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

Hybrid Type-Logical Grammars, First-Order Linear Logic and the Descriptive Inadequacy of Lambda Grammars

In this article we show that hybrid type-logical grammars are a fragment of first-order linear logic. This embedding result has several important consequences: it not only provides a simple new proof theory for the calculus, thereby clarifying the proof-theoretic foundations of hybrid type-logical grammars, but, since the translation is simple and direct, it also provides several new parsing strategies for hybrid type-logical grammars. Second, NP-completeness of hybrid type-logical grammars follows immediately. The main embedding result also sheds new light on problems with lambda grammars/abstract categorial grammars and shows lambda grammars/abstract categorial grammars suffer from problems of over-generation and from problems at the syntax-semantics interface unlike any other categorial grammar

A Labelled Analytic Theorem Proving Environment for Categorial Grammar

We present a system for the investigation of computational properties of categorial grammar parsing based on a labelled analytic tableaux theorem prover. This proof method allows us to take a modular approach, in which the basic grammar can be kept constant, while a range of categorial calculi can be captured by assigning different properties to the labelling algebra. The theorem proving strategy is particularly well suited to the treatment of categorial grammar, because it allows us to distribute the computational cost between the algorithm which deals with the grammatical types and the algebraic checker which constrains the derivation.Comment: 11 pages, LaTeX2e, uses examples.sty and a4wide.st

Complexity of Grammar Induction for Quantum Types

Most categorical models of meaning use a functor from the syntactic category to the semantic category. When semantic information is available, the problem of grammar induction can therefore be defined as finding preimages of the semantic types under this forgetful functor, lifting the information flow from the semantic level to a valid reduction at the syntactic level. We study the complexity of grammar induction, and show that for a variety of type systems, including pivotal and compact closed categories, the grammar induction problem is NP-complete. Our approach could be extended to linguistic type systems such as autonomous or bi-closed categories.Comment: In Proceedings QPL 2014, arXiv:1412.810

A Logic-based Approach for Recognizing Textual Entailment Supported by Ontological Background Knowledge

We present the architecture and the evaluation of a new system for recognizing textual entailment (RTE). In RTE we want to identify automatically the type of a logical relation between two input texts. In particular, we are interested in proving the existence of an entailment between them. We conceive our system as a modular environment allowing for a high-coverage syntactic and semantic text analysis combined with logical inference. For the syntactic and semantic analysis we combine a deep semantic analysis with a shallow one supported by statistical models in order to increase the quality and the accuracy of results. For RTE we use logical inference of first-order employing model-theoretic techniques and automated reasoning tools. The inference is supported with problem-relevant background knowledge extracted automatically and on demand from external sources like, e.g., WordNet, YAGO, and OpenCyc, or other, more experimental sources with, e.g., manually defined presupposition resolutions, or with axiomatized general and common sense knowledge. The results show that fine-grained and consistent knowledge coming from diverse sources is a necessary condition determining the correctness and traceability of results.Comment: 25 pages, 10 figure