Algebraic synchronization criterion and computing reset words
aa r X i v : . [ c s . F L ] D ec Algebraic synchronization criterionand computing reset words
Mikhail Berlinkov and Marek Szykuła Institute of Mathematics and Computer Science, Ural Federal University, Russia Institute of Computer Science, University of Wrocław, Poland
Abstract.
We refine a uniform algebraic approach for deriving upper bounds on reset thresh-olds of synchronizing automata. We express the condition that an automaton is synchronizingin terms of linear algebra, and obtain upper bounds for the reset thresholds of automata witha short word of a small rank. The results are applied to make several improvements in thearea.We improve the best general upper bound for reset thresholds of finite prefix codes (Huffmancodes): we show that an n -state synchronizing decoder has a reset word of length at most O ( n log n ) . In addition to that, we prove that the expected reset threshold of a uniformlyrandom synchronizing binary n -state decoder is at most O ( n log n ) . We also show that forany non-unary alphabet there exist decoders whose reset threshold is in Θ ( n ) .We prove the Černý conjecture for n -state automata with a letter of rank at most √ n − .In another corollary, based on the recent results of Nicaud, we show that the probabilitythat the Černý conjecture does not hold for a random synchronizing binary automaton isexponentially small in terms of the number of states, and also that the expected value of thereset threshold of an n -state random synchronizing binary automaton is at most n / o (1) .Moreover, reset words of lengths within all of our bounds are computable in polynomialtime. We present suitable algorithms for this task for various classes of automata, such as(quasi-)one-cluster and (quasi-)Eulerian automata, for which our results can be applied. Keywords:
Černý conjecture, Eulerian automaton, Huffman code, one-cluster automaton,prefix code, random automaton, reset word, reset threshold, synchronizing automaton
We deal with deterministic finite automata ( DFA ) A = ( Q, Σ, δ ) , where Q is a non-empty set ofstates , Σ is a non-empty alphabet , and δ : Q × Σ Q is the complete transition function . We extend δ to Q × Σ ∗ and Q × Σ ∗ as usual, and for the image (resp. preimage) of a set S under a word w we write shortly S.w (resp.
S.w − ). We denote Σ ≤ c = { w ∈ Σ ∗ : | w | ≤ c } , the set of all words over Σ of length at most c . The empty word is denoted by ε . Throughout the paper, by n we denote thecardinality | Q | , and by k we denote | Σ | .A word w compresses a subset S ⊆ Q if | S.w | < | S | . Then we say that S is compressible . The rank of a word w is | Q.w | . A reset word or a synchronizing word is a word w ∈ Σ ∗ of rank , thatis, w takes the automaton to a particular state no matter of the current state. An automaton iscalled synchronizing if it possesses a reset word. An example of a synchronizing automaton from theČerný series [14] is presented in Fig. 1(left). One can verify that its shortest reset word is ba ba b .The length of the shortest reset word is called the reset threshold and is denoted by rt( A ) . M. Berlinkov and M. Szykuła b bba aab a . . . . . . . . Fig. 1.
The automaton C and the associated Markov chain for P ( a ) = 0 . , P ( b ) = 0 . . Synchronizing automata serve as transparent and natural models of various systems in manyapplications in various fields, such as coding theory, DNA-computing, robotics, testing of reactivesystems, and theory of information sources. They also reveal interesting connections with symbolicdynamics, language theory, and many other parts of mathematics. For a detailed introduction tothe theory of synchronizing automata we refer the reader to the surveys [22,31], and for a reviewof relations with coding theory to [20].In various applications, reset words allow to reestablish the control under the system modeled byan automaton. The reset threshold of an automaton serves as a natural measure of synchronization.Naturally, the shorter reset word the better, so from both theoretical and practical points of viewit is important to compute the reset threshold, and shortest or short enough reset words.In 1964 Černý [14] constructed for each n > a synchronizing automaton C n with n states and2 input letters whose reset threshold is ( n − . The automaton C is shown in Fig. 1(left). Soonafter that he conjectured that those automata represent the worst possible case, thus formulatingthe following hypothesis: Conjecture (Černý).
