In this paper we consider the relations existing between four deductive systems that have been called categorial grammars and have relevant connections with linguistic investigations: the syntactic calculus, bilinear logic, compact bilinear logic and Curry''s semantic calculus.

We present a categorical/denotational semantics for the Lambek Syntactic Calculus , indeed for a λlD-typed version Curry-Howard isomorphic to it. The main novelty of our approach is an abstract noncommutative construction with right and left adjoints, called sequential product. It is defined through a hierarchical structure of categories reflecting the implicit permission to sequence expressions and the inductive construction of compound expressions. We claim that Lambek's noncommutative product corresponds to a noncommutative bi-endofunctor into a category, which (...) encloses all categories of such hierarchical structure. A soundness theorem for LSC is shown with respect to this semantical framework. (shrink)

This work is devoted to the relations between Lambek’s Syntactic Calculus and noncommutative variants of Girard’s Linear Logic; in particular the paper will consider: the geometrical representation of the laws of LC by means of proof-nets; the discovery - due to such a geometrical representation - of some laws of LC not yet considered; the discussion of possible linguistic uses of these new laws.

The aim of this study is to look at the the syntactic calculus of Bar-Hillel and Lambek, including semantic interpretation, from the point of view of constructive type theory. The syntactic calculus is given a formalization that makes it possible to implement it in a type-theoretical proof editor. Such an implementation combines formal syntax and formal semantics, and makes the type-theoretical tools of automatic and interactive reasoning available in grammar.In the formalization, the use of the (...) dependent types of constructive type theory is essential. Dependent types are already needed in the semantics of ordinary Lambek calculus. But they also suggest some natural extensions of the calculus, which are applied to the treatment of morphosyntactic dependencies and to an analysis of selectional restrictions. Finally, directed dependent function types are introduced, corresponding to the types of constructive type theory. (shrink)

An axiomatics of the product-free syntactic calculus L ofLambek has been presented whose only rule is the cut rule. It was alsoproved that there is no finite axiomatics of that kind. The proofs weresubsequently simplified. Analogous results for the nonassociativevariant NL of L were obtained by Kandulski. InLambek's original version of the calculus, sequent antecedents arerequired to be nonempty. By removing this restriction, we obtain theextensions L 0 and NL 0 ofL and NL, respectively. Later, the finiteaxiomatization (...) problem for L 0 andNL 0 was partially solved, viz., for formulas withoutleft (or, equivalently, right) division and an (infinite) cut-ruleaxiomatics for the whole of L 0 has been given. Thepresent paper yields an analogous axiomatics forNL 0. Like in the author's previous work, the notionof rank of an axiom is introduced which, although inessentialfor the results given below, may be useful for the expectednonfinite-axiomatizability proof. (shrink)

In Zielonka (1981a, 1989), I found an axiomatics for the product-free calculus L of Lambek whose only rule is the cut rule. Following Buszkowski (1987), we shall call such an axiomatics linear. It was proved that there is no finite axiomatics of that kind. In Lambek's original version of the calculus (cf. Lambek, 1958), sequent antecedents are non empty. By dropping this restriction, we obtain the variant L 0 of L. This modification, introduced in the (...) early 1980s (see, e.g., Buszkowski, 1985; Zielonka, 1981b), did not gain much popularity initially; a more common use of L 0 has only occurred within the last few years (cf. Roorda, 1991: 29). In Zielonka (1988), I established analogous results for the restriction of L 0 to sequents without left (or, equivalently, right) division. Here, I present a similar (cut-rule) axiomatics for the whole of L 0. This paper is an extended, corrected, and completed version of Zielonka (1997). Unlike in Zielonka (1997), the notion of rank of an axiom is introduced which, although inessential for the results given below, may be useful for the expected non-finite-axiomatizability proof. (shrink)

