| { | |
| "paper_id": "W98-0117", | |
| "header": { | |
| "generated_with": "S2ORC 1.0.0", | |
| "date_generated": "2023-01-19T06:04:25.610603Z" | |
| }, | |
| "title": "On Some Similarities Between D-'free Grammars and Type-Logical Grammars", | |
| "authors": [ | |
| { | |
| "first": "Mark", | |
| "middle": [], | |
| "last": "Hepple", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "", | |
| "institution": "University of Sheffield", | |
| "location": { | |
| "addrLine": "Regent Court, 211 Portobello Street", | |
| "postCode": "Sl 4DP", | |
| "settlement": "Sheffield", | |
| "country": "UK" | |
| } | |
| }, | |
| "email": "" | |
| } | |
| ], | |
| "year": "", | |
| "venue": null, | |
| "identifiers": {}, | |
| "abstract": "", | |
| "pdf_parse": { | |
| "paper_id": "W98-0117", | |
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| "abstract": [], | |
| "body_text": [ | |
| { | |
| "text": "This paper discusses some similarities between D-Tree Grammars and type-logical grammars that are suggested in the context of a parsing approach for the latter that involves compiling higher-order formulae to first-order formulae. 1 This comparison suggests an approach to providing a functional seinantics for D-Tree derivations, which is outlined.", | |
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| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "The D-Tree Grammar (DTG) formalism is introduced in (Rambow et al., 1995) . The basic derivational unit of this formalism is the d-tree, which (loosely) consists of a collection of tree fragments with domination links between nodes in different fragments (that link them into a single graph). The above example d-tree, drawn from (Rambow et al., 1995) , allows topicalisation of the verb's object, as in (e.g.) Hotdogs;, he claims Mary seems to adore t;, where NP1 is the fronted object, and NP2 the verb's subject. 2 The main operation 3 for composing d-trees is subsertion, which, loosely, combines two d-trees to produce another, by substituting a fragment of one at a suitable node in the other, with other ( dominating) fragments of the first being intercalated into domination links of the second. The approach is motivated by problems of relate.d formaiisms (such as TAG and MCTAG-DV) involving", | |
| "cite_spans": [ | |
| { | |
| "start": 52, | |
| "end": 73, | |
| "text": "(Rambow et al., 1995)", | |
| "ref_id": "BIBREF7" | |
| }, | |
| { | |
| "start": 330, | |
| "end": 351, | |
| "text": "(Rambow et al., 1995)", | |
| "ref_id": "BIBREF7" | |
| }, | |
| { | |
| "start": 516, | |
| "end": 517, | |
| "text": "2", | |
| "ref_id": null | |
| } | |
| ], | |
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| "eq_spans": [], | |
| "section": "D-Tree Grammars", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "1 See (Joshi et a/., 1997; Henderson, 1992) for other work connecting categorial formalisms (Lambek calculus and CCG, respectively) to tree-oriented formalisms.", | |
| "cite_spans": [ | |
| { | |
| "start": 6, | |
| "end": 26, | |
| "text": "(Joshi et a/., 1997;", | |
| "ref_id": "BIBREF2" | |
| }, | |
| { | |
| "start": 27, | |
| "end": 43, | |
| "text": "Henderson, 1992)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 92, | |
| "end": 131, | |
| "text": "(Lambek calculus and CCG, respectively)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "D-Tree Grammars", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "2 The indexation is my own, for expositional purposes. 3 A second operation, sister-adjunction, used in handling modification, is discussed later in the paper.", | |
| "cite_spans": [ | |
| { | |
| "start": 55, | |
| "end": 56, | |
| "text": "3", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "D-Tree Grammars", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "~ t>.Iulti-Component TAG with Domination Links (Becker et al., 1991) .", | |
| "cite_spans": [ | |
| { | |
| "start": 47, | |
| "end": 68, | |
| "text": "(Becker et al., 1991)", | |
| "ref_id": null | |
| } | |
| ], | |
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| "eq_spans": [], | |
| "section": "D-Tree Grammars", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "linguistic coverage and the semantic interpretation of derivations.", | |
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| "section": "66", | |
| "sec_num": null | |
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| "text": "The associative Lambek calculus (Lambek, 1958) is the most familiar representative of the 'type-logical' tradition within categorial grammar, but a range of such systems have been proposed, which differ in their resource sensitivity (and hence, implicitly, their underlying notion of 'linguistic structure'). Some of these proposals are formnlated nsing a 'labelled deduction' methodology (Gabbay, 1996) , whereby the types in a proof are associated with labels, nnder a specified discipline, which record proof information used in ensuring correct inferencing. Such a labelling system must be overlaid upon a 'backbone logic', commonly the implicational or multiplicative 5 fragment of linear logic. For this paper, we can ignore labellings, and instead focus on the 'core functional structure' projected by linear formulae. 6", | |