Each synchronizing automaton A with n states has a reset word of lengthat most ( n − , i.e. rt( A ) ≤ ( n − . By now, this simply looking conjecture is arguably the most longstanding open problem in thecombinatorial theory of finite automata. Moreover, the best upper bound known so far for the resetthreshold of a synchronizing n -state automaton is equal to n − n − (for n ≥ ) and so is cubic in n (see Pin [28]). Thus it is of certain importance to prove better specific upper bounds for variousimportant classes of synchronizing automata.In this paper, we improve several results concerning reset thresholds. First, we express thecondition that an automaton is synchronizing in terms of linear algebra, and derive upper boundsfor automata with a word of a small rank (Section 2). Then, we apply the results to improve upperbounds in several cases.We apply the results to improve upper bounds in several cases. In Section 4 we show that theČerný conjecture holds for automata with a letter of rank √ n − , which improves the previouslogarithmic bound [27]. Also, basing on the recent results of Nicaud [25], we show that the Černýconjecture holds for a random synchronizing binary automaton with probability exponentially (in n ) close to 1, and that the expected reset threshold is at most n / o (1) .The next important application of our results is an upper bound for the length of the shortestreset words of decoders of finite prefix codes (Huffman codes), which are one of the most popularmethods of data compression. One of the problems with compressed data is the reliability in case of lgebraic synchronization criterion and computing reset words 3 presence of errors in the compressed text. Eventually, a single error may possibly destroy the wholeencoded string. One of the solutions to this problem (for Huffman codes) is a use of codes whosedecoders can be synchronized by a reset word, regardless of the possible errors. Then, by insertingreset words into compressed data, we make the data error-resistant to some extent.The reset thresholds of binary Huffman codes was first studied by Biskup and Plandowski [10,11],who proved a general upper bound of order O ( n log n ) . They also showed that a word of this lengthcan be computed in polynomial time. The bound was later improved to O ( n ) for a wider classof one-cluster automata [3]. In Section 5 we prove an upper bound of order O ( n log n ) . Next,we consider random decoders, and show that the expected reset threshold of a uniformly randomsynchronizing n -state binary decoder is at most O ( n log n ) . We also show a series of decoders withlinear reset thresholds over any alphabet of size at least 3. Such series were known only for a binaryalphabet [11].Unlike the general case, the Černý conjecture has been approved for various classes of automatasuch as circular [15], Eulerian [21], and one-cluster automata with prime length cycle [30]. Later,specific quadratic upper bounds for some generalizations of these classes were obtained in [3,4,7].However, no efficient algorithm for finding reset words with lengths within the specified boundshas been presented for these classes. Moreover, there is no hope to get a polynomial algorithmfor finding the shortest reset words in the general case, since this problem has been shown to be FP NP[log] -hard [26]. Also, unless
P = NP , there is no polynomial algorithm for computing the resetthreshold for a given automaton within the approximation ratio n − ε for any ε > , even in thecase of a binary alphabet [18] (cf. also [8,9,19]).In Section 3 we present polynomial algorithms for finding reset words of length guaranteed to bewithin the proven bounds. In particular, our algorithms can be applied to the classes of decoders offinite prefix codes, and also to generalized classes of quasi-Eulerian and quasi-one-cluster automata.Since from our results it is possible to derive the bounds from [3,4,7,12,21,29,30]), our algorithmsapply to them as well.A preliminary version of some of these results previously appeared in [5]. In this section we refine some results from [7], formulate the algebraic synchronization criterion,and derive upper bounds for reset thresholds of automata with a word of a small rank. For thispurpose, we associate a natural linear structure with an automaton A . By R n we denote the real n -dimensional linear space of row vectors. Without loss of generality, we assume that Q = { , , . . . , n } and then assign to each subset K ⊆ Q its characteristic vector [ K ] ∈ R n , whose i -th entry is 1 if i ∈ K , and 0, otherwise. For q ∈ Q we write [ q ] instead of [ { q } ] to simplify the notation. By h S i wedenote the linear span of S ⊆ R n . The n × n identity matrix is denoted by I n .Each word w ∈ Σ ∗ corresponds to a linear transformation of R n . By [ w ] we denote the matrix ofthis transformation in the standard basis [1] , . . . , [ n ] of R n . For instance, if A = C from Figure 1(left), then [ a ] = (cid:18) (cid:19) , [ b ] = (cid:18) (cid:19) , [ ba ] = (cid:18) (cid:19) . Clearly, the matrix [ w ] has exactly one non-zero entry in each row. In particular, [ w ] is row stochastic ,that is, the sum of entries in each row is equal to . In virtue of row-vector notation (apart from [7]),we get that [ uv ] = [ u ][ v ] for every two words u, v ∈ Σ ∗ . By [ w ] T we denote the transpose of the M. Berlinkov and M. Szykuła matrix [ w ] . One easily verifies that [ S.w − ] = [ S ][ w ] T . Let us also notice that within this definitionthe (adjacency) matrix of the underlying digraph of A is equal to P a ∈ Σ [ a ] .Recall that a word w is a reset word if q.w − = Q , for some state q ∈ Q . Thus, in the language oflinear algebra, we can rewrite this fact as [ q ][ w ] T = [ Q ] . For two vectors g , g ∈ R n , we denote theirusual inner (scalar) product by ( g , g ) . We say that a vector (matrix) is positive ( non-negative )if it contains only positive (non-negative) entries. Let p ∈ R n + be a positive row stochastic vector.Then ([ Q ] , p ) = 1 , and a word w is a reset word if and only if ([ q.w − ] , p ) = ([ q ][ w ] T , p ) = ([ q ] , p [ w ]) = 1 . Now we need to recall a few properties of Markov chains. A
Markov chain of an automaton A is the random walk process of an agent on the underlying digraph of A where each time anedge labeled by a i is chosen according to a given probability distribution P : Σ R . The matrix S ( A , P ) = P ki =1 P ( a i )[ a i ] is called the transition matrix of this Markov chain. An example of aMarkov chain associated with the automaton A = C is presented in Figure 1 (right) for P ( a ) =0 . , P ( b ) = 0 . . A non-negative square matrix M is primitive if for some d > , the matrix M d is positive. Call a finite set of words W primitive if the sum of the matrices of words from W isprimitive. It is well known that if A is strongly connected and synchronizing, then the matrix ofthe underlying digraph of A is primitive, and so is the matrix of a Markov chain of A for anypositive probability distribution P (see e.g. [1,2,7]). Proposition 1.
