[Paper Review] A Real-Valued Modal Logic

Denisa Diaconescu, George Metcalfe+1 more|Bern Open Repository and Information System (University of Bern)|Jun 9, 2017

Logic, Reasoning, and Knowledge4 citations

TL;DR

This paper introduces a real-valued modal logic, K(A), which extends Abelian logic by interpreting propositional connectives using lattice and group operations over the real numbers. It presents a sound and complete labelled tableau calculus for K(A), establishing a coNEXPTIME upper bound for validity checking, and further shows that the modal-multiplicative fragment admits an EXPTIME complexity bound and a cut-free sequent calculus with a finitary axiomatization, overcoming limitations of prior infinitary systems.

ABSTRACT

A many-valued modal logic is introduced that combines the usual Kripke frame semantics of the modal logic K with connectives interpreted locally at worlds by lattice and group operations over the real numbers. A labelled tableau system is provided and a coNEXPTIME upper bound obtained for checking validity in the logic. Focussing on the modal-multiplicative fragment, the labelled tableau system is then used to establish completeness for a sequent calculus that admits cut-elimination and an axiom system that extends the multiplicative fragment of Abelian logic.

Motivation & Objective

• To develop a proof-theoretic framework for continuous many-valued modal logics based on real-valued semantics.
• To address the lack of finitary axiomatizations in infinite-valued modal logics, particularly for Łukasiewicz-based systems.
• To establish complexity bounds for validity checking in a real-valued modal logic with arithmetical connectives.
• To provide a complete labelled tableau calculus for K(A) and demonstrate its utility in deriving cut-elimination and finitary axiomatizations for fragments.
• To lay the groundwork for algebraic semantics and relational semantics relationships in continuous modal logics.

Proposed method

• The logic K(A) is defined by extending Abelian logic with modal operators, interpreting connectives via min, max, +, and − on real numbers.
• A labelled tableau calculus is developed for K(A), with labels representing possible worlds and formulae annotated with world indices.
• Completeness of the tableau calculus is established via a finite model property, enabling complexity analysis.
• The modal-multiplicative fragment is isolated by restricting to connectives closed under multiplication and additive inverses.
• A cut-free sequent calculus is constructed for the fragment using the tableau completeness, enabling proof-theoretic analysis.
• Complexity bounds are derived by analyzing the size and structure of tableaux, leading to coNEXPTIME for K(A) and EXPTIME for the multiplicative fragment.

Experimental results

Research questions

• RQ1 Can a finitary, complete, and complexity-bounded proof system be developed for a continuous modal logic based on real-valued connectives?
• RQ2 What is the computational complexity of validity checking in a modal logic with real-valued lattice and group operations?
• RQ3 Can the modal-multiplicative fragment of such a logic admit cut-elimination and a finitary axiomatization?
• RQ4 How can proof-theoretic methods like labelled tableaux be used to derive algebraic completeness results in infinite-valued settings?
• RQ5 Is it possible to extend the proof-theoretic framework to other frame classes (e.g., reflexive, transitive) while preserving completeness and complexity bounds?

Key findings

• The labelled tableau calculus for K(A) is sound and complete, enabling effective validity checking.
• The complexity of checking K(A)-validity is bounded above by coNEXPTIME, matching known bounds for Łukasiewicz description logics.
• The modal-multiplicative fragment of K(A) has an EXPTIME upper bound for validity checking, significantly improving upon the general coNEXPTIME bound.
• A cut-free sequent calculus is constructed for the modal-multiplicative fragment, demonstrating cut-elimination and proof-theoretic harmony.
• A finitary axiom system is provided for the modal-multiplicative fragment, avoiding infinitary rules present in prior systems for Łukasiewicz modal logic.
• The logic K(A) interprets Łukasiewicz modal logic K(L) as a fragment when extended with a constant, linking it to established continuous modal systems.