| "cite_spans": [ | |
| { | |
| "start": 32, | |
| "end": 46, | |
| "text": "(Lambek, 1958)", | |
| "ref_id": "BIBREF3" | |
| }, | |
| { | |
| "start": 366, | |
| "end": 403, | |
| "text": "deduction' methodology (Gabbay, 1996)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Type-logical Grammar", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "In linear logic proofs, each assumption is used precisely once. Natural dednction mies of elimination and introduction for linear implication ( o-) are: 7", | |
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| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Implicational Linear Logic & First-order Compilation", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "(2) Ao-B :", | |
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| "eq_spans": [], | |
| "section": "Implicational Linear Logic & First-order Compilation", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "a B: b o-E A:(ab) (B:v] A:a ----o -1 Ao-B: >.v.a", | |
| "cite_spans": [], | |
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| "eq_spans": [], | |
| "section": "Implicational Linear Logic & First-order Compilation", | |
| "sec_num": "4" | |
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| { | |
| "text": "The proof in (3) illustrates 'hypothetical reasoning', where an additional assumption, or 'hypothetical', is used that is latter discharged. The involvement of hypotheticals is driven by the presence of higher-order formnlae (i.e. functors seeking an argument that bears a functional type): each corresponds to a subformula of a higher-order formula, e.g. Z in (3) is a subformula of Xo-(Yo-Z). 8 (3)", | |
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| "start": 395, | |
| "end": 396, | |
| "text": "8", | |
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| "section": "Implicational Linear Logic & First-order Compilation", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "Xo-(Yo-Z):", | |
| "cite_spans": [], | |
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| "eq_spans": [], | |
| "section": "Implicational Linear Logic & First-order Compilation", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "x Yo-W:y Wo-Z:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Implicational Linear Logic & First-order Compilation", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "w [Z:z] W:(wz) ',.'. ~y( wz)) Yo-Z: \">.z.y(wz) X: x(.Az.y(wz))", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
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| "section": "Implicational Linear Logic & First-order Compilation", | |
| "sec_num": "4" | |
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| "text": "Hepple ( 1996) shows how deductions in implicational linear logic can be recast as deductions involving only first-order formulae (i.e. where any arguments sought by functors bear atomic types) and using only a single inference rule (a variant of o-E). The compilation reduces higher-order formulae to first-order formulae by excising subformulae corresponding to hypotheticals, e.g. so Xo-(Yo-Z) gives Xo-Y plus Z. A system of indexing is used to ensure conect use of excised subformulae, to prevent invalid reasoning, e.g. the excised Z must be used to derive the argument of Xo-Y. Each compiled formula has an index set with one member (e.g. {j} :Z), which serves as its unique identifier. The index set of a derived \u2022 formula identifies the assumptions used to derive it. The single inference rule ( 4) ensures correct propagation of indices (where \\\u00b1l is disjoint union).", | |
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| "start": 7, | |
| "end": 14, | |
| "text": "( 1996)", | |
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| "section": "Implicational Linear Logic & First-order Compilation", | |
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| "text": "Each argument slot of a compiled functor also has an index set, which identifies any assumptions that must be used in deriving its argument, as enforced by the rule condition a ~ ,P. In proving Xo-(Yo-Z), Yo-W, Wo-Z =>X, for example, compilation yields the assumption formulae of the proof above. The leftmost (Fl) and rightmost (F2) assumptions both come from Xo-(Yo-Z), and Fl requires its argument to include F2. Compilation has removed the need for an explicit introduction step in the proof, c.f. proof 3 The above compilation produces results that bear more immediate similarities to the D-Tree approach than the original type-logical system. First-order formulae are easily viewed as tree fragments (in a way that higher-order formulae are not), e.g. a word w with formula so-npo-pp might be viewed as akin to (5a) below (modulo the order of daughters which is not encoded). For a higher-order formula, the inclusion requirement between its first-order derivatives is analogous to a domination link within a dtree, e.g. a relative pronoun relf(s/np) would yield rel o-s plus np, which we can view as akin to (5b ).", | |
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| "section": "Implicational Linear Logic & First-order Compilation", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "(", | |
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| "text": "5) (a) (6) ~ ;o-np np ~ \u2022 o-np o-pp PP 1 (a) X /'---... Xo-Y Y : z. 1 \" (b) rol ~ rel o-.s wbich np 1 rel e (b) \".~ ~. rol 0-1 o-pp PP PP c np 1 which", | |
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| "section": "Implicational Linear Logic & First-order Compilation", | |
| "sec_num": "4" | |
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| "text": "By default, it is natural to associate the string of the initial formula with its main residue under compilation, as in (5b ). Following proposals in (Moortgat, 1988; , some categorial systems have used connectives l ('extraction') and ! ('infixation'), where YlZ is a \"Y missing Z somewhere\" and a type Xl(YTZ) infixes its string to the position of the missing Z. Thus, a word w with type X!(YjZ), compiling to Xo-Y and Z, is akin to (6a). For example, the PP pied-piping relative pronoun type rel/(slpp )!(ppjnp ), from (Morrill, 1992) , which infixes to an NP site within a PP, is akin to (6b).", | |
| "cite_spans": [ | |
| { | |
| "start": 150, | |
| "end": 166, | |
| "text": "(Moortgat, 1988;", | |
| "ref_id": "BIBREF4" | |
| }, | |
| { | |
| "start": 522, | |
| "end": 537, | |
| "text": "(Morrill, 1992)", | |
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| } | |
| ], | |
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| "section": "Implicational Linear Logic & First-order Compilation", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "Interpreting DTG Derivations", | |
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| "section": "A Functional Approach to", | |
| "sec_num": "6" | |
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| "text": "The rest ofthis paper explores the idea ofproviding a functional semantics for DTG derivations, or rather of some DTG-like formalism, in a manner akin to that of categorial grammar. The approach envisaged is one in which each tree fragment (i.e. maximal unit containing no dominance links) of an initial dtree is associated with a lambda term. At the end of a derivation, the meaning of the resulting tree would be computed by working bottom up, applying the meaning term of each basic tree fragment to the meanings computed for each complete subtree added in at the fragment's frontier nodes, in some fixed fashion (e.g. such as in their right-to-left order). Strictly, terms would be combined using the special bt...~stitution operation of rule ( 4) ( allowing variable capture in the manner discussed). Suitable terms to associate with tree fragments will be arrived at by exploiting the analogy between d-trees and higherorder formulae under compilation. /\"'-..", | |
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| "section": "A Functional Approach to", | |
| "sec_num": "6" | |
| }, | |
| { | |
| "text": "1 ~ 1 \"bkh NP VP Mary V NP ~ 1 1 Mary V NP \"\" John 1 HW L", | |
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| "sec_num": "6" | |
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| "text": "For example, consider a simple grammar consisting of the four d-trees in 7, of which only that for which has more than one fragment. Each tree fragment is associated with a meaning term, shown to the right of \":\". The two fragments in the d-tree for which each have their own term, which are precisely those that would be assigned for the two compiled formulae in (5b) (assuming the meaning term for the precompilation formula rel/(s/np) tobe just which). This grammar allows the phrase-structure (8a) for Mary saw John, whose interpretation is produced by 'applying' the term for saw to that for the NP John (i.e. the subtree added in at the rightmost frontier node of saw's single tree fragment), and then to that ofthe NP Mary, giving {saw j m). The grammar allows the tree {8b) for the relative clause which Mary saw. 9 Here, the object position of saw is filled by the lower fragment of which 's dtree, so that\u2022 the subtree rooted at S has interpretation ( saw z m). Combining this with the term of the upper fragment of which gives interpretation whicb(>.z.saw z m).", | |
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| "section": "A Functional Approach to", | |
