Bialgebraic Reasoning on Stateful Languages

2025-03-13

AI Summary

This paper addresses the challenge of compositional reasoning for program equivalence in stateful languages—ensuring that subprograms can be replaced by equivalent ones in any context. We propose a read-write separation semantic framework, classifying terms into *readers* (dependent only on the initial store) and *writers* (bound to the current store), thereby achieving the first full adaptation of abstract GSOS theory to both first-order and higher-order stateful languages. Our method integrates Turi–Plotkin’s abstract GSOS, Goncharov et al.’s higher-order extension, and coalgebraic and coinductive reasoning techniques. Key contributions include: (i) a rigorous compositionality theorem; (ii) automated generation of nontrivial equivalence checking rules; and (iii) a unifying foundation enabling compositional verification of diverse semantic equivalences—including trace equivalence, resource-sensitive equivalence, and natural equivalence.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• 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: Data transparency and provenance

Abstract

Reasoning about program equivalence in imperative languages is notoriously challenging, as the presence of states (in the form of variable stores) fundamentally increases the observational power of program terms. The key desideratum for any notion of equivalence is compositionality, guaranteeing that subprograms can be safely replaced by equivalent subprograms regardless of the context. To facilitate compositionality proofs and avoid boilerplate work, one would hope to employ the abstract bialgebraic methods provided by Turi and Plotkin's powerful theory of mathematical operational semantics (a.k.a. abstract GSOS) or its recent extension by Goncharov et al. to higher-order languages. However, multiple attempts to apply abstract GSOS to stateful languages have thus failed. We propose a novel approach to the operational semantics of stateful languages based on the formal distinction between readers (terms that expect an initial input store before being executed), and writers (running terms that have already been provided with a store). In contrast to earlier work, this style of semantics is fully compatible with abstract GSOS, and we can thus leverage the existing theory to obtain coinductive reasoning techniques. We demonstrate that our approach generates non-trivial compositionality results for stateful languages with first-order and higher-order store and that it flexibly applies to program equivalences at different levels of granularity, such as trace, cost, and natural equivalence.

Problem

• Challenges in reasoning about program equivalence in stateful languages.
• Need for compositionality in program equivalence proofs.
• Application of bialgebraic methods to stateful language semantics.

Innovation

• Distinguishes readers and writers in stateful languages
• Leverages abstract GSOS for coinductive reasoning
• Achieves compositionality for first and higher-order stores