"last": "Braune",
"suffix": ""
},
{
"first": "Andreas",
"middle": [],
"last": "Maletti",
"suffix": ""
}
],
"year": 2012,
"venue": "Proc. 2nd Workshop Applications of Tree Automata in Natural Language Processing",
"volume": "",
"issue": "",
"pages": "1--10",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Nina Seemann, Daniel Quernheim, Fabienne Braune, and Andreas Maletti. 2012. Preservation of recognizabil- ity for weighted linear extended top-down tree trans- ducers. In Proc. 2nd Workshop Applications of Tree Automata in Natural Language Processing, pages 1- 10. Association for Computational Linguistics.",
"links": null
},
"BIBREF34": {
"ref_id": "b34",
"title": "Synchronous grammars as tree transducers",
"authors": [
{
"first": "M",
"middle": [],
"last": "Stuart",
"suffix": ""
},
{
"first": "",
"middle": [],
"last": "Shieber",
"suffix": ""
}
],
"year": 2004,
"venue": "Proc. 7th Int. Workshop Tree Adjoining Grammars and Related Formalisms",
"volume": "",
"issue": "",
"pages": "88--95",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Stuart M. Shieber. 2004. Synchronous grammars as tree transducers. In Proc. 7th Int. Workshop Tree Adjoining Grammars and Related Formalisms, pages 88-95.",
"links": null
},
"BIBREF35": {
"ref_id": "b35",
"title": "A non-contiguous tree sequence alignment-based model for statistical machine translation",
"authors": [
{
"first": "Jun",
"middle": [],
"last": "Sun",
"suffix": ""
},
{
"first": "Min",
"middle": [],
"last": "Zhang",
"suffix": ""
},
{
"first": "Chew Lim",
"middle": [],
"last": "Tan",
"suffix": ""
}
],
"year": 2009,
"venue": "Proc. 47th Ann. Meeting Association for Computational Linguistics",
"volume": "",
"issue": "",
"pages": "914--922",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Jun Sun, Min Zhang, and Chew Lim Tan. 2009. A non-contiguous tree sequence alignment-based model for statistical machine translation. In Proc. 47th Ann. Meeting Association for Computational Linguistics, pages 914-922. Association for Computational Lin- guistics.",
"links": null
},
"BIBREF36": {
"ref_id": "b36",
"title": "Generalized 2 sequential machine maps",
"authors": [
{
"first": "James",
"middle": [
"W"
],
"last": "Thatcher",
"suffix": ""
}
],
"year": 1970,
"venue": "J. Comput. System Sci",
"volume": "4",
"issue": "4",
"pages": "339--367",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "James W. Thatcher. 1970. Generalized 2 sequential ma- chine maps. J. Comput. System Sci., 4(4):339-367.",
"links": null
},
"BIBREF37": {
"ref_id": "b37",
"title": "A tree sequence alignment-based tree-to-tree translation model",
"authors": [
{
"first": "Min",
"middle": [],
"last": "Zhang",
"suffix": ""
},
{
"first": "Hongfei",
"middle": [],
"last": "Jiang",
"suffix": ""
},
{
"first": "Aiti",
"middle": [],
"last": "Aw",
"suffix": ""
},
{
"first": "Haizhou",
"middle": [],
"last": "Li",
"suffix": ""
},
{
"first": "Sheng",
"middle": [],
"last": "Chew Lim Tan",
"suffix": ""
},
{
"first": "",
"middle": [],
"last": "Li",
"suffix": ""
}
],
"year": 2008,
"venue": "Proc. 46th Ann. Meeting Association for Computational Linguistics",
"volume": "",
"issue": "",
"pages": "559--567",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Min Zhang, Hongfei Jiang, Aiti Aw, Haizhou Li, Chew Lim Tan, and Sheng Li. 2008a. A tree se- quence alignment-based tree-to-tree translation model. In Proc. 46th Ann. Meeting Association for Compu- tational Linguistics, pages 559-567. Association for Computational Linguistics.",
"links": null
},
"BIBREF38": {
"ref_id": "b38",
"title": "Grammar comparison study for translational equivalence modeling and statistical machine translation",
"authors": [
{
"first": "Min",
"middle": [],
"last": "Zhang",
"suffix": ""
},
{
"first": "Hongfei",
"middle": [],
"last": "Jiang",
"suffix": ""
},
{
"first": "Haizhou",
"middle": [],
"last": "Li",
"suffix": ""
},
{
"first": "Aiti",
"middle": [],
"last": "Aw",
"suffix": ""
},
{
"first": "Sheng",
"middle": [],
"last": "Li",
"suffix": ""
}
],
"year": 2008,
"venue": "Proc. 22nd Int. Conf. Computational Linguistics",
"volume": "",
"issue": "",
"pages": "1097--1104",
"other_ids": {},
"num": null,
"urls": [],
"raw_text": "Min Zhang, Hongfei Jiang, Haizhou Li, Aiti Aw, and Sheng Li. 2008b. Grammar comparison study for translational equivalence modeling and statistical ma- chine translation. In Proc. 22nd Int. Conf. Computa- tional Linguistics, pages 1097-1104. Association for Computational Linguistics.",
"links": null
}
},
"ref_entries": {
"FIGREF0": {
"num": null,
"text": "Example rotation. In principle, such rotations are required in the translation from English to Arabic.",
"uris": null,
"type_str": "figure"
},
"FIGREF1": {
"num": null,
"text": "Rewrite step using rule \u03c1 ofFigure 3.",
"uris": null,
"type_str": "figure"
},
"FIGREF2": {
"num": null,
"text": "Example derivation using the XTOP M ex with weight 1 3 \u2022 2 4 = 16.",
"uris": null,
"type_str": "figure"
},
"FIGREF3": {
"num": null,
"text": "Overview of the proof steps.",
"uris": null,
"type_str": "figure"
},
"FIGREF4": {
"num": null,
"text": "Illustration of the derivation in Definition 4.",
"uris": null,
"type_str": "figure"
},
"FIGREF5": {
"num": null,
"text": "Example MBOT rule \u03c1 [left] and its use in a rewrite step [right].",
"uris": null,
"type_str": "figure"
}
}
}
}