The Černý Conjecture for Aperiodic Automata
Avraham N. Trahtman
Abstract
A word w is called a synchronizing
(recurrent, reset, directable) word of a deterministic finite automaton
(DFA) if w brings all states of the automaton to some specific state; a DFA
that has a synchronizing word is said to be synchronizable. Cerny
conjectured in 1964 that every n-state synchronizable DFA possesses
a synchronizing word of length at most (n-1)2. We consider automata with
aperiodic transition monoid (such automata are called aperiodic).
We show that every synchronizable n-state aperiodic DFA has a synchronizing
word of length at most n(n-1)/2. Thus, for aperiodic automata
as well as for automata accepting only star-free languages,
the Cerny conjecture holds true.
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