Tree algebras and bisimulation-invariant MSO on finite graphs

TL;DR

This paper proves that the bisimulation-invariant fragment of MSO over finite transition systems is equivalent to the modal mu-calculus, using algebraic methods involving tree algebras, resolving a long-standing open problem.

Contribution

It establishes an algebraic characterization of the bisimulation-invariant MSO fragment over finite graphs, connecting it to modal mu-calculus through finitary tree algebras.

Findings

Bisimulation-invariant MSO equals modal mu-calculus on finite graphs.

Finitary tree algebras recognize exactly the regular tree languages.

Resolved a 20-year open problem in algebraic theory of infinite tree languages.

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.