In Zielonka (1981a, 1989), I found an axiomatics for the product-free calculus L of Lambek whose only rule is the cut rule. Following Buszkowski (1987), we shall call such an axiomatics linear. It was proved that there is no finite axiomatics of that kind. In Lambek's original version of the calculus (cf. Lambek, 1958), sequent antecedents are non empty. By dropping this restriction, we obtain the variant L0 of L. This modification, introduced in the early (...) 1980s (see, e.g., Buszkowski, 1985; Zielonka, 1981b), did not gain much popularity initially; a more common use of L0 has only occurred within the last few years (cf. Roorda, 1991: 29). In Zielonka (1988), I established analogous results for the restriction of L0 to sequents without left (or, equivalently, right) division. Here, I present a similar (cut-rule) axiomatics for the whole of L0.This paper is an extended, corrected, and completed version of Zielonka (1997). Unlike in Zielonka (1997), the notion of rank of an axiom is introduced which, although inessential for the results given below, may be useful for the expected non-finite-axiomatizability proof.The paper follows the same way of subject exposition as Zielonka (2000) but it is technically much less complicated. I restrict myself to giving bare results; all the ideological background is exactly the same as in case of the non-associative calculusNL0 and those who are interested in it are requested to consult the introductory section of Zielonka (2000). (shrink)

van Benthem [1] introduces a variant of Lambek Syntactic Calculus , proposed by Lambek [6], we call the variant Lambek-van Benthem Calculus . As proved by van Benthem, LBC is complete with respect to a semantics of λ-terms. In this note we indicate other relevant properties of LBC , just supporting some expectations of van Benthem. Given a countable set P r, of primitive types, the set T p, of types, is the smallest one (...) such that: T p contains P r, if x, y are in T p then is in T p. The variables x, y, z will range over types. Expressions X → x are to be called reduction formulas. (shrink)

In 1958 J. Lambek introduced a calculusL of syntactic types and defined an equivalence relation on types: x y means that there exists a sequence x=x1,...,xn=y (n 1), such thatx i x i+1 or xi+ x i (1 i n). He pointed out thatx y if and only if there is joinz such thatx z andy z. This paper gives an effective characterization of this equivalence for the Lambeck calculiL andLP, and for the multiplicative fragments of Girard's and (...) Yetter's linear logics. Moreover, for the non-directed Lambek calculusLP and the multiplicative fragment of Girard's linear logic, we present linear time algorithms deciding whether two types are equal, and finding a join for them if they are. (shrink)

One way of coping with agreement of features in French is to perform two parallel computations, one in the free pregroup of syntactic types, the other in that of feature types. Technically speaking, this amounts to working in the direct product of two free pregroups.

Models for the Lambek calculus of syntactic categories surveyed here are based on frames that are in principle of the same type as Kripke frames for intuitionistic logic. These models are extracted from the literature on models for relevant logics, in particular the ternary relationed models introduced in the early seventies. The purpose of this brief survey is to locate some open completeness problems for variants of the Lambek calculus in the context of completeness results (...) based on various types of ternary relational models. (shrink)

As an undergraduate I was taught to multiply two numbers with the help of log tables, using the formulaHaving graduated to teach calculus to Engineers, I learned that log tables were to be replaced by slide rules. It was then that Imade the fateful decision that there was no need for me to learn how to use this tedious device, as I could always rely on the students to perform the necessary computations. In the course of time, slide rules (...) were replaced by pocket calculators and personal computers, but I stuck to my original decision.My computer phobia did not prevent me from taking an interest in the theoretical question: what is a computation? This question goes back to Hilbert's 10th problem, which asks for an algorithm, or computational procedure, to decide whether any given polynomial equation is solvable in integers. It quickly leads to the related question: which numerical functionsf: ℕn→ ℕ are computable?While Hilbert's 10th problem was only resolved in 1970, this related question had some earlier answers, of which I shall single out the following three:f is recursive,f is computable on an abstract machine,f is definable in the untyped λ-calculus.These tentative answers were shown to be equivalent by Church [1936] and Turing [1936–7].I shall discuss here some of the more recent developments of these notions of computability and their relevance to linguistics and logic. I hope to be forgiven for dwelling on some of the work I have been involved with personally, with greater emphasis than is justified by its historical significance. (shrink)

This paper develops an inference system for natural language within the ‘Natural Logic’ paradigm as advocated by van Benthem, Sánchez and others. The system that we propose is based on the Lambek calculus and works directly on the Curry-Howard counterparts for syntactic representations of natural language, with no intermediate translation to logical formulae. The Lambek -based system we propose extends the system by Fyodorov et~al., which is based on the Ajdukiewicz/Bar-Hillel calculus Bar Hillel,. This enables (...) the system to deal with new kinds of inferences, involving relative clauses, non-constituent coordination, and meaning postulates that involve complex expressions. Basing the system on the Lambek calculus leads to problems with non-normalized proof terms, which are treated by using normalization axioms. (shrink)

