| { |
| "paper_id": "E93-1044", |
| "header": { |
| "generated_with": "S2ORC 1.0.0", |
| "date_generated": "2023-01-19T10:54:37.952171Z" |
| }, |
| "title": "Categorial grammar, modalities and algebraic semantics", |
| "authors": [ |
| { |
| "first": "Koen", |
| "middle": [], |
| "last": "Versmissen", |
| "suffix": "", |
| "affiliation": { |
| "laboratory": "", |
| "institution": "Universiteit Utrecht Trails", |
| "location": { |
| "addrLine": "10", |
| "postCode": "3512 JK", |
| "settlement": "Utrecht", |
| "country": "Netherlands" |
| } |
| }, |
| "email": "koen.versmissen@let@ruu.nl" |
| }, |
| { |
| "first": "Onderzoeksinstituut", |
| "middle": [], |
| "last": "Voor Taal En Spraak", |
| "suffix": "", |
| "affiliation": { |
| "laboratory": "", |
| "institution": "Universiteit Utrecht Trails", |
| "location": { |
| "addrLine": "10", |
| "postCode": "3512 JK", |
| "settlement": "Utrecht", |
| "country": "Netherlands" |
| } |
| }, |
| "email": "" |
| } |
| ], |
| "year": "", |
| "venue": null, |
| "identifiers": {}, |
| "abstract": "This paper contributes to the theory of substructural logics .that are of interest to categorial grammarians. Combining semantic ideas of Hepple [1990] and Morrill [1990], proof-theoretic ideas of Venema [1993b; 1993a] and the theory of equational specifications, a class of resource-preserving logics is defined, for which decidability and completeness theorems are established.", |
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| "paper_id": "E93-1044", |
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| "abstract": [ |
| { |
| "text": "This paper contributes to the theory of substructural logics .that are of interest to categorial grammarians. Combining semantic ideas of Hepple [1990] and Morrill [1990], proof-theoretic ideas of Venema [1993b; 1993a] and the theory of equational specifications, a class of resource-preserving logics is defined, for which decidability and completeness theorems are established.", |
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| "section": "Abstract", |
| "sec_num": null |
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| "body_text": [ |
| { |
| "text": "The last decade has seen a keen revival of investigations into the suitability of using categorial grammars as theories of natural language syntax and semantics. Initially, this research was for the larger part confined to the classical categorial calculi of Ajdukiewicz [1935] and Bar-Hillel [1953] , and, in particular, the Lambek cMculus L [Lambek, 1958] , [Moortgat, 1988 ] and some of its close relatives.", |
| "cite_spans": [ |
| { |
| "start": 271, |
| "end": 277, |
| "text": "[1935]", |
| "ref_id": null |
| }, |
| { |
| "start": 282, |
| "end": 299, |
| "text": "Bar-Hillel [1953]", |
| "ref_id": null |
| }, |
| { |
| "start": 343, |
| "end": 357, |
| "text": "[Lambek, 1958]", |
| "ref_id": null |
| }, |
| { |
| "start": 360, |
| "end": 375, |
| "text": "[Moortgat, 1988", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Introduction", |
| "sec_num": "1" |
| }, |
| { |
| "text": "Although it turned out to be easily applicable to fairly large sets of linguistic data, one couldn't realistically expect the Lambek calculus to be able to account for all aspects of grammar. The reason for this is the diversity of the constructions found in natural language. The Lambek calculus is good at reflecting surface phrase structure, but runs into problems when other linguistic phenomena are to be described. Consequently, recent work in categorial grammar has shown a trend towards diversification of the ways in which the linguistic algebra is structured, with an accompanying ramification of proof theory.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Introduction", |
| "sec_num": "1" |
| }, |
| { |
| "text": "One of the main innovations of the past few years has been the introduction of unary type connectives, usually termed modalities, that are used to reflect certain special features linguistic entities may possess. This strand of research originates with Morrill [1990] , who adds to L a unary connective O with the following proof rules:", |
| "cite_spans": [ |
| { |
| "start": 253, |
| "end": 267, |
| "text": "Morrill [1990]", |
| "ref_id": null |
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| "section": "Introduction", |
| "sec_num": "1" |
| }, |
| { |
| "text": "OF~-A [oR]", |
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| "ref_spans": [], |
| "eq_spans": [], |
| "section": "F,B,F'FA [mL]", |
| "sec_num": null |
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| "text": "F, OB, F ~ b A OF b DA OF here denotes a sequence of types all of which have O as their main connective. The S4-1ike modality o is introduced with the aim of providing an appropriate means of dealing with certain intensional phenomena. Consequently, O inherits Kripke's possible world semantics for modal logic. The proof system which arises from adding Morrill's left and right rules for [] to the Lambek calculus L will be called Lb.", |
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| "section": "F,B,F'FA [mL]", |
