Induction in a Subset of Linear Logic, Its Soundness and Completeness

Abstract

Linear logic is introduced by Girard, and is sensitive to quantity. Quantitative rules are obtained by induction in this logic. This paper proposes quantitative semantics for linear logical formulas, and proposes a method of induction. Soundness and completeness of this method with respect to the semantics are also shown. The semantics of linear logical formula is given as a set whose elements are multi-set. Multi-set is a set whose elements can repeatedly appear, i.e.the number of element is sensitive. Quantity is represented in this manner. Entailment relation between formulas is given as inclusion relation between the sets which are the semantics of these formulas. Our logic system is a sub-system of full propositional linear logic. This system is sound with respect to the proposed semantics. The method of induction from positive only example is proposed. In linear logic, normal form of a formula, such as clause in predicate logic, is hard to introduce. Thus, the inductive operations are defined by substitution of sub formulas. Soundness of the inductive method is proved straightforwardly from soundness of the logic system. Positive only example allws that the model of hypothesis has too much elements. Thus, minimal hypothesis is introduced which is defined as a hypothesis whose proper subset connot be a hypotheses. Finally completeness of the inductive method with respect to minimal hypothesis is proved.