A Bitopological Approach to Finite Reduction and Bounded Exact-Value Certificates for Fitting's Finite Heyting-valued Modal Logic

2026-08-05

AI Summary

This study addresses the problem of finite-state reduction in finitely-valued Heyting modal logics that preserves the exact truth values of formulas. Building on relational bi-topological duality, the work proposes a minimality-preserving reduction method by constructing an observational quotient structure via evaluation maps induced by modal subalgebras, ensuring that all formulas retain their precise truth values in the reduced model. The main contributions include proving that this observational quotient is isomorphic to a finite image within its bi-topological dual; constructing tree-shaped certificates of exact truth values for any formula and state, whose depth is bounded by modal depth and whose branching depends on the height of the truth-value algebra and the number of boxed subformulas; and, for the first time, providing bounded counterexample certificates that preserve exact falsity values in cases of truth-value failure.

Abstract

Fitting's finite Heyting-valued modal logic interprets modal formulas over a finite Heyting algebra. We use a relational bitopological representation to obtain a finite-state reduction. For a finite model and a finite vocabulary, the modal subalgebra generated by the atomic valuations determines a state-evaluation map. We prove that the observational quotient is isomorphic to its finite image in the bitopological dual and that the quotient relation is the restriction of the canonical dual relation. Hence every formula over the vocabulary preserves its exact truth value, and the quotient is minimal among surjective reductions through which all generated observations factor. In addition, for any formula and state, we construct a finite tree-like exact-value certificate whose depth is bounded by modal depth and whose branching depends only on the height of the truth-value algebra and the number of boxed subformulas. Failed formulas therefore admit bounded reduced counterexamples preserving their precise failure values.

Problem

• finite reduction
• exact-value certificates
• Heyting-valued modal logic
• bitopological representation
• observational quotient

Innovation

• bitopological representation
• finite reduction
• exact-value certificates
• Heyting-valued modal logic
• observational quotient