The Degree of a Finite Set of Words

Authors Dominique Perrin, Andrew Ryzhikov

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Dominique Perrin
  • Université Gustave Eiffel, LIGM, Marne-la-Vallée, France
Andrew Ryzhikov
  • Université Gustave Eiffel, LIGM, Marne-la-Vallée, France


We thank Jean-Eric Pin and Jacques Sakarovitch for references concerning the composition of automata and transducers.

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Dominique Perrin and Andrew Ryzhikov. The Degree of a Finite Set of Words. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We generalize the notions of the degree and composition from uniquely decipherable codes to arbitrary finite sets of words. We prove that if X = Y∘Z is a composition of finite sets of words with Y complete, then d(X) = d(Y) ⋅ d(Z), where d(T) is the degree of T. We also show that a finite set is synchronizing if and only if its degree equals one. This is done by considering, for an arbitrary finite set X of words, the transition monoid of an automaton recognizing X^* with multiplicities. We prove a number of results for such monoids, which generalize corresponding results for unambiguous monoids of relations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • synchronizing set
  • degree of a set
  • group of a set
  • monoid of relations


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