Equivalence Testing of Weighted Automata over Partially Commutative Monoids

Authors V. Arvind, Abhranil Chatterjee, Rajit Datta, Partha Mukhopadhyay

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V. Arvind
  • Institute of Mathematical Sciences (HBNI), Chennai,India
Abhranil Chatterjee
  • Institute of Mathematical Sciences (HBNI), Chennai, India
Rajit Datta
  • Chennai Mathematical Institute, India
Partha Mukhopadhyay
  • Chennai Mathematical Institute, India


We thank the anonymous reviewers for their helpful feedback.

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V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. Equivalence Testing of Weighted Automata over Partially Commutative Monoids. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Motivated by equivalence testing of k-tape automata, we study the equivalence testing of weighted automata in the more general setting, over partially commutative monoids (in short, pc monoids), and show efficient algorithms in some special cases, exploiting the structure of the underlying non-commutation graph of the monoid. Specifically, if the edge clique cover number of the non-commutation graph of the pc monoid is a constant, we obtain a deterministic quasi-polynomial time algorithm for equivalence testing. As a corollary, we obtain the first deterministic quasi-polynomial time algorithms for equivalence testing of k-tape weighted automata and for equivalence testing of deterministic k-tape automata for constant k. Prior to this, the best complexity upper bound for these k-tape automata problems were randomized polynomial-time, shown by Worrell [James Worrell, 2013]. Finding a polynomial-time deterministic algorithm for equivalence testing of deterministic k-tape automata for constant k has been open for several years [Emily P. Friedman and Sheila A. Greibach, 1982] and our results make progress. We also consider pc monoids for which the non-commutation graphs have an edge cover consisting of at most k cliques and star graphs for any constant k. We obtain a randomized polynomial-time algorithm for equivalence testing of weighted automata over such monoids. Our results are obtained by designing efficient zero-testing algorithms for weighted automata over such pc monoids.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Theory of computation
  • Weighted Automata
  • Automata Equivalence
  • Partially Commutative Monoid


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