Abstract
The concept of a synchronizing word is a very important notion in the theory of finite automata. We consider the associated decision problem to decide if a given DFA possesses a synchronizing word of length at most k, where k is the standard parameter. We show that this problem DFA-SW is equivalent to the problem Monoid Factorization introduced by Cai, Chen, Downey and Fellows. Apart from the known W[2]-hardness results, we show that these problems belong to A[2], W[P] and WNL. This indicates that DFA-SW is not complete for any of these classes and hence, we suggest a new parameterized complexity class \(\textsf {W}[\textsf {Sync}]\) as a proper home for these (and more) problems.
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