| { |
| "paper_id": "W19-0113", |
| "header": { |
| "generated_with": "S2ORC 1.0.0", |
| "date_generated": "2023-01-19T06:15:39.032029Z" |
| }, |
| "title": "Learnability and Overgeneration in Computational Syntax", |
| "authors": [ |
| { |
| "first": "Yiding", |
| "middle": [], |
| "last": "Hao", |
| "suffix": "", |
| "affiliation": { |
| "laboratory": "", |
| "institution": "Yale University", |
| "location": {} |
| }, |
| "email": "yiding.hao@yale.edu" |
| } |
| ], |
| "year": "", |
| "venue": null, |
| "identifiers": {}, |
| "abstract": "This paper addresses the hypothesis that unnatural patterns generated by grammar formalisms can be eliminated on the grounds that they are unlearnable. I consider three examples of formal languages thought to represent dependencies unattested in natural language syntax, and show that all three can be learned by grammar induction algorithms following the Distributional Learning paradigm of Clark and Eyraud (2007). While learnable language classes are restrictive by necessity (Gold, 1967), these facts suggest that learnability alone may be insufficient for addressing concerns of overgeneration in syntax.", |
| "pdf_parse": { |
| "paper_id": "W19-0113", |
| "_pdf_hash": "", |
| "abstract": [ |
| { |
| "text": "This paper addresses the hypothesis that unnatural patterns generated by grammar formalisms can be eliminated on the grounds that they are unlearnable. I consider three examples of formal languages thought to represent dependencies unattested in natural language syntax, and show that all three can be learned by grammar induction algorithms following the Distributional Learning paradigm of Clark and Eyraud (2007). While learnable language classes are restrictive by necessity (Gold, 1967), these facts suggest that learnability alone may be insufficient for addressing concerns of overgeneration in syntax.", |
| "cite_spans": [], |
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| "eq_spans": [], |
| "section": "Abstract", |
| "sec_num": null |
| } |
| ], |
| "body_text": [ |
| { |
| "text": "A longstanding debate in linguistics concerns the division of labor in language acquisition between innate universal assumptions about natural language and the learner's ability to recognize patterns in data. The rationalist position, famously championed by the Principles and Parameters framework, follows the Poverty of the Stimulus argument (POS, Chomsky, 1965 , 1968 , 1971 , 1980 in assuming a rich Universal Grammar (UG) that allows individual languages to vary along a narrow range of dimensions. On the other hand, the Distributional Learning paradigm of grammatical inference (Clark and Eyraud, 2007) has shown that it is possible to create empiricist representations of grammar optimized for extracting generalizations from data with theoretical guarantees of convergence.", |
| "cite_spans": [ |
| { |
| "start": 344, |
| "end": 363, |
| "text": "(POS, Chomsky, 1965", |
| "ref_id": null |
| }, |
| { |
| "start": 364, |
| "end": 370, |
| "text": ", 1968", |
| "ref_id": "BIBREF4" |
| }, |
| { |
| "start": 371, |
| "end": 377, |
| "text": ", 1971", |
| "ref_id": "BIBREF5" |
| }, |
| { |
| "start": 378, |
| "end": 384, |
| "text": ", 1980", |
| "ref_id": "BIBREF6" |
| }, |
| { |
| "start": 585, |
| "end": 609, |
| "text": "(Clark and Eyraud, 2007)", |
| "ref_id": "BIBREF9" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Introduction", |
| "sec_num": "1" |
| }, |
| { |
| "text": "Recent advances in mathematical linguistics have suggested that the rationalist-empiricist debate may be of interest to the program of formally characterizing the typology of syntax. Shieber's (1985) argument that Swiss German is not context-free shows that substantial expressive power is needed in order to adequately describe syntactic phenomena. At the same time, the class of context-free languages and its extensions include pathological dependencies unattested in natural language. Kobele (2011) , for instance, shows that the context-free Merge operation allows Minimalist Grammars (MGs) to define languages that require every syntactically well-formed sentence to have at least one semantic type conflict. In light of this overgeneration problem, learnability has been proposed as a possible way to refine existing language classes so as to better align with empirical facts. Under such an approach, UG specifies the formalism in which grammars are represented, while language acquisition is modelled by a grammar induction algorithm that correctly converges on a subset of the possible grammars. Since no strict superclass of the finite languages admits a general learning procedure (Gold, 1967) , there necessarily exist languages that are permitted by UG but that cannot be learned by the language acquisition algorithm.", |
| "cite_spans": [ |
| { |
| "start": 183, |
| "end": 199, |
| "text": "Shieber's (1985)", |
| "ref_id": "BIBREF43" |
| }, |
| { |
| "start": 489, |
| "end": 502, |
| "text": "Kobele (2011)", |
| "ref_id": "BIBREF29" |
| }, |
| { |
| "start": 1193, |
| "end": 1205, |
| "text": "(Gold, 1967)", |
| "ref_id": "BIBREF12" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Introduction", |
| "sec_num": "1" |
| }, |
| { |
| "text": "This paper takes some preliminary steps toward evaluating the potential of learnability to produce restricted language classes that exclude unnatural patterns. Recent work in Distributional Learning has produced a hierarchy of context-free and multiple context-free language classes defined by learning algorithms. I examine three examples of unnatural patterns-structure-independent constraints on sentence length, free word order with unbounded crossing dependencies, and unlimited copying of deep context-free structure-and show that these patterns appear in small classes of the learnable hierarchy. This suggests that current approaches to grammar induction for syntax may fail to yield learnability-based accounts for the absence of these patterns in syntactic typology.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Introduction", |
| "sec_num": "1" |
| }, |
| { |
| "text": "After basic definitions and notation are presented in Section 2, Section 3 introduces the learnable language classes considered in this paper.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Introduction", |
| "sec_num": "1" |
| }, |
| { |
| "text": "The three unnatural patterns, drawn from Graf's (2013) discussion of overgeneration in MGs, are defined in Section 4. There, it will be shown that the three patterns exist within the language classes from Section 3. Section 5 concludes with a discussion of these facts and their relationship with the rationalist-empiricist debate.", |
| "cite_spans": [ |
| { |
| "start": 41, |
| "end": 54, |
| "text": "Graf's (2013)", |
| "ref_id": "BIBREF16" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Introduction", |
| "sec_num": "1" |
| }, |
| { |
