Foundations of regular coinduction

TL;DR

This paper introduces a regular interpretation of inference systems that combines inductive and coinductive reasoning, providing new proof techniques and extending to systems with corules.

Contribution

It defines a regular interpretation that bridges inductive and coinductive reasoning, with an equivalent inductive characterization and algorithms for regular derivations.

Findings

The regular interpretation coincides with a rational fixed point.

An algorithm for finding regular derivations is developed.

Proof techniques for regular reasoning are established.

Abstract

Inference systems are a widespread framework used to define possibly recursive predicates by means of inference rules. They allow both inductive and coinductive interpretations that are fairly well-studied. In this paper, we consider a middle way interpretation, called regular, which combines advantages of both approaches: it allows non-well-founded reasoning while being finite. We show that the natural proof-theoretic definition of the regular interpretation, based on regular trees, coincides with a rational fixed point. Then, we provide an equivalent inductive characterization, which leads to an algorithm which looks for a regular derivation of a judgment. Relying on these results, we define proof techniques for regular reasoning: the regular coinduction principle, to prove completeness, and an inductive technique to prove soundness, based on the inductive characterization of the regular interpretation. Finally, we show the regular approach can be smoothly extended to inference systems with corules, a recently introduced, generalised framework, which allows one to refine the coinductive interpretation, proving that also this flexible regular interpretation admits an equivalent inductive characterisation.