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Synchronizing Automata andČerný’s Conjecture
Narad Rampersad
Department of MathematicsUniversity of Liège
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A finite automaton
1 2
4 3
a,b
b b
b
a
aa
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Formal definition
I Here a finite automaton is a directed multigraph where
I every vertex has constant out-degree k , andI the outgoing arcs of each vertex are labeled by distinct
elements of a fixed k-element set.
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Terminology
I We call the vertices states and denote the set of statesby Q.
I We call the arcs transitions.I Arcs are labeled by letters.I A sequence of letters is called a word.
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The transition function
I The transition function δ(p, a) = q denotes a transitionfrom p to q labeled by a.
I If w = w1w2 · · ·wn is a word then δ(q,w) is the statereached by starting at q and following the sequence ofarcs labeled w1,w2, . . . ,wn.
I If A ⊆ Q then
δ(A,w) =⋃
q∈A
δ(q,w).
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Synchronizing automata
I A word w such that δ(q,w) = δ(q′,w) for all q, q′ ∈ Qis a reset word.
I An automaton with a reset word is synchronizing.I Equivalently, there exists a state p and a word w such
that δ(Q,w) = {p}.I Given an automaton, can we decide if it is
synchronizing?I If so, can we find the shortest reset word?
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A synchronizing automaton
Reset word: abbbabbba.
1 2
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a,b
b b
b
a
aa
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Applications
I Moore’s Gedanken-experiments (1950’s):I Imagine a satellite orbiting the moon.I Its behaviour while on the dark side of the moon cannot
be observed.I When control is reestablished, we wish to reset the
system to a particular configuration.
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Applications
I Robotics (Natarajan 1980’s):I Imagine parts arriving on an assembly line with
arbitrary orientations.I The parts must be manipulated into a fixed orientation
before proceeding with assembly.
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Černý’s Conjecture
Černý’s Conjecture (1964)
The shortest reset word of any synchronizing automatonwith n states has length at most (n− 1)2.
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Černý’s construction
Reset word: (abn−1)n−2a (length (n− 1)2).
1 2
n 3
a,b
b b
b
a
aab
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Partial results
I E.g., Kari (2003) verified the conjecture forsynchronizing automata whose underlying digraphs areEulerian.
I Conjecture verified for several other classes ofsynchronizing automata.
I Steinberg (preprint) unified and simplified many ofthese proofs.
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Best known upper bound
I M is a synchronizing automaton:I There are sets Q = P1,P2, . . . ,Pt , and words
w1,w2, . . . ,wt−1, such thatI δ(Pi,wi) = Pi+1, for i = 1, . . . , t − 1;I |Pi| > |Pi+1|, for i = 1, . . . , t − 1;I |Pt | = 1.
I w = w1w2 · · ·wt−1 is a reset word for M.
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An example
Reset word: a bbba bbba.
1 2
4 3
a,b
b b
b
a
aa
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An example
Reset word: a bbba bbba.
1 2
4 3
a,b
b b
b
a
aa
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An example
Reset word: a bbba bbba.
1 2
4 3
a,b
b b
b
a
aa
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An example
Reset word: a bbba bbba.
1 2
4 3
a,b
b b
b
a
aa
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An example
Reset word: a bbba bbba.
1 2
4 3
a,b
b b
b
a
aa
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An example
Reset word: a bbba bbba.
1 2
4 3
a,b
b b
b
a
aa
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An example
Reset word: a bbba bbba.
1 2
4 3
a,b
b b
b
a
aa
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An example
Reset word: a bbba bbba.
1 2
4 3
a,b
b b
b
a
aa
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An example
Reset word: a bbba bbba.
1 2
4 3
a,b
b b
b
a
aa
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An example
Reset word: a bbba bbba.
1 2
4 3
a,b
b b
b
a
aa
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An example
Reset word: a bbba bbba.
1 2
4 3
a,b
b b
b
a
aa
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The greedy algorithm
Algorithm to find reset word wSet P1 = Q and t = 1.
While |Pt | > 1:Find a smallest word wt such that |δ(Pt ,wt)| < |Pt |.Set Pt+1 = δ(Pt ,wt) and increment t .
Return w = w1w2 · · ·wt−1.
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Length of the reset word found
I What is the maximum length of w found by the greedyalgorithm?
I In the worst case, |Pi| − |Pi+1| = 1, so that t = n.I Consider a generic step k : i.e., Pk and wk such that|δ(Pk ,wk )| < |Pk |.
I What is the longest that wk can be?
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Length of the reset word found
I Let wk = a1a2 · · · am+1 (the a’s letters).I There are sets Pk = A1,A2, . . . ,Am+2 such that
I δ(Ai, a1) = Ai+1 for i = 1, . . . ,m + 1;I |Ai| = |Ai+1| for i = 1, . . . ,m;I |Am+1| > |Am+2|.
