On Finite Monoids over Nonnegative Integer Matrices and Short Killing Words

Authors Stefan Kiefer, Corto Mascle

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Author Details

Stefan Kiefer
  • University of Oxford, UK
Corto Mascle
  • ENS Paris-Saclay, France

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Stefan Kiefer and Corto Mascle. On Finite Monoids over Nonnegative Integer Matrices and Short Killing Words. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 43:1-43:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Let n be a natural number and M a set of n x n-matrices over the nonnegative integers such that M generates a finite multiplicative monoid. We show that if the zero matrix 0 is a product of matrices in M, then there are M_1, ..., M_{n^5} in M with M_1 *s M_{n^5} = 0. This result has applications in automata theory and the theory of codes. Specifically, if X subset Sigma^* is a finite incomplete code, then there exists a word w in Sigma^* of length polynomial in sum_{x in X} |x| such that w is not a factor of any word in X^*. This proves a weak version of Restivo’s conjecture.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • matrix semigroups
  • unambiguous automata
  • codes
  • Restivo’s conjecture


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