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| "text": "Hepple [1990] presents a detailed investigation into the possibilities of using the calculus L\u2022 to account for purely syntactic phenomena, notably the wellknown Island Constraints of Ross [1967] . Starting from the usual interpretation of the Lambek calculus in semigroups L, where types are taken to denote subsets of L, he proposes to let D refer to a fixed subsemigroup Lo of L, which leads to the following definition of its semantics:", |
| "cite_spans": [ |
| { |
| "start": 7, |
| "end": 13, |
| "text": "[1990]", |
| "ref_id": null |
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| { |
| "start": 183, |
| "end": 194, |
| "text": "Ross [1967]", |
| "ref_id": "BIBREF2" |
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| "section": "F,B,F'FA [mL]", |
| "sec_num": null |
| }, |
| { |
| "text": "[oAf = [A]n Lo", |
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| "section": "F,B,F'FA [mL]", |
| "sec_num": null |
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| "text": "As we have shown elsewhere [Versmissen, 1992] 1 the calculus LD is sound with respect to this semantics, but not complete. This can be remedied by 1This paper discusses semigroup semmatics for L and LO in detail, and is well-suited as an easy-going introduction to the ideas presented here. It is available by anonymous ftp from ftp.let.ruu.nl in directory /pub/ots/papexs/versmissen, files adding.dvi.Z and adding, ps. Z. replacing the rule [OR] with the following stronger version:", |
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| "start": 27, |
| "end": 45, |
| "text": "[Versmissen, 1992]", |
| "ref_id": "BIBREF3" |
| }, |
| { |
| "start": 442, |
| "end": 446, |
| "text": "[OR]", |
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| "section": "F,B,F'FA [mL]", |
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| "text": "FlbOB1 ... F1, ..., FnbA rl, ..., F, Hepple [1990] also investigates the benefits of using the so-called structural modalities originally proposed in [Morrill et al., 1990] , for the description of certain discontinuity 'and dislocality phenomena. The idea here is that such modalities allow a limited access to certain structural rules. Thus, we could for example have a permutation modality rap with the following proof rule (in addition to [rapL] and [OpR'] as before):", |
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| "start": 11, |
| "end": 14, |
| "text": "F1,", |
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| "start": 15, |
| "end": 19, |
| "text": "...,", |
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| "start": 20, |
| "end": 28, |
| "text": "FnbA rl,", |
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| "start": 29, |
| "end": 33, |
| "text": "...,", |
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| "start": 34, |
| "end": 36, |
| "text": "F,", |
| "ref_id": null |
| }, |
| { |
| "start": 44, |
| "end": 50, |
| "text": "[1990]", |
| "ref_id": null |
| }, |
| { |
| "start": 150, |
| "end": 172, |
| "text": "[Morrill et al., 1990]", |
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| }, |
| { |
| "start": 443, |
| "end": 449, |
| "text": "[rapL]", |
| "ref_id": null |
| } |
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| "section": "F,B,F'FA [mL]", |
| "sec_num": null |
| }, |
| { |
| "text": "r[oeA, B] ~ C r[8, opA] ~-C", |
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| "section": "F,B,F'FA [mL]", |
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| "text": "The symbol ~ here indicates that the inference is valid in both directions. The interpretation of OR would then be taken care of by a subsemigroup Lop of L having the property that x", |
| "cite_spans": [], |
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| "section": "F,B,F'FA [mL]", |
| "sec_num": null |
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| "text": "\u2022 y = y \u2022 x whenever z\u2022Lnpory\u2022Lop.", |
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| "section": "F,B,F'FA [mL]", |
| "sec_num": null |
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| "text": "Alternatively, one could require all types in such an inference to be boxed:", |
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| "eq_spans": [], |
| "section": "F,B,F'FA [mL]", |
| "sec_num": null |
| }, |
| { |
| "text": "F[rapA, DpB] I-C I r[opB, OpA] ~-C", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "F,B,F'FA [mL]", |
| "sec_num": null |
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| "text": "In this case, Lop would have to be such that z. y = y-x whenever z, y \u2022 Lop.", |
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| "section": "F,B,F'FA [mL]", |
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| "text": "Closely related to the use of structural modalities is the trend of considering different kinds of product connectives, sometimes combined into a single system. For example, Moortgat & Morrill [1992] present an account of dependency structure in terms of headed prosodic trees, using a calculus that possesses two product operators instead of just one. On the basis of this, Moortgat [1992] sketches a landscape of substructural logics parametrized by properties such as commutativity, associativity and dependency. He then goes on to show how structural modalities can be used to locally enhance or constrain the possibilities of type combination. Morrill [1992] has a non-associative prosodic calculus, and uses a structural modality to reintroduce associativity at certain points.", |