In [4], I proved that the product-free fragment L of Lambek's syntactic calculus (cf. Lambek [2]) is not finitely axiomatizable if the only rule of inference admitted is Lambek's cut-rule. The proof (which is rather complicated and roundabout) was subsequently adapted by Kandulski [1] to the non-associative variant NL of L (cf. Lambek [3]). It turns out, however, that there exists an extremely simple method of non-finite-axiomatizability proofs which works uniformly for different subsystems of (...) L (in particular, for NL). We present it below to the use of those who refer to the results of [1] and [4]. (shrink)

In [2], Bar-Hillel, Gaifman, and Shamir prove that the simple phrase structure grammars dened by Chomsky are equivalent in a cer- tain sense to Bar-Hillel's bidirectional categorial grammars . On the other hand, Cohen [3] proves the equivalence of the latter ones to what he calls free categorial grammars . They are closely related to Lambek's syntactic calculus which is, in turn, based on the idea due to Ajdukiewicz [1]. For some reasons, Cohen's proof seems to be (...) at least in- complete. Here, I give a short outline of the direct proof of the equivalence between SF G's and F CG's. (shrink)

Nonassociative Lambek Calculus (NL) is a syntactic calculus of types introduced by Lambek [8]. The polynomial time decidability of NL was established by de Groote and Lamarche [4]. Buszkowski [3] showed that systems of NL with finitely many assumptions are decidable in polynomial time and generate context-free languages; actually the P-TIME complexity is established for the consequence relation of NL. Adapting the method of Buszkowski [3] we prove an analogous result for Nonassociative Lambek (...) class='Hi'>Calculus with unit (NL1). Moreover, we show that any Lambek grammar based on NL1 (with assumptions) can be transformed into an equivalent context-free grammar in polynomial time. (shrink)

We prove completeness for some language-theoretic models of the full Lambek calculus and its various fragments. First we consider syntactic concepts and syntactic concepts over regular languages, which provide a complete semantics for the full Lambek calculus \. We present a new semantics we call automata-theoretic, which combines languages and relations via closure operators which are based on automaton transitions. We establish the completeness of this semantics for the full Lambek calculus via (...) an isomorphism theorem for the syntactic concepts lattice of a language and a construction for the universal automaton recognizing the same language. Finally, we use automata-theoretic semantics to prove completeness of relation models of binary relations and finite relation models for the Lambek calculus without and with empty antecedents and \), thus solving a problem left open by Pentus. (shrink)

Two axiomatizations of the nonassociative and commutative Lambek syntactic calculus are given and their equivalence is proved. The first axiomatization employs Permutation as the only structural rule, the second one, with no Permutation rule, employs only unidirectional types. It is also shown that in the case of the Ajdukiewicz calculus an analogous equivalence is valid only in the case of a restricted set of formulas. Unidirectional axiomatizations are employed in order to establish the generative power of (...) categorial grammars based on the nonassociative and commutative Lambek calculus with product. Those grammars produce CF-languages of finite degree generated by CF-grammars closed with respect to permutations. (shrink)

If all dependent expressions were adjacent some variety of immediate constituent analysis would suffice for grammar, but syntactic and semantic mismatches are characteristic of natural language; indeed this is a, or the, central problem in grammar. Logical categorial grammar reduces grammar to logic: an expression is well-formed if and only if an associated sequent is a theorem of a categorial logic. The paradigmatic categorial logic is the Lambek calculus, but being a logic of concatenation the Lambek (...) calculus can only capture discontinuous dependencies when they are peripheral. In this paper we present the displacement calculus, which is a logic of intercalation as well as concatenation and which subsumes the Lambek calculus. On the empirical side, we apply the new calculus to discontinuous idioms, quantification, VP ellipsis, medial extraction, pied-piping, appositive relativisation, parentheticals, gapping, comparative subdeletion, cross-serial dependencies, reflexivization, anaphora, dative alternation, and particle shift. On the technical side, we prove that the calculus enjoys Cut-elimination. (shrink)

