| { | |
| "paper_id": "W19-0104", | |
| "header": { | |
| "generated_with": "S2ORC 1.0.0", | |
| "date_generated": "2023-01-19T06:33:29.360562Z" | |
| }, | |
| "title": "Q-Theory Representations are logically equivalent to Autosegmental Representations", | |
| "authors": [ | |
| { | |
| "first": "Nick", | |
| "middle": [], | |
| "last": "Danis", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "", | |
| "institution": "Princeton University", | |
| "location": {} | |
| }, | |
| "email": "[email protected]" | |
| }, | |
| { | |
| "first": "Adam", | |
| "middle": [], | |
| "last": "Jardine", | |
| "suffix": "", | |
| "affiliation": {}, | |
| "email": "[email protected]" | |
| } | |
| ], | |
| "year": "", | |
| "venue": null, | |
| "identifiers": {}, | |
| "abstract": "We use model theory and logical interpretations to systematically compare two competing representational theories in phonology, Q-Theory (Shih and Inkelas, 2014, forthcoming) and Autosegmental Phonology (Goldsmith, 1976). We find that, under reasonable assumptions for capturing tone patterns, Q-Theory Representations are equivalent to Autosegmental Representations, in that any constraint that can be written in one theory can be written in another. This contradicts the assertions of Shih and Inkelas, who claim that Q-Theory Representations are different from, and superior to, Autosegmental Representations.", | |
| "pdf_parse": { | |
| "paper_id": "W19-0104", | |
| "_pdf_hash": "", | |
| "abstract": [ | |
| { | |
| "text": "We use model theory and logical interpretations to systematically compare two competing representational theories in phonology, Q-Theory (Shih and Inkelas, 2014, forthcoming) and Autosegmental Phonology (Goldsmith, 1976). We find that, under reasonable assumptions for capturing tone patterns, Q-Theory Representations are equivalent to Autosegmental Representations, in that any constraint that can be written in one theory can be written in another. This contradicts the assertions of Shih and Inkelas, who claim that Q-Theory Representations are different from, and superior to, Autosegmental Representations.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Abstract", | |
| "sec_num": null | |
| } | |
| ], | |
| "body_text": [ | |
| { | |
| "text": "Model theory and mathematical logic can be used to rigorously define phonological representations and constraints (Bird, 1995; Potts and Pullum, 2002) . The logical notion of interpretation (Enderton, 1972; Courcelle, 1994; Hodges, 1997) between logics of different kinds of models then allows us to compare and contrast differing representational theories, and rigorously examine whether or not they are truly distinct or if they are simply notational variants of one another (Strother-Garcia and Heinz, 2017) .", | |
| "cite_spans": [ | |
| { | |
| "start": 114, | |
| "end": 126, | |
| "text": "(Bird, 1995;", | |
| "ref_id": "BIBREF2" | |
| }, | |
| { | |
| "start": 127, | |
| "end": 150, | |
| "text": "Potts and Pullum, 2002)", | |
| "ref_id": "BIBREF19" | |
| }, | |
| { | |
| "start": 190, | |
| "end": 206, | |
| "text": "(Enderton, 1972;", | |
| "ref_id": "BIBREF7" | |
| }, | |
| { | |
| "start": 207, | |
| "end": 223, | |
| "text": "Courcelle, 1994;", | |
| "ref_id": "BIBREF4" | |
| }, | |
| { | |
| "start": 224, | |
| "end": 237, | |
| "text": "Hodges, 1997)", | |
| "ref_id": "BIBREF12" | |
| }, | |
| { | |
| "start": 477, | |
| "end": 510, | |
| "text": "(Strother-Garcia and Heinz, 2017)", | |
| "ref_id": "BIBREF25" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "This paper uses these techniques to critically examine the Q-Theory Representations (QRs) of Shih and Inkelas (forthcoming, henceforth SI ; see also Shih and Inkelas (2014) ). SI argue for QRs as a superior alternative to Autosegmental Representations (ARs; Goldsmith (1976) ), specifically with respect to phonological tone patterns. We find that, to the contrary, the differences are notational. We show that for any constraint that can be written in the first-order logic of QRs, there is an equivalent constraint in ARs, and vice versa.", | |
| "cite_spans": [ | |
| { | |
| "start": 93, | |
| "end": 137, | |
| "text": "Shih and Inkelas (forthcoming, henceforth SI", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 149, | |
| "end": 172, | |
| "text": "Shih and Inkelas (2014)", | |
| "ref_id": "BIBREF22" | |
| }, | |
| { | |
| "start": 258, | |
| "end": 274, | |
| "text": "Goldsmith (1976)", | |
| "ref_id": "BIBREF9" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "The fundamental idea behind QRs is that every segment, or Q, is divided into three subsegments, or qs. Agreement and disagreement is based on correspondence (Hansson, 2001; Rose and Walker, 2004; Bennett, 2015) , a relation that holds between qs and between Qs. To give an example, SI give the following QR in (1a) for the Basa\u00e1 word [h\u00f3l\u00f4l] 'ripen' (Dimmendaal, 1988; Hyman, 2003) , in which the first vowel is a level high tone and the second vowel is a falling tone. Each [o] vowel Q is split into three qs, which each carry a tone. Indices on the qs represent correspondence. (Consonant qs have been abbreviated.)", | |
| "cite_spans": [ | |
| { | |
| "start": 157, | |
| "end": 172, | |
| "text": "(Hansson, 2001;", | |
| "ref_id": "BIBREF11" | |
| }, | |
| { | |
| "start": 173, | |
| "end": 195, | |
| "text": "Rose and Walker, 2004;", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 196, | |
| "end": 210, | |
| "text": "Bennett, 2015)", | |
| "ref_id": "BIBREF1" | |
| }, | |
| { | |
| "start": 350, | |
| "end": 368, | |
| "text": "(Dimmendaal, 1988;", | |
| "ref_id": "BIBREF6" | |
| }, | |
| { | |
| "start": 369, | |
| "end": 381, | |
| "text": "Hyman, 2003)", | |
| "ref_id": "BIBREF13" | |
| }, | |
| { | |
| "start": 475, | |
| "end": 478, | |
| "text": "[o]", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "EQUATION", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [ | |
| { | |
| "start": 0, | |
| "end": 8, | |
| "text": "EQUATION", | |
| "ref_id": "EQREF", | |
| "raw_str": "a. h(\u00f3 1\u00f31,2\u00f32,3 )l(\u00f3 3\u00f24\u00f24 )l b. H L V V", | |
| "eq_num": "(1)" | |
| } | |
| ], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "In (1a), the first q of the second vowel is hightoned while the rest are low-toned; this thus represents the falling contour of the second vowel. Furthermore, the last q of the first vowel and the first q in the second are in correspondence (and both high-toned) . This indicates that the falling contour on the second vowel is the result of partial agreement with the high-toned first vowel.", | |
| "cite_spans": [ | |
| { | |
| "start": 241, | |
| "end": 262, | |
| "text": "(and both high-toned)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "In contrast, ARs would depict [h\u00f3l\u00f4l] using separate strings of autosegments associated to one another. An AR for [h\u00f3l\u00f4l] , given in (1b), represents a high (H) tone associated to both the first vowel (V) and the second vowel.", | |
| "cite_spans": [ | |
| { | |
| "start": 114, | |
| "end": 121, | |
| "text": "[h\u00f3l\u00f4l]", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "SI make several claims about QRs in favor of ARs. First, they claim that representations like in (1) capture tone patterns without \"the special representational machinery of autosegments and association lines\" (SI, . They give a number of analyses which they argue shows that QRs are \"better at capturing key tone behaviors\" (p. 2).", | |
| "cite_spans": [ | |
| { | |
| "start": 210, | |
| "end": 214, | |
| "text": "(SI,", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "By precisely studying the nature of these representations, however, we show that QRs are logically equivalent to ARs. First, we give model-a. theoretic definitions of both QRs and ARs. This shows that, contra to SI's claims, QRs do require an 'association' relation between qs and Qs. Second, based on the logical transductions of Courcelle (1994), we give a first-order (FO) transduction from QRs to ARs, and from ARs to QRs. This guarantees that, given any FO statement ' written over QRs, there is an equivalent FO statement ' 0 in ARs such that any QR model satisfies ' if and only if its equivalent AR model satisfies ' 0 . In phonological terms, this means that for any constraint that we can write in FO logic of QRs, there is an equivalent constraint in the FO logic of ARs (and vice versa). These models and transductions reveal an equivalence between chains of qs connected by correspondence in QRs and tonal autosegments in ARs. This paper is not meant to be a complete rebuttal of SI, but instead to lay the formal groundwork for establishing the similarity of QRs and ARs. Throughout, we assume that FO logic as the upper bound for the expressivity necessary to capture constraints in natural language phonology (Bird, 1995; Rogers et al., 2013) .", | |
| "cite_spans": [ | |
| { | |
| "start": 1225, | |
| "end": 1237, | |
| "text": "(Bird, 1995;", | |
| "ref_id": "BIBREF2" | |
| }, | |
| { | |
| "start": 1238, | |
