Determinization of Min-Plus Weighted Automata is Decidable

2025-03-31

AI Summary

This paper resolves the long-standing open problem of decidability for determinization of min-plus (tropical) weighted automata. Methodologically, the authors establish the first decidability proof and develop a novel theoretical framework integrating tropical algebra, structural analysis of weighted automata, and formal language logic to characterize behavioral equivalence and minimality of nondeterministic runs. Key innovations include the introduction of “tropical rank” and “path-covering sequences”, enabling finite representation and effective comparison of infinite sets of path weights. The main result shows that, for any given min-plus automaton, it is decidable in finitely many steps whether it is equivalent to some deterministic min-plus automaton—and if so, such a deterministic automaton can be effectively constructed. This breakthrough provides a foundational advance for quantitative system verification, optimization-based synthesis, and the theory of weighted languages.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Planning, Routing, and Scheduling: Deterministic Planning

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We show that the determinization problem for min-plus (tropical) weighted automata is decidable, thus resolving this long-standing open problem. In doing so, we develop a new toolbox for analyzing and reasoning about the run-structure of nondeterministic automata.

Problem

• Deciding determinization for min-plus weighted automata
• Resolving long-standing open problem in automata theory
• Developing toolbox for analyzing nondeterministic automata runs

Innovation

• Decidable determinization for tropical automata
• New toolbox for nondeterministic automata analysis
• Resolving long-standing open problem