Approximate minimization of weighted tree automata

Abstract

This paper studies the following approximate minimization problem: given a minimal weighted tree automaton A with n states recognizing a weighted tree language f, can we construct a smaller automaton Aˆ with nˆ<n states recognizing a language fˆ that is a good approximation of f? The corresponding problem for weighted automata on strings was recently studied by Balle et al. [16,15], where the authors introduced a new canonical form for weighted automata called singular value automata inspired by spectral methods, and showed that truncating such canonical form yields a solution for the problem satisfying a certain approximation criteria. In this paper we take a similar approach and show that in the tree case one can obtain an analogous canonical form that we call singular value tree automata, and use it to study the approximate minimization problem for weighted tree automata. We first establish the existence of this canonical form for weighted tree automata and then provide bounds on the quality of the resulting approximation method based on truncation. We also study the problem of computing the canonical form given a minimal weighted tree automaton and show that in the tree case this task is considerably more complicated than in the string case. In particular, computing the canonical form reduces to solving a system of polynomial equations. By further reducing this problem to the computation of generalized partition functions for weighted tree automata, we propose and analyze two methods for computing the canonical form based on iterative methods: fixed point iteration and Newton's method. Our analysis of Newton's method unveils a connection between iterative methods and sequences of sets of trees satisfying a certain technical condition that might be of independent interest.