| "cite_spans": [ |
| { |
| "start": 185, |
| "end": 199, |
| "text": "Morrill [1992]", |
| "ref_id": "BIBREF1" |
| }, |
| { |
| "start": 649, |
| "end": 663, |
| "text": "Morrill [1992]", |
| "ref_id": "BIBREF1" |
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| "text": "The picture that emerges is the following. Instead of the single product operator of L, one considers a range of different product operators, reflecting different modes of linguistic structuring. This results in a landscape of substructural logics, which are ultimately to be combined into a single system. Specific linguistic phenomena are given an account in terms of type constructors that are specially tailored for their description. On certain occasions it is necessary for entities to 'escape' the rules of the type constructor that governs their behaviour. This is achieved by means of structural modalities, which license controlled travel through the substructural landscape. Venema [1993a] proves a completeness theorem, with respect to the mentioned algebraic interpretation, for the Lambek calculus extended with a permutation modality. He modifies the proof system by introducing a type constant Q which refers explicitily to the subalgebra Lo. This proof system is adapted to cover a whole range of substructural logics in [Venema, 1993b] . However, the semantics given in that paper, which is adopted from Dogen [1988; 1989] , differs in several respects from the one discussed above. Most importantly, models are required to possess a partial order with a well-behaved interaction with the product operation. In the remainder of this paper we will give a fairly general definition of the notion of a resource-preserving logic. The proof theory of these logics is based on that of Venema, while their semantics, with respect to which a completeness theorem will be established, is similar to that of Hepple and Morrill.", |
| "cite_spans": [ |
| { |
| "start": 686, |
| "end": 700, |
| "text": "Venema [1993a]", |
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| { |
| "start": 1038, |
| "end": 1053, |
| "text": "[Venema, 1993b]", |
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| "end": 1140, |
| "text": "[1988; 1989]", |
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| "text": "2.1 Syntax The languages of the logics that will be considered here are specified by the following parameters: t~ Three finite, disjoint index sets Z, J and/C; A finite set B of basic types. Given these, we define the following sets of expressions:", |
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| "eq_spans": [], |
| "section": "Resource-preserving logics with structural modalities", |
| "sec_num": "2" |
| }, |
| { |
| "text": "The The division of the unary type connectives into two sets Ma and M v reflects the alternatives mentioned in Section 1. Modalities/Xj are those whose structural rules only apply when all types involved are prefixed with them, whereas only a single type prefixed with XTk needs to be involved in order for the accompanying structural rules to be applicable.", |
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| "section": "Resource-preserving logics with structural modalities", |
| "sec_num": "2" |
| }, |
| { |
| "text": "We will use equational specifications to describe the structural behaviour of connectives and modalities, as well as the algebraic structures in which these are interpreted. To start with, we recall several important definitions and results.", |
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| "eq_spans": [], |
| "section": "Equational specifications", |
| "sec_num": "2.2" |
| }, |
| { |
| "text": "A signature E is a collection of function symbols, each of which has a fixed arity. Let V be a countably infinite set of variables. The term algebra T(E, 1)) is defined as the inductive closure of l; under ~. An equational specification is a pair (~,,~) where ~ is a signature and E is a set of equations s = t of terms s,t E T(~,12). A ~-algebra .4 is a set A together with functions F A : A\" --* A for all n-ary function symbols F E ~. A E-algebra .4 is a model for a set of equations E over T(~, N), written as .4 ~ \u00a3, if every equation of ~ holds in A. A (E, g)-algebra is a ~-algebra that is a model for \u00a3.", |
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| "section": "Equational specifications", |
| "sec_num": "2.2" |
| }, |
| { |
| "text": "Let E be an equational specification. Then we define Ezxi to be the equational specification obtained from E by prefixing each variable occurrence with A~. The equational specification Ev~ is defined as follows (where V(F = G) denotes the set of variables occurring in F = G):", |
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| "section": "Equational specifications", |
| "sec_num": "2.2" |
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| "text": "(F=G)lx*'-Vkz] mD FI~*-vkxl=G[x*\"Vkx] (F=G)v/, =O UzCV(F=o) (F=G)[z*-Vk*] \u00a3vk --D UEE~r Ev~", |
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| "section": "Equational specifications", |
| "sec_num": "2.2" |
| }, |
| { |