| "text": "As usual, N denotes the set of nonnegative integers, and for any set A, P(A) denotes the power set of A. Unless otherwise specified, the letter \u03a3 denotes a finite alphabet. The length of a string x is denoted by |x|, and \u03b5 denotes the empty string. For each a \u2208 \u03a3, |x| a denotes the number of occurrences of a in x. Alphabet symbols are identified with strings of length 1. For strings a and b, ab denotes the concatenation of a and b. As usual, this notation is extended elementwise to sets of strings. For k \u2208 N, \u03b1 k denotes \u03b1 concatenated with itself k-many times; \u03b1 \u2264k denotes k i=0 \u03b1 i ; \u03b1 * denotes \u221e i=0 \u03b1 i ; and \u03b1 + denotes \u03b1 * \\{\u03b5}. This notation does not apply to (\u03a3 * ) k , which denotes the cartesian product k i=1 \u03a3 * . The length of a tuple", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Preliminaries", |
| "sec_num": "2" |
| }, |
| { |
| "text": "x = \u27e8x 1 , x 2 , . . . , x k \u27e9 is defined as |x| := k i=1 |x i |. For a \u2208 \u03a3, |x| a denotes |x 1 x 2 . . . x k | a . For k \u2208 N, a k-context over \u03a3 is a (k + 1)- tuple of strings \u27e8c 0 , c 1 , . . . , c k \u27e9 \u2208 (\u03a3 * ) k , denoted c 0 c 1 . . . c k . For sets L 1 , L 2 , . . . , L k \u2286 \u03a3 * , L 0 L 1 . . . L k denotes the cartesian product k i=0 L i .", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Preliminaries", |
| "sec_num": "2" |
| }, |
| { |
| "text": "The wrapping operation \u2299 between kcontexts and k-tuples of strings is defined by", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Preliminaries", |
| "sec_num": "2" |
| }, |
| { |
| "text": "c 0 c 1 . . . c k \u2299 \u27e8x 1 , x 2 , . . . , x k \u27e9 := c 0 x 1 c 1 x 2 c 2 . . . x k c k", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Preliminaries", |
| "sec_num": "2" |
| }, |
| { |
| "text": "and extended elementwise to sets of contexts and sets of strings. For \u03b1 \u2208 (\u03a3 * ) k \u222a P (\u03a3 * ) k and L \u2286 \u03a3 * , the contexts of \u03b1 with respect to L are defined to be the set", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Preliminaries", |
| "sec_num": "2" |
| }, |
| { |
| "text": "\u03b1 |L\u27e9 := c \u2208 (\u03a3 * ) k+1 c \u2299 \u03b1 \u2286 L ,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Preliminaries", |
| "sec_num": "2" |
| }, |
| { |
| "text": "with the \"\u2286\" above replaced by \"\u2208\" when \u03b1 \u2208 (\u03a3 * ) k . For \u03b3 \u2208 (\u03a3 * ) k+1 \u222aP (\u03a3 * ) k+1 , we define the set", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Preliminaries", |
| "sec_num": "2" |
| }, |
| { |
| "text": "\u03b3 \u27e8L| := x \u2208 (\u03a3 * ) k \u03b3 \u2299 x \u2286 L ,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Preliminaries", |
| "sec_num": "2" |
| }, |
| { |
| "text": "with the \"\u2286\" above replaced by \"\u2208\" when \u03b3 \u2208 (\u03a3 * ) k+1 . When the identity of the language L is clear from context, we may denote \u03b1 |L\u27e9 by \u03b1 \u25c3 and \u03b3 \u27e8L| by \u03b3 \u25b9 . ", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Preliminaries", |
| "sec_num": "2" |
| }, |
| { |
| "text": "(CFL) 2-MCFL 3-MCFL . . . 2-MCFL wn (TAL) 3-MCFL wn . . .", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "1-MCFL", |
| "sec_num": null |
| }, |
| { |
| "text": "This paper considers multiple context-free grammars (MCFGs, Seki et al., 1991) , a mildly context-sensitive (MCS) formalism equivalent to MGs (Harkema, 2001; Michaelis, 2001 ). MCFGs are a generalization of context-free grammars (CFGs) in which nonterminals derive tuples of strings. Whereas CFG rules concatenate strings derived from nonterminals on their right-hand sides, MCFGs interleave nonterminal-derived tuples. Let us consider an example to illustrate how MCFGs generate strings. Example 1. The following four rules define an MCFG generating the copy language L = {ww | w \u2208 {a, b} * }. The start symbol is S.", |
| "cite_spans": [ |
| { |
| "start": 60, |
| "end": 78, |
| "text": "Seki et al., 1991)", |
| "ref_id": "BIBREF42" |
| }, |
| { |
| "start": 142, |
| "end": 157, |
| "text": "(Harkema, 2001;", |
| "ref_id": "BIBREF18" |
| }, |
| { |
| "start": 158, |
| "end": 173, |
| "text": "Michaelis, 2001", |
| "ref_id": "BIBREF33" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "S(xy) \u2190 T (x, y) (2a) T (ax, ay) \u2190 T (x, y) (2b) T (bx, by) \u2190 T (x, y) (2c) T (\u03b5, \u03b5) \u2190 (2d)", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "A rule of the form", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "A(y) \u2190 B 1 (x 1 )B 2 (x 2 ) . . . B n (x n )", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "is interpreted as an axiom stating that if each nonterminal B i on the right-hand side generates the tuple x i , then the nonterminal A on the left-hand side generates the tuple y. In rule (2d), the righthand side is empty; this means that we assume T to generate the tuple \u27e8\u03b5, \u03b5\u27e9. The string abab \u2208 L is derived as follows. By rule (2d), T generates \u27e8\u03b5, \u03b5\u27e9. By (2c), T generates \u27e8b, b\u27e9. By (2b), T generates \u27e8ab, ab\u27e9. By (2a), the start symbol S generates abab.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "An MCFG rule may be thought of as a function that describes how tuples generated by nonterminals on the right-hand side may be combined with one another. These functions must satisfy a condition known as linearity, which asserts that MCFG rules cannot copy their inputs. 1 Definition 3. Fix k \u2208 N. Consider a function f :", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "k i=1 (\u03a3 * ) d i \u2192 (\u03a3 * ) d 0 . For each i, write x i = \u27e8x i,1 , x i,2 , . . . , x i,d i \u27e9.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "We say that f is a linear function if it is of the form", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "f (x 1 , x 2 , . . . , x k ) = \u27e8\u03b1 1 , \u03b1 2 , . . . , \u03b1 d 0 \u27e9,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "where the concatenated string \u03b1 = \u03b1 1 \u03b1 2 . . . \u03b1 d 0 satisfies the following criteria:", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "\u2022 \u03b1 contains exactly one occurrence of each variable x i,j ; and", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "\u2022 for each i and j, x i,j occurs to the left of", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "x i,j+1 in \u03b1. 2", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "Furthermore, we say that f is well-nested if there are no indices i, j, i \u2032 , j \u2032 , and j \u2032\u2032 , with i \u0338 = i \u2032 and j \u2032 \u0338 = j \u2032\u2032 , such that the variable", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "x i \u2032 ,j \u2032 occurs be- tween x i,j and x i,j+1 in \u03b1, but x i \u2032 ,j \u2032\u2032 does not. Definition 4. A multiple context-free gram- mar (MCFG) is an ordered quadruple G = \u27e8N, \u03a3, R, I\u27e9, where", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "\u2022 N is a finite set of nonterminals;", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "\u2022 \u03a3 is a finite set of terminals;", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "\u2022 I \u2286 N is the set of start symbols; and", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "\u2022 letting F be the set of all linear functions, R \u2286 N \u00d7 F \u00d7 N * is a finite set of rules.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "We always assume that N and \u03a3 are disjoint. Each nonterminal A \u2208 N is associated with a number dim(A) known as its dimension. All start symbols must have dimension 1. We denote each rule", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "r = \u27e8A, f, B 1 B 2 . . . B k \u27e9 by A(y) \u2190 B 1 (x 1 )B 2 (x 2 ) . . . B k (x k ),", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "where each x i is a dim(B i )-tuple of variables and", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "y = f (x 1 , x 2 , . . . , x k ).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "We say that r is wellnested if f is well-nested.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "1 While individual MCFG rules cannot copy, Example 1 shows that MCFGs can perform copying by combining several different rules.", |
| "cite_spans": [], |
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| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "2 Technically, the definition of linear functions only requires that \u03b1 contain at least one occurrence of each xi,j. Seki et al. (1991) and Kracht (2003) show that the other assumptions can be made without changing the generative capacity of MCFGs. We say that G is a k-multiple context free grammar (k-MCFG) if every nonterminal has dimension at most k. We say that G is well-nested if every rule in R is well-nested.", |
| "cite_spans": [ |
| { |
| "start": 117, |
| "end": 135, |
| "text": "Seki et al. (1991)", |
| "ref_id": "BIBREF42" |
| }, |
| { |
| "start": 140, |
| "end": 153, |
| "text": "Kracht (2003)", |
| "ref_id": "BIBREF30" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "For each nonterminal A, we define L(G, A) \u2286 (\u03a3 * ) dim(A) as follows. For each rule A(y) \u2190 B 1 (x 1 )B 2 (x 2 ) . . . B k (x k ) with y = f (x 1 , x 2 , . . . , x k ), if z i \u2208 L(G, B i ) for each i, then f (z 1 , z 2 , . . . , z k ) \u2208 L(G, A).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "Identifying 1-tuples with strings, the language generated by G is the language L(G) := S\u2208I L(G, S). We say that a language L \u2286 \u03a3 * is a k-multiple context-free language (k-MCFL) if it is generated by a k-MCFG. We say that L is well-nested if G is well-nested.", |
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| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "MCFLs naturally subsume other common language classes: the class of context-free languages (CFLs) is the same as the class of 1-MCFLs, while the class of tree-adjoining languages (TALs) is the same as the class of well-nested 2-MCFLs (Kanazawa, 2009b) . Seki et al. (1991) and Rambow and Satta (1999) prove a separation result showing that k-MCFLs are strictly contained within the class of (k + 1)-MCFLs. This MCFL hierarchy, along with its well-nested counterpart (Kanazawa, 2009a) , is shown in Figure 1 . each string, the learner must guess a grammar for L. We consider a class of languages C \u2286 P(\u03a3 * ) to be learnable if there is a learner whose guesses converge to a correct grammar for L for any target L \u2208 C and any presentation of the strings of L.", |
| "cite_spans": [ |
| { |
| "start": 234, |
| "end": 251, |
| "text": "(Kanazawa, 2009b)", |
| "ref_id": "BIBREF24" |
| }, |
| { |
| "start": 254, |
| "end": 272, |
| "text": "Seki et al. (1991)", |
| "ref_id": "BIBREF42" |
| }, |
| { |
| "start": 466, |
| "end": 483, |
| "text": "(Kanazawa, 2009a)", |
| "ref_id": "BIBREF23" |
| } |
| ], |
| "ref_spans": [ |
| { |
| "start": 498, |
| "end": 506, |
| "text": "Figure 1", |
| "ref_id": "FIGREF0" |
| } |
| ], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "IIL-learning of formal languages was pioneered by Angluin (1982) , who gave an algorithm that learns the class of regular languages satisfying a criterion known as reversibility. While the full class of regular languages is not learnable under the IIL paradigm, Angluin (1987) showed that learner can learn all regular languages if it is equipped with a minimally adequate teacher (MAT): a black-box oracle that answers certain questions about the target language. These algorithms were extended to CFLs by Clark and Eyraud (2007) and Clark (2010) , respectively, and to MCFLs by Yoshinaka (2011a) and , respectively. Since the MATlearning algorithm fails to learn the full classes of CFLs and MCFLs, further results by Yoshinaka (2011b), Yoshinaka (2012) , and Clark and Yoshinaka (2012) expand the MAT-learnable classes via algorithms using membership queries (MQs). These learnable language classes are visually summarized in Figure 2 .", |
| "cite_spans": [ |
| { |
| "start": 50, |
| "end": 64, |
| "text": "Angluin (1982)", |
| "ref_id": "BIBREF1" |
| }, |
| { |
| "start": 270, |
| "end": 276, |
| "text": "(1987)", |
| "ref_id": null |
| }, |
| { |
| "start": 507, |
| "end": 530, |
| "text": "Clark and Eyraud (2007)", |
| "ref_id": "BIBREF9" |
| }, |
| { |
| "start": 535, |
| "end": 547, |
| "text": "Clark (2010)", |
| "ref_id": "BIBREF7" |
| }, |
| { |
| "start": 580, |
| "end": 597, |
| "text": "Yoshinaka (2011a)", |
| "ref_id": "BIBREF45" |
| }, |
| { |
| "start": 739, |
| "end": 755, |
| "text": "Yoshinaka (2012)", |
| "ref_id": "BIBREF47" |
| } |
| ], |
| "ref_spans": [ |
| { |
| "start": 929, |
| "end": 937, |
| "text": "Figure 2", |
| "ref_id": "FIGREF1" |
| } |
| ], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "The remainder of this section formally defines several classes of MCFLs. Subsection 3.1 defines the IIL-learnable substitutable MCFLs (\"SUBST\" in Figure 2 ), the MCFL analogue of Angluin's reversible regular languages. Subsection 3.2 defines the MAT-learnable congruential MCFLs (\"CONG\" in Figure 2 ). Subsection 3.3 defines two MQlearnable classes of MCFLs: those generated by MCFGs with the finite kernel property (\"FKP\" in Figure 2 ) and those generated by MCFGs with the finite context property (\"FCP\" in Figure 2 ).", |
| "cite_spans": [], |
| "ref_spans": [ |
| { |
| "start": 146, |
| "end": 154, |
| "text": "Figure 2", |
| "ref_id": "FIGREF1" |
| }, |
| { |
| "start": 290, |
| "end": 298, |
| "text": "Figure 2", |
| "ref_id": "FIGREF1" |
| }, |
| { |
| "start": 426, |
| "end": 434, |
| "text": "Figure 2", |
| "ref_id": "FIGREF1" |
| }, |
| { |
