Decision problems on unary probabilistic and quantum automata

TL;DR

This paper explores the decidability of the emptiness problem for unary probabilistic and quantum automata, connecting it with classical problems like Skolem's and positivity, and introduces linear recurrence automata for broader analysis.

Contribution

It introduces linear recurrence automata and discusses their relation to undecidability issues in unary probabilistic and quantum automata.

Findings

Emptiness problem for unary probabilistic automata remains undecidable.

Connections established between automata problems and classical mathematical problems.

Proposed generalizations of linear recurrence relations and automata on vectors.

Abstract

It is well known that the emptiness problem for binary probabilistic automata and so for quantum automata is undecidable. We present the current status of the emptiness problems for unary probabilistic and quantum automata with connections with Skolem's and positivity problems. We also introduce the concept of linear recurrence automata in order to show the connection naturally. Then, we also give possible generalizations of linear recurrence relations and automata on vectors.