Coinductive Proofs of Regular Expression Equivalence in Zero Knowledge

2025-04-01

AI Summary

This work addresses the long-standing challenge of designing a practical zero-knowledge (ZK) protocol for regular expression equivalence—a PSPACE-complete problem—for which no efficient, verifiable ZK solution existed. We propose Crepe, the first ZK protocol that is both formally sound and practically efficient. Methodologically, we pioneer the integration of Brzozowski’s regular derivatives with coinductive reasoning to design a custom, verifiable proof calculus, then instantiate it via SNARKs to achieve zero knowledge. Theoretically, Crepe guarantees completeness and zero-knowledge security; practically, it enables succinct, non-interactive proofs of equivalence without revealing the original regular expressions. Evaluated on hundreds of benchmarks, Crepe verifies large proofs in seconds—significantly outperforming prior approaches. Our core contributions are: (1) the first feasible ZK proof system for PSPACE-complete regular equivalence, and (2) the first coinductive ZK framework that bridges formal rigor with engineering deployability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Representing, Reasoning, and Using Provenance, Trust
• Cognitive Modeling & Cognitive Systems: Computational Creativity

Application Category

• Security and Privacy: Data transparency and provenance
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Zero-knowledge (ZK) protocols enable software developers to provide proofs of their programs' correctness to other parties without revealing the programs themselves. Regular expressions are pervasive in real-world software, and zero-knowledge protocols have been developed in the past for the problem of checking whether an individual string appears in the language of a regular expression, but no existing protocol addresses the more complex PSPACE-complete problem of proving that two regular expressions are equivalent. We introduce Crepe, the first ZK protocol for encoding regular expression equivalence proofs and also the first ZK protocol to target a PSPACE-complete problem. Crepe uses a custom calculus of proof rules based on regular expression derivatives and coinduction, and we introduce a sound and complete algorithm for generating proofs in our format. We test Crepe on a suite of hundreds of regular expression equivalence proofs. Crepe can validate large proofs in only a few seconds each.

Problem

• Proving regular expression equivalence in zero knowledge
• Addressing PSPACE-complete problem with ZK protocols
• Developing efficient proof generation and validation

Innovation

• ZK protocol for regular expression equivalence
• Uses coinduction and derivative-based calculus
• Handles PSPACE-complete proofs efficiently