Synchronizing Automata with Random Inputs

Transcription

1 Synchronizing Automt with Rndom Inputs Vldimir Gusev Url Federl University, Ekterinurg, Russi 9 August, 14 Vldimir Gusev (UrFU) Synchronizing Automt with Rndom Input 9 August, 14 1 / 13

2 Introduction Synchronizing Automt We consider DFA: A = Q,Σ,δ Q the stte set Σ the input lphet δ : Q Σ Q the trnsition function A is clled synchronizing if there exists word w Σ whose ction resets A, tht is, leves the utomton in one prticulr stte no mtter which stte in Q it strted t: δ(q,w) = δ(q,w) for ll q,q Q Any w with this property is synchronizing word for A The minimum length of synchronizing words for A is clled the reset threshold of A nd denoted y rt(a) Vldimir Gusev (UrFU) Synchronizing Automt with Rndom Input 9 August, 14 / 13

3 Introduction Synchronizing Automt, 1 The word is synchronizing It resets the utomton to the stte 1 Vldimir Gusev (UrFU) Synchronizing Automt with Rndom Input 9 August, 14 3 / 13

4 Min results Clssicl synchroniztion Clssicl synchroniztion prolem: Sttes of ojects re unknown, ut we control environment,,, Environment:,,, Finlly, ll ojects re in the sme stte Vldimir Gusev (UrFU) Synchronizing Automt with Rndom Input 9 August, 14 4 / 13

5 Min results Rndomized synchroniztion Rndomized synchroniztion prolem: We do not control environment. It is rndom.,,, 1 Environment: 1 1 Sequence of rndom letters,,, How long do we hve to wit until ll ojects re in the sme stte? Vldimir Gusev (UrFU) Synchronizing Automt with Rndom Input 9 August, 14 5 / 13

6 Min results Rndomized synchroniztion Let Σ e inry lphet {,} Model for the environment: Bernoulli process B(p, q) Letters re drwn independently nd sequentilly letter with the proility p letter with the proility q = 1 p We re interested in the expected numer of letters drwn from B(p,q) until n utomton A is synchronized Is it finite? Sure. Let w e synchronizing word of n utomton A. If (l,l ) is the Prikh vector of w then the expected numer of letters until synchroniztion of A is t most w p l q l Vldimir Gusev (UrFU) Synchronizing Automt with Rndom Input 9 August, 14 6 / 13

7 Min results Automton U n Let n e n odd integer, nd Σ = {,} U n is the miniml utomton of the lnguge Σ n+1 n 1 Σ , Figure: Automton U 7 Lemm A word w is synchronizing if nd only if it rings the stte 1 to the sink stte. Note, the reset threshold of U n is equl to n 1 Vldimir Gusev (UrFU) Synchronizing Automt with Rndom Input 9 August, 14 7 / 13

8 Min results Automton U n Synchroniztion in rndom environment is equivlent to rndom wlk from the stte 1 until sorption in the sink stte Recll, P() = p nd P() = q q p q q p p p p q q q q p Figure: Mrkov chin for U 7 Let µ i e the expected numer of steps until sorption in the sink stte µ 8 = µ 7 = pµ +qµ 8 +1 p µ 6 = pµ +qµ Vldimir Gusev (UrFU) Synchronizing Automt with Rndom Input 9 August, 14 8 / 13

9 Min results Automton U n System of liner equtions in the generl cse: Theorem µ 1 = pµ +qµ 1 +1 (1) µ i = pµ i+1 +qµ 1 +1, if 1 i n+1 () +1 (3) µn+3 = pµn+3 +qµn+5 µ i = pµ +qµ i+1 +1, if n+5 i n 1 (4) µ n = pµ +qµ n+1 +1 (5) µ n+1 = (6) The expected numer of letters, tht re drwn from B(p,q), until U n is 1 1 synchronized, is equl to if n is odd, nd is equl to p n+1 q n 1 p n q n if n is even. Vldimir Gusev (UrFU) Synchronizing Automt with Rndom Input 9 August, 14 9 / 13

10 Min results Automton C n The Černý utomton C n hs the lrgest known reset threshold, Figure: The utomton C 7 We hve to del with pirs A pir {s,t} is synchronized y w if δ(s,w) = δ(t,w) Vldimir Gusev (UrFU) Synchronizing Automt with Rndom Input 9 August, 14 1 / 13

11 Min results Automton C n 1,5 1,4 1,3 1, 1,1 9,5 9,4 9,3 9, 9,1 8,5 8,4 8,3 8, 8,1 7,5 7,4 7,3 7, 7,1 6,5 6,4 6,3 6, 6,1 5,5 5,4 5,3 5, 5,1 4,5 4,4 4,3 4, 4,1 3,5 3,4 3,3 3, 3,1,5,4,3,,1 1,5 1,4 1,3 1, 1,1,5,4,3,,1 z Figure: Pir utomton of C 11 Vldimir Gusev (UrFU) Synchronizing Automt with Rndom Input 9 August, / 13

12 Min results Automton C n Theorem (odd) Let n e positive odd integer. The expected numer of letters, tht re drwn from B(p,q), until the pir {1, n 1 } of C n is synchronized, is equl to (n 1)((n 1) +q(3n 5)+4q ) 8pq. Theorem (even) Let n e positive even integer. The expected numer of letters, tht re drwn from B(p,q), until the pir {1, n } of C n is synchronized, is equl to n((n 1)(n )+q(3n 6)+4q ) 8pq. Expected numer of letters until C n is synchronized is t most (n 1)4 8pq Vldimir Gusev (UrFU) Synchronizing Automt with Rndom Input 9 August, 14 1 / 13

13 Conclusion Outline : Notion of rndomized synchroniztion of utomt Automton U n is esy to synchronize in the clssicl setting (n 1 letters) nd is hrd to synchronize in the rndom setting (exponentil numer). Opposite sitution for the Černý utomt Future work : Cler picture of this notion Similr notion for two-sided idel lnguge L: the expected numer of letters you need to drw until you get word from the lnguge L Vldimir Gusev (UrFU) Synchronizing Automt with Rndom Input 9 August, / 13