Weakly-unambiguous Parikh automata and their link to holonomic series

2025-12-10

AI Summary

This work investigates the deep connection between weakly unambiguous Parikh automata (WUPA) and holomorphic multivariate generating series. Method: We establish a semantic equivalence between WUPA and unambiguous two-way reversal-bounded counter machines, enabling a rigorous analysis of generating functions for WUPA languages. Contribution/Results: We prove, for the first time, that the multivariate generating series of every WUPA language is holomorphic. Conversely, we construct a counterexample—a language whose generating series is algebraic (hence holomorphic) yet inherently weakly ambiguous—thereby refuting the converse. Furthermore, we settle the decidability of the inclusion problem for WUPA languages, providing an EXPSPACE upper bound and proving its tightness. These results unify formal language theory, automata semantics, and analytic combinatorics, yielding a key sufficient condition—and a sharp boundary counterexample—for characterizing holomorphic generating functions via automata.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Web Mining and Content Analysis: Web data generation and simulation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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.

Problem

• Prove equivalence between weakly-unambiguous Parikh automata and unambiguous two-way reversal bounded counter machines
• Show that holonomic generating series do not guarantee weak-unambiguity in Parikh automata
• Establish decidability and complexity bounds for inclusion in weakly-unambiguous Parikh automata

Innovation

• Proving equivalence between weakly-unambiguous Parikh automata and unambiguous counter machines
• Showing holonomic series property for these automata's generating functions
• Providing decidability and complexity bounds for inclusion problem