Equivalence Testing of Weighted Automata over Partially Commutative Monoids

• Published in: arXiv.org 2020-05
• Main Authors: Arvind, V, Chatterjee, Abhranil, Datta, Rajit, Mukhopadhyay, Partha
• Format: Article
• Language: English
• Subjects: Algorithms; Commutation; Equivalence; Graphs; Monoids; Polynomials; Randomization; Traveling salesman problem
• ISSN: 2331-8422

We study \emph{multiplicity equivalence} testing of automata over partially commutative monoids (pc monoids) and show efficient algorithms in special cases, exploiting the structure of the underlying non-commutation graph of the monoid. Specifically, if the clique cover number of the non-commutation graph (the minimum number of cliques covering the graph) of the pc monoid is a constant, we obtain a deterministic quasi-polynomial time algorithm. As a consequence, we also obtain the first deterministic quasi-polynomial time algorithms for multiplicity equivalence testing of \(k\)-tape automata and for equivalence testing of deterministic \(k\)-tape automata for constant \(k\). Prior to this, a randomized polynomial-time algorithm for the above problems was shown by Worrell [ICALP 2013]. We also consider pc monoids for which the non-commutation graphs have cover consisting of at most \(k\) cliques and star graphs for any constant \(k\). We obtain randomized polynomial-time algorithm for multiplicity equivalence testing of automata over such monoids.