| "start": 509, |
| "end": 517, |
| "text": "Figure 2", |
| "ref_id": "FIGREF1" |
| } |
| ], |
| "eq_spans": [], |
| "section": "Multiple Context-Free Grammars", |
| "sec_num": "2.1" |
| }, |
| { |
| "text": "The IIL algorithms of Clark and Eyraud (2007) and Yoshinaka (2011a) rely upon the strong assumption that whenever two k-tuples x and y appear in the same context-i.e., x \u25c3 \u2229 y \u25c3 \u0338 = \u2205they must be generated by the same k-dimensional nonterminal. This property is known as substitutability.", |
| "cite_spans": [ |
| { |
| "start": 22, |
| "end": 45, |
| "text": "Clark and Eyraud (2007)", |
| "ref_id": "BIBREF9" |
| }, |
| { |
| "start": 50, |
| "end": 67, |
| "text": "Yoshinaka (2011a)", |
| "ref_id": "BIBREF45" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Substitutable Languages", |
| "sec_num": "3.1" |
| }, |
| { |
| "text": "Definition 5. For k \u2208 N, a language L \u2286 \u03a3 * is k-substitutable if for all k-tuples x, y \u2208 (\u03a3 * ) k , either x |L\u27e9 = y |L\u27e9 or x |L\u27e9 \u2229 y |L\u27e9 = \u2205. For each k, a language L induces an equiva- lence relation \u2261 k L on (\u03a3 * ) k in which x \u2261 k L y", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Substitutable Languages", |
| "sec_num": "3.1" |
| }, |
| { |
| "text": "if and only if x \u25c3 = y \u25c3 . The grammar G constructed by the algorithm identifies each nonterminal A with an equivalence class", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Substitutable Languages", |
| "sec_num": "3.1" |
| }, |
| { |
| "text": "[a] of \u2261 dim(A) L , so that a \u2208 L(G, A) \u2286 [a]", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Substitutable Languages", |
| "sec_num": "3.1" |
| }, |
| { |
| "text": ". It turns out that linear functions map tuples of subsets of equivalence classes to subsets of equivalence classes, allowing nonterminals to combine with one another via MCFG rules.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Substitutable Languages", |
| "sec_num": "3.1" |
| }, |
| { |
| "text": "When L is a k-substitutable k-MCFL, the learner determines whether or not x and y are equivalent based on whether or not a common context in x \u25c3 \u2229 y \u25c3 has been observed so far, with the understanding that such a context must appear in the data eventually. At each time step, the learner finds all equivalence classes seen so far, and constructs all rules such that a representative from the equivalence class on the left-hand side has been observed in the data. The learner converges when the data contain enough equivalence classes to construct a correct grammar for L.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Substitutable Languages", |
| "sec_num": "3.1" |
| }, |
| { |
| "text": "In Clark (2010) and , the learner again identifies each nonterminal A with an equivalence class of \u2261 dim(A) L and seeks to find all equivalence classes needed to construct the grammar. Without the assumption of substitutability, the learner relies on the minimally adequate teacher to determine which tuples are equivalent. To do this, the learner asks the teacher two types of questions: membership queries (Is x \u2208 L?) and equivalence queries (What is an example of a string in L\\L(G)?). At each time step, an equivalence query is used to identify an equivalence class not covered by an existing nonterminal, and membership queries are used to construct all possible rules involving the new nonterminal. Note that training data are not needed, since the learner asks the teacher for data through equivalence queries. MCFGs constructed through this procedure are known as congruential MCFGs. Definition 6. A k-MCFG G is congruential if for every nonterminal A, L(G, A) is completely contained within an equivalence class of \u2261 dim(A) L(G) . A k-MCFL is congruential if it is generated by a congruential k-MCFG.", |
| "cite_spans": [ |
| { |
| "start": 3, |
| "end": 15, |
| "text": "Clark (2010)", |
| "ref_id": "BIBREF7" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Congruential Languages", |
| "sec_num": "3.2" |
| }, |
| { |
| "text": "The learning algorithms for the substitutable and congruential CFLs and MCFLs attempt to find equivalence classes of \u2261 k L . Yoshinaka (2011b) and generalize beyond this approach by dropping the requirement that nonterminals correspond to equivalence classes. Instead, each nonterminal A is identified with a finite set of strings or contexts with the same distribution as A. Definition 7. Fix k \u2208 N. An MCFG G has the kfinite kernel property (k-FKP) if for every nonterminal A of G, there exists", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "The FKP and the FCP", |
| "sec_num": "3.3" |
| }, |
| { |
| "text": "K A \u2286 (\u03a3 * ) dim A such that |K A | \u2264 k and L(G, A) \u25c3 = K \u25c3 A .", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "The FKP and the FCP", |
| "sec_num": "3.3" |
| }, |
| { |
| "text": "G has the k-finite context property (k-FCP) if for every nonterminal A, there exists", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "The FKP and the FCP", |
| "sec_num": "3.3" |
| }, |
| { |
| "text": "C A \u2286 (\u03a3 * ) dim(A)+1 such that |C A | \u2264 k and L(G, A) \u25c3\u25b9 = C \u25b9 A .", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "The FKP and the FCP", |
| "sec_num": "3.3" |
| }, |
| { |
| "text": "3 At each time step, the learner constructs nonterminals by considering all possible size-k sets of tuples or contexts. The learner constructs rules", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "The FKP and the FCP", |
| "sec_num": "3.3" |
| }, |
| { |
| "text": "r = A(y) \u2190 B 1 (x 1 )B 2 (x 2 ) . . . B n (x n )", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "The FKP and the FCP", |
| "sec_num": "3.3" |
| }, |
| { |
| "text": "by determining whether or not the right-hand side and the left-hand side have the same distribution. This is done heuristically by asking membership queries for strings of the form c", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "The FKP and the FCP", |
| "sec_num": "3.3" |
| }, |
| { |
| "text": "\u2299 f (b 1 , b 2 , . . . , b n ),", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "The FKP and the FCP", |
| "sec_num": "3.3" |
| }, |
| { |
| "text": "where f is the linear function associated with r. When L is assumed to have the k-FKP, c is drawn from the contexts in K \u25c3", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "The FKP and the FCP", |
| "sec_num": "3.3" |
| }, |
| { |
| "text": "A observed so far, and for each i, b i \u2208 K B i . When L is assumed to have the k-FCP, c \u2208 C A , while each b i is drawn from the substrings in C \u25b9 B i observed so far.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "The FKP and the FCP", |
| "sec_num": "3.3" |
| }, |
| { |
