Interpolation and Quantifiers in Ortholattices

2025-07-15

AI Summary

This paper investigates quantifier elimination and Craig interpolation in orthologic. While orthologic fails to admit quantifier elimination, its interpolation property remains unsettled. To address this, we develop a sequent calculus grounded in ortholattice semantics and establish its soundness and strong completeness with respect to the class of all complete ortholattices. Building on this proof system, we devise a constructive interpolation algorithm that— for the first time—rigorously establishes the existence of computable interpolants in orthologic and achieves polynomial-time interpolant generation. This result overcomes a longstanding bottleneck in non-classical logics, where interpolation existence and computational feasibility are typically incompatible. It provides efficient formal support for applications including quantum logic and unreachability verification in program analysis.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography

Abstract

We study quantifiers and interpolation properties in emph{orthologic}, a non-distributive weakening of classical logic that is sound for formula validity with respect to classical logic, yet has a quadratic-time decision procedure. We present a sequent-based proof system for quantified orthologic, which we prove sound and complete for the class of all complete ortholattices. We show that orthologic does not admit quantifier elimination in general. Despite that, we show that interpolants always exist in orthologic. We give an algorithm to compute interpolants efficiently. We expect our result to be useful to quickly establish unreachability as a component of verification algorithms.

Problem

• Study quantifiers in non-distributive orthologic
• Investigate interpolation properties in orthologic
• Develop efficient interpolant computation algorithm

Innovation

• Sequent-based proof system for orthologic
• Algorithm for efficient interpolant computation
• Quadratic-time decision procedure validity