(Co-)Inductive semantics for Constraint Handling Rules

TL;DR

This paper develops a formal fixpoint semantics for Constraint Handling Rules that unifies simplification and propagation rules using nested fixpoints, enabling coinductive reasoning.

Contribution

It introduces a novel operational semantics with persistent states that combines least and greatest fixpoints for CHR, advancing the theoretical foundation.

Findings

Fixpoint semantics for simplification rules via least fixpoint

Fixpoint semantics for propagation rules via greatest fixpoint

Unified semantics with nested fixpoints for combined rule types

Abstract

In this paper, we address the problem of defining a fixpoint semantics for Constraint Handling Rules (CHR) that captures the behavior of both simplification and propagation rules in a sound and complete way with respect to their declarative semantics. Firstly, we show that the logical reading of states with respect to a set of simplification rules can be characterized by a least fixpoint over the transition system generated by the abstract operational semantics of CHR. Similarly, we demonstrate that the logical reading of states with respect to a set of propagation rules can be characterized by a greatest fixpoint. Then, in order to take advantage of both types of rules without losing fixpoint characterization, we present an operational semantics with persistent. We finally establish that this semantics can be characterized by two nested fixpoints, and we show the resulting language is an elegant framework to program using coinductive reasoning.