| "text": "To give a concrete example of these definitions, let E consist of the following two equations: x+y = y+x", |
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| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Equational specifications", |
| "sec_num": "2.2" |
| }, |
| { |
| "text": "x+(y+z) = (x+y)+z", |
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| "section": "Equational specifications", |
| "sec_num": "2.2" |
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| "text": "Then ~ contains these two: We will call a term equation resource-preserving if each variable occurs the same number of times on both sides of the equality sign. An equational specification is resource-preserving if all of its member equations are. Note that this definition encompasses the important cases of commutativity and associativity. On the other hand, well-known rules such as weakening and contraction can't be modelled by resource-preserving equations. Not only do they fail to be resource-preserving in the strict sense introduced here, but also they are one-way rules that would have to be described by means of rewrite rules rather than equations.", |
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| "section": "Equational specifications", |
| "sec_num": "2.2" |
| }, |
| { |
| "text": "Ajz+Ajy = Ajy + Ajx A~x+(A~y+Aiz ) = (Ajx+A~y)", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Equational specifications", |
| "sec_num": "2.2" |
| }, |
| { |
| "text": "A resource-preserving logic is determined by the following: Instantiation of the language parameters B, Z, ,7 and K;", |
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| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Resource-preserving logics", |
| "sec_num": "2.3" |
| }, |
| { |
| "text": "t, An equational specification E over the signature {+~}iEz;", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Resource-preserving logics", |
| "sec_num": "2.3" |
| }, |
| { |
| "text": "Two sets of indices {ij}j\u00a2,7, {ik}~er C_ Z; t> Two sets of equational specifications {Ej}jej and {Ek}ke/c, where Et is specified over the signature {+i, } (I E ,7 U K). Of course, all equational specifications occurring in the above list are required to be resource-preserving. The operator + is intended as a generic one, which is to be replaced by a specific connective of the language on each separate occasion. We will write \u00a3* for the equational specification obtained by substituting \u2022 for + in E, but will drop this superscript when it is clear from the context. (Ej)zxi will be abbreviated as \u00a3~j, and (\u00a3k)Vk as \u00a3W\"", |
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| "section": "Resource-preserving logics", |
| "sec_num": "2.3" |
| }, |
| { |
| "text": "Henceforth, we assume that we are dealing with a fixed resource-preserving logic \u00a3.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Resource-preserving logics", |
| "sec_num": "2.3" |
| }, |
| { |
| "text": "For \u00a3 we have the following rules of inference:", |
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| "eq_spans": [], |
| "section": "Proof system", |
| "sec_num": "2.4" |
| }, |
| { |
| "text": "AFA FI-A A(B) I-fi roiAt-B A[(BIiA) ol r] ~ c [/,L] r k aliA [/,R] FI-A A(B) I-C Aoirl-B [\\~L] [~iR] A[r ol (A\\iB)] ~-C r I-A\\iB F, FQt r~l-Ot r[Q,]l-A [Qd r[rl oi, r2] ~-A r[Al i-B F[AjA] F B rtQjl ~ e r[A/A] k B [~jL2] r[A] F B r[o,] F a r[VkA] I-B [vkLq r[VkA] k-B [vkL2l FI-A FI-Q/ [A#R] FFA rl--Ok [VkR] r b AjA r b VkA rI-A rI-A r,t-Q, ... r,F-0al[~d A ~ A ! [E] 'A k A rI-A A[A] I-B [Caq A[F] b B", |
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| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Proof system", |
| "sec_num": "2.4" |
| }, |
| { |
| "text": "In these rules i, j and k range over I, `7 and JC, respectively, and 1 ranges over `7 U/U. As before, a I indicates that we have a two-way inference rule. The [\u00a3(0]-rule schemata are subject to the following condition: there exist an equation s = t E E(' 0 and a substitution a : V --T such that A can be obtained from r by replacing a substructure s ~ of r with ft. On [Ell we put the further restriction that the ri's are exactly the elementary substructures of s a. For example, for gj = {x + y = y + z} we would obtain the following rule: ", |
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| "section": "Proof system", |
| "sec_num": "2.4" |
| }, |
| { |
| "text": "We will address the logical aspects of the calculi defined in the last section shortly, but first we pause for a brief intermezzo, illustrating how they are applied in linguistic practice. As our first example we look at how the Lambek calculus deals with extraction. Suppose we have the following type assignments: John, Mary : NP loves : NP\\S/NP madly : (NP\\S)\\(NP\\S)", |
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| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Some sample applications", |
| "sec_num": "2.5" |
| }, |
| { |
| "text": "We would like to find type assignments to who such that we can derive type REL for the following phrases:", |
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| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Some sample applications", |
| "sec_num": "2.5" |
| }, |
| { |
