GKAT with Hoare Hypotheses

2026-06-29

AI Summary

This work addresses the challenge of effectively incorporating and reasoning about pre- and post-condition assumptions in Guarded Kleene Algebra with Tests (GKAT) to enable domain-knowledge-driven verification of program equivalence. It presents the first complete Hoare-style axiomatic system for GKAT and generalizes it to arbitrary word hypotheses. By leveraging automata-theoretic techniques, the authors construct a sound and complete axiomatization, demonstrating that program equivalence in GKAT remains decidable in near-linear time even in the presence of such hypotheses—matching the complexity of plain GKAT. Bridging Hoare logic, automata theory, and formal verification, this study substantially extends the expressiveness and decidability boundaries of GKAT for reasoning about programs under assumptions.

Abstract

Guarded Kleene Algebra with Tests (GKAT) is a variant of Kleene algebra which allows for reasoning about simple imperative programs, and which features a decision procedure for program equivalence in nearly linear time. In the current paper, we address the challenge of reasoning under assumptions about these programs. In particular, we develop a form of Hoare hypotheses, which allow modelling basic domain knowledge on pre- and post-conditions of uninterpreted basic programs, and which are well-developed for classical Kleene algebra but not yet for GKAT. We show that the resulting axiomatisation is sound and complete. We then extend Hoare hypotheses to the more general form of word hypotheses. Based on an automata-theoretic approach, we show that equivalence of GKAT under word hypotheses is as efficiently decidable as for plain GKAT.

Problem

• GKAT
• Hoare hypotheses
• program equivalence
• word hypotheses
• reasoning under assumptions

Innovation

• GKAT
• Hoare hypotheses
• program equivalence
• automata-theoretic approach
• word hypotheses