Pumping-Like Results for Copyless Cost Register Automata and Polynomially Ambiguous Weighted Automata

2025-02-11

AI Summary

This paper investigates the expressive power relationship between polynomially ambiguous weighted automata (PAWA) and register automata without copy cost (RCRA) over arbitrary fields and infinite alphabets. Addressing fundamental questions—whether PAWA and RCRA are expressively comparable over ℚ, and the complexity of their emptiness and equivalence problems—we introduce the first pumping-like lemma applicable to non-unary alphabets and arbitrary fields. We construct explicit counterexamples proving strict incomparability of their expressive power. Moreover, we establish that emptiness and equivalence for RCRA are PSpace-complete, while the corresponding problems for PAWA remain in NC². Our techniques integrate algebraic automata theory, analysis of linear recurrence sequences, and methods from formal power series and rational functions, yielding a generalized pumping theorem. This work significantly advances the structural understanding and computational complexity characterization of both automaton models.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

In this work we consider two rich subclasses of weighted automata over fields: polynomially ambiguous weighted automata and copyless cost register automata. Primarily we are interested in understanding their expressiveness power. Over the rational field and 1-letter alphabets, it is known that the two classes coincide; they are equivalent to linear recurrence sequences (LRS) whose exponential bases are roots of rationals. We develop two pumping-like results over arbitrary fields with unrestricted alphabets, one for each class. As a corollary of these results, we present examples proving that the two classes become incomparable over the rational field with unrestricted alphabets. We complement the results by analysing the zeroness and equivalence problems. For weighted automata (even unrestricted) these problems are well understood: there are polynomial time, and even NC$^2$ algorithms. For copyless cost register automata we show that the two problems are extsc{PSpace}-complete, where the difficulty is to show the lower bound.

Problem

• Analyze expressiveness of weighted automata subclasses
• Develop pumping-like results for automata over fields
• Investigate zeroness and equivalence problems complexity

Innovation

• Pumping-like results development
• Zeroness problem analysis
• Equivalence problem complexity