Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

On semantic resolution with lemmaizing and contraction and a formal treatment of caching

Reducing redundancy in search has been a major concern for automated deduction. Subgoal-reduction strategies, such as those based on model elimination and implemented in Prolog technology theorem provers, prevent redundant search by using lemmaizing and caching, whereas contraction-based strategies prevent redundant search by using contraction rules, such as subsumption. In this work we show that lemmaizing and contraction can coexist in the framework of semantic resolution. On the lemmaizing side, we define two meta-level inference rules for lemmaizing in semantic resolution, one producing unit lemmas and one producing non-unit lemmas, and we prove their soundness. Rules for lemmaizing are meta-rules because they use global knowledge about the derivation, e.g. ancestry relations, in order to derive lemmas. Our meta-rules for lemmaizing generalize to semantic resolution the rules for lemmaizing in model elimination. On the contraction side, we give contraction rules for semantic strategies, and we define a purity deletion rule for first-order clauses that preserves completeness. While lemmaizing generalizes success caching of model elimination, purity deletion echoes failure caching. Thus, our approach integrates features of backward and forward reasoning. We also discuss the relevance of our work to logic programming