Categorial Grammar Research Articles

Complexity of the Lambek Calculus with One Division and a Negative-Polarity Modality for Weakening

In this paper, we consider a variant of the Lambek calculus allowing empty antecedents. This variant uses two connectives: the left division and a unary modality that occurs only with negative polarity and allows weakening in antecedents of sequents. We define the notion of a proof net for this calculus, which is similar to those for the ordinary Lambek calculus and multiplicative linear logic. We prove that a sequent is derivable in the calculus under consideration if and only if there exists a proof net for it. We present a polynomial-time algorithm for deciding whether an arbitrary given sequent is derivable in this calculus.

Some results on relation algebra reducts: Residuated and semilattice-ordered semigroups

Abstract In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is representable over a finite base. This result gives a positive solution to Hirsch and Hodkinson (2002, Relation Algebras by Games). The finite representation property for residuated semigroups also implies that the Lambek calculus has the finite model property with respect to relational models, the so-called $R$-models. We also show that the class of representable join semilattice-ordered semigroups is pseudo-universal and it has a recursively enumerable axiomatization. For this purpose, we introduce representability games for join semilattice-ordered semigroups.

Versions of a Local Contraction Subexponential in the Lambek Calculus

Versions of a Local Contraction Subexponential in the Lambek Calculus

Coordination of unlike grammatical cases (and unlike categories)

It is often claimed that conjuncts in coordinate structures must be alike in various ways, in particular, that they should have the same syntactic category and the same grammatical case, if any. This article aims to refute such claims. On the basis of data from Polish, Estonian, and other languages, it demonstrates that there is no universal requirement that conjuncts be alike. Any appearances of such a requirement result from the fact that each conjunct must satisfy all functional constraints on the coordinate structure. The article discusses ways of formalizing such distributive satisfaction of constraints within four major linguistic frameworks: lexical-functional grammar, categorial grammar, head-driven phrase structure grammar, and minimalism.

Tractable Parsing for CCGs of Bounded Degree

Abstract Unlike other mildly context-sensitive formalisms, Combinatory Categorial Grammar (CCG) cannot be parsed in polynomial time when the size of the grammar is taken into account. Refining this result, we show that the parsing complexity of CCG is exponential only in the maximum degree of composition. When that degree is fixed, parsing can be carried out in polynomial time. Our finding is interesting from a linguistic perspective because a bounded degree of composition has been suggested as a universal constraint on natural language grammar. Moreover, ours is the first complexity result for a version of CCG that includes substitution rules, which are used in practical grammars but have been ignored in theoretical work.

Ambiguous Definite Pronoun Resolution via Lambek Calculus with Soft Sub-Exponentials and Machine Learning

Ambiguous Definite Pronoun Resolution via Lambek Calculus with Soft Sub-Exponentials and Machine Learning

Hypergraph Lambek grammars

In the graph grammar theory, there is a natural generalization of the context-free grammar approach to graphs called hyperedge replacement grammar (HRG). The latter is based on the hyperedge replacement procedure, and it preserves the main principles and properties of context-free grammars (e.g. the pumping lemma). In the string case, there is an alternative approach to describing formal languages, which was introduced by J. Lambek; it is based on a substructural logic called the Lambek calculus. In this paper, we generalize this calculus and this grammar approach to hypergraphs, which results in defining the hypergraph Lambek calculus (HL) and hypergraph Lambek grammars (HL-grammars). We show how to embed the Lambek calculus in HL justifying that the presented generalization is appropriate. After that, we study several structural properties of HL and turn to the question of expressive power of its grammars. It appears that HL-grammars are more powerful than HRGs while having the same algorithmic complexity with them; in particular, we show that intersection of a language generated by an HRG and of a language generated by an HL-grammar can be generated by some HL-grammar as well. Consequently, HL-grammars might be considered as enhancement of HRGs.

A categorial grammar of Spanish auxiliary chains

Spanish auxiliary sequences as in Juan puede haber tenido que estar empezando a trabajar hasta tarde ‘Juan may have had to be starting to work until late’, traditionally termed auxiliary chains, have two properties that are not naturally captured in phrase-structure approaches to syntax: (i) they follow no a priori fixed order; auxiliary permutations have different meanings, none of which is any more basic than any other (cf. Juan puede estar trabajando ‘Juan may be working’ and Juan está pudiendo trabajar ‘Juan is currently able to work’); and (ii) the syntactic and semantic relations established within a chain go beyond strict monotonicity or cumulative influence; rather, they present different kinds of syntactic relations in distinct local domains. We show that an alternative to syntax grounded in a modification of the categorial grammar introduced in Ajdukiewicz (1935) that closely follows Montague (1973), Dowty (1978, 1979, 2003), and Schmerling (1983a, b, 2019) provides effective tools for subsuming Spanish auxiliary chains in an explicit and explanatory grammar.

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).

A van Benthem Theorem for Atomic and Molecular Logics

After recalling the definitions of atomic and molecular logics, we show how notions of bisimulation can be automatically defined from the truth conditions of the connectives of any of these logics. Then, we prove a generalization of van Benthem modal characterization theorem for molecular logics. Our molecular connectives should be uniform and contain all conjunctions and disjunctions. We use modal logic, the Lambek calculus and modal intuitionistic logic as case study and compare in particular our work with Olkhovikov's work.

Proof Nets for the Lambek Calculus with One Division and a Negative-Polarity Modality for Weakening

In this paper, we introduce a variant of the Lambek calculus allowing empty antecedents. This variant uses two connectives: the left division and a unary modality that occurs only with negative polarity and allows weakening in antecedents of sequents. We define the notion of a proof net for this calculus, which is similar to those for the ordinary Lambek calculus and multiplicative linear logic. We prove that a sequent is derivable in the calculus under consideration if and only if there exists a proof net for it. Thus, we establish a derivability criterion for this calculus in terms of the existence of a graph with certain properties. The size of the graph is bounded by the length of the sequent.

