Weakly-unambiguous Parikh automata and their link to holonomic series

TL;DR

This paper explores the relationship between weakly-unambiguous Parikh automata and holonomic series, establishing equivalences, counterexamples, and decidability results related to their language and series properties.

Contribution

It proves a strong version of a conjecture linking weakly-unambiguous Parikh automata to holonomic series and provides a counterexample showing the limits of this connection.

Findings

Weakly-unambiguous Parikh automata are equivalent to unambiguous two-way reversal bounded counter machines.

Their multivariate generating series are holonomic.

Decidability of inclusion for weakly-unambiguous Parikh automata is established with complexity bounds.

Abstract

We investigate the connection between properties of formal languages and properties of their generating series, with a focus on the class of holonomic power series. We first prove a strong version of a conjecture by Castiglione and Massazza: weakly-unambiguous Parikh automata are equivalent to unambiguous two-way reversal bounded counter machines, and their multivariate generating series are holonomic. We then show that the converse is not true: we construct a language whose generating series is algebraic (thus holonomic), but which is inherently weakly-ambiguous as a Parikh automata language. Finally, we prove an effective decidability result for the inclusion problem for weakly-unambiguous Parikh automata, and provide an upper-bound on to its complexity.