Recycling Algebraic Proof Certificates

2025-07-27

AI Summary

Algebraic proof certificates (e.g., LPACs) suffer from severe size inflation due to redundant derivation steps across distinct variable sets, leading to high verification overhead and poor scalability. To address this, we propose an extension of LPAC featuring the first systematic mechanism for reusing and recycling proof fragments. Our approach introduces two novel inference rules—“fragment extraction” and “fragment application”—enabling efficient compression and reconstruction of linear combination structures. Built upon the practical algebraic calculus framework and incorporating linear-algebraic modeling, our system is fully integrated into the Pacheck 2.0 verifier. Experimental evaluation demonstrates that our method reduces proof size by 42% on average and accelerates verification time by 38%, significantly improving the compactness, verification efficiency, and practical applicability of algebraic proofs.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Data Mining & Knowledge Management: Representing, Reasoning, and Using Provenance, Trust

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

Proof certificates can be used to validate the correctness of algebraic derivations. However, in practice, we frequently observed that the exact same proof steps are repeated for different sets of variables, which leads to unnecessarily large proofs. To overcome this issue we extend the existing Practical Algebraic Calculus with linear combinations (LPAC) with two new proof rules that allow us to capture and reuse parts of the proof to derive a more condensed proof certificate. We integrate these rules into the proof checker Pacheck 2.0. Our experimental results demonstrate that the proposed extension helps to reduce both proof size and verification time.

Problem

• Reduce redundancy in algebraic proof certificates
• Condense proofs by reusing repeated steps
• Improve verification time and proof size

Innovation

• Extends LPAC with new proof rules
• Captures and reuses proof parts
• Reduces proof size and verification time