Complexity of Equivalence and Learning for Multiplicity Tree Automata

TL;DR

This paper explores the computational complexity of equivalence and learning problems for multiplicity tree automata, revealing connections to polynomial identity testing and proposing a more efficient learning algorithm with near-optimal query complexity.

Contribution

It establishes the equivalence problem's complexity as logspace equivalent to polynomial identity testing and introduces a new learning algorithm with quadratic query complexity.

Findings

Equivalence problem is logspace equivalent to polynomial identity testing.

Lower bounds on the number of queries needed for learning automata.

A new learning algorithm with quadratic query complexity, nearly optimal.

Abstract

We consider the complexity of equivalence and learning for multiplicity tree automata, i.e., weighted tree automata over a field. We first show that the equivalence problem is logspace equivalent to polynomial identity testing, the complexity of which is a longstanding open problem. Secondly, we derive lower bounds on the number of queries needed to learn multiplicity tree automata in Angluin's exact learning model, over both arbitrary and fixed fields. Habrard and Oncina (2006) give an exact learning algorithm for multiplicity tree automata, in which the number of queries is proportional to the size of the target automaton and the size of a largest counterexample, represented as a tree, that is returned by the Teacher. However, the smallest tree-counterexample may be exponential in the size of the target automaton. Thus the above algorithm does not run in time polynomial in the size of the target automaton, and has query complexity exponential in the lower bound. Assuming a Teacher that returns minimal DAG representations of counterexamples, we give a new exact learning algorithm whose query complexity is quadratic in the target automaton size, almost matching the lower bound, and improving the best previously-known algorithm by an exponential factor.