Weakly-unambiguous Parikh automata and their link to holonomic series

TL;DR

The paper strengthens the bridge between formal languages and holonomic multivariate series by proving that languages recognized by weakly-unambiguous Parikh automata are holonomic and equate to unambiguous reversal-bounded counter machines, while showing that the converse fails via a holonomic counterexample. It extends the framework to pushdown Parikh automata and establishes a precise equivalence with RCM and unambiguous RBCMs, enriching the semantic landscape of automata-with-count constraints. The work also yields practical algorithmic consequences, providing effective bounds for language-inclusion decisions and illustrating analytic criteria for inherent weak-ambiguity using holonomic properties and Hadamard products. Overall, the results connect analytic combinatorics techniques with automata theory to classify and decide properties of languages beyond regular and context-free classes, with implications for verification and language design.

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.

Figures (8)

• Figure 1: Representation of a tree of root $v$ with $r$ children, labeled by a semilinear set $L$
• Figure 2: Decomposition of a $V$-context under the form $t=t'[(B,L,V,t_1, \ldots, t_r)]$
• Figure 3: Action of $\Phi$ on $t$ an $a$-context, where $left(t)=a\in\Sigma$, and $t'$ is not reduced to $root(t)=S$
• Figure 4: Action of $\Phi$ on an $a$-context, where $a\in\Sigma$, and $t'$ is reduced to $S$
• Figure 5: Action of $\Phi$ on a $V$-context, where $V\in N$, and $t'$ is not reduced to $S$

Theorems & Definitions (2)

• Claim 39
• Claim 41