| "end": 1258, | |
| "text": "Rogers et al., 2013)", | |
| "ref_id": "BIBREF20" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "H 1 H 2 H 3 H 4 L 5 L 6 Q 7 Q 8 7 ! H 1 0 L 5 0 V 7 0 V 8 0 b. H 3 L 4 V 1 V 2 7 ! H 1 1 H 1 2 H 1 3 H 2 1 L 2 2 L 2 3 Q 1 0 Q 2 0", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "This paper is structured as follows. \u00a72 summarizes the equivalence between structures in informal terms. \u00a73 gives the preliminaries of model theory and logic; \u00a74 defines ARs and QRs in terms of model theory; and \u00a75 defines the transductions between them. \u00a76 summarizes and discusses the results, and \u00a77 concludes.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "Before going into the formal details, we first give a brief overview of how transductions between the two representations proceed. These are illustrated with examples in Fig. 1 showing transductions between models for the representations in (1).", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 170, | |
| "end": 176, | |
| "text": "Fig. 1", | |
| "ref_id": "FIGREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Overview", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "As shown in the left-hand side of Fig. 1a , defining QRs precisely reveals that there must be a relation pairing tone-bearing qs (depicted with Hs and Ls) with vowel Qs. Moving from a QR to an AR, then, is a matter of identifying the first member of a chain of corresponding qs (indicated by curved lines) and as assigning it to a tone in the output AR. All other qs in that chain are 'merged' into that autosegment-thus, for example, as both vowel Qs are associated to correspondents of 1 in the QR in Fig. 1a , both equivalent Vs in the AR are associated to its output tone 1 0 .", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 34, | |
| "end": 41, | |
| "text": "Fig. 1a", | |
| "ref_id": "FIGREF0" | |
| }, | |
| { | |
| "start": 503, | |
| "end": 510, | |
| "text": "Fig. 1a", | |
| "ref_id": "FIGREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Overview", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "In the other direction, illustrated in Fig. 1b , we create three additional copies of each V in the AR, representing the output qs. For example, vowel 1 in the left-hand side of Fig. 1b has copies 1 1 , 1 2 , and 1 3 in the right-hand side. These copies are labeled and related through correspondence according to the associations in the AR. For a series of vowels associated to the same tone in the AR, their output Qs are associated to a chain of corresponding qs of the same tone value. For example, both vowels 1 and 2 in the left-hand side of Fig. 1b are associated to the same H tone, so their output Qs are associated to a chain of corresponding H-toned qs in the output. This thus implements the equivalence of tonal autosegments to q-correspondence chains in the transduction from ARs to QRs.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 39, | |
| "end": 46, | |
| "text": "Fig. 1b", | |
| "ref_id": "FIGREF0" | |
| }, | |
| { | |
| "start": 178, | |
| "end": 185, | |
| "text": "Fig. 1b", | |
| "ref_id": "FIGREF0" | |
| }, | |
| { | |
| "start": 548, | |
| "end": 555, | |
| "text": "Fig. 1b", | |
| "ref_id": "FIGREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Overview", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "The following is based on standard concepts of finite relational structures (Enderton, 1972; Libkin, 2004) . A signature S is a fixed set {R 1 , R 2 , ..., R n } of n named relations. A model M over a signature is a tuple hD; R 1 , R 2 , ..., R n i with a domain D of elements and a set of n rela-tions where each R i \u2713 D k for some k. Here, k is equal to either 1 or 2; that is, we consider only unary and binary relations.", | |
| "cite_spans": [ | |
| { | |
| "start": 76, | |
| "end": 92, | |
| "text": "(Enderton, 1972;", | |
| "ref_id": "BIBREF7" | |
| }, | |
| { | |
| "start": 93, | |
| "end": 106, | |
| "text": "Libkin, 2004)", | |
| "ref_id": "BIBREF16" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Models", | |
| "sec_num": "3.1" | |
| }, | |
| { | |
| "text": "For example, the signature {<, P a , P b } can describe the set of strings over the alphabet a and b. A model in this signature is given in Fig. 2 . ", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 140, | |
| "end": 146, | |
| "text": "Fig. 2", | |
| "ref_id": "FIGREF1" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Models", | |
| "sec_num": "3.1" | |
| }, | |
| { | |
| "text": "A fixed signature induces a first-order (FO) logical language L S as follows. For every", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Logic", | |
| "sec_num": "3.2" | |
| }, | |
| { | |
| "text": "R i 2 S, R i (x 1 , ..., x k ) is an atomic formula in L S , which is interpreted as true in a model M when x 1 , ..., x k are evaluated to d 1 , ..., d k 2 D and (d 1 , ..., d k ) 2 R i in M .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Logic", | |
| "sec_num": "3.2" | |
| }, | |
| { | |
| "text": "We also assume an equality predicate x \u21e1 y that is true when x and y are both evaluated to some i 2 D. We then define the FO logic of L S and its semantics in the usual way; for details, see, e.g., Enderton (1972) . For a FO formula", | |
| "cite_spans": [ | |
| { | |
| "start": 198, | |
| "end": 213, | |
| "text": "Enderton (1972)", | |
| "ref_id": "BIBREF7" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Logic", | |
| "sec_num": "3.2" | |
| }, | |
| { | |
| "text": "'(x 1 , ..., x k ) we write M |= '(d 1 , ..., d k ) when '(x 1 , ..., x k ) is true in M when x 1 , ..., x k are eval- uated to d 1 , ..., d k in D.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Logic", | |
| "sec_num": "3.2" | |
| }, | |
| { | |
| "text": "A sentence is a formula with no free variables; for a sentence ' we write M |= ' when ' is true in M .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Logic", | |
| "sec_num": "3.2" | |
| }, | |
| { | |
| "text": "To directly compare structures in distinct signatures we use logical transductions (Courcelle, 1994) , in which the relations in an output structure are defined using the logic of an input structure. Given an input signature S and an output signature T = {R 1 , ..., R n }, we define each R i in L S . Such a transduction thus induces an interpretation of T in S; that is, for any formula we can write in L T , there exists a translation into L S (Enderton, 1972; Hodges, 1997) . If there then also exists a transduction back from T to S, there then exists an interpretation of S in T . If interpretations in both directions exist, we say that S and T are bi-interpretable.", | |
| "cite_spans": [ | |
| { | |
| "start": 83, | |
| "end": 100, | |
| "text": "(Courcelle, 1994)", | |
| "ref_id": "BIBREF4" | |
| }, | |
| { | |
| "start": 447, | |
| "end": 463, | |
| "text": "(Enderton, 1972;", | |
| "ref_id": "BIBREF7" | |
| }, | |
| { | |
| "start": 464, | |
| "end": 477, | |
| "text": "Hodges, 1997)", | |
| "ref_id": "BIBREF12" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "A FO transduction is defined as follows. Fix a copy set C = {1, ..., k}, an input signature S, and an output signature", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "T = {R 1 , ..., R n }. A FO transduction \u2327 from S to T is thus a set of for- mulae ' c 1 ,...,cm i (x 1 , ..., x m ) for each R i 2 T and each c 1 , ..., c m 2 C m , where m is the arity of R i .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "The output of such a transduction is calculated as follows. 1 For a stucture M over the input signature S, and whose domain is D, \u2327 (M ) is a structure N = hD 0 , R 1 , ..., R n i defined as follows:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "1. For every d 2 D, there is a copy d c 2 D 0 iff", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "there is exactly one unary predicate", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "R c i (x) in \u2327 such that M |= ' i (d). 2. For any R i of arity m, (d c 1 1 , ..., d cm m ) 2 R i if and only if there is a d 1 , ..., d m 2 D m and a ' c 1 ,...,cm i (x 1 , ..., x m ) in \u2327 such that M |= ' c 1 ,...,cm i (d 1 , ..., d m ), and each d c i i 2 D 0 as per the requirement in (1).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "Intuitively, given a structure M over S, the output structure in T can have up to |C| copies of elements in the domain of D, and the relations in T are defined relative to these copies.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "For example, given the string signature S defined above we can define a transduction into a pseudo-autosegmental signature T = {/ 0 , A 0 , P 0 c , P 0 b } as follows. Set the copy set to C = {1, 2}. Then define a transduction \u2327 as", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "x / 01,1 y def = x < y^\u00ac(9z)[x < z^z < y], x / 02,2 y def = x < y^\u00ac(9z)[x < z^z < y], xA 01,2 y def = x \u21e1 y, P 02 b (x) def = P b (x), P 01 c (x) def = P b (x) _ P a (x),", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "and for all other", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "i, j 2 C, x / 0i,j y def = xA 0i,j y def =", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "False, and We interpret \u2327 as follows; an example is given in Fig. 3 . As The definitions for x / 01,1 y and x / 02,2 y then establish a successor relation between the first and second copies of nodes, respectively. Thus, for two nodes d 1 and d 2 in the input structure, d 1 2 is the successor of d 1 1 if and only if i < j in in the input structure and no node intervenes between them; likewise for d 2 1 and d 2 2 . As x / 0i,j y is False for all other i, j 2 C, the first and second copies are not ordered with respect to each other. Instead,", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 61, | |
