A Proof Search System for a Modal Substructural Logic Based on Labelled Deductive Systems

This paper describes a proof search system for a non{classical logic (modal concatenation (substructural) logic) based on Gabbay's Labelled Deductive System (LDS). The logic concerned is treated as a case study. It has some unusual features which combine resource (linear, Lambek Calculus or relevance logics) with modality (intensional, temporal, or epistemic logics), and may have some useful applications in AI and natural language processing. We present axiomatic and LDS style proof theories (two-dimensional label structure) and semantics for the logic. Soundness and completeness results are proved. We show that, for non{classical logic theorem proving, LDS is more exible than the existing methods, and is suitable for mechanisation directly. It also bridges the gap between proof theory and implementation. This paper also veriies Gabbay's open claims that LDS is a good framework for handling logics which combine diierent features, such as modality and resource restriction. We believe our approach can be extended to any variant which combines substructurality and modality.