W. Buszkowski, Completeness Results for Lambek Syntactic Calculus, Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (1986), 13-28.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that we have failed to include the set of product con nectives i)iz in the language of the resource preserving 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 ....

Wojciech Buszkowski. Com- pleteness results for Lambek syntactic calculus. Zeitschrift f6r mathematische Logik und Grund- !agen der Mathematik, 32:13-28, 1986.

....introduced a calculus for deriving reduction laws of syntactic types. The inteneded syntactic string models, i.e. free semigroup models (also called language models or L models) for this calculus were considered in [2] 3] and [4] The more general class of groupoid models has been studied in [5], 6] and [7] In [3] W. Buszkowski established that the product free fragment of the Lambek calculus is L complete (i.e. complete w.r.t. free semigroup models) using the canonical model. The question of L completeness of the full Lambek calculus remained open (cf. 1] At the end of 1992 the ....

....of associative ternary frames [7] It is known that the Lambek calculus is sound w.r.t. associative ternary frames. Thus it is also sound w.r.t. all partial semigroup models, i.e. w(#) # w(B) for any partial semigroup model #W, #, w#, whenever L # ##B. On the other hand, W. Buszkowski [5] has proved that the Lambek calculus is complete w.r.t. models over arbitrary semigroups (Example 1 (a) The completeness w.r.t. models over binary relational frames (Example 1 (d) has been proved by Sz. Mikulas [9] In this paper we prove that the Lambek calculus is also complete w.r.t. smaller ....

W. Buszkowski. Completeness Results for Lambek Syntactic Calculus. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, 32:13--28, 1986.

....and formal languages. The intended models for these calculi are free semigroup models (also called language models or string models) where each syntactic category is interpreted as a set of non empty strings over some alphabet of symbols. Models for Lambek calculus were studied in [2] 3] [4], 5] 6] etc. Completeness of the Lambek calculus with respect to string models was proved in [9] 10] and [11] Closely related is the result about completeness with respect to relational semantics [8] There is a natural modification of the original Lambek calculus, which we call the Lambek ....

W. Buszkowski. Completeness results for Lambek syntactic calculus. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 32:13--28, 1986.

....A is duplicated in the premises and the rule fails to be complexity reducing in the sense of the logical rules. However, the 2 Officially the antecedent is non empty, a detail we gloss over. 3 Regarding completeness with respect to semigroup, free semigroup, and relational interpretation see Buszkowski (1986), Pentus (1994) and Andr eka and Mikul as (1994) respectively; see also Kurtonina (1995) Buszkowski (1996) gives a survey. calculus has the property of Cut elimination: for every proof there is an equivalent Cut free proof. This means that naive Cut free backward chaining proof search ....

Buszkowski, Wojciech: 1986, `Completeness results for Lambek Syntactic Calculus', Zeitschr. f. math. Logik und Grundlagen d. Math. Bd. 32, S. 13--28 (1986).

....a=ff 0 g. The result of the semantic trip is the simplified semantic form (15b) 3 Discontinuity Calculus Associative Lambek calculus was presented as a logic of concatenation, which role is strengthened by the result of Pentus (1994) improving to free semigroups the completeness result of Buszkowski (1986) for semigroups. However, natural grammar includes non concatenative phenomena, leading to the search for calculi of discontinuity serving for non concatenative adjunction as the Lambek calculus serves for concatenative adjunction. Thus Versmissen (1991) proposes to treat discontinuity via split ....

Buszkowski, Wojciech: 1986, `Completeness results for Lambek syntactic calculus', Zeitschrift fur mathematische Logik und Grundlage der Mathematik 32, 13--28.

....is a Cut free counterpart, so that fully representative proof search can be carried out on the basis of the Cut free presentation. Since each refinement step decrements the total number of connective occurrences by one, this provides a decision procedure for the calculus (cf. Moortgat 1988b) Buszkowski (1986) provides a summary of completeness results; in fact few such results have currently been secured for categorial calculi. The , fragment is complete for the interpretation given above, but even for just , completeness is still unproved for this interpretation. 3 , 4 3 In ....

Buszkowski, W. (1986) `Completeness Results for Lambek Syntactic Calculus', Zeitschrift fr mathematische Logik und Grundlagen der Mathematik 32, 13-28.

.... one directional structural postulates (e.g. one half 7 of the Associativity postulate, A . B . C) # (A . B) C, allowing restructuring of left branching structures) unless one is willing to reintroduce abstractness in the form of a partial order on the resources W. See[Do sen 92, Buszkowski 86] and Chapter Twelve for discussion. Even more concrete are the language models or free semigroup semantics for L. In the language models, one takes W as V (non empty strings over the vocabulary) and as string concatenation. This type of semantics turns out to be too specialized for our ....