Let M be a row stochastic n × n matrix. Then there exists a stationary distribution α ∈ R n , that is, a non-negative stochastic vector satisfying αM = α . Moreover, if M is primitivethen α is unique and positive. Call a set of words W ⊆ Σ ∗ complete for a subspace V ≤ R n , with respect to a vector g ∈ V , if h g [ w ] | w ∈ W i = V. For a subset S ⊆ Q we define V S = h [ p ] | p ∈ S i ≤ R n .We aim to strengthen [7, Theorem 9]. Namely, we show that the condition that A is synchro-nizing is not necessary if we require completeness for the corresponding set of words, and thatonly completeness with respect to the stationary distribution of A is required. As in [7] we con-struct an auxiliary automaton. We fix two positive integers d , d and two non-empty sets of words W ⊆ Σ ≤ d , W ⊆ Σ ≤ d . Consider the automaton A c ( W , W ) = ( R, W W , δ A c ) , where R = { q.w | q ∈ Q, w ∈ W } and W W = { w w ∈ Σ ∗ | w ∈ W , w ∈ W } . The transitionfunction δ A c is defined in compliance with the actions of words in A , i.e. δ A c ( q, w ) = δ ( q, w ) , forall q ∈ R and w ∈ W W . Note that δ A c is well defined because q.w ∈ R for all q ∈ Q and w ∈ Σ A c .Without loss of generality we may assume that R = { , , . . . , | R |} where r = | R | .Let P and P be some positive probability distributions on the sets W and W , respectively,and denote [ P i ] = P w ∈ W i P i ( w )[ w ] for i = 1 , . Then the r × r submatrix formed by the first r rowsand the first r columns of the matrix S ( B , P P ) = [ P ][ P ] = X w ∈ W ,w ∈ W P ( w ) P ( w )[ w ][ w ] lgebraic synchronization criterion and computing reset words 5 is the transition matrix of the Markov chain on A c . By Proposition 1 there exists a steady statedistribution α = α ( A c ) ∈ V R , that is, a stochastic vector (with first r non-negative entries) satisfying αS ( A c , P P ) = α .For a vector g ∈ R n + , by DS( g ) we denote the number of different positive sums of entries of g ,i.e. DS( g ) = |{ ( g, z ) | z ∈ { , } n }| − . Theorem 1.
Let A = ( Q, Σ, δ ) be an automaton and let B = A c ( W , W ) = ( R, W W , δ B ) , be the automaton defined as above. If W W is complete for V R with respect to α , and w ∈ Σ ∗ isa word with Q.w = R , then:1. If x ∈ V R \ h [ R ] i , then there exists w ∈ W W such that ( x, α [ w ]) > ( x, α ) ;2. B is synchronizing and rt( B ) ≤ DS( α ) − ;3. A is synchronizing and rt( A ) ≤ ( | w | + rt( B )( d + d ) ≤ | w | + (DS( α ) − d + d ) if R = Q , α ) − d + d ) if R = Q .Proof. Let x ∈ V R \ h [ R ] i . We have ( x, [ q ]) = ( x, α ) for some q ∈ R. (1)Since [ q ] ∈ V R and W W is complete for V R with respect to α , we can represent it as follows: [ q ] = X w ∈ W ,w ∈ W λ w ,w α [ w ][ w ] for some λ w ,w ∈ R . (2)Multiplying (2) by the vector [ Q ] we obtain q ] , [ Q ]) = X w ∈ W ,w ∈ W λ w ,w α [ w ][ w ] , [ Q ] = X w ∈ W ,w ∈ W λ w ,w . (3)Multiplying (2) by the vector x we obtain ([ q ] , x ) = X w ∈ W ,w ∈ W λ w ,w α [ w ][ w ] , x . (4)Arguing by contradiction, suppose ( x, α [ u ][ u ]) = ( x, α ) for every u ∈ W , u ∈ W . Then by(3) and (4) we get that ([ q ] , x ) = ( x, α ) contradicts (1). Hence ( x, α [ u ][ u ]) = ( x, α ) , for some u ∈ W , u ∈ W .Since α [ P ][ P ] = α , we have either ( x, α [ u ][ u ]) > ( x, α ) or ( x, α [ v ][ v ]) > ( x, α ) for someother v ∈ W , v ∈ W . Thus Claim 1 follows.The proof of Claims 2 and 3 follows from an application of the greedy extension algorithm fromSection 3. ⊓⊔ M. Berlinkov and M. Szykuła
The following properties are easily verified and will be useful:
Remark 1. If W is complete for R n with respect to some vector g , then W W is complete for V R with respect to g . Remark 2.
