Tree algebras and bisimulation-invariant MSO on finite graphs

2024-07-17

arXiv.org

AI Summary

This paper resolves a long-standing open problem in modal logic: whether the bisimulation-invariant fragment of monadic second-order logic (MSO) over finite transition systems is expressively equivalent to the modal μ-calculus. The authors introduce the novel framework of finite tree algebras—specifically, finite algebras for the variety of zero- and unary-ranked trees—and prove that they precisely characterize the regular tree languages definable in the μ-calculus, thereby filling a two-decade gap in the algebraic theory of regular infinite-tree languages. By integrating algebraic semantics, tree automata theory, and Wilke algebra techniques, and by establishing a model-theoretic translation between MSO and the μ-calculus, they rigorously show that, over finite graphs, bisimulation-invariant MSO formulas are exactly those expressible in the modal μ-calculus. This result not only settles the classical problem but also establishes a fundamental bridge between algebraic classifications of tree languages and expressive power theory in modal logic.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Description Logics
• Machine Learning: Calibration & Uncertainty Quantification

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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We establish that the bisimulation invariant fragment of MSO over finite transition systems is expressively equivalent over finite transition systems to modal mu-calculus, a question that had remained open for several decades. The proof goes by translating the question to an algebraic framework, and showing that the languages of regular trees that are recognized by finitary tree algebras whose sorts zero and one are finite are the regular ones, ie. the ones expressible in mu-calculus. This corresponds for trees to a weak form of the key translation of Wilke algebras to omega-semigroup over infinite words, and was also a missing piece in the algebraic theory of regular languages of infinite trees for twenty years.

Problem

• Equivalence between bisimulation-invariant MSO and modal mu-calculus
• Translation of the problem to an algebraic framework
• Connection between tree algebras and regular tree languages

Innovation

• Translates MSO to algebraic framework
• Uses finitary tree algebras
• Links to modal mu-calculus