Computational Complexity of Synchronization under Regular Constraints

On a Class of Constrained Synchronization Problems in NP

“…As shown in [11], for regular commutative constraint languages, we only find constraint problems that are NP-complete, PSPACE-complete, or solvable in polynomial time. In all the mentioned work [8,11,12], it was noted that the constraint automata for which the corresponding constraint synchronization problem is NP-complete admit a special form, which we generalize in this work.…”

“…In [10] the notion of constraint synchronization was introduced in connection with a reduction procedure for synchronizing automata. The paper [8] introduced the computational problem of constraint synchronization. In this problem, we search for a synchronizing word coming from some specific subset of allowed input sequences.…”

“…In this problem, we search for a synchronizing word coming from some specific subset of allowed input sequences. For further motivation and applications we refer to the aforementioned paper [8]. Let us mention that restricting the solution space by a regular language has also been applied in other areas, for example to topological sorting [1], solving word equations [4,5], constraint programming [17], or shortest path problems [19].…”

“…Let us mention that restricting the solution space by a regular language has also been applied in other areas, for example to topological sorting [1], solving word equations [4,5], constraint programming [17], or shortest path problems [19]. In [8] it was shown that the smallest partial constraint automaton for which the problem becomes PSPACE-complete has two states and a ternary alphabet. Also, the smallest constraint automaton for which the problem is NP-complete needs three states and a binary alphabet.…”

“…Our contribution: Here, we generalize a Theorem from [8] to give a wider class of constrained synchronization problems in NP. As noted in [12], the constraint automata that yield problems in NP admit a special form and our class encompasses all known cases of constraint problems in NP.…”