If for some positive probability distributions on W and W , the set W W is completefor V R with respect to each stationary distribution, then B = A c ( W , W ) is strongly connectedand synchronizing. Remark 3. If B = A c ( W , W ) is strongly connected and W W is complete for V R with respect toa stationary distribution induced by some positive probability distributions on W and W , then W W is complete for V R with respect to any stochastic vector. Criterion 1
Let α be a stationary distribution of the Markov chain associated with a stronglyconnected n -state automaton A by a given positive probability distribution P on the alphabet Σ .Then A is synchronizing if and only if there exists a set of words W which is complete for R n withrespect to α .Proof. If A is synchronizing then for each state q ∈ Q there is a reset word w q such that Q.w q = q .Hence, W = { w q | q ∈ Q } is complete for R n with respect to α , because α [ w q ] = [ q ] .Let us prove the opposite direction. Set W = { ε } , W = Σ ≤ n − , and [ P ] = 1 n n − X i =0 [ P ] i . Then α [ P ] = α , and W is complete for R n with respect to α . Hence A is synchronizing byTheorem 1. ⊓⊔ It is worth mentioning that an equivalent criterion with respect to our case has been indepen-dently obtained in [32] in terms of affine operators via a so called fixed point approach.Now we can provide an upper bound for the reset threshold, if we can find a short word of asmall rank.
Theorem 2.
Let A = ( Q, Σ, δ ) be a synchronizing automaton. Then there is a unique (stronglyconnected) sink component S = ( S, Σ, δ ) . Let w be a word and denote r = | Q.w | . Let < d < n bethe smallest positive integer such that Σ ≤ d is complete for V S with respect to any stochastic vector g ∈ V S and for each q ∈ Q there is a word u q ∈ Σ ≤ d such that q.u q ∈ S ∩ Q.w . Then rt( A ) ≤ ( ( | w | + d )( r − r ) − d if r ≥ ; | w | + ( | w | + d )( r − if r ≤ .Moreover, any pair of states from Q is compressible by a word of length at most | w | + ( | w | + d ) r − r .Proof. Let W = { w } , W = Σ ≤ d , w = w , and let P , P be arbitrary positive distributions on W and W , respectively. We define B = A c ( W , W ) as in Theorem 1, and consider its sink component C = S c ( W , W ) = ( Q C , Σ, W W ) . Clearly Q C = Q.w ∩ S , and W W is complete for V Q C ≤ V S with respect to any stochastic vector g ∈ V Q C . By Criterion 1 we obtain that C is synchronizing.Since for each q ∈ Q.w there is a word u q ∈ W and so w q ∈ W W (a letter of B ) which takes q to Q C , the automaton B is synchronizing. lgebraic synchronization criterion and computing reset words 7 Since B is synchronizing, | Q.w | = r , and | u | ≤ | w | + d for each u ∈ W W , we have that rt( A ) ≤ | w | + rt( B )( | w | + d ) . By Pin’s bound for the reset threshold in the general case [28], rt( B ) ≤ r − r − for r ≥ .Since B is synchronizing and there are r − r pairs in Q.w , any pair of states in Q can becompressed by a word of length at most | w | + ( | w | + d ) r − r . ⊓⊔ Throughout this section suppose we are given a strongly connected automaton A , a word w such that Q.w = R for some R ⊆ Q , a non-empty polynomial set of words W with a positivedistribution P , and a set of words W with a positive distribution P , which satisfy Theorem 1.Consider the case when W is of polynomial size. Then we can calculate the dominant eigenvector α ∈ R n of the matrix [ P ][ P ] . Under certain assumptions on rationality of the distributions, it canbe done in polynomial time. Next, depending on whether the bound is obtained by Theorem 2or Claim 2 of Theorem 1, we use either a greedy compressing algorithm (such as in [16]), or thefollowing greedy extension algorithm , respectively. The Greedy Extension Algorithm . We start from x = [ q ] for q ∈ R and by Claim 2of Theorem 1 find u ∈ W W such that ( x , α [ u ]) > ( x , α ) . For i = 0 , , . . . following this wayuntil x i ∈ h [ R ] i , find for x i +1 = x i [ u i ] t a word u i +1 ∈ W W such that ( x i +1 , α [ u i +1 ]) > ( x i +1 , α ) .Since x i is a 1-0 vector, we need at most DS( α ) − steps until x i = [ q ]([ u i u i − . . . u ]) t = [ R ] . Asthe result we return the word w u i u i − . . . u . Notice that in the case when R = Q we can choose q such that for some letter a ∈ Σ , we have | q.a − | > and set u = a . ⊓⊔ The problem is that usually W is given by Σ ≤ d for some d = poly( n ) . The following reductionprocedure allows to replace potentially exponential set W with a polynomial set of words W , whosethe longest words are not longer than those of W . The Reduction