Morphisms and minimisation of weighted automata

TL;DR

This paper introduces a unified framework for automaton minimisation using morphisms, extending bisimulation, and compares two partition refinement algorithms with quadratic complexity for weighted automata.

Contribution

It generalizes bisimulation to weighted automata, introduces minimal quotient algorithms, and analyzes their complexity and potential improvements.

Findings

Both algorithms have quadratic complexity.

Partition refinement strategies are equivalent in complexity.

Potential improvements can reduce complexity similar to Hopcroft's algorithm.

Abstract

This paper studies the algorithms for the minimisation of weighted automata. It starts with the definition of morphisms-which generalises and unifies the notion of bisimulation to the whole class of weighted automata-and the unicity of a minimal quotient for every automaton, obtained by partition refinement. From a general scheme for the refinement of partitions, two strategies are considered for the computation of the minimal quotient: the Domain Split and the Predecesor Class Split algorithms. They correspond respectivly to the classical Moore and Hopcroft algorithms for the computation of the minimal quotient of deterministic Boolean automata. We show that these two strategies yield algorithms with the same quadratic complexity and we study the cases when the second one can be improved in order to achieve a complexity similar to the one of Hopcroft algorithm.