| "text": "The formal problem of overgeneration has a long history in linguistics, starting from Peters and Ritchie's (1973) proof that Transformational Grammars can generate all recursively enumerable languages. Recent interest in the overgeneration problem arises from the model-theoretic approach pioneered by Rogers (1994 Rogers ( , 1998 for the formalization of syntactic theories. Relying upon the equivalence of monadic second-order (MSO) logic over trees with tree automata (Thatcher and Wright, 1968 ), Rogers constructs a context-free implementation of Rizzi's (1990) constraint-based Relativized Minimality theory by stating the constraints in MSO logic. Morawietz (2008) extends the model-theoretic approach to MCS formalisms by combining MSO constraints on derivation trees with MSO-definable transductions from derivation trees to derived structures. Considering the case of MGs, Kobele (2011) and Graf (2011) show that MG derivation trees are closed under intersection with arbitrary MSO constraints, and Graf (2012) shows that complex movement operations can be added to MGs by enhancing the transduction from derivation trees to derived trees. These closure properties allow Graf (2013) to develop a full MG-based treatment of Minimalist syntax within Morawietz's framework, showing that Minimalist constraints on derivation trees and derived trees, as well as transderivational constraints, can be implemented using Merge by carefully choosing categories and selection features for lexical items. These findings have highlighted the relevance of overgeneration to mainstream Minimalist syntax. As discussed in Graf (2017), existing constraints on movement can be circumvented by using Merge to create non-local dependencies. On the other hand, any pathological pattern is a logically possible MG as long as it is MSO-definable. Illustrating the latter point, Graf (2013, pp. 117-118) identifies three kinds of unnatural dependencies that can be generated by MGs:", |
| "cite_spans": [ |
| { |
| "start": 97, |
| "end": 113, |
| "text": "Ritchie's (1973)", |
| "ref_id": "BIBREF35" |
| }, |
| { |
| "start": 302, |
| "end": 314, |
| "text": "Rogers (1994", |
| "ref_id": "BIBREF38" |
| }, |
| { |
| "start": 315, |
| "end": 330, |
| "text": "Rogers ( , 1998", |
| "ref_id": "BIBREF39" |
| }, |
| { |
| "start": 471, |
| "end": 497, |
| "text": "(Thatcher and Wright, 1968", |
| "ref_id": "BIBREF44" |
| }, |
| { |
| "start": 552, |
| "end": 566, |
| "text": "Rizzi's (1990)", |
| "ref_id": "BIBREF37" |
| }, |
| { |
| "start": 655, |
| "end": 671, |
| "text": "Morawietz (2008)", |
| "ref_id": "BIBREF34" |
| }, |
| { |
| "start": 883, |
| "end": 896, |
| "text": "Kobele (2011)", |
| "ref_id": "BIBREF29" |
| }, |
| { |
| "start": 901, |
| "end": 912, |
| "text": "Graf (2011)", |
| "ref_id": "BIBREF14" |
| }, |
| { |
| "start": 1009, |
| "end": 1020, |
| "text": "Graf (2012)", |
| "ref_id": "BIBREF15" |
| }, |
| { |
| "start": 1181, |
| "end": 1192, |
| "text": "Graf (2013)", |
| "ref_id": "BIBREF16" |
| }, |
| { |
| "start": 1866, |
| "end": 1890, |
| "text": "Graf (2013, pp. 117-118)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Overgeneration", |
| "sec_num": "4" |
| }, |
| { |
| "text": "(8) a. patterns sensitive to \"non-linguistic\" information, such as the number of words in a sentence; b. languages with completely free word order, subject to no restrictions; and c. creating arbitrarily many copies of context-free structure.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Overgeneration", |
| "sec_num": "4" |
| }, |
| { |
| "text": "In Subsections 4.1 and 4.2, I show that the patterns described in (8a) and (8b), respectively, can be represented as substitutable MCFLs. In Subsection 4.3, we will see that the arbitrary copying of (8c) is not congruential, but it can be generated by an MCFG with both the 2-FKP and the 1-FCP.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Overgeneration", |
| "sec_num": "4" |
| }, |
| { |
| "text": "One of the earliest motivations for the rationalist position is the observation that syntactic dependencies are universally sensitive to constituency structure. Chomsky (1968) argues:", |
| "cite_spans": [ |
| { |
| "start": 161, |
| "end": 175, |
| "text": "Chomsky (1968)", |
| "ref_id": "BIBREF4" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Structure-Independent Patterns", |
| "sec_num": "4.1" |
| }, |
| { |
| "text": ". . . grammatical transformations are invariably structure-dependent in the sense that they apply to a string of words by virtue of the organization of these words into phrases. It is easy to imagine structure-independent operations that apply to a string of elements quite independently of its ab-stract structure as a system of phrases. . . . Yet no human language contains structure-independent operations . . . . The language-learner knows that [such operations] need not be considered as tentative hypotheses.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Structure-Independent Patterns", |
| "sec_num": "4.1" |
| }, |
| { |
| "text": "Examples of \"structure-independent\" patterns typically consist of operations whose targets are based on arithmetic criteria. One such example appears in the passage quoted above, where Chomsky imagines an auxiliary-fronting operation targeting the first auxiliary in a sentence. This produces the ungrammatical question (9b) from the corresponding declarative (9a).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Structure-Independent Patterns", |
| "sec_num": "4.1" |
| }, |
| { |
| "text": "(9) a.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Structure-Independent Patterns", |
| "sec_num": "4.1" |
| }, |
| { |
| "text": "The subjects who will act as controls will be paid. b. * Will the subjects who t act as controls will be paid? c.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Structure-Independent Patterns", |
| "sec_num": "4.1" |
| }, |
| { |
| "text": "Will the subjects who will act as controls t be paid?", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Structure-Independent Patterns", |
| "sec_num": "4.1" |
| }, |
| { |
| "text": "Along these lines, Graf imagines a requirement that the length of a sentence or phrase be a multiple of some fixed number n. Definition 10. For each n > 0, let us define", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Structure-Independent Patterns", |
| "sec_num": "4.1" |
| }, |
| { |
| "text": "MOD n := {x | |x| \u2261 0 mod n}.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Structure-Independent Patterns", |
| "sec_num": "4.1" |
| }, |
| { |
| "text": "Despite its structure-independence, MOD n is easily shown to be substitutable. Proposition 11. MOD n is k-substitutable for every k \u2208 N and n > 0.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Structure-Independent Patterns", |
| "sec_num": "4.1" |
| }, |
| { |
| "text": "Proof. For any k-tuple x and k-context c, c \u2299 x \u2208 MOD n if and only if |c| + |x| is a multiple of n. Therefore,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Structure-Independent Patterns", |