Procedure . The procedure takes a number d , and returns a polynomialsubset W ⊆ Σ ≤ d such that h W i = h Σ ≤ d i and the maximum length of words from W is the shortestpossible.We start with V = { I n } and W = { ε } . In each iteration i ∈ { , , . . . } we first set V i +1 = V i .Then we subsequently check each letter a ∈ Σ and each word u ∈ W of length i : If the matrix [ ua ] does not belong to the subspace V i +1 , we add the word ua to W and the matrix [ ua ] to the basisof V i +1 . We stop the procedure at the first iteration where nothing is added.Since in an i -th iteration we have considered a ∈ Σ and u ∈ W of length less than i in theprevious iterations, by induction we get V i = h I n ( W ∩ Σ ≤ i ) i = h I n Σ ≤ i i . It follows from the ascending chain argument (see e.g. [30,21]) that for some j < n we have V j = V j +1 = . . . . Thus the procedure is stopped at the first such j , and j ≤ min { d, n − } . We get that h W i = V j = h Σ d i . Since in each step we add only independent matrices as the basis of V i +1 , we get | W | = dim( V j ) . Also the lengths of words in W are at most j ≤ min { d, n − } . ⊓⊔ Using the reduction procedure for total completeness we can replace Σ d from Theorem 2 bya polynomial W , which is also complete for V S with respect to any stochastic vector g ∈ V S . M. Berlinkov and M. Szykuła
Hence, this yields a polynomial time algorithm finding reset words of lengths within the bound ofTheorem 2.In some situations we are interested only in completeness with respect to a given vector α .Then we can find a reduced set W of potentially shorter words than that obtained by the generalreduction procedure. The Reduction Procedure for α -Completeness . The procedure takes a number d and avector α , and returns a polynomial subset W ⊆ Σ ≤ d such that h αW i = h αΣ ≤ d i and the maximumlength of words from W is the shortest possible.We just follow the general reduction procedure, where instead of matrix spaces we considervector spaces. It is enough to replace I by α , and we obtain h αW i = V j = h αΣ ≤ d i . ⊓⊔ Remark 4.
Instead of Σ ≤ d the reduction procedures can also reduce any set of words W ′ ⊂ Σ ∗ that is factor-closed. A set of words W ′ is factor-closed if uvw ∈ W ′ implies that uw ∈ W ′ , for each u, v, w ∈ Σ ∗ .This follows since the ascending chain argument still holds. If V i = V i +1 then also V i = h W ′ i .Assume for a contrary that V i < h W ′ i , and let ua ∈ W ′ be a shortest word such that I n [ ua ] isindependent to I n W . Then there is some prefix w ∈ W of u , and u = wva . Since u was a shortestword, wv is dependent to I n W , so h I n W i = h I n ( W ∪ [ wv ]) i < h I n ( W ∪ [ wva ]) = h I n ( W ∪ [ wa ]) . Since W ′ is factor-closed, wa ∈ W ′ , and it was considered in the reduction procedure and added to W – a contradiction. ⊓⊔ The following procedure finds a polynomial subset W ⊆ W such that W W is still primitiveunder the restriction to V R , and the words in W are as short as possible. The Reduction Procedure for Primitive Sets . As the input the procedure takes a setof words W and a number d > such that Σ ≤ d W is primitive when restricted to V R where R = Q.W , and returns a polynomial subset W ⊆ Σ ≤ d such that W W is also primitive for R .We follow the reduction procedure with the following modification: Instead of adding a word ua to W if [ ua ] does not belong to the current subspace V i +1 , we add ua if for some w ∈ W there isa non-zero entry in [ ua ][ w ] such that this entry is zero in all matrices [ w ] for w ∈ W W . We stopthe procedure as soon as the set of words W W restricted to V R becomes primitive.To check whether W W is primitive, since the exponent of r × r primitive matrix is at most ( r − + 1 (see [2]), it is enough to check that the (( r − + 1) -th power of the sum of all matrices [ w ] for w ∈ W W is positive. Since in each step we make positive at least one entry in this sum, weneed at most ( r − + 1 steps in total. ⊓⊔ Now, given some sets W and W = Σ ≤ d , we can first find W ⊆ W such that W W is primitivefor V R . Then, we can choose some positive probability distribution on W , which induces a uniquestationary distribution β . We can also find W ′ ⊆ W complete with respect to β . The problem hereis that, for the set ( W ∪ W ′ ) W there is possibly no positive probability distribution inducing thestationary distribution β . In order to apply Theorem 1, we need to show that ( W ∪ W ′ ) W can becomplete with respect to its stationary distribution. The following theorem solves this problem. Theorem 3.
