Completeness for Ancestral Logic via a Computationally-Meaningful Semantics

摘要

First-order logic (FOL) is evidently insufficient for the many applications of logic in computer science, mainly due to its inability to provide inductive definitions. Therefore, only an extension of FOL which allows finitary inductive definitions can be used as a framework for automated reasoning. The minimal logic that is suitable for this goal is Ancestral Logic (AL), which is an extension of FOL by a transitive closure operator. In order for AL to be able to serve as a reasonable (and better) substitute to the use of FOL in computer science, it is crucial to develop adequate, user-friendly proof systems for it. While the expressiveness of AL renders any effective proof system for it incomplete with respect to the standard semantics, there are useful approximations. In this paper we show that such a Gentzen-style approximation is both sound and complete with respect to a natural, computationally-meaningful Henkin-style semantics for AL.