Bracket induction for Lambek calculus with bracket modalities

Relativisation involves dependencies which, although unbounded, are constrained with respect to certain island domains. The Lambek calculus L can provide a very rudimentary account of relativisation limited to unbounded peripheral extraction; the Lambek calculus with bracket modalities Lb can further condition this account according to island domains. However in naïve parsing/theorem-proving by backward chaining sequent proof search for Lb the bracketed island domains, which can be indefinitely nested, have to be specified in the linguistic input. In realistic parsing word order is given but such hierarchical bracketing structure cannot be assumed to be given. In this paper we show how parsing can be realised which induces the bracketing structure in backward chaining sequent proof search with Lb

On the generative capacity of multi-modal Categorial Grammars

In Moortgat 1996 the Lambek Calculus L (Lambek 1958) is extended by a pair of residuation modalities ◊ and □↓. Categorial Grammars based on the resulting logic L◊ are attractive for linguistic purposes since they offer a compromise between the strict constituent structures imposed by context free grammars and related formalisms on the one hand, and the complete absence of hierarchical information in Lambek grammars on the other hand. The paper contains some results on the generative capcity of Categorial Grammars based on L◊. First it is shown that adding residuation modalities does not extend the weak generative capacity. This is proved by extending the proof for the context freeness of L-grammars from Pentus 1993 to L◊. Second the strong generative capacity of L◊-grammars is compared to context free grammars. The results are mainly negative. The set of tree languages generated by L◊-grammars neither contains nor is contained in the class of context free tree languages

A Polynomial-Time Algorithm for the Lambek Calculus with Brackets of Bounded Order

Lambek calculus is a logical foundation of categorial grammar, a linguistic paradigm of grammar as logic and parsing as deduction. Pentus (2010) gave a polynomial-time algorithm for determining provability of bounded depth formulas in L*, the Lambek calculus with empty antecedents allowed. Pentus\u27 algorithm is based on tabularisation of proof nets. Lambek calculus with brackets is a conservative extension of Lambek calculus with bracket modalities, suitable for the modeling of syntactical domains. In this paper we give an algorithm for provability in Lb*, the Lambek calculus with brackets allowing empty antecedents. Our algorithm runs in polynomial time when both the formula depth and the bracket nesting depth are bounded. It combines a Pentus-style tabularisation of proof nets with an automata-theoretic treatment of bracketing

Parsing/theorem-proving for logical grammar CatLog3

CatLog3 is a 7000 line Prolog parser/theorem-prover for logical categorial grammar. In such logical categorial grammar syntax is universal and grammar is reduced to logic: an expression is grammatical if and only if an associated logical statement is a theorem of a fixed calculus. Since the syntactic component is invariant, being the logic of the calculus, logical categorial grammar is purely lexicalist and a particular language model is defined by just a lexical dictionary. The foundational logic of continuity was established by Lambek (Am Math Mon 65:154–170, 1958) (the Lambek calculus) while a corresponding extension including also logic of discontinuity was established by Morrill and Valentín (Linguist Anal 36(1–4):167–192, 2010) (the displacement calculus). CatLog3 implements a logic including as primitive connectives the continuous (concatenation) and discontinuous (intercalation) connectives of the displacement calculus, additives, 1st order quantifiers, normal modalities, bracket modalities, and universal and existential subexponentials. In this paper we review the rules of inference for these primitive connectives and their linguistic applications, and we survey the principles of Andreoli’s focusing, and of a generalisation of van Benthem’s count-invariance, on the basis of which CatLog3 is implemented.Peer ReviewedPostprint (author's final draft

Grammar logicised: relativisation

Many variants of categorial grammar assume an underlying logic which is associative and linear. In relation to left extraction, the former property is challenged by island domains, which involve nonassociativity, and the latter property is challenged by parasitic gaps, which involve nonlinearity. We present a version of type logical grammar including ‘structural inhibition’ for nonassociativity and ‘structural facilitation’ for nonlinearity and we give an account of relativisation including islands and parasitic gaps and their interaction.Peer ReviewedPostprint (published version

Non-associative, Non-commutative Multi-modal Linear Logic

Adding multi-modalities (called subexponentials) to linear logic enhances its power as a logical framework, which has been extensively used in the specification of e.g. proof systems, programming languages and bigraphs. Initially, subexponentials allowed for classical, linear, affine or relevant behaviors. Recently, this framework was enhanced so to allow for commutativity as well. In this work, we close the cycle by considering associativity. We show that the resulting system (acLLΣ ) admits the (multi)cut rule, and we prove two undecidability results for fragments/variations of acLLΣ