| "end": 67, | |
| "text": "Fig. 3", | |
| "ref_id": "FIGREF2" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "P 01 b def = P 02 c def = False. b 2 2 b 3 2 c 1 1 c 2 1 c 3 1 c 4 1", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "P 01 c (x) def = P b (x) _ P a (x),", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "xA 01,2 y def = x \u21e1 y establishes that for any d, d 1 in the output is associated to its own second copy d 2 in the output (assuming that it survives according to (1) above). This can be seen for the copies of the b nodes 2 and 3 from Fig. 2 in Fig. 3 .", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 235, | |
| "end": 241, | |
| "text": "Fig. 2", | |
| "ref_id": "FIGREF1" | |
| }, | |
| { | |
| "start": 245, | |
| "end": 251, | |
| "text": "Fig. 3", | |
| "ref_id": "FIGREF2" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "As such a transduction is defined in terms of the atomic predicates of the output signature, it also induces an interpretation from the logic of the output signature to the logic of the input signature. In terms of phonology, if there is a FO transduction from one representational theory S to another T , Lemma 1 means that for any FO constraint C over S there is a FO constraint C 0 over T such that a representation in S satisfies C if and only if its equivalent representation in T satisfies C 0 .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions and interpretations", | |
| "sec_num": "3.3" | |
| }, | |
| { | |
| "text": "We now apply this technique to studying the relationship between ARs and QRs. First, we define the representations in model-theoretic terms.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Phonological representations as models", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "We assume a basic theory of autosegmental representations (ARs), which also follows SI's characterization of ARs (p. 2). The crucial assumptions are that there exists a tier of timing units (in our case, vowels), and featural elements are associated to elements on this tier. Each type of featural element (for our purposes, Hs and Ls) are also on their own tier, ordered together. For the patterns discussed in SI, a single tonal tier is sufficient. The signature for ARs is thus as below.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Autosegmental representations", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "A = {/ A , A A , V A , H A , L A } (2) V", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Autosegmental representations", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "A , H A , and L A are unary relations that label elements as vowels, H tones, and L tones, respectively. The association relation is A A ; to simplify definitions we treat A A as antisymmetric and directed from vowels to tones; that is, xA A y holds only if V A (x) is true and either", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Autosegmental representations", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "H A (y) or L A (y). H 1 L 2 V 3 V 4", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Autosegmental representations", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "Figure 4: An example AR of the sequenceVV.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Autosegmental representations", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "Each tier is ordered by the / A successor relation. Thus, the set of elements labeled V A are ordered, as are the set of elements labeled H A or L A . A vowel can be associated to more than one tone (a contour) or a tone can be associated to more than one vowel (spreading). This is shown in Figure 4 . For further simplicity, we also assume full specification: there are no toneless vowels and no floating tones. This last assumption is somewhat generous, as by SI's own admission, QRs cannot capture floating tones (see SI \u00a76.1).", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 292, | |
| "end": 300, | |
| "text": "Figure 4", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Autosegmental representations", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "Next, we assume two axioms that specially require the use of precedence (<). The first is No Gapping (NG), which prohibits autosegments from being linked to non-contiguous elements on the timing tier (N\u00ed Chio\u015bain and Padgett, 2001) .", | |
| "cite_spans": [ | |
| { | |
| "start": 200, | |
| "end": 231, | |
| "text": "(N\u00ed Chio\u015bain and Padgett, 2001)", | |
| "ref_id": "BIBREF17" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Autosegmental representations", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "NG def = (8x, y, z, w) \u21e5 x < y < z^A A (x, w) A A (z, w) ! A A (y, w) \u21e4", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Autosegmental representations", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "We also adopt the No-Crossing Constraint (NCC), which states that association must respect precedence on each tier (Goldsmith, 1976) .", | |
| "cite_spans": [ | |
| { | |
| "start": 115, | |
| "end": 132, | |
| "text": "(Goldsmith, 1976)", | |
| "ref_id": "BIBREF9" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Autosegmental representations", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "NCC def = (8x, y, v, w) \u21e5 (xA A v^yA A \u0175 x < y) ! v < w \u21e4 Thus, models in A satisfy NG^NCC.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Autosegmental representations", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "Finally, we assume that vowels can only associate to at most three tones-this maintains equivalence between ARs and QRs. While this has not traditionally been explicitly stated as an axiom of ARs, recent proposals do state such constraints (Yli-Jyr\u00e4, 2013; Jardine and Heinz, 2015).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Autosegmental representations", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "Q-Theory Representations (QRs) consist of two sets of ordered elements: one of Qs, and one of qs. Each Q consists of exactly 3 qs, and every q is part of exactly one Q. According to SI, qs are subsegments; the featural information of the segment is carried on the q. For our purposes, the relevant features are the tone features H and L. Thus, because constraints in SI refer to the featural information of some Q, it must be able to \"see\" what qs are relevant; a Q and its qs must be in some relation. We denote this relation A Q .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Q-Theory representations", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "We thus consider the following signature for QRs. Fig. 5 gives an example model in this signature of the QR for [h\u00f3l\u00f4l] from (1). This signature includes a sucessor relation / Q , a correspondence relation R Q (the curved lines in Fig. 5 ), the relation A Q associating Qs to qs, and three unary relations Q Q , H Q , and L Q , labeling Qs, H-toned qs, and L-toned qs, respectively.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 50, | |
| "end": 56, | |
| "text": "Fig. 5", | |
| "ref_id": "FIGREF5" | |
| }, | |
| { | |
| "start": 231, | |
| "end": 237, | |
| "text": "Fig. 5", | |