Let A be a strongly connected automaton, and the sets W , W be chosen so that thematrix of the underlying digraph of B = A c ( W , W ) is primitive. Let α be a stationary distributionof the Markov chain associated with B for arbitrary positive distributions P , P on W , W , respec-tively. Then, for each set of words W which is complete for R n with respect to α , the automaton C = A c ( W , W ∪ W ) is synchronizing. lgebraic synchronization criterion and computing reset words 9 Proof.
For each ≤ δ < we define S ( δ ) = ((1 − δ )[ P ] + δ | W | X w ∈ W [ w ])[ P ] . Clearly, S ( δ ) is a positive probability distribution on ( W ∪ W ) W for each < δ < , and on W W for δ = 0 . Because the matrix of the underlying digraph of B is primitive, for each ≤ δ < there is a unique stationary distribution β ( δ ) such that β ( δ ) S ( δ ) = β ( δ ) or, equivalently, β ( δ ) isthe unique stochastic solution x of the equation x ( S ( δ ) − I n ) = (0 , , . . . , . Therefore β ( δ ) = ˜ S − ( δ )(1 , , . . . , , where ˜ S ( δ ) is the invertible matrix obtained from the matrix S ( δ ) − I n by replacing the first row by the vector of all -s. Note that β (0) = α , and β ( δ ) is(component wise) continuous in [0 , .Since W is complete with respect to α , there are words w , w , . . . , w n ∈ W such that the squarematrix D = ( αw i ) i ∈{ , ,...,n } has rank n . For ≤ δ < define the matrix D δ = ( β ( δ ) w i ) i ∈{ , ,...,n } and consider the function φ ( δ ) = det( D δ ) . Since β ( δ ) is continuous in [0 , , φ ( δ ) is also continuousin [0 , . Since φ (0) = det( D ) = 0 , we get that φ ( δ ′ ) = 0 for some < δ ′ < . Hence W is completefor R n with respect to β ( δ ′ ) . Since β ( δ ′ ) is the stationary distribution of the Markov chain definedon ( W ∪ W ) W by the positive probability distribution S ( δ ′ ) , by Theorem 1 we obtain that theautomaton C is synchronizing. ⊓⊔ Let α be the probability distribution on Σ ≤ d induced by a probability distribution P : Σ R + onthe alphabet, that is, [ P ] = n P di =0 [ P ] i . Suppose that d < poly( n ) is such that Σ ≤ d is completefor R n with respect to α . Using the reduction procedure, we can construct a set U of at most n words such that h αU i = h αΣ ≤ d i = R n . However, α is not necessarily the stationary distribution for some positive probability distributionon U . The following lemma solves this problem. Lemma 1.
Let W = { au | u ∈ Suff( U ) , a ∈ Σ } , where Suff( U ) is the set of proper suffixes of U .Then there exists a positive probability distribution on W such that α is the corresponding stationarydistribution.Proof. Since W is complete with respect to α , following the proof of Theorem 1 for each x ∈ R n \ h [ Q ] i , there exists w ∈ W such that ( x, α [ w ]) = ( x, α ) . Suppose that w is a shortest word from W with this property. If ( x, α [ w ]) > ( x, α ) then we have found an extension word from W . Supposethat ( x, α [ w ]) < ( x, α ) . Clearly ≤ | w | ≤ d , and w = au for a ∈ Σ and u ∈ Σ ≤ d − . Since ( x, α [ u ]) = ( x, α [ P ][ u ]) = P ( a )( x, α [ w ]) + X b ∈ Σ,b = a P ( b )( x, α [ bu ]) , we get that either ( x, α [ u ]) < ( x, α ) or ( x, α [ bu ]) > ( x, α ) for some b = a . Since w ∈ W , we have u ∈ Suff( U ) and so bu ∈ W . If ( x, α [ u ]) < ( x, α ) then u = ε , so u ∈ W , and u is a shorterword with ( x, α [ u ]) = ( x, α ) , which contradicts the choice of w . Therefore by [7, Theorem 13]the automaton B = A c ( { ε } , W ) is synchronizing and α is the stationary distribution for someprobability distribution on W . ⊓⊔ As an application we get a polynomial algorithm for finding a reset word for the class of quasi-Eulerian automata, a generalization of Eulerian automata. We call an automaton A quasi-Eulerian with respect to an integer c ≥ if it satisfies the following two conditions:1. There is a subset E c ⊆ Q containing n − c states such that only one of these states, say s , canhave incoming edges from the set Q \ E c .2. There exists a positive probability distribution P on Σ such that the columns of the matrix [ P ] that correspond to the states from E c \ { s } sum up to 1.Within this definition, for c = 0 we get so-called pseudo-Eulerian automata, and if additionally P is uniform on Σ , then we get Eulerian automata. The upper bound n − n − on thereset thresholds of Eulerian automata was found by Kari [21], and extended to the class of pseudo-Eulerian automata by Steinberg [29]. These results were generalized in [7, Corollary 11] by showingthe upper bound c ( n − c + 1)( n − for the class of quasi-Eulerian automata with respect to anon-negative integer c . The following theorem gives a polynomial time algorithm for finding resetwords satisfying these bounds. Theorem 4.
