Natasha Kurtonina

Simulating without Negation

Although negation-free languages are widely used in logic and computer science, relatively little is known about their expressive power. To address this issue we consider kinds of non-symmetric bisimulations called directed simulations, and use these to analyse the expressive power and model theory of negation-free modal and temporal languages. We first use them to obtain preservation, safety and definability results for a simple negation-fret modal language. We then obtain analogous results for stronger negation-free languages. Finally, we extend our methods to deal with languages with non-Boolean negation.

Categorial Inference and Modal Logic

This paper establishes a connection between structure sensitive categorial inference and classical modal logic. The embedding theorems for non-associative Lambek Calculus and the whole class of its weak Sahlqvist extensions demonstrate that various resource sensitive regimes can be modelled within the framework of unimodal temporal logic. On the semantic side, this requires decomposition of the ternary accessibility relation to provide its correlation with standard binary Kripke frames and models.

An LTAG Perspective on Categorial Inference

This paper describes a system of categorial inference based on insights from Lexicalized Tree Adjoining Grammar (LTAG). LTAG is a tree-rewriting system and therefore deals with structural, not string, adjacency. When looked at from the logic perspective, the nodes of the trees become types as in a categorial grammar, with corresponding deductive connections between parent and daughter nodes. The resulting system is based on a hybrid logic, with one logic for building Partial Proof Trees, and the other for composing the partial proofs. We reexamine the use of structural modalities in categorial grammar from this perspective, concluding that the use of structural modalities can be considerably simplified, or even eliminated in some cases. The generative power of the hybrid logic system is beyond context-free, as we demonstrate with a derivation of the cross-serial dependencies in Dutch. The system also inherits polynomial parsing from LTAG.