A fixed-point problem for theories of meaning

In this paper I argue that it’s impossible for there to be a single universal theory of meaning for a language. First, I will consider some minimal expressiveness requirements a language must meet to be able to express semantic claims. Then I will argue that in order to have a single unified theory of meaning, these expressiveness requirements must be satisfied by a language which the semantic theory itself applies to. That is, we would need a language which can express its own meaning. It has been well-known since Tarski that theories of meaning whose central notion is truth can’t be expressed in a language which they apply to. Here, I develop Quine’s formulation of the Liar Paradox in grammatical terms and use this to extend Tarski’s result to all theories of meaning. This general version of the paradox can be formalised as a special case of the Lawvere Fixed-Point Theorem applied to a categorial grammar. Taken together with the initial arguments, I infer that a universal theory of meaning is impossible and conclude the paper with a brief discussion on what alternatives are available.

Grammars Based on a Logic of Hypergraph Languages

The hyperedge replacement grammar (HRG) formalism is a natural and well-known generalization of context-free grammars. HRGs inherit a number of properties of context-free grammars, e.g. the pumping lemma. This lemma turns out to be a strong restriction in the hypergraph case: it implies that languages of unbounded connectivity cannot be generated by HRGs. We introduce a formalism that turns out to be more powerful than HRGs while having the same algorithmic complexity (NP-complete). Namely, we introduce hypergraph Lambek grammars; they are based on the hypergraph Lambek calculus, which may be considered as a logic of hypergraph languages. We explain the underlying principles of hypergraph Lambek grammars, establish their basic properties, and show some languages of unbounded connectivity that can be generated by them (e.g. the language of all graphs, the language of all bipartite graphs, the language of all regular graphs).

Proof‐theoretic semantics of natural deduction based on inversion

AbstractThe article presents a full proof‐theoretic semantics for natural deduction based on an extended inversion principle: the elimination rule for an operator q may invert the introduction rule for q, but also vice versa, the introduction rule for a connective q may invert the elimination rule for q. Such an inversion—extending Prawitz' concept of inversion—gives the following theorem: Inversion for two rules of operator q (intro rule, elim rule) exists iff a reduction of a maximum formula for q exists. The inversion theorem is specified to two logics, Lambek calculus (LC) and intuitionistic linear logic (ILL), with four propositional connectives: two multiplicatives (implication and conjunction) and two additives (conjunction and disjunction), with two quantifiers and two modals. LC is defined by using elimination rules by composition. ILL is defined by using usual general elimination rules. Elimination rules by composition are an exciting alternative to general elimination rules; in some cases, they do not need permutations.

Complexity of Lambek calculi with modalities and of total derivability in grammars

Complexity of Lambek calculi with modalities and of total derivability in grammars

The tree-generative capacity of combinatory categorial grammars

The generative capacity of combinatory categorial grammars (CCGs) as generators of tree languages is investigated. It is demonstrated that the tree languages generated by CCGs can also be generated by simple monadic context-free tree grammars. However, the important subclass of pure combinatory categorial grammars cannot even generate all regular tree languages. Additionally, the tree languages generated by combinatory categorial grammars with limited rule degrees are characterized: If only application rules are allowed, then these grammars can generate only a proper subset of the regular tree languages, whereas they can generate exactly the regular tree languages once first-degree composition rules are permitted.

Complexity of Lambek Calculi with Modalities and of Total Derivability in Grammars

Complexity of Lambek Calculi with Modalities and of Total Derivability in Grammars

Infinitary action logic with exponentiation

We introduce infinitary action logic with exponentiation—that is, the multiplicative-additive Lambek calculus extended with Kleene star and with a family of subexponential modalities, which allow some of the structural rules (contraction, weakening, permutation). The logic is presented in the form of an infinitary sequent calculus. We prove cut elimination and, in the case where at least one subexponential allows non-local contraction, establish exact complexity boundaries in two senses. First, we show that the derivability problem for this logic is Π11-complete. Second, we show that the closure ordinal of its derivability operator is ω1CK. In the case where no subexponential allows contraction, we show that complexity is the same as for infinitary action logic itself. Namely, the derivability problem in this case is Π10-complete and the closure ordinal is not greater than ωω.

On embedding Lambek calculus into commutative categorial grammars

AbstractWe consider tensor grammars, which are an example of ‘commutative’ grammars, based on the classical (rather than intuitionistic) linear logic. They can be seen as a surface representation of abstract categorial grammars (ACG) in the sense that derivations of ACG translate to derivations of tensor grammars and this translation is isomorphic on the level of string languages. The basic ingredients are tensor terms, which can be seen as encoding and generalizing proof nets. Using tensor terms makes the syntax extremely simple and a direct geometric meaning becomes transparent. Then we address the problem of encoding noncommutative operations in our setting. This turns out possible after enriching the system with new unary operators. The resulting system allows representing both ACG and Lambek grammars as conservative fragments, while the formalism remains, as it seems to us, rather simple and intuitive.

Case alternation in lexicalized grammar

In this paper, we propose to extend the Przepiórkowski's 2000 analysis of Long Distance Genitive of Negation to the same phenomenon in Lithuanian. We discuss the features that have their origin in Categorial Grammar. We then develop a novel analysis of the case alternation in Categorial Grammar incorporating features of the HPSG analysis. The two accounts show a surprising convergence in basic assumptions and predictions.