Synchronising Word Problem for DFAs

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Synchronising Word Prolem for DFAs Automt Theory nd Computility Rghunndn M. A. Deprtment of Computer Science nd Automtion Indin Institute of Science, nglore rghunndn.m@gmil.com August 26, 2011

Tle of Contents 1 Synchroniztion Synchroniztion - Definition nd Exmples Rod Colouring Prolem Rod Colouring Theorem Synchronizing Word Prolem for DFAs Find Synchronising Word 2 Computtion of Sync Words Length of Sync Words Computtion of Sync Words Power Automton Algorithm using the Power Automton Polynomil Time Algorithm 3 Summry References Rghu (IISc) Sync-Word Prolem August 2011 2 / 16

Synchroniztion - Definition nd Exmples Definition Bringing system into desired known stte. Bringing group of ojects into the sme stte. Exmples Spceship round the moon Rootics Single root in unknown stte Group of roots in different sttes Prts orienttion Circuit testing Rghu (IISc) Sync-Word Prolem August 2011 3 / 16

Rod Colouring Prolem Given directed grph G, colour the edges such tht for every vertex, there re set of insturctions tht led to tht vertex, from every other vertex. Instructions to rech yellow vertex: B-R-R B-R-R B-R-R Imge source: uplod.wikimedi.org/wikipedi/commons /6/69/Rod coloring conjecture.svg Rghu (IISc) Sync-Word Prolem August 2011 4 / 16

Rod Colouring Theorem Adler nd Weiss, 1970 (Conjecture) Every finite strongly connected periodic directed grph of uniform out-degree hs synchronizing coloring. O Brien, 1981 Grphs contining cycle of prime length. Kri, 2003 Eulerin digrphs (every vertex hs the sme in-degree nd out-degree) Trhtmn, 2009 Generl cse 1 2 0 3 1 2 0 3 Rghu (IISc) Sync-Word Prolem August 2011 5 / 16

Synchronizing Word Prolem for DFAs Sync Word Given DFA A = (Q, Σ, δ) Find w Σ nd q Q : q Q, δ(q, w) = q 1 2 Exmple For q = 1; w = For q = 2; w =, Rghu (IISc) Sync-Word Prolem August 2011 6 / 16

Find Synchronising Word - 1 1 2, Rghu (IISc) Sync-Word Prolem August 2011 7 / 16

Find Synchronising Word - 1 1 2, Synchronising word:, Rghu (IISc) Sync-Word Prolem August 2011 7 / 16

Find Synchronising Word - 2 0 2 1 Rghu (IISc) Sync-Word Prolem August 2011 8 / 16

Find Synchronising Word - 2 0 2 1 This utomton cnnot e synchronised. Rghu (IISc) Sync-Word Prolem August 2011 8 / 16

Find Synchronising Word - 3 0, 2 1 Rghu (IISc) Sync-Word Prolem August 2011 9 / 16

Find Synchronising Word - 3 0, 2 1 Synchronising word: 2 Rghu (IISc) Sync-Word Prolem August 2011 9 / 16

Find Synchronising Word - 4 00,d 01,c c c d d 11,c 10,d Rghu (IISc) Sync-Word Prolem August 2011 10 / 16

Find Synchronising Word - 4 00,d 01,c c c d d 11,c 10,d Synchronising word: c Rghu (IISc) Sync-Word Prolem August 2011 10 / 16

Length of Sync Words Given synchronising DFA, how long cn its shortest synchronising word e? Cerny, 1964 Fmily of utomt C n whose shortest sync words hve (n 1) 2 letters. Cerny, 1971 Conjecture tht (n 1) 2 is the upper ound for ny synchronising utomton. Klychko, 1987 Upper ound of (n3 n) 6. Rghu (IISc) Sync-Word Prolem August 2011 11 / 16

Computtion of Sync Words Questions: Given DFA, determine the shortest sync word, if one exists. Given DFA, determine if it is synchronising. Procedure: Decidle? Polynomil? Rghu (IISc) Sync-Word Prolem August 2011 12 / 16

Power Automton 012 0, 01 12 02 2 1 1 2, 0 Rghu (IISc) Sync-Word Prolem August 2011 13 / 16

Algorithm using the Power Automton 012 12 01 02 1 2, 0 Shortest-Sync-Word(A) Construct P(A). Determine shortest pth from Q to ny singleton. If no such pth exists, return FALSE. Else, output the lels long the pth found. Rghu (IISc) Sync-Word Prolem August 2011 14 / 16

Algorithm using the Power Automton 012 12 01 02 1 2, 0 Shortest-Sync-Word(A) Construct P(A). Determine shortest pth from Q to ny singleton. If no such pth exists, return FALSE. Else, output the lels long the pth found. Time Complexity: 2 Q Σ Rghu (IISc) Sync-Word Prolem August 2011 14 / 16

Polynomil Time Algorithm Theorem (Cerny, 1964) A DFA A is synchronising if nd only if q, q Q, w Σ : δ(q, w) = δ(q, w) Rghu (IISc) Sync-Word Prolem August 2011 15 / 16

Polynomil Time Algorithm Theorem (Cerny, 1964) A DFA A is synchronising if nd only if q, q Q, w Σ : δ(q, w) = δ(q, w) Sync-Word(A) Construct suutomton P 2 (A) of P(A). Compute pths from ech qq to singleton. If ll pths exist, then output TRUE. If ny pth does not exist, then output FALSE. Rghu (IISc) Sync-Word Prolem August 2011 15 / 16

References I R. L. Adler, B. Weiss, Similrity of utomorphisms of the torus, Memoires of the Americn Mthemticl Society, 98. (1970) H. Jurgensen, Synchroniztion, Informtion nd Computtion 206 (910): 10331044, (2008) Rod coloring prolem, http://en.wikipedi.org/wiki/rod coloring prolem Synchronizing word, http://en.wikipedi.org/wiki/synchronizing word A. Trhtmn, The rod coloring prolem, Isrel Journl of Mthemtics 172 (1): 5160 (2009) M. Volkov, Synchronizing Automt nd the Cerny Conjecture, Proc. 2nd Int l. Conf. Lnguge nd Automt Theory nd Applictions (LATA 2008), LNCS, 5196, Springer-Verlg, pp. 1127 Rghu (IISc) Sync-Word Prolem August 2011 16 / 16

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