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\(^\mathrm{c}\)s). The study of GWM\(^\mathrm{c}\)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\(^\mathrm{c}\)s involves fundamental notions from the theory of finite dimensional algebra. We posit that the properties of GWM\(^\mathrm{c}\)s unraveled in this paper willprove useful for the study of GWMs defined over richer families of graphs.

Highlights

  • Et al [2] proposed a computational model for functions mapping labeled graphs to values in a field: Graph Weighted Models (GWMs)

  • We proposed polynomial time algorithms to handle both the minimization and the equivalence problems for Graph weighted models (GWMs) defined over circular strings

  • One promising direction we are currently investigating relies on extending the central notion of semi-simple GWM A over circular strings (GWMc) to GWMs defined over arbitrary families of labeled graphs: by opening any edge e in a graph G one obtains a graph Ge with two free ports which would be mapped by a d-dimensional GWM A to a matrix AGe ∈ Md(F)

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Summary

Introduction

Functions defined over syntactical structures such as strings, trees and graphs are ubiquitous in computer science. While the problem of minimizing a GWM defined over the simple family of circular strings is central to this paper, we see it as a test bed for developing the theory of general GWMs: beyond the minimization and equivalence algorithms we propose, we believe that one of our main contributions is to illustrate how the theory of GWMs will rely on advanced concepts from algebra theory and to unravel fundamental properties that will surely be central to the study of GWMs defined over more general families of graphs (such as the one of semi-simple GWMc)

Notations
Preliminaries
Weighted Automata and GWMs over Circular Strings
Finite-Dimensional Algebras
Semi-Simple GWMs and the Equivalence Problem
Semi-Simplicity, Nilpotent Matrices and Traces
Equivalence of GWMs
Minimization of GWMs over Circular Strings
Conclusion
Full Text
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