Towards Coinductive Theory Exploration in Horn Clause Logic: Position Paper

TL;DR

This paper introduces a unified proof-theoretic framework for coinductive reasoning in Horn clause logic, addressing two types of coinduction and establishing its soundness relative to coinductive models.

Contribution

It provides the first systematic analysis and a coinductive extension of Miller et al's uniform proofs framework for Horn clause logic.

Findings

Unified framework for two coinductive proof types

Proof of soundness relative to coinductive models

Addresses a gap in systematic analysis of coinductive proofs

Abstract

Coinduction occurs in two guises in Horn clause logic: in proofs of self-referencing properties and relations, and in proofs involving construction of (possibly irregular) infinite data. Both instances of coinductive reasoning appeared in the literature before, but a systematic analysis of these two kinds of proofs and of their relation was lacking. We propose a general proof-theoretic framework for handling both kinds of coinduction arising in Horn clause logic. To this aim, we propose a coinductive extension of Miller et al's framework of uniform proofs and prove its soundness relative to coinductive models of Horn clause logic.