Proof nets for display logic

This paper explores several extensions of proof nets for the Lambek calculus in order to handle the different connectives of display logic in a natural way. The new proof net calculus handles some recent additions to the Lambek vocabulary such as Galois connections and Grishin interactions. It concludes with an exploration of the generative capacity of the Lambek-Grishin calculus, presenting an embedding of lexicalized tree adjoining grammars into the Lambek-Grishin calculus

Learnability of type-logical grammars

AbstractA procedure for learning a lexical assignment together with a system of syntactic and semantic categories given a fixed type-logical grammar is briefly described. The logic underlying the grammar can be any cut-free decidable modally enriched extension of the Lambek calculus, but the correspondence between syntactic and semantic categories must be constrained so that no infinite set of categories is ultimately used to generate the language. It is shown that under these conditions various linguistically valuable subsets of the range of the algorithm are classes identifiable in the limit from data consisting of sentences labeled by simply typed lambda calculus meaning terms in normal form. The entire range of the algorithm is shown to be not a learnable class, contrary to a mistaken result reported in a preliminary version of this paper. It is informally argued that, given the right type logic, the learnable classes of grammars include members which generate natural languages, and thus that natural languages are learnable in this way

Parsing Corpus-Induced Type-Logical Grammars

International audienceType-logical grammars which have been automatically extracted from linguistic corpora provide parsers for these grammars with a considerable challenge. The size of the lexicon and the combinatory possibilities of the lexical entries both call for rethinking of the traditional type-logical parsing strategies. We show how methods from statistical natural language processing can be incorporated into a type-logical parser, give some preliminary data and sketch some new experiments we expect to produce better results

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

The Grail theorem prover: Type theory for syntax and semantics

As the name suggests, type-logical grammars are a grammar formalism based on logic and type theory. From the prespective of grammar design, type-logical grammars develop the syntactic and semantic aspects of linguistic phenomena hand-in-hand, letting the desired semantics of an expression inform the syntactic type and vice versa. Prototypical examples of the successful application of type-logical grammars to the syntax-semantics interface include coordination, quantifier scope and extraction.This chapter describes the Grail theorem prover, a series of tools for designing and testing grammars in various modern type-logical grammars which functions as a tool . All tools described in this chapter are freely available

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

Perspectives on neural proof nets

In this paper I will present a novel way of combining proof net proof search with neural networks. It contrasts with the 'standard' approach which has been applied to proof search in type-logical grammars in various different forms. In the standard approach, we first transform words to formulas (supertagging) then match atomic formulas to obtain a proof. I will introduce an alternative way to split the task into two: first, we generate the graph structure in a way which guarantees it corresponds to a lambda-term, then we obtain the detailed structure using vertex labelling. Vertex labelling is a well-studied task in graph neural networks, and different ways of implementing graph generation using neural networks will be explored.Comment: This is an extended version of an invited talk for the workshop End-to-End Compositional Models of Vector-Based Semantic