Given a synchronizing automaton A which is quasi-Eulerian with respect to an in-teger c ≥ , there is a polynomial time algorithm for finding a reset word of length at most: ( c ( n − c + 1) d if c > n − d if c = 0 , where d ≤ n − is the smallest integer such that Σ ≤ d is complete.Proof. First we need to calculate a stationary distribution α , which has n − c equal entries. Forthis purpose, for each of the (cid:0) nn − c (cid:1) ways of choosing the set E c containing n − c states, we find asolution of the following task of linear programming: α [ P ] = α, ([ Q ] , α ) = 1 ,α p = α q for each p ∈ E c ,P ( a ) > for each a ∈ Σ ; with the variable set { P ( a ) | a ∈ Σ } , { α p | p ∈ Q } , and q is an arbitrary state from E c . If there is a solution ( α, P ) , then α is the stationary distributionfor the positive probability distribution P on the alphabet and it has at least n − c equal entries.Since (cid:0) nn − c (cid:1) is polynomial and linear programming is solvable in polynomial time, such a solutioncan be found in polynomial time.Next, according to the reduction procedure for α -completeness we can find a polynomial set ofwords W ′ ⊆ Σ ≤ d which is complete for R n with respect to α . Due to Lemma 1 we can change theset W ′ to a set W of polynomial size preserving the stationary distribution α and then use thegreedy extension algorithm to find a reset word of the proposed lengths. ⊓⊔ lgebraic synchronization criterion and computing reset words 11 The underlying digraph of a letter a ∈ Σ is the digraph with edges labeled by a . Every connectedcomponent, called cluster , in the underlying digraph of a letter has exactly one cycle, and possiblesome trees rooted on this cycle. An automaton A = ( Q, Σ, δ ) is called one-cluster if there is aletter a ∈ Σ whose underlying digraph has only one cluster. An automaton A is quasi-one-cluster with respect to an integer c ≥ if it has a letter whose underlying digraph has a cluster suchthat there are at most c states in the cycles of all other clusters. Clearly, one-cluster automataare quasi-one-cluster with respect to c = 0 . An automaton A is circular is it has a letter whoseunderlying digraph consists of only one cycle of length n .The Černý conjecture was proved for circular automata [15], and for one-cluster automata withprime length cycle [30]. Also, quadratic bounds for the reset thresholds in the general case of one-cluster automata were presented [4,3,29,12]. In [7] the upper bound c (2 n − c − n − c + 1) wasproved for quasi-one-cluster with respect to c .The following theorem gives a polynomial algorithm finding a reset word for quasi-one-clusterautomata, whose length is of the mentioned bounds. Theorem 5.
Let A be a synchronizing automaton that is quasi-one-cluster with respect to a letter a and c ≥ . Let C be the largest cycle of a and h be the maximal height of the trees labeled by a .Let W = { a h + i | i ∈ { , . . . , | C | − }} . Then there is a polynomial algorithm for finding a resetword for A of length at most ( c (2 n − c )( n − c + 1) if c > n − r )( n − if c = 0 , where r is the smallest dimension of h W β i for β ∈ V C \ h [ C ] i . In particular, if | C | is prime then r = | C | .Proof. We can assume that A is strongly connected; otherwise, we can use the same technique asin Theorem 2.Let us define W = Σ ≤ n − r +1 for c = 0 and W = Σ ≤ n − otherwise. It is proved in [30] that forone-cluster automata each non-trivial subset of S ⊆ C can be extended to a bigger one by a wordfrom W W . Hence due to the greedy extension algorithm the induced automaton is synchronizingand W W is complete for V C with respect to any stochastic vector from V C . Thus in both caseswe get that W W is complete for V Q.a h . Using the reduction procedure W can be replaced witha polynomial set of words W while keeping the maximal length of words.Let β be the stationary distribution for some positive distribution on W W . Then β p > if andonly if p is a cycle state and β p = β q for each p, q ∈ C . Clearly DS( β ) ≤ c ( | C | + 1) if c > , and DS( β ) = | C | − if c = 0 . According to Theorem 1 the automaton B = A c ( W , W ) is synchronizingand we get that rt( A ) ≤ ( h + 2 c ( | C | + 1)( h + | C | + n ) if c > h + | C | + n − r )( n − if c = 0 . Since the worst case appears when | C | = n − c and h = 0 , the bound follows. Since W and W have polynomial size, a reset word of this bound can be found by the greedy extension algorithmin polynomial time. ⊓⊔ Remark 5.
