Weighted Linear Dynamic Logic

TL;DR

This paper introduces weighted linear dynamic logic (weighted LDL), establishing its expressive equivalence to weighted rational expressions and expanding the understanding of recognizable series within formal language theory.

Contribution

The paper presents weighted LDL and proves its expressive equivalence to weighted rational expressions, providing a new characterization for recognizable series without restrictions.

Findings

Weighted LDL is expressively equivalent to weighted rational expressions.

The equivalence holds over arbitrary semirings for finite words and totally complete semirings for infinite words.

Decidability of the equivalence problem for weighted LDL formulas over fields is established in doubly exponential time.

Abstract

We introduce a weighted linear dynamic logic (weighted LDL for short) and show the expressive equivalence of its formulas to weighted rational expressions. This adds a new characterization for recognizable series to the fundamental Sch\"utzenberger theorem. Surprisingly, the equivalence does not require any restriction to our weighted LDL. Our results hold over arbitrary (resp. totally complete) semirings for finite (resp. infinite) words. As a consequence, the equivalence problem for weighted LDL formulas over fields is decidable in doubly exponential time. In contrast to classical logics, we show that our weighted LDL is expressively incomparable to weighted LTL for finite words. We determine a fragment of the weighted LTL such that series over finite and infinite words definable by LTL formulas in this fragment are definable also by weighted LDL formulas.