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Length of the reset word found
I For i = 1, . . . ,m + 1,
|δ(Ai, ai · · · am+1)| < |Ai|.
I Thus there exists qi, q′i ∈ Ai such that
δ(qi, ai · · · am+1) = δ(q′i, ai · · · am+1).
I To each Ai, associate the set Bi = {qi, q′i}.
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Length of the reset word found
I Note that Bi ⊆ Ai.I Furthermore, for i < j , Bj 6⊆ Ai.I Otherwise, we would have a shorter word
w ′k = a1 · · · ai−1aj · · · am+1 such that |δ(Pk ,w ′k )| < |Pk |.
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Length of the reset word found
I Let Ai denote the complement of Ai, i.e., the set Q \ Ai.I We thus have
I Bi ∩ Ai = ∅ for i = 1, . . . ,m;I Bj ∩ Ai 6= ∅ for i < j .
I What is the largest that m can be subject to theseconstraints?
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A result from extremal set theory
Theorem (Frankl 1982)
Let A1, . . . ,Am be sets of size r and let B1, . . . ,Bm be sets ofsize s such that
(a) Ai ∩ Bi = ∅ for i = 1, . . . ,m;
(b) Ai ∩ Bj 6= ∅ if i < j .
Then m ≤(r+s
s
).
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A bound on the length of the reset word
I Let |Q| = n. Then |Ai| = n− k (since |Ai| = k) and|Bi| = 2 for i = 1, . . . ,m.
I By Frankl’s result, m ≤(n−k+2
2).
I Total length of the reset word at mostn∑
k=2
(n− k + 2
2
)= n3 − n
6 .
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Running time of the algorithm
I Originally conjectured by Fischler and Tannenbaum(1970) and (independently) by Pin (1981).
I After hearing Pin’s 1981 talk, Frankl proved thenecessary combinatorial result (independentlyrediscovered by Klyachko, Rystsov, and Spivak (1987)).
I Eppstein (1990) showed how to implement the greedyalgorithm in O(n3 + kn2) time.
I Greedy algorithm does not find a shortest reset word.
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Finding a reset word of a given length
SYNCWORDGiven an automaton A and a positive integer k , does A havea reset word of length at most k?
I Clearly in NP since it suffices to “guess” a reset wordof length at most min{(n3 − n)/6, k}.
I Eppstein showed it is NP-complete.
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Finding a shortest reset word
MIN-SYNCWORDGiven an automaton A and a positive integer k , does A havea shortest reset word of length k?
I Olschewski and Ummels (preprint) showed it isDP-complete.
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The class DP
I DP consists of all languages L such that L = L1 \ L2 forsome languages L1, L2 in NP.
I A DP-complete problem is both NP-hard andcoNP-hard.
I The canonical DP-complete problem is:
SAT-UNSATGiven CNF formulae φ and ψ, is φ satisfiable and ψunsatisfiable?
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DP-completeness
I MIN-SYNCWORD clearly in DP, since it is the differenceof SYNCWORD and
{(A, k) : k > 0 and (A, k − 1) ∈ SYNCWORD}.
I To show DP-hardness, reduce from SAT-UNSAT.
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Approximating the shortest reset word
Thereom (Berlinkov (preprint))
Unless P = NP, there is no polynomial-time algorithm toapproximate the minimum length of a reset word for a givenautomaton within a constant factor.
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Synchronizing colouring
I Start with a strongly connected directed multigraph Gwhere every vertex has constant out-degree k .
I Is it possible to assign labels to the arcs so that Gbecomes synchronizing?
I If so, then G has a synchronizing colouring.
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The road colouring problem
I Can graphs with synchronizing colourings becharacterized?
I A graph is aperiodic if the gcd of the lengths of all ofits cycles is 1.
I It is not hard to show that aperiodicity is a necessarycondition.
I Adler and Weiss (1970) conjectured that it is also asufficient condition.
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The resolution of the problem
Theorem (Trahtman 2007)
Let G be a strongly connected directed multigraph whereevery vertex has constant out-degree k . Then G has asynchronizing coloring if and only if the the gcd of thelengths of all of its cycles is 1.
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For further reading
I The literature on synchronizing automata is huge. Formore information, see:
I Volkov’s 2008 survey:http://csseminar.kadm.usu.ru/tarragona_volkov2008.pdf
I Jean-Eric Pin’s webpage:http://www.liafa.jussieu.fr/~jep/Problemes/Cerny.html
I Avraham Trahtman’s webpage:http://u.cs.biu.ac.il/~trakht/syn.html
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The End