On Propositional Program Equivalence (extended abstract)

2025-07-10

AI Summary

This paper addresses the problem of deciding program equivalence at the propositional logic level—i.e., abstracting away concrete semantic details (e.g., the actual evaluation of condition predicates or branch bodies) and focusing solely on control-flow structure to determine behavioral equivalence (e.g., whether `if b then e else f` is equivalent to `if ¬b then f else e`). To this end, the authors develop a unified algebraic framework based on (Guarded) Kleene Algebra with Tests ((G)KAT), modeling programs as symbolic expressions driven by Boolean tests and formally defining propositional equivalence within this setting. Crucially, while full program equivalence is undecidable, this abstraction renders the equivalence problem decidable. The framework enables automated verification of common equivalence patterns—including predicate negation, branch swapping, and conditional reordering—thereby enhancing both the theoretical rigor and practical applicability of program equivalence analysis.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

General program equivalence is undecidable. However, if we abstract away the semantics of statements, then this problem becomes not just decidable, but practically feasible. For instance, a program of the form "if $b$ then $e$ else $f$" should be equivalent to "if not $b$ then $f$ else $e$" - no matter what $b$, $e$ and $f$ are. This kind of equivalence is known as propositional equivalence. In this extended abstract, we discuss recent developments in propositional program equivalence from the perspective of (Guarded) Kleene Algebra with Tests, or (G)KAT.

Problem

• Deciding propositional program equivalence feasibility
• Abstracting statement semantics for equivalence analysis
• Exploring equivalence via Guarded Kleene Algebra with Tests

Innovation

• Uses propositional equivalence for program analysis
• Leverages Guarded Kleene Algebra with Tests
• Abstracts statement semantics to ensure decidability