Buszkowski, W. (1986), `Completeness results for Lambek syntactic calculus'. Zeitschrift f ur mathematische Logik und Grundlagen der Mathematik ,32, 13--28.

....types corresponding to) the left and right sides of a Lambek sequent. The result follows by a Curry Howard style types as formulas observation: the rules for building Sigma terms mimic the proof rules exactly. Completeness of the Lambek calculus for a different notion of semantics is found in [6]. 13 Thus the theory of syntax outlined here is first order, even equational (and decidable, as shown by Lambek) Passing to viewing strings as arrows in a category provides a refined point of view on the type shifting of categorial grammar, in that the canonical type shifts whose existence is ....

Buszkowski, W. Completeness Results for Lambek Syntactic Calculus, Zeitschr. fur math. Logik und Grunclagen d. Math. 32, 13-28, 1986.

....of formulas. 1 A sequent is valid if and only if in all interpretations the ordered adjunction of elements inhabiting the antecedent formulas always yields an element inhabiting the succedent formula. The following Gentzen style sequent presentation is sound and complete for this interpretation (Buszkowski 1986, Dosen 1992) and indeed for free semigroups (Pentus 1994) hence the Lambek calculus 1 Officially the antecedent is non empty, a detail we gloss over. Morrill Processing and Acceptability can make an impressive claim to be the logic of concatenation; the parenthetical notation Gamma( Delta) ....

Buszkowski, Wojciech: 1986, `Completeness results for Lambek syntactic calculus', Zeitschrift f ur mathematische Logik und Grundlage der Mathematik 32, 13--28.

....to choose the functor type B A resp. A B from the subformulas of the sequence U 2 . 2.3. 2 Soundness and Completeness Various completeness results for the Lambek calculus and its variants have been reported in the literature, see e.g. Buszkowski 1982 [7] Dosen 1985 [12] Buszkowski 1986 [8], Dosen 1990 [13] Pentus 1994 [28] Versmissen 1993 [36] Definition 6 The denotation of a sequence U 1 U 2 is the cross product of the denotations of the 5 i.e. all labels of nodes in a proof tree are subformulas of the original proof task Journal of Logic, Language and Information ....

Wojciech Buszkowski. Completeness results for Lambek syntactic calculus. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, 32:13--28, 1986.

.... one directional structural postulates (e.g. one half of the Associativity postulate, A ffl (B ffl C) A ffl B) ffl C, allowing restructuring of left branching structures) unless one is willing to reintroduce abstractness in the form of a partial order on the resources W . See [Dosen 92, Buszkowski 86] and Chapter Twelve for discussion. Even more concrete are the language models or free semigroup semantics for L. In the language models, one takes W as V (non empty strings over the vocabulary) and Delta as string concatenation. This type of semantics turns out to be too specialized for our ....

....of proof, especially in adaptations of Girard s proof nets to various forms of categorial inference. We discuss this use in x7. Another focus of work on labeled deduction has been the investigation of how various logical systems are related to particular model structures. Already in 1986, Buszkowski 86] employed term labels in his study of completeness proofs for various members of the Lambek family of substructural logics. An LDS presentation of NL in [Venema 94] forms the basis for completeness proofs for tree based models. We have seen in x2 that the unlabeled system NL is incomplete for ....

Buszkowski, W. (1986), `Completeness results for Lambek syntactic calculus'. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik ,32, 13--28.

.... have been much studied recently, particularly since the article of Ades and Steedman (1982) and the re discovering of previous works done by Lambek (1958, 1961) The most comprehensive form taken by Categorial Grammars is the Lambek Calculus, studied by many authors like Moortgat (1988, 1990) Buszkowski (1986, 1988) Descl s (1990) etc. Since the recent work by J Y Girard (see for instance Girard 1987) which led to the framework of Linear Logic, it has become apparent that the Lambek Calculus amounts to a non commutative version of a sub system of Linear Logic, where a structural rule forbids ....

....Girard 1987) which led to the framework of Linear Logic, it has become apparent that the Lambek Calculus amounts to a non commutative version of a sub system of Linear Logic, where a structural rule forbids sequents with an empty antecedent. Semantic properties of this system have been studied by Buszkowski (1986, 1988) and Wansing (1990) Two models are often given: one consists of residuation semigroups spread over free semigroups, and another one is given by the directional typed lambda calculus. 1 I am indebted to Dirk Roorda for fruitful discussions during a brief visit I made in Amsterdam in Spring ....

