Degree of sequentiality of weighted automata

TL;DR

This paper investigates the degree of sequentiality of weighted automata with set semantics, providing characterizations, decision procedures, and translations for automata with bounded union sizes, and establishing complexity results.

Contribution

It introduces a novel twinning property and characterizations for multi-sequential weighted automata, enabling minimization and decision procedures for their degree of sequentiality.

Findings

Characterization of relations realized by unions of k sequential automata.

A decision procedure for the twinning property in commutative groups.

PSPACE-completeness of the associated decision problem.

Abstract

Weighted automata (WA) are an important formalism to describe quantitative properties. Obtaining equivalent deterministic machines is a longstanding research problem. In this paper we consider WA with a set semantics, meaning that the semantics is given by the set of weights of accepting runs. We focus on multi-sequential WA that are defined as finite unions of sequential WA. The problem we address is to minimize the size of this union. We call this minimum the degree of sequentiality of (the relation realized by) the WA. For a given positive integer k, we provide multiple characterizations of relations realized by a union of k sequential WA over an infinitary finitely generated group: a Lipschitz-like machine independent property, a pattern on the automaton (a new twinning property) and a subclass of cost register automata. When possible, we effectively translate a WA into an equivalent union of k sequential WA. We also provide a decision procedure for our twinning property for commutative computable groups thus allowing to compute the degree of sequentiality. Last, we show that these results also hold for word transducers and that the associated decision problem is Pspace-complete.