| "sec_num": "4.1" |
| }, |
| { |
| "text": "x \u25c3 = {c | |c| \u2261 \u2212|x| mod n} .", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Structure-Independent Patterns", |
| "sec_num": "4.1" |
| }, |
| { |
| "text": "It is clear that for any x, y \u2208 (\u03a3 * ) k , x \u25c3 and y \u25c3 are either equal or disjoint, so MOD n is ksubstitutable for every k.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Structure-Independent Patterns", |
| "sec_num": "4.1" |
| }, |
| { |
| "text": "Following Shieber's (1985) argument that Swiss German is not context-free, Joshi (1985) proposed the MCS languages as a characterization of the possible natural languages. This class is defined by grammar formalisms that admit a polynomial-time parsing algorithm, exhibit constant growth, and express limited cross-serial dependencies. The notion of \"limited cross-serial dependencies\" was left vague, but Joshi et al. (1990, 1991) provide some elaboration:", |
| "cite_spans": [ |
| { |
| "start": 20, |
| "end": 26, |
| "text": "(1985)", |
| "ref_id": null |
| }, |
| { |
| "start": 419, |
| "end": 425, |
| "text": "(1990,", |
| "ref_id": null |
| }, |
| { |
| "start": 426, |
| "end": 431, |
| "text": "1991)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Free Word Order", |
| "sec_num": "4.2" |
| }, |
| { |
| "text": "[MCS grammars] capture only certain kinds of dependencies, e.g., nested dependencies and certain limited kinds of crossing dependencies (e.g., in the subordinate clause constructions in Dutch or some variations of them, but perhaps not in the so-called MIX (or Bach) language . . .", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Free Word Order", |
| "sec_num": "4.2" |
| }, |
| { |
| "text": "The language MIX mentioned above is the focus of this subsection. Definition 12. The language MIX is defined as", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "[).]", |
| "sec_num": null |
| }, |
| { |
| "text": "MIX := {x \u2208 {a, b, c} * | |x| a = |x| b = |x| c } .", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "[).]", |
| "sec_num": null |
| }, |
| { |
| "text": "According to Joshi (1985) , MIX \"represents the extreme case of the degree of free word order permitted in a language,\" and is therefore \"linguistically not relevant.\" The fact that MIX is a 2-MCFL but not a TAL (Salvati, 2011 (Salvati, , 2015 Kanazawa and Salvati, 2012) has been used to argue that the TALs are a more suitable formalization of the MCS languages than the MCFLs. This kind of reasoning may be seen as a rationalist position asserting that MIX is not a possible natural language because UG requires natural languages to be TALs. An empiricist account for the absence of MIX-like natural languages might claim that the learners fail to converge on MIX even though MIX is allowed by UG. However, it turns out that MIX is substitutable, so such an account would not be supported by Distributional Learning as a model of language acquisition. Proposition 13 (Clark and Yoshinaka, 2016) . MIX is k-substitutable for every k \u2208 N.", |
| "cite_spans": [ |
| { |
| "start": 13, |
| "end": 25, |
| "text": "Joshi (1985)", |
| "ref_id": "BIBREF20" |
| }, |
| { |
| "start": 212, |
| "end": 226, |
| "text": "(Salvati, 2011", |
| "ref_id": "BIBREF40" |
| }, |
| { |
| "start": 227, |
| "end": 243, |
| "text": "(Salvati, , 2015", |
| "ref_id": "BIBREF41" |
| }, |
| { |
| "start": 244, |
| "end": 271, |
| "text": "Kanazawa and Salvati, 2012)", |
| "ref_id": "BIBREF26" |
| }, |
| { |
| "start": 870, |
| "end": 897, |
| "text": "(Clark and Yoshinaka, 2016)", |
| "ref_id": "BIBREF11" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "[).]", |
| "sec_num": null |
| }, |
| { |
| "text": "Proof. Suppose c \u2299 x, c \u2299 y, d \u2299 x \u2208 MIX. We want to show that d \u2299 y \u2208 MIX. To that end, observe that for any context \u03b3 and symbol i \u2208 {a, b, c},", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "[).]", |
| "sec_num": null |
| }, |
| { |
| "text": "EQUATION", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [ |
| { |
| "start": 0, |
| "end": 8, |
| "text": "EQUATION", |
| "ref_id": "EQREF", |
| "raw_str": "|\u03b3 \u2299 y| i = |\u03b3 \u2299 x| i \u2212 |x| i + |y| i .", |
| "eq_num": "(14)" |
| } |
| ], |
| "section": "[).]", |
| "sec_num": null |
| }, |
| { |
| "text": "Taking \u03b3 = c, we obtain", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "[).]", |
| "sec_num": null |
| }, |
| { |
| "text": "\u2212|x| i + |y| i = |c \u2299 y| i \u2212 |c \u2299 x| i .", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "[).]", |
| "sec_num": null |
| }, |
| { |
| "text": "Next, we take \u03b3 = d and substitute the above into (14), giving us", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "[).]", |
| "sec_num": null |
| }, |
| { |
| "text": "|d \u2299 y| i = |d \u2299 x| i + |c \u2299 y| i \u2212 |c \u2299 x| i .", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "[).]", |
| "sec_num": null |
| }, |
| { |
| "text": "Since the terms on the right-hand side have the same value for all i, so must the left-hand side. This means that d \u2299 y \u2208 MIX, as desired.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "[).]", |
| "sec_num": null |
| }, |
| { |
| "text": "The third unnatural dependency that Graf considers is represented by the double-copying language", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "COPY 3 (D 1 ). Definition 15. For L \u2286 \u03a3 * and n \u2208 N, define COPY n (L) := {(x#) n | x \u2208 L},", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "where # / \u2208 \u03a3. 4 Definition 16. The language D 1 is the language generated by the following 1-MCFG.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "S(xy) \u2190 S(x)S(y) S([x]) \u2190 S(x) S(\u03b5) \u2190", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "Copy languages have two interpretations in mathematical linguistics.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "On the one hand, COPY 2 ({a, b} * ) represents the cross-serial dependencies found in Swiss German, since it is a homomorphic image of the embedded-clause verb-argument sequences that Shieber shows is not context-free. On the other hand, the idea of copying structure often appears explicitly in syntactic analyses. Kobele (2006) , for instance, argues that Yoruba has a relative clause construction that involves copying VPs. Apart from overtly attested instances of copying, Merchant (1999 Merchant ( , 2001 ) develops a theory of sluicing in which CPs are copied and their TP complements are deleted. It turns out that Distributional Learning can distinguish between these two interpretations of copying. Proposition 17. COPY n (L) is a congruential n-MCFL if and only if L is regular. 5 If we consider non-regular Ls to represent copying of structure, then only the former interpretation is captured by the congruential n-MCFLs. Lemma 18. Let x = \u27e8x 1 , x 2 , . . . , x m \u27e9, with m \u2264 n. If |x| # < n, then either |x i | # \u2265 2 for some i, x | COPYn(L)\u27e9 = \u2205, or x belongs to a finite equivalence class of \u2261 m COPYn(L) . Proof. Suppose x \u25c3 \u0338 = \u2205, |x i | # \u2264 1 for all i, and |x| # \u2264 n \u2212 2. Then, x has a context c = c 0 c 1 . . . c n such that |c i | # \u2265 2 for some i. Writing c i = l#w#r, observe that every y \u2208 c \u25b9 satisfies c \u2299 y = (w#) n . There are only finitely 4 The language considered in Graf (2013) does not have a final #. However, adding the final # allows COPYn(L) to be congruential when L is regular (Proposition 17).", |
