Verified and Optimized Implementation of Orthologic Proof Search (2501.09418v3)

Abstract: We report on the development of an optimized and verified decision procedure for orthologic equalities and inequalities. This decision procedure is quadratic-time and is used as a sound, efficient and predictable approximation to classical propositional logic in automated reasoning tools. We formalize, in the Coq proof assistant, a proof system in sequent-calculus style for orthologic. We then prove its soundness and completeness with respect to the algebraic variety of ortholattices, and we formalize a cut-elimination theorem (in doing so, we discover and fix a missing case in a previously published proof). We then implement and verify a complete proof search procedure for orthologic. A naive implementation is exponential, and to obtain an optimal quadratic runtime, we optimize the implementation by memoizing its results and simulating reference equality testing. We leverage the resulting correctness theorem to implement a reflective Coq tactic. We present benchmarks showing the procedure, under various optimizations, matches its theoretical complexity. Finally, we develop tactics including normalization with respect to orthologic and a boolean solver, which we also benchmark. We make tactics available as a standalone Coq plugin.

• The paper presents a novel proof search algorithm for orthologic that formalizes the decision process using Coq’s typeclasses and dependent types.
• It rigorously demonstrates soundness, completeness, and cut-elimination of the sequent-calculus styled proof system in ortholattices.
• The authors optimize performance by incorporating memoization and reference equality techniques to reduce computational complexity effectively.

Verified and Optimized Implementation of Orthologic Proof Search

The paper presents a comprehensive treatise on the formalization, verification, and optimization of a decision procedure for orthologic—an algebraic logic encapsulated within ortholattices—using the Coq proof assistant. This work builds upon the sequent-calculus styled proof system, affirming its soundness and completeness against the algebraic structure of ortholattices, and notably contributes a detailed proof of cut-elimination, potentially correcting earlier deficiencies in the literature.

Formalization and Verification

This paper begins with the formal definition of ortholattices using a typeclass approach in Coq. A novel induction principle aids in proving that all ortholattices satisfy properties extending Boolean algebras, thus situating orthologic as a computationally attractive fragment. The paper formulates orthologic proofs via Coq's dependent types, enforcing proof correctness by construction. Cut-elimination is established through an intricate case analysis, ensuring no use of cut rules outside essential transformations, hence guaranteeing the decidability of sequents.

Decision Procedure

A decision procedure for orthologic by proof search is constructed, hinging on orthologic's subformula property which ensures a termination and completeness congruent to the capabilities of sequent calculus. This procedure recursively searches for proofs by attempting applicable proof rules, leveraging a simplification heuristic that drastically reduces unnecessary rule applications.

Optimization via Memoization

The authors effectively mitigate the exponential complexity typically associated with proof search by employing memoization. By caching results of recursive calls, the overall complexity is reduced to a quadratic time scale, effectively rendering the procedure viable for larger and more complex expressions. Two implementations of memoization—using list-based and AVL-tree-based storage—demonstrate varied scaling, with AVL-trees providing superior efficiency through logarithmic lookup time.

Reference Equality Optimization

Further optimization is achieved using reference equality to sidestep costly structural comparisons. By annotating syntactic nodes with pointers, the equality of two terms can be resolved to pointer equivalence—a requisite for optimal performance. This extension demanded rigorous proofs to ensure pointer assignments' correctness without structural collisions, encapsulated in nearly 800 lines of proof in their formalization.

Empirical Evaluation

The empirical evaluation showcases the algorithm's efficiency across a variety of test cases, surpassing existing tactics such as btauto in solving equivalence for formulas up to 60 variables. The quantifiable impact of each optimization layer is critically assessed, underscoring the potential for reflective tactics in practical applications.

Implications and Future Directions

The implications for theorem proving and formal verification are significant. This work not only refines the decision procedure for a subset of propositional logic but also presents a scalable methodology for optimal proof search, which could extend to various logical formalisms. Future investigations could extend these optimizations to more expressive logical forms or integrate them into broader verification systems, offering expansive utility in automated reasoning contexts.

Through meticulous formalization and strategic algorithmic improvements, the paper succeeds in not only advancing the computational efficiency of orthologic decision procedures but also in tightly interlinking logic theory with practical verification tools. Such contributions enhance both the theoretical framework and practical capabilities in logical computation and verification domains.