Representing One Letter Weighted Automata Over the Tropical Semiring

2026-06-24

AI Summary

This work investigates the determinization of weighted automata over tropical semirings on a unary alphabet and the computational complexity of related problems. By refining Gaubert’s result as a weighted generalization of Chrobak normal form, the study presents the first polynomial-time algorithm that transforms a given automaton into an equivalent union of deterministic automata of quadratic size. Building on this construction, it establishes that the problems of determinization, register minimization, and boundedness are all coNP-complete. Furthermore, the paper demonstrates that these problems remain inherently difficult even in the parameterized setting, proving them to be coW[1]-hard under the framework of parameterized complexity. These results collectively characterize the intrinsic hardness of the considered problems both in classical and parameterized computational complexity.

Abstract

We consider weighted automata over the tropical semiring $\mathbb{Z}_\infty(min, +)$. Recently, it was shown that determinisation is decidable; in this paper we focus on the complexity when the alphabet is unary. In 2001, Lombardy showed this problem is decidable, a close inspection of his proof yields a coNP upper bound on the complexity. Earlier Gaubert showed that every weighted automaton in this setting can be effectively turned into an equivalent union of deterministic weighted automata. We prove Gaubert's result efficiently, presenting it as a generalisation of Chrobak's normal form for unary NFA. In particular, we prove that the equivalent union of deterministic weighted automata can be represented by a weighted automaton of quadratic size in the size of the original one, and this representation can be computed in polynomial time. Building on this, we show that determinisation, and even register minimisation (which generalises determinisation), is coNP-complete. We complete the paper with observations that the boundedness problem is also coNP-complete by reductions with determinisation. Lastly, we provide evidence that all of these problems are not FPT (by proving $coW_1$-hardness) when parametrised by the number of deterministic automata in the union.

Problem

• weighted automata
• tropical semiring
• unary alphabet
• determinisation
• computational complexity

Innovation

• weighted automata
• tropical semiring
• determinisation
• Chrobak normal form
• parameterized complexity