Minimization of graph weighted models over circular strings

Abstract

Graph weighted models (GWMs) have recently been proposed as a natural generalization of weighted automata over strings, trees and 2-dimensional words to arbitrary families of labeled graphs (and hypergraphs). In this paper, we propose polynomial time algorithms for minimizing and deciding the equivalence of GWMs defined over the family of circular strings on a finite alphabet (GWM c s). The study of GWM c s is particularly relevant since circular strings can be seen as the simplest family of graphs with cycles. Despite the simplicity of this family and of the corresponding computational model, the minimization problem is considerably more challenging than in the case of weighted automata over strings and trees: while linear algebra tools are overall sufficient to tackle the minimization problem for classical weighted automata (defined over a field), the minimization of GWM c s involves fundamental notions from the theory of finite dimensional algebra. We posit that the properties of GWM c s unraveled in this paper willprove useful for the study of GWMs defined over richer families of graphs.