synchronizing words and carefully synchronizing words

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Workshop Dynamical Aspects of Automata and Semigroup Theories, 25-26 November 2010, Wien , Austria. Synchronizing Words and Carefully Synchronizing Words. Pavel Martyugin Ural State University, Ekaterinburg , Russia. - PowerPoint PPT Presentation

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Synchronizing Words and Carefully Synchronizing WordsPavel MartyuginUral State University, Ekaterinburg, RussiaWorkshop Dynamical Aspects of Automata and Semigroup Theories, 25-26 November 2010, Wien, AustriaSynchronizing automataa,abab1234bbaThe example of a synchronizing DFA with 4 states and 2 letters.Synchronizing automataSynchronizing automataSynchronizing automataern conjectureThis conjecture is unproved.Computational problemsComputation problems:Is a given DFA synchronizing or not? What is the length of the shortest synchronizing word for a given DFA? Computational problemsComputational problemsD.Eppstein(1990) The problem BOUNDED SYNis NP-complete. This problem remains NP-complete for automata over a 2-letter alphabet.Computational problemsJ.Olschewski, M.Ummels (2010) The problem MIN SYN is complete for the complexity class DPSubclasses of DFAThe erns problem and the complexities problems can be considered for some special cases of DFA.We consider here classes of cyclical, Eulerian, monotonic, cyclically monotonic, commutative DFA and DFA with a zero state.0DFA with a zero state1203n-1n-2DFA with a zero stateIn the previous example the input alphabet size grows with number of states, while in the ern example the alphabet has two elements for every number of states.0Cyclical DFA~ (2008) The problem BOUNDED SYN is NP-complete for cyclical and one-cluster DFA Eulerian DFAThe DFA is called Eulerian if its digraph is Eulerian.~ (2008) The problem BOUNDED SYN is NP-complete for Eulerian DFA Monotonic DFACommutative DFA~ (2008) The problem BOUNDED SYN is NP-complete for the class of all commutative DFA Careful synchronizationIf the PFA is a DFA then careful synchronization = synchronizationThe example of a carefully synchronizing PFA with 4 states and 2 letters.Careful synchronizationa,bab1234bbaCareful synchronizationCareful synchronizationExponential length012Exponential lengthThe lengths of cycles are consecutive prime numbers3 LettersThe shortest carefully synchronizing word has nonpolynomial length.2 LettersThe shortest carefully synchronizing word has nonpolynomial length.The lengths of blocks are consecutive prime numbersComputational problems~ (2010)The problem CARSIN is PSPACE-completeThe problem CARSIN remain PSPACE-complete for automata with 2-letter alphabet Thank you!