Focused sequent calculi are a refinement of sequent calculi, where additional side-conditions on the applicability of inference rules force the implementation of a proof search strategy. Focused cut-free proofs exhibit a special normal form that is used for defining identity of sequent calculi proofs. We introduce a novel focused display calculus fD.LG and a fully polarized algebraic semantics FP.LG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {FP.LG}$$\end{document} for Lambek–Grishin logic by generalizing the theory of multi-type calculi (...) and their algebraic semantics with heterogenous consequence relations. The calculus fD.LG has strong focalization and it is sound and complete w.r.t. FP.LG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {FP.LG}$$\end{document}. This completeness result is in a sense stronger than completeness with respect to standard polarized algebraic semantics, insofar as we do not need to quotient over proofs with consecutive applications of shifts over the same formula. We also show a number of additional results. fD.LG is sound and complete w.r.t. LG-algebras: this amounts to a semantic proof of the so-called completeness of focusing, given that the standard (display) sequent calculus for Lambek–Grishin logic is complete w.r.t. LG-algebras. fD.LG and the focused calculus fLG of Moortgat and Moot are equivalent with respect to proofs, indeed there is an effective translation from fLG-derivations to fD.LG-derivations and vice versa: this provides the link with operational semantics, given that every fLG-derivation is in a Curry–Howard correspondence with a directional λ¯μμ~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\lambda }\mu \widetilde{\mu }$$\end{document}-term. (shrink)

The Lambek-Grishin calculus is a symmetric version of categorial grammar obtained by augmenting the standard inventory of type-forming operations (product and residual left and right division) with a dual family: coproduct, left and right difference. Interaction between these two families is provided by distributivity laws. These distributivity laws have pleasant invariance properties: stability of interpretations for the Curry-Howard derivational semantics, and structure-preservation at the syntactic end. The move to symmetry thus offers novel ways of reconciling the demands (...) of natural language form and meaning. (shrink)

We present an embedding of the Lambek–Grishin calculus into an extension of the nonassociative Lambek calculus with negation. The embedding is based on the De Morgan interpretation of the dual Grishin connectives.

Several Gentzen-style syntactic type calculi with product are considered. They form a hierarchy in such a way that one calculus results from another by imposing a new condition upon the sequent-forming operation. It turns out that, at some steps of this process, two different functors collapse to a single one. For the remaining stages of the hierarchy, analogues of Wajsbergs's theorem on non-mutual-definability are proved.

Some formal properties of enriched systems of Lambek calculus with analogues of conjunction and disjunction are investigated. In particular, it is proved that the class of languages recognizable by the Lambek calculus with added intersective conjunction properly includes the class of finite intersections of context-free languages.

A formulation of Full Lambek Calculus in the framework of natural deduction is given.

We study an expansion of the Distributive Non-associative Lambek Calculus with conjugates of the Lambek product operator and residuals of those conjugates. The resulting logic is well-motivated, under-investigated and difficult to tackle. We prove completeness for some of its fragments and establish that it is decidable. Completeness of the logic is an open problem; some difficulties with applying the usual proof method are discussed.

The Distributional Compositional Categorical model is a mathematical framework that provides compositional semantics for meanings of natural language sentences. It consists of a computational procedure for constructing meanings of sentences, given their grammatical structure in terms of compositional type-logic, and given the empirically derived meanings of their words. For the particular case that the meaning of words is modelled within a distributional vector space model, its experimental predictions, derived from real large scale data, have outperformed other empirically validated methods that (...) could build vectors for a full sentence. This success can be attributed to a conceptually motivated mathematical underpinning, something which the other methods lack, by integrating qualitative compositional type-logic and quantitative modelling of meaning within a category-theoretic mathematical framework. The type-logic used in the DisCoCat model is Lambekʼs pregroup grammar. Pregroup types form a posetal compact closed category, which can be passed, in a functorial manner, on to the compact closed structure of vector spaces, linear maps and tensor product. The diagrammatic versions of the equational reasoning in compact closed categories can be interpreted as the flow of word meanings within sentences. Pregroups simplify Lambekʼs previous type-logic, the Lambek calculus. The latter and its extensions have been extensively used to formalise and reason about various linguistic phenomena. Hence, the apparent reliance of the DisCoCat on pregroups has been seen as a shortcoming. This paper addresses this concern, by pointing out that one may as well realise a functorial passage from the original type-logic of Lambek, a monoidal bi-closed category, to vector spaces, or to any other model of meaning organised within a monoidal bi-closed category. The corresponding string diagram calculus, due to Baez and Stay, now depicts the flow of word meanings, and also reflects the structure of the parse trees of the Lambek calculus. (shrink)