| "text": "As is easily seen, assignment of REL/(S/NP) to who works for the first sentence, while REL/(N P\\S) is the appropriate type to assign to who to get the second case right. However, the third case can't he done in the Lambek calculus, since we have no way of referring to gaps occuring inside larger constituents; we only have access to the periphery. This can be handled by adding a permutation modality VP and assigning to who the type REL/(S/VP NP) to who. This single type assignment works for all three cases. For the third sentence, this is worked out in Figure 1 .", |
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| "start": 558, |
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| "text": "Figure 1", |
| "ref_id": null |
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| "section": "who John loves madly", |
| "sec_num": "3." |
| }, |
| { |
| "text": "As a second example, consider the following noun phrase:", |
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| "section": "who John loves madly", |
| "sec_num": "3." |
| }, |
| { |
| "text": "the man at the desk For the nouns and the determiner we make the usual type assignments: the : NP/N man, desk : N From a prosodic point of view, at should be assigned type (N\\N)/NP. However, semantically at combines not just with the noun it modifies, but with the entire noun phrase headed by that noun. Moortgat & Morrill [1992] show how both these desiderata can be fulfilled. First, the type assignment to at is lifted to ((NP/(N\\N))\\NP)/NP in order to force the required semantic combination. This is not the end of the story, because due to the non-associativity of the prosodic algebra we still can't derive a type NP for the man at the desk. To enable this, they add a structural modality VA to the type assignment for at to make it ((NP/VA (N\\N))\\NP)/NP, after which things work out nicely, as is shown by the derivation in Figure 2 .", |
| "cite_spans": [ |
| { |
| "start": 316, |
| "end": 330, |
| "text": "Morrill [1992]", |
| "ref_id": "BIBREF1" |
| } |
| ], |
| "ref_spans": [ |
| { |
| "start": 833, |
| "end": 841, |
| "text": "Figure 2", |
| "ref_id": null |
| } |
| ], |
| "eq_spans": [], |
| "section": "who John loves madly", |
| "sec_num": "3." |
| }, |
| { |
| "text": "Before turning to the semantics of/~ we will prove the Cut-elimination theorem and subformula property for it, since the latter is essential for the completeness proof, and a corollary to the former.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Cut-elimination and the subformula property", |
| "sec_num": "2.6" |
| }, |
| { |
| "text": "As we remarked earlier, our proof rules are adapted from [Venema, 1993b] . Therefore, we can refer the reader to that paper for most of the Cutelimination proof. The only notable difference between both systems lies in the structural rules they allow. Note that resource-preservation implies that for any [E(j)]-inference we have the following two simple but important properties (where the complexity of a type is defined as the number of connectives occurring in it):", |
| "cite_spans": [ |
| { |
| "start": 57, |
| "end": 72, |
| "text": "[Venema, 1993b]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Cut-elimination and the subformula property", |
| "sec_num": "2.6" |
| }, |
| { |
| "text": "1. Each type occurring in r occurs also in A, and vice versa;", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Cut-elimination and the subformula property", |
| "sec_num": "2.6" |
| }, |
| { |
| "text": "2. The complexity of r equals that of A. The subformula property says that any provable sequent has a proof in which only subformulas of that sequent occur. Under the proviso that Qj is considered a subtype of AiA, and QI, of wkA, the subformula property follows from Cut-elimination, since in each inference rule other than [Cut], the premises are made up of subformulas of the conclusion.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Cut-elimination and the subformula property", |
| "sec_num": "2.6" |
| }, |
| { |
| "text": "Let \u00a3. be the logic obtained from \u00a3 by adding a set of product connectives {*i}iez to the language, and the following inference rules to the proof system:", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Cut-elimination and the subformula property", |
| "sec_num": "2.6" |
| }, |
| { |
| "text": "roiAI-A [,~L] rFh AFB [.~a] r.i A P A roi A F A*i B", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Cut-elimination and the subformula property", |
| "sec_num": "2.6" |
| }, |
| { |
| "text": "Like \u00a3, the system \u00a3, enjoys Cut-elimination and the subformula property. Note that this implies that if an \u00a3-sequent is/:.-derivable, then it is \u00a3-derivable. This property will be used several times in the course of the completeness proof.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Cut-elimination and the subformula property", |
| "sec_num": "2.6" |
| }, |
| { |
| "text": "Now consider a naive top-down 2 proof search strategy. At every step, we have a finite choice of possible applications of an inference rule, and every such application either removes a connective occurence, thus diminishing the complexity of the sequent to he proved, or rewrites the sequent's antecedent to a term of equal complexity. Therefore, if we make sure that a search path is relinquished whenever a sequent reappears on it (which prevents the procedure from entering into an infinite loop), the proof search tree will be finite. This implies that the calculus is decidable.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Cut-elimination and the subformula property", |
| "sec_num": "2.6" |
| }, |
| { |
| "text": "The basis for any model of \u00a3 is a (E, C)-algebra ,4, where I] = {+i}iex and the product operation interpreting oi is denoted as \"i. We say that 3 C ,4", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Semantics", |
| "sec_num": null |
| }, |
| { |
| "text": "is an Fd-subalgebra of ,4 if it is closed under .~j, and 2Note that we use the term top-down in the usual sense, i.e. for a proof search procedure that works back from the goal to the axioms. Visually, top-down proofs actually proceed bottom-up! s \u00b0 = t \u00a2 whenever s = t E gj and a : V --, 8. An easy Ck-subalgebra of`4 is a subset of ,4 that is closed under \"ik, and such that s \u00b0 = t \u00b0 whenever s = t E gk and a : V --* ,4 assigns an element of $ to at last one of the variables occurring in the equation. A model for \u00a3 is a 4-tuple (,4, {,4j}jeJ, {,4k}ke~:, i.I) such that: t> ,4 is a (~, C)-algebra;", |
| "cite_spans": [ |
| { |
| "start": 535, |
| "end": 565, |
| "text": "(,4, {,4j}jeJ, {,4k}ke~:, i.I)", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Semantics", |
| "sec_num": null |
| }, |
| { |
| "text": "Aj is an Ci-subalgebra of `4 (j E if); t> `4k is an easy gk-subalgebra of`4 (k E/C); t, [.] is a function B --* 7)(`4). ", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Semantics", |
| "sec_num": null |
| }, |
| { |
| "text": "As usual, the soundness proof boils down to a straightforward induction on the length of a derivation, and we omit it.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Soundness and completeness", |
| "sec_num": "2.8" |
| }, |
| { |
| "text": "For completeness, we start by defining the canonical model .A4. Its carrier is the set S/--, where = is the equivalence relation defined by r _--A iff VA : r F A \u00a2~ A F A. The ----equivalence class containing F will be denoted as [r] . On the set S/_= we define products \"i (i E 27) by stipulating that By definition of --we know that r ' F\" r* and A ~ }\" A*. Now, r' ol A' I-A by the derivation below:", |
| "cite_spans": [ |
| { |
| "start": 230, |
| "end": 233, |
| "text": "[r]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Soundness and completeness", |
| "sec_num": "2.8" |
| }, |
| { |
| "text": "r ol A I-A [.L]* r' r\" r \u00b0 oi A \u00b0 I-A t- r' oi A\" }-A [Cut] A' I-A\" r' ol A' I-A [Cut]", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Soundness and completeness", |
| "sec_num": "2.8" |
| }, |
| { |
| "text": "Evidently, .A4 = (S/=, {.i}icz) is a (E, ~)-algebra.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Soundness and completeness", |
| "sec_num": "2.8" |
| }, |
| { |
| "text": "Next, we define \u00a2~41 = {IF] [ F ~-Qz} (! e ,] u/C). It must be shown that these have the desired properties. Since it would be notationally awkward to have to refer to an arbitrary equational specification, we do this by means of an example. Let Then, by the induction hypothesis, A I-A. We now have the following derivation: Also, by the definition of A4~, I\" t-Qj.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Soundness and completeness", |
| "sec_num": "2.8" |
| }, |
| { |
| "text": "rl oi# r~ \u2022 A Q# \u2022 Q# r~ oi# r~ I-A [.L] AjF~,2 \u2022 Q# [A#L2] Air~ oi# AiF~ t-A [A#L1] AjF~ oi~ A#F~ \u2022 A [t:Aj] r~ oi# A#F~ b A r~ \u2022 zx#rt [Z~#R] [Cut] r2 oi# rl \u2022 A r2 F r~ F2 F Qj r2 F Air** [AIR]", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Soundness and completeness", |
| "sec_num": "2.8" |
| }, |
| { |
| "text": "AI-A BI-B [/,L] r ~ B/,A (B/,A) o, A I-B A F A r oi A I-B [Cut] r oi A F B [Cut]", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Soundness and completeness", |
| "sec_num": "2.8" |
| }, |
| { |
| "text": "Applying the [Aj R]-rule two these two sequents, we find that I\" I-AjA. 2. Conversely, suppose r I-AjA. Then r I-A:", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Soundness and completeness", |
| "sec_num": "2.8" |
| }, |
| { |
| "text": "A k A [A~L1] FFAjA AjA I-A r F A [cut]", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Soundness and completeness", |
| "sec_num": "2.8" |
| }, |
| { |
| "text": "From this we conclude by the induction hy- []", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Soundness and completeness", |
| "sec_num": "2.8" |
| }, |
| { |
| "text": "It will not have escaped the reader's attention that we have failed to include the set of product connectives {.i}iEz in the language of the resourcepreserving logics. The reason for this is that a completeness proof along the above lines runs into problems for such extended logics. This is already the case for the full Lambek calculus. Buszkowski [1986] presents a rather complicated completeness proof for that logic. It remains to be seen whether his approach also works in the present setting. Although we've tried to give a liberal definition of what constitutes a resource-preserving logic, some choices had to be made in order to keep things manageable. There is room for alternative definitions, especially concerning the interaction of the modalities with the different product operators. It would seem to be worthwile to study some of the systems that have occurred in practice in detail on the basis of the ideas presented in this paper.", |
| "cite_spans": [ |
| { |
| "start": 339, |
| "end": 356, |
| "text": "Buszkowski [1986]", |
| "ref_id": "BIBREF0" |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Further research", |
| "sec_num": "3" |
| }, |
| { |
| "text": "Finally, it is important to realize that we limited ourselves to resource-preserving logics in order to obtain relatively easy proofs of Cut-elimination and decidability. Since such results tend also to hold for many systems with rules that are not resourcepreserving, such as weakening and contraction, it is probably possible to characterize a larger class of equational theories for which these properties can be proved. We hope to address this point on a later occassion.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Further research", |
| "sec_num": "3" |
| } |
| ], |
| "back_matter": [ |
| { |
| "text": "The task of preparing this paper was alleviated considerably thanks to enlightening discussions with, and comments on earlier versions by Kees Vermeulen, Yde Venema, Erik Aarts, Marco Hollenberg and Michael Moortgat.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Acknowledgements", |
| "sec_num": null |
| } |
| ], |
| "bib_entries": { |
| "BIBREF0": { |
| "ref_id": "b0", |
| "title": "Hepple, 1990] Mark Hepple. The Grammar and Processing of Order and Dependency: A Categorial Approach", |
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| "first": "Ajdukiewicz ; Kazimierz Ajdukiewicz ; Wojciech", |
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| "last": "Buszkowski", |
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| "venue": "Bar-Hillel, 1953] Yehoshua Bar-Hillel. A quasiarithmetical notation for syntactic description. Language", |
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| "raw_text": "Ajdukiewicz, 1935] Kazimierz Ajdukiewicz. Die syntaktische Konnexit~it. Studia Philosophica, 1:1-27, 1935. [Bar-Hillel, 1953] Yehoshua Bar-Hillel. A quasi- arithmetical notation for syntactic description. Language, 29:47-58, 1953. [Buszkowski, 1986] Wojciech Buszkowski. Com- pleteness results for Lambek syntactic calculus. Zeitschrift f~r mathematische Logik und Grund- lagen der Mathematik, 32:13-28, 1986. [Do~en, 1988] Kosta Do~en. Sequent systems and groupoid models, I. Studia Logica, 47:353-385, 1988. [Do~en, 1989] Kosta Do~en. Sequent systems and groupoid models, II. Studia Logica, 48:41-65, 1989. [Hepple, 1990] Mark Hepple. The Grammar and Processing of Order and Dependency: A Catego- rial Approach. PhD thesis, Centre for Cognitive Science, University of Edinburgh, 1990. [Lambek, 1958] Joachim Lambek. The mathemat- ics of sentence structure. American Mathematical Monthly, 65:154-170, 1958. [Moortgat and Morrill, 1992] Michael Moortgat and Glyn Morrill. Heads and phrases. Type calculus for dependency and constituent structure. 1992. [Moortgat, 1988] Michael Moortgat. Categorial In- vestigations. Logical and Linguistic Aspects of the Lambek Calculus. Foris, Dordrecht, 1988. [Moortgat, 1992] Michael Moortgat. Labelled de- ductive systems for categorial theorem proving. 1992. [Morrill et aL, 1990] Glyn Morrill, Neil Leslie, Mark l/epple, and Guy Barry. Categorial deductions and structural operations. In Guy Barry and Glyn Morrill, editors, Studies in Categorial Grammar (Edinburgh Working Papers in Cognitive Science, Vol. 5), pages 1-21. Centre for Cognitive Science, University of Edinburgh, 1990. [Morrill, 1990] Glyn Morrill. Intensionality and boundedness. Linguistics and Philosophy, 13:699- 726, 1990.", |
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| "title": "Type-logical grammar", |
| "authors": [ |
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| "first": "; ] Glyn", |
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| "last": "Morrill", |
| "suffix": "" |
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| { |
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| "last": "Morrill", |
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| "raw_text": "Morrill, 1992] Glyn Morrill. Type-logical grammar. OTS Working Paper OTS-WP-CL-92-002, Re- search Institute for Language and Speech, Utrecht University, 1992.", |
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| "title": "Venema, 1993a] Yde Venema. Meeting a modality? Restricted permutation for the Lambek calculus", |
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| "venue": "Venema, 1993b] Yde Venema. Meeting strength in substructural logics", |
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| "raw_text": "Ross, 1967] J. R. Ross. Constraints on Variables in Syntax. PhD thesis, MIT, 1967. [Venema, 1993a] Yde Venema. Meeting a modality? Restricted permutation for the Lambek calculus. OTS Working Paper (to appear), Research Insti- tute for Language and Speech, Utrecht University, 1993. [Venema, 1993b] Yde Venema. Meeting strength in substructural logics. Logic Group Preprint Series No. 86, Department of Philosophy, Utrecht Uni- versity, 1993.", |
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| "last": "Versmissen ; Koen", |
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| "venue": "OTS Yearbook", |
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| "raw_text": "Versmissen, 1992] Koen Versmissen. Adding a uni- versal modality to the Lambek calculus. In OTS Yearbook 1992. Research Institute for Language and Speech, Utrecht University, 1992.", |
| "links": null |
| } |
| }, |
| "ref_entries": { |
| "FIGREF0": { |
| "num": null, |
| "uris": null, |
| "text": "set of binary type connectives c = {/i, \\0~z; Two sets of unary type connectives M~ = {Aj}je.~ and M v = {~Tk}~\u00a2Jc; ~, The set of type constants q = {Qj}j~ u {Qk}kE~; The set of types T, being the inductive closure of B U Q under C U Mz~ U M e; The set of structural connectives SC = {oi}iez; The set of slructures S, being the inductive closure of T under SC; c, The set of sequents {F b A I r \u2022 S,A \u2022 T}.", |
| "type_str": "figure" |
| }, |
| "FIGREF1": { |
| "num": null, |
| "uris": null, |
| "text": "+A~z whereas gw is comprised of five equations in all: Vkz+Y = y+~7kz z+Wky = VkY+X w~+(y+z) = (wx+y)+z x+(Vky+z) = (x+Wy)+z x+(y+Vkz) = (x+y)+Vkz", |
| "type_str": "figure" |
| }, |
| "FIGREF2": { |
| "num": null, |
| "uris": null, |
| "text": "r~ k oi r~ k Oi r[r, % r2] k A I [6] r[r2 % r,] f-A NP I-NP NPI-NP SI-S [\\L] NP\\S I-NP\\S NP, NP\\S I-S NP, NP\\S, (NP\\S)\\(NP\\S) I-S [~L] [/LI NP, NP\\S/NP, NP, (NP\\S)\\(NP\\S) I-S NP, NP\\SINP, vpNP, (NP\\S)\\(NP\\S) I-S [VpLI] NP, NP\\S/NP, (NP\\S)\\(NP\\S), VpNP I-S [evp] [IR] NP, NP\\S/NP, (NP\\S)\\(NP\\S) I-Sl V~' NP REL I-REL RELI(S/VP NP), NP, NP\\S/NP, (NP\\S)\\(NP\\S) NPI-NP [/L] (NP/N) o N I-NP NI-N NPI-NP I/L] /N) o N) o VA(N\\N) I-NP [IR] (NP/N) o N I-NP/VA (N\\N) NP I-NP [\\L] ((NP/N) o N) o ((NP/VA (N\\N))\\NP) I-NP [ILl ((NP/N) o N) o ((((NP/VA (N\\N))\\NP)/NP) o ((NP/N) o N)) F NP Figure 2", |
| "type_str": "figure" |
| }, |
| "FIGREF3": { |
| "num": null, |
| "uris": null, |
| "text": "Here, :P(,4) denotes the set of all subsets of,4. The interpretation function [.] is extended to arbitrary types and structures as follows: [Od = ,4t (l e y u Ic) t> IB/,A] = {c e ,4 I Va e [A]: c., a e [[3]} > [A\\iB] = {c E ,4 I Va e [A] : a \"i c E [13]} z> EAoiB]--{cE,4[~aE[A],bE[Bl:c=a.+b} A sequent F k A is said to be valid with respect to a given model, if ir] g [A]. A sequent is gene~lly valid if it is valid in all models. The proof system is said to be sound with respect to the semantics if all derivable sequents are generally valid. It is complete if the converse holds, i.e. if all generally valid sequents are derivable.", |
| "type_str": "figure" |
| }, |
| "FIGREF4": { |
| "num": null, |
| "uris": null, |
| "text": "[r] .i [A] = [r oi A]. We need to prove that this is well-defined. So suppose r -r', A -A' and r oi A F A. For a structure O, let O* be the \u00a3.-type obtained from O by replacing each oi with oi. The sequent O* [-A can be derived from O ~-A by a sequence of [.L]-rules.", |
| "type_str": "figure" |
| }, |
| "FIGREF5": { |
| "num": null, |
| "uris": null, |
| "text": "{Aim +i~ Ajy = Ajy +ij Ajx}. Supposing that[rl], [r2] \u2022 .N4Aj we must prove that [rl] \"ij IF2] = [r2] \"ij IF1], i.e. that VA : r, % F2 F A ,## r~. oij rl F A. This follows from the derivation inFigure 3. The proof for A4Vk is similar.Finally, we set[B l -{[r]l r e B} for B \u2022 B, which completes our definition of the canonical model.We proceed to prove the so-called canonical lemma:Lemma IT] = {[r] I r F T} for all T \u2022 T.Proof We prove this by induction on the complexity of the type T. ~, For basic types T it is true by the definition of [.]; ~, For Qt (1 \u2022 3\" U/C) it is true by the definition of A4a; ~, For T = B/iA: 1. First, suppose [r] \u2022 ~\"]] --~'B/ia]. Then for any [A] \u2022 [A] we have that [F]., [A] = [r oi A] \u2022 [B]. By the induction hypothesis we deduce from this that r oi A I-B. In particular, since [A] \u2022 [[A], we have that r ol A I-B, whence, by [/iR], it follows that r I-B/iA. 2. Conversely, suppose that r F B/iA, and let [A] \u2022 ~A].", |
| "type_str": "figure" |
| }, |
| "FIGREF6": { |
| "num": null, |
| "uris": null, |
| "text": "this we conclude by the induction hypothesis that IF oi A] = [r] .i [A] \u2022 [B] for all [A] \u2022 [A]. That is, [F] \u2022 [B/IA]I, and we're done. For the other binary connectives, the proof is similar. t> For T = AjA: I. First, suppose [r] \u2022 [AjA] = [A]n A41. Then, by the induction hypothesis, r F A.", |
| "type_str": "figure" |
| }, |
| "TABREF1": { |
| "text": "From this we find by the definition of .A4j that [r] \u2022 [Qj] = .A4j. So [r] \u2022 lAin [Qfll = IAjA]. For ~7k, the proof is similar. Now suppose that the sequent r I-A is not derivable. Then in the canonical model we have, by the lemma we just proved, that [r] \u00a2 [[A]. Since IF] \u2022 [r], this implies that IF] ~ [[A]. That is, r I-A is not valid in the canonical model, and hence is not generally valid.", |
| "num": null, |
| "html": null, |
| "content": "<table><tr><td colspan=\"2\">pothesis that [F] \u2022 [Al. Also, r</td></tr><tr><td>FFA~A AjA I-Qj</td><td>[&#L2]</td></tr><tr><td>r F Q~</td><td>[Cut]</td></tr></table>", |
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