Buszkowski 1986; 'Completeness Results for Lambek Syntactic Calculus', Zeitschr. f. math. Logik und Grundlagen d. Math. 32, 13-28.

....admits a complete denotational semantics [81] this should help in finding a linguistic meaning for denotational semantics. Truth value semantics, for which logic completeness results usually hold, have been substantially studied in categorial grammar, mainly by Buskowski and van Benthem, [22,23,16 18,24]. For instance, they established completeness with regards to relational interpretations and with regards to monoidal interpretations see e.g. the section 1 of Martin Emms paper in this volume. These latter models are very similar to the so called phase semantics of linear logic [53,2,118] ....

Wojciech Buszkowski. Completeness results for lambek syntactic calculus. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 32:13--28, 1986.

....sets of labels. Their roles are twofold. Firstly, the labels allow us to generate countermodels in terms of a sufficiently simple semantics. Labels are often used to prove a completeness theorem with respect to a certain simple semantics to which the Lindenbaum Tarski method cannot apply (e.g. [Bus86], Kur94] Pan94] OT98] Our labels play the same role as in those works; indeed, our sets of labels are direct descendants of Buszkowski s one ( Bus86] Secondly, the labels establish a tight correspondence between our labelled tableaux and bottom up proof search trees of the sequent ....

.... used to prove a completeness theorem with respect to a certain simple semantics to which the Lindenbaum Tarski method cannot apply (e.g. Bus86] Kur94] Pan94] OT98] Our labels play the same role as in those works; indeed, our sets of labels are direct descendants of Buszkowski s one ([Bus86]) Secondly, the labels establish a tight correspondence between our labelled tableaux and bottom up proof search trees of the sequent calculus. The idea is to introduce a certain condition on the labels and to identify a set of labelled formulas whose labels satisfy the condition with a sequent. ....

[Article contains additional citation context not shown here]

W. Buszkowski. Completeness results for Lambek syntactic calculus. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, 32:13--28, 1986.

No context found.

W. Buszkowski, Completeness Results for Lambek Syntactic Calculus, Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (1986), 13-28.

....A i.e. the rule of associativity and the rule of permutation. ASS) acts up down and down up. Associative Lambek Calculus (L) is NL plus (ASS) it is due to Lambek [16] NL plus (PER) will be denoted by NLP. L is complete with respect to residuated semigroups, powerset structures over semigroups [3] and over free semigroups [23] Cut elimination and decidability hold for both systems. They, however, need not hold, if we ax new nonlogical axioms of the form A B. For a set , of formulas A B, NL( denotes the system NL with all formulas from as new axioms, and similarly for L( ....

....of the situation for L( 2 NL with nonlogical axioms Since (CUT) is a legal rule in NL( it is not obvious that the system has the subformula property. A slightly generalized form of this property can be shown by a re nement of standard proofs of the completeness theorem. Actually, in [3] the subformula property for L( has been proven in this way. Here we give a simpler proof, based on more general algebraic models of NL. A residuated groupoid is a structure M = M; n; such that (M; is a poset, M; is a groupoid, and n; are binary operations on M , satisfying ....

W. Buszkowski, Completeness results for Lambek Syntactic Calculus, Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (1986), 13-28.

....semigroup; it will be called the powerset frame over M. Models (P (M) will be called powerset models. In powerset models, one interprets formulas with as follows: A B) A) B) It is obvious that L[ Omega ; is sound with respect to powerset models, and it has been shown in [5] that this calculus is also strongly complete with respect to powerset models. Standard linguistic frames for the Lambek calculus are powerset frames over (finitely generated) free semigroups. Thus, M = V ; Delta) where V is a nonempty finite set (the alphabet or lexicon) V denotes the ....

....functors from category (A) to category (B) and A B (resp. B A) is a type of these functors. For instance, a verb phrase is a functor of type n s, and a transitive verb phrase is a functor of type (n s) n. L[ is strongly complete with respect to powerset models over free semigroups [5], but L[ Omega ] is merely weakly complete with respect to this semantics (see Pentus [15] 3 Let M = M; Delta) be a semigroup, and let K be a nonempty subset of M , satisfying the condition: K) for all x; y; z 2 K, if xyz 2 K, then xy 2 K iff yz 2 K. On the powerset P (K) we define ....

