Coaxioms: flexible coinductive definitions by inference systems

TL;DR

This paper introduces coaxioms, a generalized inference system concept that enables flexible recursive definitions by allowing axioms to be applied only at infinite depth, expanding the interpretative capabilities beyond classical fixed points.

Contribution

It proposes coaxioms as a new element in inference systems, enabling interpretations of recursive definitions that are neither purely inductive nor coinductive, thus broadening formal reasoning tools.

Findings

Supports fixed points beyond least and greatest

Unifies inductive and coinductive reasoning

Enables reasoning in mixed recursive cases

Abstract

We introduce a generalized notion of inference system to support more flexible interpretations of recursive definitions. Besides axioms and inference rules with the usual meaning, we allow also coaxioms, which are, intuitively, axioms which can only be applied "at infinite depth" in a proof tree. Coaxioms allow us to interpret recursive definitions as fixed points which are not necessarily the least, nor the greatest one, whose existence is guaranteed by a smooth extension of classical results. This notion nicely subsumes standard inference systems and their inductive and coinductive interpretation, thus allowing formal reasoning in cases where the inductive and coinductive interpretation do not provide the intended meaning, but are rather mixed together.