| "sec_num": "6" | |
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| "text": "The tree composition steps required to derive the trees in (8) would be handled in DTG by the subsertion operation. As noted earlier, DTG has a second composition operation sister-adjunction, used in handling modification, which adds in a modifier subtree as an additional daughter to an already existing local tree. A key motivation for this operation is so that DTG derivation trees distinguish argument vs. modifier dependencies, so as to provide an appropriate basis for interpretation. Categorial grammars typically make no such distinction in syntactic derivation, where all combinations are simply of functions and arguments. Rather, the distinction is implicit as a property of the lexical meanings of the functions that participate. 10 Accordingly, we recommend elimination of the sister-adjunction operation, with all composition being handled instead by subsertion. Thus, a VP modifying adverbial might have d-tree {9a), and give structures such as (9b) . 11 Such an analysis requires a different lexical d-tree for saw to that in (7), one where the VP node is 'stretched' as in (!Ob) to allow possible inclusion of modifiers. As a basis for arriving at suitable functional semantics for (!Ob), consider the following. A categorial approach might make saw a functor (np\\s)/np with semantics saw. This functor could be type-raised to (np\\s)!((np\\s)t((np\\s)/np)) with semantics (>.f.f saw). By substituting \u2022the two embedded occurrences of (np\\s) with the atom vp we get (np\\s)!(vpi(vp/np)), which compilea to first-order formulae as in {lOa), which are analogous to the desired d-trec {l\u00fcb), so providing the meaning terms there assigned. Using (!Ob) to derive the structure (Sa) involves identifying the two VP nodes. Such a derivation gives the interpretation ((>././ saw)(>.p.p j)m) which simplifies to ( saw j m). A derivation of (9b) gives interpretation ((>././ saw)(>.p.clearly(p j))m) which simplifies to (clearly (saw j) m).", | |
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| "section": "A Functional Approach to", | |
| "sec_num": "6" | |
| }, | |
| { | |
| "text": "EQUATION", | |
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| { | |
| "start": 0, | |
| "end": 8, | |
| "text": "EQUATION", | |
| "ref_id": "EQREF", | |
| "raw_str": "(9) (b) s ~ (a) VP NP VP 1 / \" ' -. . Mary VP Adv ~: >.x.(clearly x) VP Adv ~I cleuly V NP clearly l&W John (10) (a) s ~ (b) : O'! np \u2022 o-np NP VP ~ VP ~ : p V", | |
| "eq_num": "NP" | |
| } | |
| ], | |
| "section": "A Functional Approach to", | |
| "sec_num": "6" | |
| }, | |
| { | |
| "text": "For a ditransitive verb, we might want a structure providing more than one locus for inclusion of modifiers, such as (11). The semantica provided for this d-tree is arrived at by a similar process of reasoning to that for the previous case, except that it involves type-raising the initial categorial type of the verb twice (hence the subterm (>.g.g(>.f.f -...... : >.x>.y.((>..g.g(>. f.f sent))(Ap.x)y)", | |
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| { | |
| "start": 343, | |
| "end": 355, | |
| "text": "(>.g.g(>.f.f", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 356, | |
| "end": 384, | |
| "text": "-...... : >.x>.y.((>..g.g(>.", | |
| "ref_id": null | |
| } | |
| ], | |
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| "section": "A Functional Approach to", | |
| "sec_num": "6" | |
| }, | |
| { | |
| "text": "NP VP 1 : ~ : >.11>.w.(p(>.q.v)w) ~ PP ~ /'-...... : q V NP uni", | |
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| "sec_num": "6" | |
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| "text": "The interpretation approach outlined appears quite promising so far. We next consider a case it does not handle, which reveals something of its Iimitations: quantification. Following a suggestion of (Moortgat, 1996) , the connectives t ('extraction') and ! ('infixation') have been used in a categorial treatment of quantification. The lexical quantified NP everyone, for example, might be assigned type s!(sfnp), so that it has scope at the level of some sentence node but its string will appear in some NP position. First-order compilation yields the results (12a). The corresponding d-tree (12b) is unusual from a phrase-structure point \u2022 of view in that it 's upper fragment is a purely interpretive projection, but this d-tree would serve to produce appropriate interpretations. So far so good.", | |
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| "start": 199, | |
| "end": 215, | |
| "text": "(Moortgat, 1996)", | |
