Combining Deduction Modulo and Logics of Fixed-Point Definitions

TL;DR

This paper introduces a new logical framework combining deduction modulo with fixed-point and recursive specifications, enabling more flexible reasoning about inductive, coinductive, and recursive definitions.

Contribution

It develops a natural deduction calculus that integrates closed-world equality into deduction modulo, supporting recursive specifications without strong monotonicity restrictions.

Findings

The calculus is strongly normalizing under general rewrite system conditions.

It effectively specifies and reasons about syntax-based descriptions.

The approach resolves proof stability issues under reduction.

Abstract

Inductive and coinductive specifications are widely used in formalizing computational systems. Such specifications have a natural rendition in logics that support fixed-point definitions. Another useful formalization device is that of recursive specifications. These specifications are not directly complemented by fixed-point reasoning techniques and, correspondingly, do not have to satisfy strong monotonicity restrictions. We show how to incorporate a rewriting capability into logics of fixed-point definitions towards additionally supporting recursive specifications. In particular, we describe a natural deduction calculus that adds a form of "closed-world" equality - a key ingredient to supporting fixed-point definitions - to deduction modulo, a framework for extending a logic with a rewriting layer operating on formulas. We show that our calculus enjoys strong normalizability when the rewrite system satisfies general properties and we demonstrate its usefulness in specifying and reasoning about syntax-based descriptions. The integration of closed-world equality into deduction modulo leads us to reconfigure the elimination principle for this form of equality in a way that, for the first time, resolves issues regarding the stability of finite proofs under reduction.