Synchronization Problems in Automata Without Non-trivial Cycles
We study the computational complexity of various problems related to synchronization of weakly acyclic automata, a subclass of widely studied aperiodic automata. We provide upper and lower bounds on the length of a shortest word synchronizing a weakly acyclic automaton or, more generally, a subset of its states, and show that the problem of approximating this length is hard. We also show inapproximability of the problem of computing the rank of a subset of states in a binary weakly acyclic automaton and prove that several problems related to recognizing a synchronizing subset of states in such automata are NP-complete.
KeywordsSynchronizing automata Computational complexity Weakly acyclic automata Subset rank
We would like to thank Vladimir Gusev and Ilia Fridman for very useful discussions, and Mikhail V. Volkov and anonymous reviewers for their great contribution to the improvement of the paper.
- 5.Cardoso, A.: The Černý Conjecture and Other Synchronization Problems. Ph.D. thesis. University of Porto, Portugal (2014)Google Scholar
- 6.Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press (2009)Google Scholar
- 9.Gerencsér, B., Gusev, V.V., Jungers, R.M.: Primitive sets of nonnegative matrices and synchronizing automata. CoRR abs/1602.07556 (2016)Google Scholar
- 11.Kozen, D.: Lower bounds for natural proof systems. In: Proceedings of the 18th Annual Symposium on Foundations of Computer Science, pp. 254–266 (1977)Google Scholar
- 14.Natarajan, B.K.: An algorithmic approach to the automated design of parts orienters. In: Proceedings of the 27th Annual Symposium on Foundations of Computer Science, pp. 132–142 (1986)Google Scholar
- 19.Ryzhikov, A.: Approximating the maximum number of synchronizing states in automata. CoRR abs/1608.00889 (2016)Google Scholar
- 21.Sipser, M.: Introduction to the Theory of Computation. Cengage Learning, 3rd edn. (2012)Google Scholar
- 22.Szykuła, M.: Improving the upper bound the length of the shortest reset words. CoRR abs/1702.05455 (2017)Google Scholar
- 24.Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2001)Google Scholar