Abstract
Let A be a finite automaton. We are concerned with the minimal length of the words that send all states on a unique state (synchronizing words). J. ČERNÝ has conjectured that, if there exists a synchronizing word in A, then there exists such a word with length ⩽(n−1)2 where n is the number of states of A. As a generalization, we conjecture that, if there exists a word of rank ⩽k in A, there exists such a word with length ⩽(n−k)2.
In this paper we deal only with automata in which a letter induces a circular permutation and prove the following results :
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1)
The second conjecture is true for (n-1)/2 ⩽k⩽n
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2)
If n is prime the first conjecture is true
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3)
IF n is prime and if there exists a letter of rank (n−1) the second conjecture is true
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Pin, J.E. (1978). Sur un cas particulier de la conjecture de Cerny. In: Ausiello, G., Böhm, C. (eds) Automata, Languages and Programming. ICALP 1978. Lecture Notes in Computer Science, vol 62. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-08860-1_25
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DOI: https://doi.org/10.1007/3-540-08860-1_25
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