[Paper Review] Recording Completion for Finding and Certifying Proofs in Equational Logic

Thomas Sternagel, René Thiemann+2 more|arXiv (Cornell University)|Aug 8, 2012

Logic, programming, and type systems9 references3 citations

TL;DR

This paper introduces recording completion, a method that extends Knuth-Bendix completion with a history component to explicitly record the derivation of rewrite rules from equational axioms. By maintaining a trace of rule derivations, it enables both the reconstruction of equational proofs and the certification of completion proofs, ensuring correctness through traceable derivations. The key contribution is a sound and complete framework for finding and certifying proofs in equational logic, with formalized results on critical pair theorems and confluence under finite variable sets.

ABSTRACT

When we want to answer/certify whether a given equation is entailed by an equational system we face the following problems: (1) It is hard to find a conversion (but easy to certify a given one). (2) Under the assumption that Knuth-Bendix completion is successful, it is easy to decide the existence of a conversion but hard to certify this decision. In this paper we introduce recording completion, which overcomes both problems.

Motivation & Objective

• To address the challenge of reconstructing and certifying equational proofs after Knuth-Bendix completion.
• To overcome the difficulty of certifying completion results, especially equivalence between the original equational system and the resulting convergent rewrite system.
• To provide a traceable, formally verifiable method for constructing and validating conversions in equational logic.
• To formalize the critical pair theorem and confluence properties under finite variable sets, where standard assumptions fail.

Proposed method

• Extends standard Knuth-Bendix completion rules with a history component H that tracks how each rule in the final TRS was derived from equations in the original equational system.
• Introduces annotated inference rules (e.g., deduce, orientl, orientr, simplifyl, simplifyr, compose, collapse, delete) that update both the equational system and the history trace.
• Uses unique indices for equations and rules, and records derivations as entries like i : s ◦1 u ◦2 t, where ◦1, ◦2 ∈ {←, →, ≈}.
• Enables proof reconstruction by tracing how terms reduce to a common normal form via the recorded history, thus reconstructing s ↔*E t from s →*R · ←*R t.
• Supports certification by verifying equivalence between the original equational system and the final TRS using the history, along with confluence and termination certificates.
• Proves that for finite completion runs, the strict encompassment condition in the collapse rule is not required, and that an infinite variable set is essential for the critical pair theorem and modularity of confluence.

Experimental results

Research questions

• RQ1 Can we reconstruct equational proofs from a convergent rewrite system without storing the full derivation?
• RQ2 How can we formally certify that a completion procedure has produced a correct and equivalent rewrite system?
• RQ3 Is the strict encompassment condition necessary in finite completion runs?
• RQ4 Does the critical pair theorem and modularity of confluence hold under finite variable sets?

Key findings

• For finite completion runs, the strict encompassment condition in the collapse rule is not required for soundness, which simplifies the completion process.
• The critical pair theorem and modularity of confluence do not hold for finite variable sets, necessitating formalization over infinite or string-based variables.
• An infinite set of variables is essential for the validity of the critical pair theorem and modularity of confluence in equational logic.
• The proposed recording completion framework enables full proof reconstruction and certification using traceable derivations, with successful certification of 94 out of 115 equational systems in experiments.
• KBCV 1.7, which uses auxiliary equations instead of full recall, achieved better certification performance than KBCV 1.6, demonstrating the practical benefit of optimized proof output.
• The method supports certification via equivalence, confluence, and termination certificates, allowing both proof existence and non-existence to be formally verified.