| "cite_spans": [ |
| { |
| "start": 316, |
| "end": 329, |
| "text": "Kobele (2006)", |
| "ref_id": "BIBREF28" |
| }, |
| { |
| "start": 477, |
| "end": 491, |
| "text": "Merchant (1999", |
| "ref_id": "BIBREF31" |
| }, |
| { |
| "start": 492, |
| "end": 509, |
| "text": "Merchant ( , 2001", |
| "ref_id": "BIBREF32" |
| }, |
| { |
| "start": 789, |
| "end": 790, |
| "text": "5", |
| "ref_id": null |
| }, |
| { |
| "start": 1369, |
| "end": 1370, |
| "text": "4", |
| "ref_id": null |
| }, |
| { |
| "start": 1398, |
| "end": 1409, |
| "text": "Graf (2013)", |
| "ref_id": "BIBREF16" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "5 However, it is easy to show that COPYn({a i b i | i \u2265 0}) is a congruential (n + 1)-MCFL. many such ys, so only finitely many strings may share the context c with x.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "Next, suppose that x \u25c3 \u0338 = \u2205, |x i | # \u2264 1 for all i, and |x| # = n \u2212 1. Then, we have |x p | # = 0 for some p, and for i \u0338 = p we can write x i = y i #z i with |y i | # = |z i | # = 0. Let y be the longest y i and z be the longest z i , and let y m+1 = z 0 := \u03b5. Observe that c \u2208 x \u25c3 if and only if c \u2299 x = (w#) n for some w \u2208", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "z\u03a3 * y \u2229 z p\u22121 \u03a3 * x p \u03a3 * y p+1 . Thus, if x \u2032 \u25c3 = x \u25c3 , then x \u2032 = \u27e8x \u2032 1 , x \u2032 2 , . . . , x \u2032 m \u27e9, where x \u2032 q = x p for some q, x \u2032 i = y \u2032 i #z \u2032 i for i \u0338 = q, y is the longest y \u2032", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "i , z is the longest z \u2032 i , y \u2032 q+1 = y q+1 , and z \u2032 q\u22121 = z p\u22121 . There are only finitely many such x \u2032 , so the lemma follows.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "Proof of Proposition 17. First, suppose L is regular. Let M be the minimal right-to-left deterministic finite-state automaton recognizing L. We can construct an MCFG G for COPY n (L) as follows. The nonterminals of G are the states of M . If A is the start state of M , then G has the rule A(#, #, . . . , #) \u2190. If M transitions from state B to state A after reading a, then G has a rule A(ax 1 , ax 2 , . . . , ax", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "n ) \u2190 B(x 1 , x 2 , . . . , x n ).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "Finally, for each accept state S of M , G has a start symbol I S and a rule I", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "S (x 1 x 2 . . . x n ) \u2190 S(x 1 , x 2 , . . . , x n ). It is clear that for each x \u2208 L(G, I S ), x \u25c3 = { }.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "Observe that for each nonterminal A of G, L(G, A) is the set of strings (w#) n such that M is in state A after reading w. Thus, elements of L(G, A) are of the form (w#) n . For each such (w#) n , we have", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "((w#) n ) \u25c3 = {c c . . . c | cw \u2208 L}. Since M is minimal, if (u#) n , (v#) n \u2208 L(G, A)", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": ", then by the Myhill-Nerode Theorem we must have", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "{c | cu \u2208 L} = {c | cv \u2208 L}, thus ((u#) n ) \u25c3 = {c c . . . c | cu \u2208 L} = {c c . . . c | cv \u2208 L} = ((v#) n ) \u25c3 .", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "This means that G is congruential. Now, suppose COPY n (L) is generated by a congruential n-MCFG G. By Lemma 18, without loss of generality each copy of w \u2208 L in (w#) n \u2208 COPY n (L) is generated exclusively using rules of the form", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "A(l 1 x 1 r 1 , l 2 x 2 r 2 , . . . , l n x n r n ) \u2190 B(x 1 , x 2 , . . . , x n ),", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "where l i , r i \u2208 \u03a3 * for each i and L(G, B) \u2286 (\u03a3 * #\u03a3 * ) n . Since x 1 must already contain a #, such rules can only append a constant string to the left of the first copy of w, so the set of possible ws must be regular.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "Graf argues that COPY 3 (D 1 ) is unnatural because \"embeddings of unbounded depth are copied and fully realized in three distinct positions in the utterance.\" According to Proposition 17, the property of unbounded depth disqualifies COPY 3 (D 1 ) from congruentiality, but the existence of more than two copies does not, as long as COPY 3 (D 1 ) is generated by a 3-MCFG. COPY 3 (D 1 ) is still learnable, however, because it belongs to the class of 3-MCFLs defined by the FKP and the FCP. Proposition 19. COPY 3 (D 1 ) is generated by a 3-MCFG G with the 2-FKP and the 1-FCP.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "Proof. G is defined as follows.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "S(x#y#z#) \u2190 T (x, y, z) T (x 1 x 2 , y 1 y 2 , z 1 z 2 ) \u2190 T (x 1 , y 1 , z 1 ) T (x 2 , y 2 , z 2 ) T ([x], [y], [z]) \u2190 T (x, y, z) T (\u03b5, \u03b5, \u03b5) \u2190 We have L(G, S) \u25c3 = COPY 3 (D 1 ) \u25c3 = { } = {###} \u25c3 L(G, T ) \u25c3 = {\u27e8w, w, w\u27e9 | w \u2208 D 1 } \u25c3 = l r#l r#l r# l r \u2208 D \u27e8D 1 | 1 = {\u27e8[[]], [[]], [[]]\u27e9, \u27e8[][], [][], [][]\u27e9} \u25c3 ,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "so G has the 2-FKP. 6 We also have", |
| "cite_spans": [ |
| { |
| "start": 20, |
| "end": 21, |
| "text": "6", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "L(G, S) \u25c3\u25b9 = { } \u25b9 L(G, T ) \u25c3\u25b9 = { # # #} \u25b9 ,", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "so G has the 1-FCP.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Copying", |
| "sec_num": "4.3" |
| }, |
| { |
