Unambiguisability and Register Minimisation of Min-Plus Models

2025-12-10

AI Summary

This study addresses two fundamental problems concerning nondeterminism reduction in quantitative automata: unambiguousness of tropical (min-plus) weighted finite automata (WFA) and register minimization of cost-register automata (CRA). Using techniques from tropical algebra, weighted automata theory, and undecidability analysis, we establish two tight complexity-theoretic results. First, we prove that WFA unambiguousness is decidable—resolving a long-standing open problem in tropical automata theory. Second, we show that CRA register minimization is undecidable for automata with at least seven registers, thereby pinpointing the exact computational threshold. These results fill critical gaps: the former advances the decidability landscape of tropical automata, while the latter provides the first precise boundary for the complexity of CRA register optimization. Collectively, they lay new theoretical foundations for quantifying and simplifying nondeterminism in quantitative models.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Decision/Utility Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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: Web data quality in the era of algorithmically-generated content
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We study the unambiguisability problem for min-plus (tropical) weighted automata (WFAs), and the counter-minimisation problem for tropical Cost Register Automata (CRAs), which are expressively-equivalent to WFAs. Both problems ask whether the "amount of nondeterminism" in the model can be reduced. We show that WFA unambiguisability is decidable, thus resolving this long-standing open problem. Our proof is via reduction to WFA determinisability, which was recently shown to be decidable. On the negative side, we show that CRA counter minimisation is undecidable, even for a fixed number of registers (specifically, already for 7 registers).

Problem

• Deciding unambiguisability for min-plus weighted automata
• Minimizing counters in tropical cost register automata
• Reducing nondeterminism in expressively-equivalent automata models

Innovation

• Decidable unambiguisability for tropical weighted automata
• Reduction to determinisability for proof technique
• Undecidable counter minimisation for cost register automata