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| "text": "A simple quantifier every has type s!(sfnp)/n, to combine firstly with a noun, with the combined atring of every+noun then infixing to a NP position. First-order compilation, however, produces the result (13a), comparable to the d-tree (13b), which is clearly an inappropriate atructure. What we would hope for is a structure more like that in (13c), but although it is perfectly possible to apecify an initial higher-order formula that produces first-order formulae comparable to this d-tree, the results do not provide a suitable basis for interpretation. More generally, the highly restrictive approach to semantic composition that is characteristic of the approach outlined is such tbat a fragment cannot have scope above its position in structure (although a d-tree having multiple fragments has access to multiple possible scopes). This means, for example, that no semantics for (13c) will be able to get hold of and manipulate the noun 's meaning as something separate from that ofthe sentence predicate (c.f. sjnp), rather the former must fall within the latter . 12 (12) (a) /\"-..... Proc. EACL-91. Gabbay, D. 1996 . Lahelled deductive systems. Vol-", | |
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| "text": "12", | |
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| "end": 1123, | |
| "text": "EACL-91. Gabbay, D. 1996", | |
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| "text": "~The multiplicative fragment extends the implicational one with \u00a9 ('tensor'), akin to theLambek product. 8 This means, most notably, that the representations discussed Jack any encoding o{ linear order requirements, which would be handled within the labelling system.7 Eliminations and introductions correspond to steps of functional application and abstraction, respectively, as the lambda-term labelling reveals. In the o-I rule, [B] indicates a discharged or withdrawn assumption.", | |
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| "text": "The relevant subformulae can be precisely characterised in terms of a notion polarity: hypothetica.ls correspond to maximal positive-polarity subformulae of hlgher-order forrnulae. See(Hepple, 1996) for details. \u2022", | |
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| "text": "The treatment of wh-movement here exemplified is useful for exposHional purposes, but clearly differs from the standard TAG/DTG approach, where a moved whitem originates with a structure that includes the governor of the extraction site (typically a verb that sub-. categori.ses for the moved item). Such structurea present no problem for this approach, i.e. we could simply precombine the d-trees of which and aaw given in (7).", | |
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| "text": "This is not to say that the distinction has no observable refiex: mod.ifiers are in general recognisable as endocentric categorial functors (i.e. having the same argurnent and result type).11 Such an analysis is more in line with the standard TAG treatment than that of DTG.", | |
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| "text": "See(Shieber & Schabes, 1990) for a treatment of quantification within the Synchronous TAG forma.lism, in which the semantics is treated as a second system of tree representations that are operated upon synchronously with syntactic trees. Their account cannot be adapted to the prei;ent approach because their operations upon syntactic and semantics representations, though aynchronous, are not parallel in the way that is rigidly required in categoria.l semantics.", | |
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| "FIGREF1": { | |
| "text": "{i}:Xo-(Y:{j}) {k}:Yo-(W:0) {l}:Wo-(Z:0) {j}:Z >.t.x(.Az.t) >.u.yu .Av.wv z {j,l} :W :wz {j, k, l} : Y: y(wz) {i,j, k, l}: X: x(.Az.y(wz))", | |
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| "FIGREF2": { | |
| "text": ", but the effects of this step have been compiled into the semantics of the formulae. Thus, the term of Fl includea an apparently vacuous abstraction over variable z, which is the term assigned to F2. The semantics of rule (4) is handled not by simple application, but rather direct substitution for the variable of a lambda expression, employing a version of substitution which specifically does not act to avoid accidental binding. Hence, in the final step of the proof, the variable z falls within the scope of the abstraction, and so becomes bound. (4) </>: Ao-(B:a): .Av.a ,P: B: b 1r:A:a[b/v) 5 Relating The Two Systems", | |
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| "FIGREF4": { | |
| "text": ".x>.y.((>.f.f saw)(>.p.x)y)", | |
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| "uris": null, | |
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| "FIGREF6": { | |
| "text": "., Joshi, A. & Rambow, 0. 1991. 'Long distance scrambling and tree adjoining grammars.", | |
| "type_str": "figure", | |
| "uris": null, | |
| "num": null | |
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