When is Containment Decidable for Probabilistic Automata?

TL;DR

This paper investigates the decidability of containment problems for probabilistic automata, revealing that ambiguity levels significantly influence decidability, with some cases becoming decidable and others remaining undecidable.

Contribution

It provides a nuanced analysis of how ambiguity levels affect decidability of containment and emptiness problems in probabilistic automata, including new decidability results under specific ambiguity constraints.

Findings

Decidability of the gap emptiness problem for polynomial ambiguity automata.

Undecidability of emptiness for automata with linear ambiguity.

Decidability of containment when one automaton is unambiguous and the other finitely ambiguous.

Abstract

The emptiness and containment problems for probabilistic automata are natural quantitative generalisations of the classical language emptiness and inclusion problems for Boolean automata. It is well known that both problems are undecidable. In this paper we provide a more refined view of these problems in terms of the degree of ambiguity of probabilistic automata. We show that a gap version of the emptiness problem (that is known be undecidable in general) becomes decidable for automata of polynomial ambiguity. We complement this positive result by showing that the emptiness problem remains undecidable even when restricted to automata of linear ambiguity. We then turn to finitely ambiguous automata. Here we show decidability of containment in case one of the automata is assumed to be unambiguous while the other one is allowed to be finitely ambiguous. Our proof of this last result relies on the decidability of the theory of real exponentiation, which has been shown, subject to Schanuel's Conjecture, by Macintyre and Wilkie.