The new area of logic and computation is now undergoing rapid development. This has affected the social pattern of research in the area. A new topic may rise very quickly with a significant body of research around it. The community, however, cannot wait the traditional two years for a book to appear. This has given greater importance to thematic collections of papers, centred around a topic and addressing it from several points of view, usually as a result of a workshop, (...) summer school, or just a scientific initiative. Such a collection may not be as coherent as a book by one or two authors yet it is more focused than a collection of key papers on a certain topic. It is best thought of as a thematic collection, a study in the area of logic and computation. The new series Studies in Logic and Computation is intended to provide a home for such thematic collections. Substructural logics are nonclassical logics, which arose in response to problems in foundations of mathematics and logic, theoretical computer science, mathematical linguistics, and category theory. They include intuitionistic logic, relevant logic, BCK logic, linear logic, and Lambek's calculus of syntactic categories. Substructural logics differ from classical logics, and from each other, in their presuppositions about Gentzen's structural rules, although their presuppositions about the deductive role of logical constants are invariant. Substructural logics have been a subject of study for logicians during the last sixty years. Specialists have often worked in isolation, however, largely unaware of the contributions of others. This book brings together new papers by some of the most eminent authorities in these varioustraditions to produce a unified view of substructural logics. (shrink)

We present sound and complete semantics and a sequent calculus for the Lambek calculus extended with intuitionistic propositional logic.

The problem of whether Lambek Calculus is complete with respect to (w.r.t.) relational semantics, has been raised several times, cf. van Benthem (1989a) and van Benthem (1991). In this paper, we show that the answer is in the affirmative. More precisely, we will prove that that version of the Lambek Calculus which does not use the empty sequence is strongly complete w.r.t. those relational Kripke-models where the set of possible worlds,W, is a transitive binary relation, while (...) that version of the Lambek Calculus where we admit the empty sequence as the antecedent of a sequent is strongly complete w.r.t. those relational models whereW=U×U for some setU. We will also look into extendability of this completeness result to various fragments of Girard's Linear Logic as suggested in van Benthem (1991), p. 235, and investigate the connection between the Lambek Calculus and language models. (shrink)

This paper deals with some strengthenings of the non-directional product-free Lambek calculus by means of additional structural rules. In fact, the rules contraction and expansion are restricted to basic types. For each of the presented systems the usual proof-theoretic notions are discussed, some new concepts especially designed for these calculi are introduced reflecting their intermediate position between the weaker and the stronger sequent-systems.

We show that the relational semantics of the Lambek calculus, both nonassociative and associative, is also sound and complete for its extension with classical propositional logic. Then, using filtrations, we obtain the finite model property for the nonassociative Lambek calculus extended with classical propositional logic.

In [2], Bar-Hillel, Gaifman, and Shamir prove that the simple phrase structure grammars (SPGs) defined by Chomsky are equivalent in a certain sense to Bar-Hillel's bidirectional categorial grammars (BCGs). On the other hand, Cohen [3] proves the equivalence of the latter ones to what the calls free categorial grammars (FCGs). They are closely related to Lambek's syntactic calculus which, in turn, is based on the idea due to Ajdukiewicz [1]. For the reasons which will be discussed in (...) the last section, Cohen's proof seems to be at least incomplete. This paper yields a direct proof of the equivalence ofFCGs andSPGs. (shrink)

We introduce non-associative linear logic, which may be seen as the classical version of the non-associative Lambek calculus. We define its sequent calculus, its theory of proof-nets, for which we give a correctness criterion and a sequentialization theorem, and we show proof search in it is polynomial.

For some modal fixed point logics, there are deductive systems that enjoy syntactic cut-elimination. An early example is the system in Pliuskevicius [15] for LTL. More recent examples are the systems by the authors of this paper for the logic of common knowledge [5] and by Hill and Poggiolesi for PDL[8], which are based on a form of deep inference. These logics can be seen as fragments of the modal mu-calculus. Here we are interested in how far this (...) approach can be pushed in general. To this end, we introduce a nested sequent system with syntactic cut-elimination which is incomplete for the modal mu-calculus, but complete with respect to a restricted language that allows only fixed points of a certain form. This restricted language includes the logic of common knowledge and PDL. There is also a traditional sequent system for the modal mu-calculus by Jäger et al. [9], without a syntactic cut-elimination procedure. We embed that system into ours and vice versa, thus establishing cut-elimination also for the shallow system, when only the restricted language is considered. (shrink)

We prove that the Lambek calculus is complete w.r.t. L-models, i.e., free semigroup models. We also prove the completeness w.r.t. relativized relational models over the natural linear order of integers.

In this paper we prove the Chomsky Conjecture (all languages recognized by the Lambek calculus are context-free) for both the full Lambek calculus and its product-free fragment. For the latter case we present a construction of context-free grammars involving only product-free types.