[Article contains additional citation context not shown here]

W. Buszkowski, Completeness results for Lambek Syntactic Calculus, ibidem 32 (1986), 13-28.

.... issues of current interest in logic, as e.g. linear logics [17] action logic [25] gaggle theory [14] labelled deductive systems [16] and in natural language semantics and computational linguistics (see van Benthem [2] Moortgat [22] A thorough logical discussion of the domain can be found in [10, 5], and many linguistic applications in [23] In terms of type theory, B is a purely applicative system, while Lambek systems employ lambda abstraction. A problem which has quite early appeared in this discipline is whether lambda abstraction affects generative capacity. Strictly, the question is ....

W. Buszkowski, Completeness Results for Lambek Syntactic Calculus, ibidem 32, (1986), 13-28.

....inclusion fails, in general. The following representation theorem: RT) each residuated semigroup is embeddable into a powerset residuated semigroup and similar results for residuated groupoids, monoids, commutative structures etc. can be proven with the aid of proof theoretic tools from [2]. We use the Lambek calculus L [8] which deals with sequents A B, where A; B are propositional formulas with logical constants ffi; The axioms and rules of L are: A0) A A; 4 (A1) A ffi B) ffi C A ffi (B ffi C) A2) A ffi (B ffi C) A ffi B) ffi C; R1) A ffi B C B A C ....

W. Buszkowski, Completeness Results for Lambek Syntactic Calculus, Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (1986), 13-28.

.... Linguistics is mainly interested in L models for these calculi which are based on algebras of languages over a finite alphabet [4] More general structures are powerset frames over arbitrary semigroups (monoids) the strong completeness of L and L1 with respect to the latter models was proved in [6] with applying a special system ND which might be considered as a Labeled 1 Deductive System (LDS) in the sense of Gabbay [11] It has been shown in [7, 8] that these methods yield, actually, representation theorems for residuated semigroups and monoids with respect to powerset frames over ....

.... a similar way (now, we assume sequents from X have nonempty antecedents) It is known that, for any set X, the sequents derivable in L1(X) are precisely the sequents true in all RS1 models in which all sequents from X are true, and an analogous strong completeness theorem holds for L and RS models [6]. Actually, we need merely the soundness, which is easy: all sequents derivable in L1(X) are true in every RS1 model in which all sequents from X are true, and similarly for L(X) and RS models. The strong completeness follows from our lemmas on canonical models. For R U 2 , we consider the ....

W. Buszkowski, Completeness results for Lambek Syntactic Calculus, Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (1986), 13-28.

....of grammatical description which result from overgeneration. From the logical point of view, that amounts to an interpretation of the Lambek calculus in more general frames than powerset frames on free semigroups (standard frames) This approach has already been announced in the author s [9, 10]; generalized frames of that form have also been used in [6] to prove the finite model property of the (product free) Lambek calculus. In the above negative information is still external to the logical systems of categorial grammar; it lives on the level of type assignments in the former setting, ....

.... has been proven by Pentus [26] that theorems of L (with product) are precisely the sequents true in all language models (for the product free Lambek Calculus, an analogous completeness result has been proven in [5] and for the full Lambek Calculus and a more general class of semigroup models, in [9]) Language models correspond to the standard interpretation of the meaning of types in categorial grammar. Below we define a more realistic notion of a restricted language model which relativizes clauses (M Theta) M ) and (Mn) to a fixed set of non ill formed expressions . We believe that ....

[Article contains additional citation context not shown here]

W. Buszkowski, Completeness Results for Lambek Syntactic Calculus, Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, 32, (1986), 13-28.

No context found.

Buszkowski, W., `Completeness Results for Lambek Syntactic Calculus', Zeitschrift f ur mathematische Logik und Grundlagen der Mathematik' 32, 1984, p. 13-28

No context found.

Buszkowski, W., `Completeness Results for Lambek Syntactic Calculus', Zeitschrift f ur mathematische Logik und Grundlagen der Mathematik' 32, 1984, p. 13-28

No context found.

Wojciech Buszkowski. Completeness results for Lambek syntactic calculus. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik. 32, 13--28, 1986.

No context found.

Wojciech Buszkowski. Completeness results for Lambek syntactic calculus. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik. 32, 13-28, 1986.

No context found.

Buszkowski, W. (1986), `Completeness results for Lambek syntactic calculus'. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik ,32, 13--28.

No context found.

Buszkowski, W. 1986. `Completeness results for Lambek syntactic calculus.' Zeitschrift fur mathematischen Logik und Grundlagen der Mathematik, 32, pp13--28.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC