Comparing and evaluating extended Lambek calculi

Lambeks Syntactic Calculus, commonly referred to as the Lambek calculus, was innovative in many ways, notably as a precursor of linear logic. But it also showed that we could treat our grammatical framework as a logic (as opposed to a logical theory). However, though it was successful in giving at least a basic treatment of many linguistic phenomena, it was also clear that a slightly more expressive logical calculus was needed for many other cases. Therefore, many extensions and variants of the Lambek calculus have been proposed, since the eighties and up until the present day. As a result, there is now a large class of calculi, each with its own empirical successes and theoretical results, but also each with its own logical primitives. This raises the question: how do we compare and evaluate these different logical formalisms? To answer this question, I present two unifying frameworks for these extended Lambek calculi. Both are proof net calculi with graph contraction criteria. The first calculus is a very general system: you specify the structure of your sequents and it gives you the connectives and contractions which correspond to it. The calculus can be extended with structural rules, which translate directly into graph rewrite rules. The second calculus is first-order (multiplicative intuitionistic) linear logic, which turns out to have several other, independently proposed extensions of the Lambek calculus as fragments. I will illustrate the use of each calculus in building bridges between analyses proposed in different frameworks, in highlighting differences and in helping to identify problems.Comment: Empirical advances in categorial grammars, Aug 2015, Barcelona, Spain. 201

Dialectica Categories for the Lambek Calculus

We revisit the old work of de Paiva on the models of the Lambek Calculus in dialectica models making sure that the syntactic details that were sketchy on the first version got completed and verified. We extend the Lambek Calculus with a \kappa modality, inspired by Yetter's work, which makes the calculus commutative. Then we add the of-course modality !, as Girard did, to re-introduce weakening and contraction for all formulas and get back the full power of intuitionistic and classical logic. We also present the categorical semantics, proved sound and complete. Finally we show the traditional properties of type systems, like subject reduction, the Church-Rosser theorem and normalization for the calculi of extended modalities, which we did not have before

Lambek vs. Lambek: Functorial Vector Space Semantics and String Diagrams for Lambek Calculus

The Distributional Compositional Categorical (DisCoCat) 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, 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 Lambek's 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 Lambek's previous type-logic, the Lambek calculus, which has been extensively used to formalise and reason about various linguistic phenomena. 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.Comment: 29 pages, pending publication in Annals of Pure and Applied Logi

Dual-Context Calculi for Modal Logic

We present natural deduction systems and associated modal lambda calculi for the necessity fragments of the normal modal logics K, T, K4, GL and S4. These systems are in the dual-context style: they feature two distinct zones of assumptions, one of which can be thought as modal, and the other as intuitionistic. We show that these calculi have their roots in in sequent calculi. We then investigate their metatheory, equip them with a confluent and strongly normalizing notion of reduction, and show that they coincide with the usual Hilbert systems up to provability. Finally, we investigate a categorical semantics which interprets the modality as a product-preserving functor.Comment: Full version of article previously presented at LICS 2017 (see arXiv:1602.04860v4 or doi: 10.1109/LICS.2017.8005089