Axiomatizations and Computability of Weighted Monadic Second-Order Logic

TL;DR

This paper provides axiomatizations and explores the computability of weighted monadic second-order logic, establishing its relation to weighted automata and analyzing decision problems' decidability and complexity.

Contribution

It offers a complete axiomatization for weighted monadic second-order logic and examines the decidability of various decision problems within this framework.

Findings

Complete axiomatization for the full logic and a fragment without general sum.

Many decision problems are decidable but have high complexity.

Weighted satisfiability becomes undecidable under abstract interpretation.

Abstract

Weighted monadic second-order logic is a weighted extension of monadic second-order logic that captures exactly the behaviour of weighted automata. Its semantics is parameterized with respect to a semiring on which the values that weighted formulas output are evaluated. Gastin and Monmege (2018) gave abstract semantics for a version of weighted monadic second-order logic to give a more general and modular proof of the equivalence of the logic with weighted automata. We focus on the abstract semantics of the logic and we give a complete axiomatization both for the full logic and for a fragment without general sum, thus giving a more fine-grained understanding of the logic. We discuss how common decision problems for logical languages can be adapted to the weighted setting, and show that many of these are decidable, though they inherit bad complexity from the underlying first- and second-order logics. However, we show that a weighted adaptation of satisfiability is undecidable for the logic when one uses the abstract interpretation.