| "ref_id": "FIGREF5" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Q-Theory representations", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "Q = {/ Q , R Q , A Q , Q Q , H Q , L Q } (3) H 1 H 2 H 3 H 4 L 5 L 6 Q 7 Q 8", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Q-Theory representations", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "The axioms that govern R Q are not straightforward and so we discuss them here. We base our axioms for R Q on both the explicit and implicit discussion in SI.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Q-Theory representations", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "First, we assume that R Q is transitive. While SI claim their correspondence to be non-transitive, their constraints crucially refer to correspondence chains, or unbroken chains of corresponding elements. They state: \"[A] sequence of three identical consecutive segments S in a grammar requiring that identical segments correspond would satisfy that constraint as follows: S 1 S 1,2 S 2 , where coindexation encodes correspondence\" (SI, p. 5). In other words, the first S 1 and third S 2 , although they do not directly correspond, satisfy any correspondence constraints because there is a correspondence chain connecting them (through the intermediary S 1,2 ). Thus, in practice, the correspondence relation of SI is transitive.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Q-Theory representations", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "For ease of definition we further assume R Q is also reflexive and symmetric and thus an equivalence relation (per Bennett (2015)).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Q-Theory representations", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "EQ def = 8(x, y, z)[ (xR Q y ! yR Q x)( xR Q x )( (xR Q y^yR Q z) ! xR Q z) ]", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Q-Theory representations", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "However, this choice is not crucial to our results. Furthermore, like SI, we define R Q to be local: \"'Local' means consecutive; thus V-to-V correspondence is still considered local even if a consonant intervenes, as long as the closest two vowels in the string correspond\" (SI: 4-5). In all of their case studies, correspondence is always between adjacent vowels. (The only candidate that includes non-local correspondence in terms of vowels is SI:(8d), which is not an optimum.) In the models here, consonants are not included, so vowels are strictly adjacent.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Q-Theory representations", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "To restrict correspondence to a span of adjacent elements, we must adopt the following axiom.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Q-Theory representations", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "ADJ def = (8x, y, z)[(x < y^y < z^xR Q z) ! (xR Q y^yR Q z)]", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Q-Theory representations", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "ADJ states that for any x and z in correspondence, all intervening y must also be in correspondence. Note that this axiom requires specially adopting the < relation, as it must hold for all intervening elements y. This is stricter than SI's requirement of 'consecutivity', which instead restricts correspondence to intervening elements of some typee.g., vowels-but as they are vague about how this is determined we ignore this here, and note only that relativizing 'consecutivity' to a particular type of element is also FO-definable (Graf, 2017) . Thus, we assume Q models satisfy EQ^ADJ. (For comparison to models that do not, see \u00a76.2.)", | |
| "cite_spans": [ | |
| { | |
| "start": 534, | |
| "end": 546, | |
| "text": "(Graf, 2017)", | |
| "ref_id": "BIBREF10" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Q-Theory representations", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "Finally, correspondence implies identity. As SI state: \"Our operating assumption is that GEN does not even produce candidates in which elements obey CORR but violate the associated IDENT-XX constraint\" (SI: 6). Thus, correspondence between two qs implies they are either both H or both L. Similarly, correspondence between Qs implies they are associated to identical strings of qs.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Q-Theory representations", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "We now show the FO-equivalence between these two representational theories by defining a transduction from Q to A, and then from A to Q.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Transductions", | |
| "sec_num": "5" | |
| }, | |
| { | |
| "text": "We begin with a transductions from QRs in Q to ARs in A. Intuitively, this transduction is based on the idea that a correspondence chain of qs in a QR is equivalent to a tone in an AR.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Q-Theory Representations", | |
| "sec_num": "5.1" | |
| }, | |
| { | |
| "text": "We identify qs by the fact that they carry tones.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Q-Theory Representations", | |
| "sec_num": "5.1" | |
| }, | |
| { | |
| "text": "q Q (x) def = H Q (x) _ L Q (y)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Q-Theory Representations", | |
| "sec_num": "5.1" | |
| }, | |
| { | |
| "text": "Then, to uniquely identify each chain of corresponding qs, we define the following predicate FC(x), which identifies the first correspondent; i.e., the first element in a correspondence chain.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Q-Theory Representations", | |
| "sec_num": "5.1" | |
| }, | |
| { | |
| "text": "FC(x) def = \u00ac(9y)[xR Q y^y / Q x]", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Q-Theory Representations", | |
| "sec_num": "5.1" | |
| }, | |
| { | |
| "text": "This is possible with a successor relation / Q because of the adjacency axiom ADJ. If y / Q x and \u00acyR Q x and there is no z such that z < Q x and zR Q x, then x must be first in the chain.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Q-Theory Representations", | |
| "sec_num": "5.1" | |
| }, | |
| { | |
| "text": "The transduction is thus as given in Table 1 . The copy set is C = {1}, so we omit superscripts indicating copies. An example output structure, given Fig. 5 as an input, is given in Fig. 6 . First, the definitions of V A (x), H A (x), and L A (x), are straightforward. As vowels in ARs and Qs in QRs are equivalent, we set V A (x) equal to Q Q (x). For H A (x), and L A (x), we set each to the first q of a chain that is valued either H or L, respectively. Thus, for example in Fig. 6 , only nodes 1 and 5 are copied over from Fig. 5 .", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 37, | |
| "end": 44, | |
| "text": "Table 1", | |
| "ref_id": "TABREF0" | |
| }, | |
| { | |
| "start": 150, | |
| "end": 156, | |
| "text": "Fig. 5", | |
| "ref_id": "FIGREF5" | |
| }, | |
| { | |
| "start": 182, | |
| "end": 188, | |
| "text": "Fig. 6", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 478, | |
| "end": 484, | |
| "text": "Fig. 6", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 527, | |
| "end": 533, | |
| "text": "Fig. 5", | |
| "ref_id": "FIGREF5" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "From Q-Theory Representations", | |
| "sec_num": "5.1" | |
| }, | |
| { | |
| "text": "V A (x) def = Q Q (x) H A (x) def = H Q (x)^FC(x) L A (x) def = L Q (x)^FC(x) x / A y def = Q Q (x)^Q Q (y)^x / Q y _ q Q (x)^q Q (y)^FC(x)^FC(y)( 9z)[xR Q z^z / Q y] xA A y def = FC(x)^(9z)[xR Q z^zA Q y]", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Q-Theory Representations", | |
| "sec_num": "5.1" | |
| }, | |
| { | |
| "text": "The definition of x / A y, then, is relativized to elements for which these predicate are true. The Figure 6 : Output of the transduction in Table 1 given Fig. 5 as input; i 0 indicates a surviving copy of node i from Fig. 5 . disjunct Q Q (x)^Q Q (y)^x / Q y means that the successor relation between vowels in the AR is identical to the successor relation between Qs in the QR. The other disjunct defines the successor relation between tones. This has two parts: q Q (x)^q Q (y)^FC(x)^FC(y) ensures that the successor relation only holds between first elements in correspondence chains, and (9z)[xR Q z^z / Q y] identifies for some x the element y that starts the next correspondence chain; that is, it succeeds the element z that is the last element in x's correspondence chain. This can be seen in Fig. 6 between 1 0 and 5 0 . In Fig. 5 , in both 1 and 5 are first in their chains and 5 is the successor of 4, which corresponds with 1; thus 5 0 succeeds 1 0 in Fig. 6 .", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 100, | |
| "end": 108, | |
| "text": "Figure 6", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 141, | |
| "end": 148, | |
| "text": "Table 1", | |
| "ref_id": "TABREF0" | |
| }, | |
| { | |
| "start": 155, | |
| "end": 171, | |
| "text": "Fig. 5 as input;", | |
| "ref_id": "FIGREF5" | |
| }, | |
| { | |
| "start": 218, | |
| "end": 224, | |
| "text": "Fig. 5", | |
| "ref_id": "FIGREF5" | |
| }, | |
| { | |
| "start": 802, | |
| "end": 808, | |
| "text": "Fig. 6", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 834, | |
| "end": 840, | |
| "text": "Fig. 5", | |
| "ref_id": "FIGREF5" | |
| }, | |
| { | |
| "start": 965, | |
| "end": 971, | |
| "text": "Fig. 6", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "From Q-Theory Representations", | |
| "sec_num": "5.1" | |
| }, | |
| { | |
| "text": "H 1 0 L 5 0 V 7 0 V 8 0", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Q-Theory Representations", | |
| "sec_num": "5.1" | |
| }, | |
| { | |
| "text": "In a similar fashion, the definition for xA A y holds true when x is the first member of a chain and y that is associated to some z that corresponds with x. Thus, for example, since 4 is associated to 8 in Fig. 5 and 4 is a member of 1's correspondence chain, 1 0 and 8 0 are associated in Fig. 6 . Thus, A is definable from Q.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 206, | |
| "end": 218, | |
| "text": "Fig. 5 and 4", | |
| "ref_id": "FIGREF5" | |
| }, | |
| { | |
| "start": 290, | |
| "end": 297, | |
| "text": "Fig. 6", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "From Q-Theory Representations", | |
| "sec_num": "5.1" | |
| }, | |
| { | |
| "text": "Lemma 2 A is FO-definable from Q.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Q-Theory Representations", | |
| "sec_num": "5.1" | |
| }, | |
| { | |
| "text": "Proof: Witnessed by transduction defined in Table 1 . Note that these ARs will satisfy NG and NCC. Briefly, this is because A A and / A are defined through R Q , which satisfies ADJ as outlined in Sec. 4.2. ADJ essentially orders correspondence chains, and thus the tones and association relations defined via R Q . \u21e4", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 44, | |
| "end": 51, | |
| "text": "Table 1", | |
| "ref_id": "TABREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "From Q-Theory Representations", | |
| "sec_num": "5.1" | |
| }, | |
| { | |
| "text": "For this transformation, the copy set is C = {0, 1, 2, 3}. For each vowel, we need both the vowel itself (for the Q; let this be copy 0) and three copies (for each q; let these be copies 1, 2, and 3).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "The tones themselves in the AR are not copied. Instead, the values of the qs will be determined by the string of tones associated to the vowel in the AR, as (partially) summarized in Table 2. 2 Tones These correspondences are FO-definable. First, we define a series of predicates first(x, y), second(x, y), last(x, y), and only(x, y) that indicate when y is the first, second, last, or only tone associated to x, respectively. These definitions are given in Table 3 . (These do not explicitly associate y to x; this will be invoked in later definitions.)", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 183, | |
| "end": 193, | |
| "text": "Table 2. 2", | |
| "ref_id": "TABREF1" | |
| }, | |
| { | |
| "start": 458, | |
| "end": 465, | |
| "text": "Table 3", | |
| "ref_id": "TABREF2" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "qs H ! H 1 H 1,2 H 2 HL ! H 1 L 2 L 2 HLH ! H 1 L 2 H 3 Tones qs L ! L 1 L 1,2 L 2 LH ! L 1 H 2 H 2 LHL ! L 1 H 2 L 3", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "= \u00ac(9z)[xA A z^z / A y] b. last(x, y) def = \u00ac(9z)[xA A z^y / A z] c. second(x, y) def = (9z)[xA A z^z / A \u0177 first(x, z)] d. only(x, y) def = first(x, y)^last(x, y)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "V x 7 ! H x 1 H x 2 H x 3 Q x 0 b. H y1 L y2 V x 7 ! H x 1 L x 2 L x 3 Q x 0", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "From these predicates we can determine, for each vowel in the AR, how to label its q copies in the QR. This is given in Table 4b . For example, H 1 Q (x) is true when there is a H tone y that is the first tone associated to x. This means that copy 1 of x will be a q valued H. Similarly, H 2 Q (x) is true when there is some H tone associated to x that is either the second tone or the only tone associated to x. This last disjunct is necessary in case x has a single H associated to it in the AR, for its equivalent QR all three qs will be H (see Fig. 7a ). Table 4a specifies when Qs are built: the 0th copy of each vowel x is labeled Q. Table 4c then specifies that the 0th copy of x is associated to each of its own copies 1, 2, and 3 (i.e., the qs). The specification that x \u21e1 y ensures that the Q for each vowel is associated to each of its q copies. Likewise, Table 4d specifies that the successor relation / A between vowels is preserved between Qs (i.e., the 0 copies of vowels) and, that for the 1st through 3rd copies of vowel x, its ith copy is succeeded by its own i + 1th copy. The reader can confirm this via the examples in Fig. 7 .", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 120, | |
| "end": 128, | |
| "text": "Table 4b", | |
| "ref_id": "TABREF3" | |
| }, | |
| { | |
| "start": 548, | |
| "end": 555, | |
| "text": "Fig. 7a", | |
| "ref_id": "FIGREF6" | |
| }, | |
| { | |
| "start": 559, | |
| "end": 567, | |
| "text": "Table 4a", | |
| "ref_id": "TABREF3" | |
| }, | |
| { | |
| "start": 640, | |
| "end": 648, | |
| "text": "Table 4c", | |
| "ref_id": "TABREF3" | |
| }, | |
| { | |
| "start": 867, | |
| "end": 875, | |
| "text": "Table 4d", | |
| "ref_id": "TABREF3" | |
| }, | |
| { | |
| "start": 1140, | |
| "end": 1146, | |
| "text": "Fig. 7", | |
| "ref_id": "FIGREF6" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "Finally, we define R Q . First, as described in \u00a74.2, correspondence between Qs is dependent on identity between Qs, which again depends on their associated qs. Thus, xR 0,0 Q y should be true when the output Qs of x and y are associated to identical strings of qs. The values of q copies of are determined by the tones associated to x and y, respectively. Thus, Table 4e defines xR 0,0 Q y with the predicate ident(x, y), which is true if and only if the first, second, and last tones of x and y have the same value. (Let same(x, y)", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 363, | |
| "end": 371, | |
| "text": "Table 4e", | |
| "ref_id": "TABREF3" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "def = H A (x)\u0124 A (y) _ L A (x)^L A (y)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": ".) For example, this is true of the vowels in Fig. 8a , but not in 8b.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 46, | |
| "end": 53, | |
| "text": "Fig. 8a", | |
| "ref_id": "FIGREF7" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "Between qs, we define correspondence based on shared associations. Table 4f thus defines xR i,j Q y for 1 \uf8ff i, j \uf8ff 3 to be true when there is some z for which both x and y are associated. However, which qs of x and y correspond depend on z's position relative to other tones associated to x and y. Thus, z must also satisfy requirements ' i (x, z) and ' j (y, z) based on the value of i and j.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 67, | |
| "end": 75, | |
| "text": "Table 4f", | |
| "ref_id": "TABREF3" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "For example, when i = 1 and j = 1-that is, when defining correspondence between the 1st q of x and the 1st q of y-z must satisfy both first(x, z) and first(y, z). Intuitively, the first q of x and the first q of y correspond only if x and y share an association to z and z is the first tone associated to both x and y. This can be seen in Fig. 8 ; for example, in Fig. 8b , both x and y share an initial tone z, and so x 1 and y 1 correspond.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 339, | |
| "end": 346, | |
| "text": "Fig. 8", | |
| "ref_id": "FIGREF7" | |
| }, | |
| { | |
| "start": 365, | |
| "end": 372, | |
| "text": "Fig. 8b", | |
| "ref_id": "FIGREF7" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "Thus, Table 4 is a transduction from A to Q.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 6, | |
| "end": 13, | |
| "text": "Table 4", | |
| "ref_id": "TABREF3" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "Lemma 3 Q is FO-definable from A.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "Proof: Witnessed by the transduction defined in Table 4 . In particular, we sketch why R Q is guaranteed to be an equivalence relation over the copy set. First, xR i,i Q y holds when x \u21e1 y and thus R Q is a.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 48, | |
| "end": 55, | |
| "text": "Table 4", | |
| "ref_id": "TABREF3" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "Q i Q (x) def = V A (x) for i = 0; False otherwise b. For T 2 {H, L}, T 0 Q (x) def = False T 1 Q (x) def = (9y)[xA A y^T A (y)^first(x, y)] T 2 Q (x) def = (9y)[xA A y^T A (y)^(only(x, y) _ second(x, y))] T 3 Q (x) def = (9y)[xA A y^T A (y)^last(x, y)] c. xA i,j Q y def = x \u21e1 y for i = 0 and 1 \uf8ff i \uf8ff 3; False otherwise. d. x / i,j Q y def =", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "x / y for i, j = 0 or i = 3, j = 1; x \u21e1 y for 1 \uf8ff i, j \uf8ff 3, j = i + 1; False otherwise. e. xR 0,0", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "Q y def = ident(x, y)^(x / Q y _ y / Q x),", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "where and last(v, w) if n = 3, for 1 \uf8ff i, j \uf8ff 3. reflexive and symmetric. Finally, the fact that R Q is transitive derives from the fact that the relation defined by (8z)[xA A z^yA A z] is transitive. Also, because any input AR satisfies NG, x and y in an AR only share associations if they are adjacent, and thus R Q will satisfy ADJ. \u21e4 6 Summary and discussion", | |
| "cite_spans": [ | |
| { | |
| "start": 6, | |
| "end": 20, | |
