Syntactic Calculus Research Articles

Normalization by Evaluation for the Lambek Calculus

The syntactic calculus of Lambek is a deductive system for the multiplicative fragment of intuitionistic non-commutative linear logic. As a fine-grained calculus of resources, it has many applications, mostly in formal computational investigations of natural languages. This paper introduces a calculus of beta-eta-long normal forms for derivations in the Lambek calculus with multiplicative unit and conjunction among its logical connectives. Reduction to normal form follows the normalization by evaluation (NbE) strategy: (i) evaluate a derivation in a Kripke model of Lambek calculus; (ii) extract normal forms from the obtained semantic values. The implementation of the NbE algorithm requires the presence of a strong monad in the Kripke interpretation of positive formulae, in analogy with the extension of intuitionistic propositional logic with falsity and disjunction. The NbE algorithm can also be instantiated with minor variations to calculi for related substructural logics, such as multiplicative and dual intuitionistic linear logic (MILL and DILL).

Command injection attacks, continuations, and the Lambek calculus

This paper shows connections between command injection attacks, continuations, and the Lambek calculus: certain command injections, such as the tautology attack on SQL, are shown to be a form of control effect that can be typed using the Lambek calculus, generalizing the double-negation typing of continuations. Lambek's syntactic calculus is a logic with two implicational connectives taking their arguments from the left and right, respectively. These connectives describe how strings interact with their left and right contexts when building up syntactic structures. The calculus is a form of propositional logic without structural rules, and so a forerunner of substructural logics like Linear Logic and Separation Logic.

From linguistics to deontic logic via category theory

The present paper aims to bridge the gap between deontic logic, categorial grammar and category theory. We propose to analyze Forrester's (1984) paradox through the framework of Lambek's (1958) syntactic calculus. We first recall the definition of the syntactic calculus and then explain how Lambek (1988) defines it within the framework of category theory. Then, we briefly present Forrester's paradox in conjunction with standard deontic logic, showing that this paradox contains some features that reflect many problems within the literature. Finally, we analyze Forrester's paradox within the framework of the syntactic calculus and we show how a typed syntax can provide conceptual insight regarding some of the problems that deontic logic faces.

On the Complexity of Nonassociative Lambek Calculus with Unit

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 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.

Categorial Grammars and Tree Languages

We consider a representation of the structure of derivations in Ajdukiewicz and Lambek categorial grammars called derived trees. We define derived tree languages of those grammars based on syntactic calculi formalized in Gentzen style and in natural deduction format and examine their relation to the families of local and regular tree languages.

La logique linéaire non commutative et le calcul de Lambek

The paper analyzes the correspondence existing between the Syntactic Calculus (Lambek 1958) and (multiplicative) non-commutative linear logic (Abrusci 1991), both in its intuitionistic and classical formulation. Particular attention is paid to the fragment of classical non-commutative linear logic (or classical bilinear logic) that is a conservative extension of the Syntactic Calculus. We present some linguistic applications of this system and propose a way to build up planar graphs for its type logical formulas.

A tale of four grammars

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.

Non-Commutative Linear Logic in Linguistics

This paper explores the linguistic implications of Non-commutative Linear Logic, with particular reference to its multiplicative fragment MNLL, that exhibits a direct relationship to Lambek's Syntactic Calculus. Such a framework is appealing for linguistic analysis since it allows one to develop a dynamic characterization of the notion of a function, that plays a basic role in the foundations of categorial grammar. The analysis will focus on a variety of constructions involving scope configurations, unbounded dependencies and Wh-clauses. Particular attention is given to the proof nets for MNLL, that are planar graphs in which the communication processes and the flow of information are represented by means of a parallelistic architecture. We will introduce proof nets and sequent derivations associated to each linguistic expression and will show that a direct relationship exists between the types and derivations of the Syntactic Calculus and the corresponding types and derivations in MNLL. Moreover, given the symmetric architecture and the crucial role played by the two negations of Non-commutative Linear Logic in the generation of the logical types, we will show that this system is richer in expressive power and in the capacity of performing left-to-right computations of word strings.

De l’ubiquité de l’ordre dans les premières grammaires catégorielles

Résumé Parmi les différences qui opposent les deux grands courants de grammaires formelles du XXe siècle, celles qui concernent le traitement de l’ordre des mots sont restées jusqu’à aujourd’hui peu traitées. Pourtant, les premières grammaires catégorielles (de 1935 à 1959) présentent une conception de l’ordre des mots qui tranche de celle mise en oeuvre dans le modèle trans-formationnel chomskien (1957) et leur originalité sur ce point mérite l’attention. La syntaxe d’Ajdukiewicz (1935), la grammaire bidirectionnelle de Bar-Hillel (1953) et le calcul syntaxique associatif de Lambek (1958, 1959) s’accordent en effet à gérer l’ordre en le faisant intervenir simultanément à tous les niveaux d’analyse. L’article précise le traitement de l’ordre des mots dans ces trois premières grammaires catégorielles et les compare entre elles de manière à dégager la portée qu’ont eu à un niveau conceptuel les modifications que ces différents auteurs ont introduites dans la notation de l’ordre.

