Verified and Optimized Implementation of Orthologic Proof Search

2025-01-16

AI Summary

This work addresses the automated decision problem for orthologic. We present the first formally verified decision procedure for equations and inequations in orthologic, with a rigorously established quadratic time complexity bound. To overcome the undecidability of classical propositional logic under orthogonal semantics, we fully formalize the sequent calculus for orthologic in Coq—correcting a critical gap in prior proofs of cut elimination—and design an optimal proof-search algorithm incorporating memoization and reference simulation. Implemented via a reflection-based tactic in Coq, the procedure delivers end-to-end correctness guarantees. Empirical evaluation confirms that its runtime strictly adheres to the theoretical O(n²) bound. Our contribution fills a foundational gap in the formalization of automated reasoning for orthologic and provides an efficient, trustworthy approximate decision procedure applicable to quantum logic and related domains.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Data transparency and provenance
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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 start by formalizing, 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.

Problem

• Optimization
• Orthogonal Logic Proofs
• Automated Reasoning

Innovation

• Orthogonal Logic
• Automated Reasoning
• Optimized Proof Search