Extending Resource-Aware Logics

This book introduces some extensions of resource- aware logics and shows some fundamental (e.g. cut- elimination and completeness) theorems for these extensions. By resource-aware logics we mean substructural logics without the structural rule of contraction in Gentzen-type sequent calculi. Typical examples of these resource-aware logics are linear logics which are known to be useful for describing concurrent systems and non-commutative logics which are known to be useful for formalizing grammatical reasoning. These logics are known to be appropriate for representing resource-sensitive reasoning. In this book we study (1) some temporal extensions (i.e. extensions by adding some temporal operators) of linear and non-commutative logics (2) some generalizations of these temporal linear and non-commutative logics (3) some paraconsistent and involutive extensions (i.e. extensions by adding a paraconsistent negation connective or an involution operator) of linear and non-commutative logics.