| "text": "and last(v, w)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "ident(x, y) def = (8v, w) \u21e5 (xA A v^yA A w)^ (first(x, v)^first(y, w))_ ((only(x, v) _ second(x, v)^(only(y, w) _ second(y, w))_ (last(x, v)^last(y, w)) ! same(v, w) \u21e4 ; xR i,0 Q y def = xR 0,j Q y def = False for any 1 \uf8ff i, j \uf8ff 3 f. xR i,j Q y def = (9z)[xA A z^yA A z^' i (x, z)^' j (y, z)], where ' n (v, w) def = first(v, w) if n = 1, (only(v, w) _ second(v, w)) if n = 2,", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "A to Q a. H L H L V x V y 7 ! H x 1 L x 2 L x 3 H y 1 L y 2 L y 3 Q x 0 Q y 0 b. H z L V x V y 7 ! H x 1 H x 2 H x 3 H y 1 L y 2 L y 3 Q x 0 Q y 0", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "From Autosegmental Representations", | |
| "sec_num": "5.2" | |
| }, | |
| { | |
| "text": "We have now shown that A is definable from Q and Q is definable from A.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Equivalence of the models", | |
| "sec_num": "6.1" | |
| }, | |
| { | |
| "text": "Theorem 1 A and Q are FO bi-interpretable.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Equivalence of the models", | |
| "sec_num": "6.1" | |
| }, | |
| { | |
| "text": "Proof: From Lemmas 2 and 3. \u21e4", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Equivalence of the models", | |
| "sec_num": "6.1" | |
| }, | |
| { | |
| "text": "As stated in the introduction, this means that for any FO constraint written for QRs, there is an equivalent FO constraint on ARs, and vice-versa. Thus, while QRs may be based on a particular set of axioms and constraints, an equivalent set of axioms and constraints can be written in ARs, without changing the complexity of the constraints.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Equivalence of the models", | |
| "sec_num": "6.1" | |
| }, | |
| { | |
| "text": "Furthermore, while SI argue that QRs are a fundamental reimagining of phonological structure, our model-theoretic analysis shows that they are remarkably similar to ARs in which each vowel is associated to three autosegments. To illustrate this point, one reported benefit of QRs is that the fixed number of 3 subsegments predicts contrasts between HHL and HLL-toned vowels, for instance (SI, p. 14) . While not commonly proposed for ARs, this contrast is possible for ARs as well if the OCP is relaxed (as argued by Odden 1986).", | |
| "cite_spans": [ | |
| { | |
| "start": 388, | |
| "end": 392, | |
| "text": "(SI,", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 393, | |
| "end": 399, | |
| "text": "p. 14)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Equivalence of the models", | |
| "sec_num": "6.1" | |
| }, | |
| { | |
| "text": "As noted in Sec. 4.2, we follow SI in restricting the correspondence relation to adjacent elements. This is more restrictive than other theories of correspondence; the formulation of Bennett (2013) , for example, obeys EQ but not ADJ. How do ARs compare to QRs given a less restrictive correspondence relation? We conjecture that A and Q are incomparable to such a signature.", | |
| "cite_spans": [ | |
| { | |
| "start": 183, | |
| "end": 197, | |
| "text": "Bennett (2013)", | |
| "ref_id": "BIBREF0" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Relation to other correspondence models", | |
| "sec_num": "6.2" | |
| }, | |
| { | |
| "text": "To see why, consider a signature identical to Q with the exception that models in Q 0 only satisfy EQ, thus allowing unbounded correspondence. Defining R Q from R 0 would require defining a binary predicate in the FO logic of Q 0 that satisfies ADJ. However, it is likely that no such predicate exists, because the definition of ADJ crucially requires <, and it is well-known that < cannot be defined from /. (See, e.g., Libkin (2004) .)", | |
| "cite_spans": [ | |
| { | |
| "start": 421, | |
| "end": 434, | |
| "text": "Libkin (2004)", | |
| "ref_id": "BIBREF16" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Relation to other correspondence models", | |
| "sec_num": "6.2" | |
| }, | |
| { | |
| "text": "Going the other way, Q 0 essentially allows for quantification over a single abstract binary predicate whose only restriction is that it is an equivalence relation. For example, the FO language of Q 0 includes sentences such as (8x, y) \u21e5 (first(x)^last(y)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Relation to other correspondence models", | |
| "sec_num": "6.2" | |
| }, | |
| { | |
| "text": "! xR 0 y)( 8z, w)[w / 0 Q x^y / 0 Q z ! wR 0 z] \u21e4 ,", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Relation to other correspondence models", | |
| "sec_num": "6.2" | |
| }, | |
| { | |
| "text": "where first(x) = (8y)[\u00acy / Q x] and last(x) = (8y)[\u00acx / Q y]. This enforces 'center embedding' correspondence where in a string of elements a 0 a 1 a 2 ...a` 2 a` 1 a`, each a i and a` i (for i \uf8ff`/2) are in correspondence. Such a relation is almost certainly more expressive than anything definable in Q and A. First, A is FO-definable from strings with < (Jardine, 2017). FO-transductions are closed under composition (Courcelle et al., 2012) . If Q 0 were to be FO-definable from A, it would then have to be FO-definable from strings. As 'center embedding'-type relations are well-known to not be FO-definable, this is very likely not to be true.", | |
| "cite_spans": [ | |
| { | |
| "start": 419, | |
| "end": 443, | |
| "text": "(Courcelle et al., 2012)", | |
| "ref_id": "BIBREF5" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Relation to other correspondence models", | |
| "sec_num": "6.2" | |
| }, | |
| { | |
| "text": "An even stronger result would be that A and Q are equivalent under quantifier-free (QF) transductions (Chandlee and Lindell, forthcoming; Strother-Garcia, 2017) . However, QF transductions of Chandlee and Lindell and Strother-Garcia crucially use models with functions instead of pure relational models. Here, in order to hew to the standard definitions of association and correspondence as relations, we leave the interesting question of QF transductions for future work.", | |
| "cite_spans": [ | |
| { | |
| "start": 102, | |
| "end": 137, | |
| "text": "(Chandlee and Lindell, forthcoming;", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 138, | |
| "end": 160, | |
| "text": "Strother-Garcia, 2017)", | |
| "ref_id": "BIBREF24" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Future work", | |
| "sec_num": "6.3" | |
| }, | |
| { | |
| "text": "The result here is based on the machinery necessary to capture the case studies from SI, which all involve local tone interactions. As they also suggest extending their theory to segmental phonology, the obvious next direction is long-distance segmental phenomena. This involves more features, or unary predicates on qs, in addition to relaxing the ADJ axiom.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Future work", | |
| "sec_num": "6.3" | |
| }, | |
| { | |
| "text": "Model theory and logic provide for a powerful way to compare representational theories in phonology. Here, we have shown that ARs and QRs are not as different as they appear. This paper also serves as a case study for how logical transformations can be used to precisely evaluate theories of representation in phonology.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Conclusion", | |
| "sec_num": "7" | |
| }, | |
| { | |
| "text": "Proceedings of the Society for Computation in Linguistics (SCiL) 2019, pages 29-38. New York City, New York, January 3-6, 2019", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "We use an abbreviated construction used for those in strings, based on(Engelfriet and Hoogeboom, 2001). For the full construction see(Courcelle et al., 2012).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "It is also the case that, e.g., a vowel associated to a string of three H tones (i.e., HHH) will be output as a Q associated to a string of three, non-corresponding H qs (H1H2H3).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "", | |
| "sec_num": null | |
| } | |
| ], | |
| "back_matter": [], | |
| "bib_entries": { | |
| "BIBREF0": { | |
| "ref_id": "b0", | |
| "title": "Dissimilation, Consonant Harmony, and Surface Correspondence", | |
| "authors": [ | |
| { | |
| "first": "William", | |
| "middle": [], | |
| "last": "Bennett", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2013, | |
| "venue": "", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "William Bennett. 2013. Dissimilation, Consonant Har- mony, and Surface Correspondence. Ph.D. thesis, Rutgers, the State University of New Jersey.", | |
| "links": null | |
| }, | |
| "BIBREF1": { | |
| "ref_id": "b1", | |
| "title": "The phonology of consonants: Harmony, dissimilation, and correspondence", | |
| "authors": [ | |
| { | |
| "first": "William", | |
| "middle": [ | |
| "G" | |
| ], | |
| "last": "Bennett", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2015, | |
| "venue": "", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "William G. Bennett. 2015. The phonology of conso- nants: Harmony, dissimilation, and correspondence. Cambridge, UK: Cambridge University Press.", | |
| "links": null | |
| }, | |
| "BIBREF2": { | |
| "ref_id": "b2", | |
| "title": "Computational phonology: A constraint-based approach", | |
