Minimisation of Multiplicity Tree Automata

TL;DR

This paper presents a polynomial-time algorithm for minimizing multiplicity tree automata over rationals, explores complexity bounds related to polynomial identity testing, and extends techniques to multiplicity word automata.

Contribution

It introduces a polynomial-time minimization algorithm for multiplicity tree automata and an NC algorithm for multiplicity word automata, along with complexity results for related decision problems.

Findings

Polynomial-time minimization algorithm for multiplicity tree automata.

NC algorithm for minimising multiplicity word automata.

Complexity results linking automaton minimization to polynomial identity testing.

Abstract

We consider the problem of minimising the number of states in a multiplicity tree automaton over the field of rational numbers. We give a minimisation algorithm that runs in polynomial time assuming unit-cost arithmetic. We also show that a polynomial bound in the standard Turing model would require a breakthrough in the complexity of polynomial identity testing by proving that the latter problem is logspace equivalent to the decision version of minimisation. The developed techniques also improve the state of the art in multiplicity word automata: we give an NC algorithm for minimising multiplicity word automata. Finally, we consider the minimal consistency problem: does there exist an automaton with $n$ states that is consistent with a given finite sample of weight-labelled words or trees? We show that this decision problem is complete for the existential theory of the rationals, both for words and for trees of a fixed alphabet rank.