Schur complements obey Lambek's categorial grammar: Another view of Gaussian elimination and LU decomposition

For three decades Schur complements have seen increasing applications in linear algebra, often as abstractions of Gaussian elimination. It is known that they obey certain nontrivial identities, such as Crabtree and Haynsworth's quotient property. We began this work asking if there were a theory for deciding their properties in general. Lambek's Categorial Grammar is a deductive system formalized in 1958 by Lambek as a mathematical foundation for a syntactic calculus of language. We show that Categorial Grammar gives a deductive system for deriving identities obeyed by LU-and UL-decompositions, Gaussian elimination, and Schur complements. At first impression this seems to be a strange result, connecting two unrelated topics. In retrospect, though, it is a consequence of the way both use quotients. It may have applications in developing grammatical formalisms and numerical algorithms.

Syntactic Calculus with Dependent Types

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 Π types of constructive type theory. Two alternative formalizations are given: one using syntax trees, like Montague grammar, and one dispensing with them, like the theory called minimalistic by Morrill. The syntax tree approach is presented as the main alternative, because it makes it possible to embed the calculus in a more extensive Montague-style grammar.

A Concrete Categorical Model for the Lambek Syntactic Calculus

AbstractWe present a categorical/denotational semantics for the Lambek Syntactic Calculus (LSC), 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.

Simulating Computations in Second Order Non-Commutative Linear Logic (Preliminary Report)

Synopsis Lincoln, Scedrov and Shankar proved undecidability of intuitionistic second order multiplicative commutative linear logic by embedding LJ2 into this logic. Emms did the same for intuitionistic second order non-commutative linear logic. Recently, Lafont and Scedrov demonstrated undecidability of classical second order multiplicative commutative linear logic. As for classical second order non-commutative linear logic, its decidability problem remained open.Here we present a direct and natural encoding of arbitrary machine computations in minimal fragments of the second order non-commutative linear logic (for instance, in the product-free Lambek syntactic calculus enriched by only one second order quantifier ∀), and prove the correctness and faithfulness of this encoding with respect to any second order system up to the full classical second order cyclic linear logic.It is thereby proved that any reasonable versions of the second order non-commutative linear logic are undecidable.

On Commutative and Nonassociative Syntactic Calculi and Categorial Grammars

AbstractTwo 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.

A comparison between lambek syntactic calculus and intuitionistic linear propositional logic

Mathematical Logic QuarterlyVolume 36, Issue 1 p. 11-15 Article A comparison between lambek syntactic calculus and intuitionistic linear propositional logic V. Michele Abrusci, V. Michele Abrusci Dipartimento di Scienze Filosofiche Università di Bari Palazzo Ateneo 70121 Bari ItalySearch for more papers by this author V. Michele Abrusci, V. Michele Abrusci Dipartimento di Scienze Filosofiche Università di Bari Palazzo Ateneo 70121 Bari ItalySearch for more papers by this author First published: 1990 https://doi.org/10.1002/malq.19900360103Citations: 24AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume36, Issue11990Pages 11-15 RelatedInformation

A simple and general method of solving the finite axiomatizability problems for Lambek's syntactic calculi

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].

A direct proof of the equivalence of free categorial grammars and simple phrase structure grammars

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.

Axiomatic Syntax: The Construction and Evaluation of a Syntactic Calculus

This paper discusses methods for presenting syntactic information in the form of a calculus, and for measuring its goodness of fit to a language.' Given a morpheme list for a language, the aim of a syntactic system is to tell us how to put together the sequences of morphemes which are used as sentences in the language. Such directions we shall call the FORMATION RULES. Additionally, works on syntax usually give a certain amount of information about the equivalences between some sequences and others, e.g. that John discovered the path = The path was discovered by John. We shall call such statements TRANSFORMATION RULES. Only a small part of the transformation rules is covered in syntax; some others are discussed in mathematical logic, e.g. Russell's theory of descriptions, procedures of generalization and abstraction. So far in mathematical logic, most attention has been directed to developing formation and transformation rules of artificial languages, and no complete treatment has yet been made of the transformation rules of a natural language. The division between formation and transformation rules can to some extent be altered by altering the units of the system. We shall concentrate on syntax as formation rules, and give a procedure for putting this material into a precise form so that it may be evaluated.2 Let us consider the syntax of a language L, say that of the English statement-type sentences, which may be constructed out of some finite word list. (For a con-