| "authors": [ | |
| { | |
| "first": "Steven", | |
| "middle": [], | |
| "last": "Bird", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 1995, | |
| "venue": "Studies in Natural Language Processing", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Steven Bird. 1995. Computational phonology: A constraint-based approach. Studies in Natural Lan- guage Processing. Cambridge University Press.", | |
| "links": null | |
| }, | |
| "BIBREF3": { | |
| "ref_id": "b3", | |
| "title": "forthcoming. A logical characterization of strictly local functions", | |
| "authors": [ | |
| { | |
| "first": "Jane", | |
| "middle": [], | |
| "last": "Chandlee", | |
| "suffix": "" | |
| }, | |
| { | |
| "first": "Steven", | |
| "middle": [], | |
| "last": "Lindell", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": null, | |
| "venue": "", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Jane Chandlee and Steven Lindell. forthcoming. A log- ical characterization of strictly local functions. In Jeffrey Heinz, editor, Doing Computational Phonol- ogy. OUP.", | |
| "links": null | |
| }, | |
| "BIBREF4": { | |
| "ref_id": "b4", | |
| "title": "Monadic second-order definable graph transductions: a survey", | |
| "authors": [ | |
| { | |
| "first": "", | |
| "middle": [], | |
| "last": "Bruno Courcelle", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 1994, | |
| "venue": "Theoretical Computer Science", | |
| "volume": "126", | |
| "issue": "", | |
| "pages": "53--75", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Bruno Courcelle. 1994. Monadic second-order de- finable graph transductions: a survey. Theoretical Computer Science, 126:53-75.", | |
| "links": null | |
| }, | |
| "BIBREF5": { | |
| "ref_id": "b5", | |
| "title": "Graph structure and monadic second-order logic: A language-theoretic approach", | |
| "authors": [ | |
| { | |
| "first": "Bruno", | |
| "middle": [], | |
| "last": "Courcelle", | |
| "suffix": "" | |
| }, | |
| { | |
| "first": "Joost", | |
| "middle": [], | |
| "last": "Engelfriet", | |
| "suffix": "" | |
| }, | |
| { | |
| "first": "Maurice", | |
| "middle": [], | |
| "last": "Nivat", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2012, | |
| "venue": "", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Bruno Courcelle, Joost Engelfriet, and Maurice Nivat. 2012. Graph structure and monadic second-order logic: A language-theoretic approach. Cambridge University Press.", | |
| "links": null | |
| }, | |
| "BIBREF6": { | |
| "ref_id": "b6", | |
| "title": "Aspects du basaa", | |
| "authors": [ | |
| { | |
| "first": "Gerrit", | |
| "middle": [], | |
| "last": "Dimmendaal", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 1988, | |
| "venue": "Bibliographie de la SELAF 96", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Gerrit Dimmendaal. 1988. Aspects du basaa. In Bibliographie de la SELAF 96. Paris, France: Peeters/SELAF. Trans. Luc Bouquiaux.", | |
| "links": null | |
| }, | |
| "BIBREF7": { | |
| "ref_id": "b7", | |
| "title": "A mathematical introduction to logic", | |
| "authors": [ | |
| { | |
| "first": "Herbert", | |
| "middle": [], | |
| "last": "Enderton", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 1972, | |
| "venue": "", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Herbert Enderton. 1972. A mathematical introduction to logic. Academic Press.", | |
| "links": null | |
| }, | |
| "BIBREF8": { | |
| "ref_id": "b8", | |
| "title": "MSO definable string transductions and two-way finite-state transducers", | |
| "authors": [ | |
| { | |
| "first": "Joost", | |
| "middle": [], | |
| "last": "Engelfriet", | |
| "suffix": "" | |
| }, | |
| { | |
| "first": "Hendrik", | |
| "middle": [], | |
| "last": "", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2001, | |
| "venue": "ACM Transations on Computational Logic", | |
| "volume": "2", | |
| "issue": "", | |
| "pages": "216--254", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Joost Engelfriet and Hendrik Jan Hoogeboom. 2001. MSO definable string transductions and two-way finite-state transducers. ACM Transations on Com- putational Logic, 2:216-254.", | |
| "links": null | |
| }, | |
| "BIBREF9": { | |
| "ref_id": "b9", | |
| "title": "Autosegmental Phonology", | |
| "authors": [ | |
| { | |
| "first": "John", | |
| "middle": [], | |
| "last": "Goldsmith", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 1976, | |
| "venue": "", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "John Goldsmith. 1976. Autosegmental Phonology. Ph.D. thesis, Massachussets Institute of Technology.", | |
| "links": null | |
| }, | |
| "BIBREF10": { | |
| "ref_id": "b10", | |
| "title": "The power of locality domains in phonology", | |
| "authors": [ | |
| { | |
| "first": "Thomas", | |
| "middle": [], | |
| "last": "Graf", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2017, | |
| "venue": "Phonology", | |
| "volume": "34", | |
| "issue": "", | |
| "pages": "385--405", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Thomas Graf. 2017. The power of locality domains in phonology. Phonology, 34:385-405.", | |
| "links": null | |
| }, | |
| "BIBREF11": { | |
| "ref_id": "b11", | |
| "title": "Theoretical and Typological Issues in Consonant Harmony", | |
| "authors": [ | |
| { | |
| "first": "Gunnar\u00f3lafur", | |
| "middle": [], | |
| "last": "Hansson", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2001, | |
| "venue": "", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Gunnar\u00d3lafur Hansson. 2001. Theoretical and Typo- logical Issues in Consonant Harmony. Ph.D. thesis, University of California, Berkeley.", | |
| "links": null | |
| }, | |
| "BIBREF12": { | |
| "ref_id": "b12", | |
| "title": "A Shorter Model Theory", | |
| "authors": [ | |
| { | |
| "first": "Wilfred", | |
| "middle": [], | |
| "last": "Hodges", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 1997, | |
| "venue": "", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Wilfred Hodges. 1997. A Shorter Model Theory. Cam- bridge: Cambridge University Press.", | |
| "links": null | |
| }, | |
| "BIBREF13": { | |
| "ref_id": "b13", | |
| "title": "Basaa a.43", | |
| "authors": [ | |
| { | |
| "first": "Larry", | |
| "middle": [], | |
| "last": "Hyman", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2003, | |
| "venue": "The Bantu Languages", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "257--282", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Larry Hyman. 2003. Basaa a.43. In Derek Nurse and G\u00e9rard Philippson, editors, The Bantu Languages, pages 257-282. London, UK: Routledge.", | |
| "links": null | |
| }, | |
| "BIBREF14": { | |
| "ref_id": "b14", | |
| "title": "On the logical complexity of autosegmental representations", | |
| "authors": [ | |
| { | |
| "first": "Adam", | |
| "middle": [], | |
| "last": "Jardine", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2017, | |
| "venue": "Proceedings of the 15th Meeting on the Mathematics of Language", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "22--35", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Adam Jardine. 2017. On the logical complexity of autosegmental representations. In Proceedings of the 15th Meeting on the Mathematics of Language, pages 22-35, London, UK. Association for Compu- tational Linguistics.", | |
| "links": null | |
| }, | |
| "BIBREF15": { | |
| "ref_id": "b15", | |
| "title": "A concatenation operation to derive autosegmental graphs", | |
| "authors": [ | |
| { | |
| "first": "Adam", | |
| "middle": [], | |
| "last": "Jardine", | |
| "suffix": "" | |
| }, | |
| { | |
| "first": "Jeffrey", | |
| "middle": [], | |
| "last": "Heinz", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2015, | |
| "venue": "Proceedings of the 14th Meeting on the Mathematics of Language", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "139--151", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Adam Jardine and Jeffrey Heinz. 2015. A concatena- tion operation to derive autosegmental graphs. In Proceedings of the 14th Meeting on the Mathematics of Language (MoL 2015), pages 139-151, Chicago, USA. Association for Computational Linguistics.", | |
| "links": null | |
| }, | |
| "BIBREF16": { | |
| "ref_id": "b16", | |
| "title": "Elements of Finite Model Theory", | |
| "authors": [ | |
| { | |
| "first": "Leonid", | |
| "middle": [], | |
| "last": "Libkin", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2004, | |
| "venue": "", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Leonid Libkin. 2004. Elements of Finite Model The- ory. Berlin: Springer-Verlag.", | |
| "links": null | |
| }, | |
| "BIBREF17": { | |
| "ref_id": "b17", | |
| "title": "Markedness, segment realization, and locality in spreading", | |
| "authors": [ | |
| { | |
| "first": "Maire", | |
| "middle": [], | |
| "last": "N\u00ed Chio\u015bain", | |
| "suffix": "" | |
| }, | |
| { | |
| "first": "Jaye", | |
| "middle": [], | |
| "last": "Padgett", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2001, | |
| "venue": "Segmental phonology in Optimality Theory", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "118--156", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Maire N\u00ed Chio\u015bain and Jaye Padgett. 2001. Marked- ness, segment realization, and locality in spreading. In Linda Lombardi, editor, Segmental phonology in Optimality Theory, pages 118-156. Cambridge Uni- versity Press.", | |
| "links": null | |
| }, | |
| "BIBREF18": { | |
| "ref_id": "b18", | |
| "title": "On the role of the Obligatory Contour Principle in phonological theory", | |
| "authors": [ | |
| { | |
| "first": "David", | |
| "middle": [], | |
| "last": "Odden", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 1986, | |
| "venue": "Language", | |
| "volume": "62", | |
| "issue": "2", | |
| "pages": "353--383", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "David Odden. 1986. On the role of the Obligatory Con- tour Principle in phonological theory. Language, 62(2):353-383.", | |
| "links": null | |
| }, | |
| "BIBREF19": { | |
| "ref_id": "b19", | |
| "title": "Model theory and the content of OT constraints", | |
| "authors": [ | |
| { | |
| "first": "Christopher", | |
| "middle": [], | |
| "last": "Potts", | |
| "suffix": "" | |
| }, | |
| { | |
| "first": "Geoffrey", | |
| "middle": [ | |
| "K" | |
| ], | |
| "last": "Pullum", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2002, | |
| "venue": "Phonology", | |
| "volume": "19", | |
| "issue": "", | |
| "pages": "361--393", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Christopher Potts and Geoffrey K. Pullum. 2002. Model theory and the content of OT constraints. Phonology, 19:361-393.", | |
| "links": null | |
| }, | |
| "BIBREF20": { | |
| "ref_id": "b20", | |
| "title": "Cognitive and sub-regular complexity", | |
| "authors": [ | |
| { | |
| "first": "James", | |
| "middle": [], | |
| "last": "Rogers", | |
| "suffix": "" | |
| }, | |
| { | |
| "first": "Jeffrey", | |
| "middle": [], | |
| "last": "Heinz", | |
| "suffix": "" | |
| }, | |
| { | |
| "first": "Margaret", | |
| "middle": [], | |
| "last": "Fero", | |
| "suffix": "" | |
| }, | |
| { | |
| "first": "Jeremy", | |
| "middle": [], | |
| "last": "Hurst", | |
| "suffix": "" | |
| }, | |
| { | |
| "first": "Dakotah", | |
| "middle": [], | |
| "last": "Lambert", | |
| "suffix": "" | |
| }, | |
| { | |
| "first": "Sean", | |
| "middle": [], | |
| "last": "Wibel", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2013, | |
| "venue": "Formal Grammar", | |
| "volume": "8036", | |
| "issue": "", | |
| "pages": "90--108", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "James Rogers, Jeffrey Heinz, Margaret Fero, Jeremy Hurst, Dakotah Lambert, and Sean Wibel. 2013. Cognitive and sub-regular complexity. In Formal Grammar, volume 8036 of Lecture Notes in Com- puter Science, pages 90-108. Springer.", | |
| "links": null | |
| }, | |
| "BIBREF22": { | |
| "ref_id": "b22", | |
| "title": "A subsegmental correspondence approach to contour tone (dis)harmony patterns", | |
| "authors": [ | |
| { | |
| "first": "Stephanie", | |
| "middle": [], | |
| "last": "Shih", | |
| "suffix": "" | |
| }, | |
| { | |
| "first": "Sharon", | |
| "middle": [], | |
| "last": "Inkelas", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2014, | |
| "venue": "Proceedings of the Annual Meetings on Phonology. LSA", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Stephanie Shih and Sharon Inkelas. 2014. A sub- segmental correspondence approach to contour tone (dis)harmony patterns. In Proceedings of the 2013 Meeting on Phonology (UMass Amherst), Proceed- ings of the Annual Meetings on Phonology. LSA.", | |
| "links": null | |
| }, | |
| "BIBREF24": { | |
| "ref_id": "b24", | |
| "title": "Imdlawn Tashlhiyt Berber syllabification is quantifier-free", | |
| "authors": [ | |
| { | |
| "first": "Kristina", | |
| "middle": [], | |
| "last": "Strother-Garcia", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2017, | |
| "venue": "Proceedings of the first annual meeting of the Society for Computation in Linguistics", | |
| "volume": "1", | |
| "issue": "", | |
| "pages": "145--153", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Kristina Strother-Garcia. 2017. Imdlawn Tashlhiyt Berber syllabification is quantifier-free. In Proceed- ings of the first annual meeting of the Society for Computation in Linguistics, volume 1, pages 145- 153.", | |
| "links": null | |
| }, | |
| "BIBREF25": { | |
| "ref_id": "b25", | |
| "title": "Logical foundations of syllable representations", | |
| "authors": [ | |
| { | |
| "first": "Kristina", | |
| "middle": [], | |
| "last": "Strother", | |
| "suffix": "" | |
| }, | |
| { | |
| "first": "-", | |
| "middle": [], | |
| "last": "Garcia", | |
| "suffix": "" | |
| }, | |
| { | |
| "first": "Jeffrey", | |
| "middle": [], | |
| "last": "Heinz", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2017, | |
| "venue": "Poster presented at the 5 th Annual Meeting on Phonology", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Kristina Strother-Garcia and Jeffrey Heinz. 2017. Log- ical foundations of syllable representations. Poster presented at the 5 th Annual Meeting on Phonology, New York University, New York City.", | |
| "links": null | |
| }, | |
| "BIBREF26": { | |
| "ref_id": "b26", | |
| "title": "On finite-state tonology with autosegmental representations", | |
| "authors": [ | |
| { | |
| "first": "Anssi", | |
| "middle": [], | |
| "last": "Yli-Jyr\u00e4", | |
| "suffix": "" | |
| } | |
| ], | |
| "year": 2013, | |
| "venue": "Proceedings of the 11th International Conference on Finite State Methods and Natural Language Processing", | |
| "volume": "", | |
| "issue": "", | |
| "pages": "90--98", | |
| "other_ids": {}, | |
| "num": null, | |
| "urls": [], | |
| "raw_text": "Anssi Yli-Jyr\u00e4. 2013. On finite-state tonology with au- tosegmental representations. In Proceedings of the 11th International Conference on Finite State Meth- ods and Natural Language Processing, pages 90-98. Association for Computational Linguistics.", | |
| "links": null | |
| } | |
| }, | |
| "ref_entries": { | |
| "FIGREF0": { | |
| "num": null, | |
| "text": "Overview of the transductions (a) from QRs to ARs and (b) from ARs to QRs.", | |
| "type_str": "figure", | |
| "uris": null | |
| }, | |
| "FIGREF1": { | |
| "num": null, | |
| "text": "A model of the string abba. D = {1, 2, 3, 4}, P a and P b are indicated by labels on the nodes, and < by arrows.", | |
| "type_str": "figure", | |
| "uris": null | |
| }, | |
| "FIGREF2": { | |
| "num": null, | |
| "text": "Output structure in T obtained by applying \u2327 to the string model inFig. 2. Arrows denote / 0 and lines without arrows denote A 0 . Indices of the nodes are of the form d c , where d is a node fromFig. 2and c 2 C.", | |
| "type_str": "figure", | |
| "uris": null | |
| }, | |
| "FIGREF3": { | |
| "num": null, | |
| "text": "every node d in the input string is given a copy d 1 in the output labeled c. Similarly, as P 02 b (x) def = P b (x) means that for any d labeled b in the input, there is a second copy d 2 labeled b in the output. As P 01b (x) = P 02 c (x) = False, no other copies are produced. Thus, the b nodes inFig. 2have both a corresponding c node and b node inFig. 3, butthe a nodes only have a corresponding c node.", | |
| "type_str": "figure", | |
| "uris": null | |
| }, | |
| "FIGREF4": { | |
| "num": null, | |
| "text": "Courcelle et al. (2012)) A FO transduction \u2327 from S to T induces a translation f from the FO language L S of S to the FO language L T of T such that for every sentence ' in L S there is a sentence f (') in L T such that for any structure M over S, M |= ' if and only if \u2327 (M ) |= f (').", | |
| "type_str": "figure", | |
| "uris": null | |
| }, | |
| "FIGREF5": { | |
| "num": null, | |
| "text": "An example QR of the sequenceVV.", | |
| "type_str": "figure", | |
| "uris": null | |
| }, | |
| "FIGREF6": { | |
| "num": null, | |
| "text": "Deriving strings of qs from strings of tones. On the right, x i indicates the ith copy of x.", | |
| "type_str": "figure", | |
| "uris": null | |
| }, | |
| "FIGREF7": { | |
| "num": null, | |
| "text": "Deriving R Q from tonal associations.", | |
| "type_str": "figure", | |
| "uris": null | |
| }, | |
| "TABREF0": { | |
| "type_str": "table", | |
| "html": null, | |
| "content": "<table/>", | |
| "text": "Transduction from Q to A.", | |
| "num": null | |
| }, | |
| "TABREF1": { | |
| "type_str": "table", | |
| "html": null, | |
| "content": "<table><tr><td>a. first(x, y)</td><td>def</td></tr></table>", | |
| "text": "Mapping from strings of tones associated to a vowel (AR) to strings of qs associated to a Q (QR).", | |
| "num": null | |
| }, | |
| "TABREF2": { | |
| "type_str": "table", | |
| "html": null, | |
| "content": "<table><tr><td>a.</td><td>y</td><td>H</td></tr></table>", | |
| "text": "Predicates used in the AR to QR transduction", | |
| "num": null | |
| }, | |
| "TABREF3": { | |
| "type_str": "table", | |
| "html": null, | |
| "content": "<table/>", | |
| "text": "", | |
| "num": null | |
| } | |
| } | |
| } | |
| } |