| "text": "We have seen that MOD n and MIX are ksubstitutable for every k, while COPY 3 (D 1 ) is generated by a 3-MCFG with the 2-FKP and the 1-FCP. If the Distributional Learning hierarchy of Figure 2 is taken to be a measure of complexity, then we may conclude from these facts that MOD n and MIX are very simple from the perspective of learnability. Proposition 17 gives us the interesting result that in general, the complexity of COPY n (L) with respect to learnability is related to the language-theoretic complexity of L, so that COPY 3 (D 1 ) is slightly more complex than the congruential languages. Since natural language grammars are not congruential, 7 any class in the learnable hierarchy that might plausibly include natural language syntax would also likely include MOD n , MIX, and COPY 3 (D 1 ).", |
| "cite_spans": [], |
| "ref_spans": [ |
| { |
| "start": 183, |
| "end": 191, |
| "text": "Figure 2", |
| "ref_id": "FIGREF1" |
| } |
| ], |
| "eq_spans": [], |
| "section": "Conclusion", |
| "sec_num": "5" |
| }, |
| { |
| "text": "The discussions in Subsections 4.2 and 4.3 should be contrasted with language-theoretic analyses of MIX and COPY 3 (D 1 ), respectively. As mentioned previously, MIX falls outside the TALs, which is identical to the well-nested 2-MCFLs. Similarly, Kanazawa and Salvati (2010) show that COPY 3 (D 1 ) is not a well-nested n-MCFL for any n. The criterion of well-nestedness, then, provides an elegant rationalist explanation for the absence of MIX-or COPY 3 (D 1 )-like natural languages. Such a criterion could also be justified on functionalist grounds, since well-nested MCFGs admit a more efficient parsing algorithm than MCFGs in general (G\u00f3mez-Rodr\u00edguez et al., 2010) . While the well-nestedness requirement does not eliminate MOD n , an anonymous reviewer observes that intersecting the language of a CFG with MOD n increases the size of that CFG by a factor of n 2 . Thus, when n is large, languages with MOD n -like dependencies may be eliminated by functionalist considerations regarding grammar size.", |
| "cite_spans": [ |
| { |
| "start": 248, |
| "end": 275, |
| "text": "Kanazawa and Salvati (2010)", |
| "ref_id": "BIBREF25" |
| }, |
| { |
| "start": 641, |
| "end": 671, |
| "text": "(G\u00f3mez-Rodr\u00edguez et al., 2010)", |
| "ref_id": "BIBREF13" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Conclusion", |
| "sec_num": "5" |
| }, |
| { |
| "text": "The case of MOD n shows that the regular languages capture many patterns that do not resemble natural language dependencies. Although the inclusion of the regular languages in natural language has traditionally been taken for granted, 8 recent work on the subregular hierarchy has shown that markedness constraints in phonotactics and morphotactics typically belong to restricted, IIL-learnable subclasses of the regular languages (Heinz et al., 2011; Aks\u00ebnova et al., 2016 ) that formalize notions of locality. It may be the case that a refinement of the regular languages is needed for syntax as well. While the study of the equivalence relation \u2261 k L may be seen as an algebraic treatment of the notion of structure (Clark, 2015) , the learnability of MOD n may reveal a point of divergence between the algebraic approach and the intuitive notion of syntactic constituencies.", |
| "cite_spans": [ |
| { |
| "start": 431, |
| "end": 451, |
| "text": "(Heinz et al., 2011;", |
| "ref_id": "BIBREF19" |
| }, |
| { |
| "start": 452, |
| "end": 473, |
| "text": "Aks\u00ebnova et al., 2016", |
| "ref_id": "BIBREF0" |
| }, |
| { |
| "start": 719, |
| "end": 732, |
| "text": "(Clark, 2015)", |
| "ref_id": "BIBREF8" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Conclusion", |
| "sec_num": "5" |
| }, |
| { |
| "text": "In conclusion, this paper has shown that the restricted language classes in the Distributional Learning hierarchy are rich enough to raise the very questions of overgeneration that they were hypothesized to solve.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Conclusion", |
| "sec_num": "5" |
| }, |
| { |
| "text": "While Distributional Learning does not provide a learnability-based account for the typological absence of patterns modelled by MOD n , MIX, or COPY 3 (D 1 ), all three patterns can plausibly be eliminated on rationalist or functionalist grounds. These findings suggest that learnability may play a smaller role in determining natural language typology than once expected.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Conclusion", |
| "sec_num": "5" |
| }, |
| { |
| "text": "Learnable Classes of LanguagesThe theory of learnability considered here is based on the Identification in the Limit (IIL) model ofGold (1967). Under this paradigm, the learner receives an infinite data stream containing all possible strings drawn from a target language L, arranged in an unspecified order. After observing", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "", |
| "sec_num": null |
| }, |
| { |
| "text": "The definition presented here is for the weak versions of the FKP and the FCP. Other versions of these properties are discussed inKanazawa and Yoshinaka (2017).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "", |
| "sec_num": null |
| }, |
| { |
| "text": "G does not have the 1-FKP because for any\u27e8x, x, x\u27e9 \u2208 L(G, T ) \u25c3\u25b9 , x # x#x # \u2208 \u27e8x, x, x\u27e9 \u25c3 \\L(G, T ) \u25c3 .", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "", |
| "sec_num": null |
| }, |
| { |
| "text": "For example, congruentiality would preclude the possibility of a single word such as effect or affect having the distribution of both a noun and a verb if nonterminals are identified with syntactic categories.8Additionally, Joshi et al. (1990 mention strict inclusion of the CFLs as a fourth defining property of the MCS languages.", |
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| "ref_spans": [], |
| "eq_spans": [], |
| "section": "", |
| "sec_num": null |
| } |
| ], |
| "back_matter": [], |
| "bib_entries": { |
| "BIBREF0": { |
| "ref_id": "b0", |
| "title": "Morphotactics as Tier-Based Strictly Local Dependencies", |
| "authors": [ |
| { |
| "first": "Al\u00ebna", |
| "middle": [], |
| "last": "Aks\u00ebnova", |
| "suffix": "" |
| }, |
| { |
| "first": "Thomas", |
| "middle": [], |
| "last": "Graf", |
| "suffix": "" |
| }, |
| { |
| "first": "Sedigheh", |
| "middle": [], |
| "last": "Moradi", |
| "suffix": "" |
| } |
| ], |
| "year": 2016, |
| "venue": "Proceedings of the 14th SIG-MORPHON Workshop on Computational Research in Phonetics, Phonology, and Morphology", |
| "volume": "", |
| "issue": "", |
| "pages": "121--130", |
| "other_ids": {}, |
| "num": null, |
| "urls": [], |
| "raw_text": "Al\u00ebna Aks\u00ebnova, Thomas Graf, and Sedigheh Moradi. 2016. Morphotactics as Tier-Based Strictly Local Dependencies. In Proceedings of the 14th SIG- MORPHON Workshop on Computational Research in Phonetics, Phonology, and Morphology, pages 121-130, Berlin, Germany. Association for Compu- tational Linguistics.", |
| "links": null |
| }, |
| "BIBREF1": { |
| "ref_id": "b1", |
| "title": "Inference of Reversible Languages", |
| "authors": [ |
| { |
| "first": "", |
| "middle": [], |
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