Characterizing Weighted MSO for Trees by Branching Transitive Closure Logics

TL;DR

This paper introduces a branching transitive closure operator for weighted monadic second-order logic on trees, establishing equivalences with certain logical formulas over arbitrary commutative semirings.

Contribution

It defines a new branching transitive closure operator and proves its equivalence to existing logical frameworks for weighted MSO logic on trees.

Findings

Weighted MSO logic on trees is equivalent to formulas starting with branching transitive closure or specific quantifier patterns.

The operator applies to formulas over Boolean logic with modulo counting or monadic second-order logic.

The results hold for arbitrary commutative semirings.

Abstract

We introduce the branching transitive closure operator on weighted monadic second-order logic formulas where the branching corresponds in a natural way to the branching inherent in trees. For arbitrary commutative semirings, we prove that weighted monadic second order logics on trees is equivalent to the definability by formulas which start with one of the following operators: (i) a branching transitive closure or (ii) an existential second-order quantifier followed by one universal first-order quantifier; in both cases the operator is applied to step-formulas over (a) Boolean first-order logic enriched by modulo counting or (b) Boolean monadic-second order logic.