Abstract

To express fine-grained resource-sensitive reasoning, a temporal soft linear logic (TSLL) is introduced as an extension of both Girard's (propositional classical) linear logic (CLL) and Lafont's (propositional classical) soft linear logic (SLL). It is known that the linear exponential operator in CLL can express a specific infinitely reusable resource, i.e. it is reusable not only for any number, but also many times. In contrast, the soft exponential operator in SLL, which is a weak version of the linear exponential operator, can express a specific usable resource, i.e. it is usable in any number, but only once (i.e. it is consumed after use). In TSLL, the resource operators (i.e. linear and soft exponentials) and some temporal operators are combined based on the interpretation that “time” is regarded as a “resource”. The completeness theorem (with respect to phase semantics) for TSLL and the cut-elimination and decidability theorems for some subsystems of TSLL are proved as the main results of this paper. A decidable subsystem, called a bounded soft linear logic (BSLL), which has a restricted soft exponential operator, can represent a specific finitely usable resource.