The algorithm of Theorem 5 works also for the bounds from [12] for one-cluster au-tomata. This follows in the same way as referring to [30] in the proof of the theorem.
Using the new bound, we can extend the class of automata for which the Černý conjecture isproven. In particular, we can improve the result from [27], where the Černý conjecture is provenfor automata with a letter of rank at most n . Corollary 1.
Let A = ( Q, Σ, δ ) be a synchronizing automaton. If there is a letter of rank r ≤ √ n − , then A satisfies the Černý conjecture.Proof. Assume that r ≥ . Using Theorem 2 with d = n − and | w | = 1 we obtain the bound rt( A ) ≤ n ( r − r −
1) + 1 . Then using r ≤ √ n − we obtain rt( A ) < n (cid:18) r − (cid:19) + 1 ≤ n (cid:18) n − − (cid:19) + 1 = ( n − . If r ≤ then the bound of Theorem 2 is n ( r − , which is not larger than ( n − for n ≥ .For n ≤ the Černý conjecture has been verified [23]. ⊓⊔ Another corollary concerns random synchronizing automata. We consider the uniform dis-tribution P s on all synchronizing binary automata with n states, which is formally defined by P s ( A ) = P ( A ) /P n , where P is the uniform distribution on all n n binary automata, and P n is theprobability that a uniformly random binary automaton is synchronizing. It is known that P n tendsto as n goes to infinity ([6,25]).Given an arbitrary small ε > and n large enough, Nicaud [25] proved that a random binaryautomaton with n states has a word of1. length n / ε (1 + o (1)) and rank at most n / ε with probability at least − O (exp( − n ε / ,2. length n / ε (1 + o (1)) and rank at most n / ε with probability at least − O ( n − / ε ) ,The following corollary is a consequence of these results and our Theorem 2. Corollary 2.
For any ε > and n large enough, with probability at least − O (exp( n − ε/ )) , arandom n -state automaton with at least two letters has a reset word of length at most n / ε (1 + o (1)) , and so satisfies the Černý conjecture. Moreover, the expected value of the reset threshold ofa random synchronizing binary automaton is at most n / o (1) .Proof. Because a random binary automaton is synchronizing with high probability, the probabilitiesin (1) and (2) remain asymptotically at least the same for a random binary synchronizing automaton.Now, by applying our Theorem 2 (with d = n − ) to (1) and (2) we get that a random binarysynchronizing automaton has a reset word of1. length n / ε (1 + o (1)) with probability at least − O (exp( − n ε / ,2. length n / ε (1 + o (1)) with probability at least − O ( n − / ε ) . lgebraic synchronization criterion and computing reset words 13 Claim (1) is the first statement of the corollary.Calculating an upper bound of the average of (1), (2) and the general cubic bound applied tothe rest of automata we get: n / ε (1 + o (1)) + n / ε (1 + o (1)) · O ( n − / ε ) +( n − n ) / · O (exp( − n ε / O ( n / ε ) + O ( n / ε )= O ( n / ε ) . ⊓⊔ A finite prefix code (Huffman code) T is a set of N ( N > ) non-empty words { w , . . . , w N } from Σ ∗ , such that no word in T is a prefix of another word in T . A finite prefix code T is maximal ifadding any word w ∈ Σ ∗ to T does not result in a finite prefix code. We consider only maximalprefix codes. A reset word for the code T is a word w such that for any u ∈ Σ ∗ the word uw is asequence of words from T .One can easily see that a finite prefix code corresponds naturally to a DFA called the de-coder , whose states are proper prefixes of words from this code [11]. Formally, for a finite pre-fix code T we have the corresponding decoder A T , which is the DFA ( Q, Σ, δ ) with Q = { q v | v is a proper prefix of a word in T } , and δ defined as follows: δ ( q v , a ) = ( q va if va
6∈ T ; q ε otherwise.If for an edge from a state q v to the root q ε we assign an output symbol associated with the word q v ,the decoder can read a compressed input string and produce the decompressed output according tothe code T . Observe that a reset word w for T is a reset word for the decoder A T , and Q.w = { q ε } .The decoder A T naturally corresponds to a rooted k -ary tree. We say that q ε is the root state. The level of a state q v ∈ Q is | v | , which is also the length of the shortest path from q ε to q v in thedecoder DFA. The height of A T is the maximal level of the states in Q ; this is also the maximallength of words from T . Remark 6. If N = |T | and k = | Σ | , then the number n of states of A T is N − k − . Note that it doesnot depend on the length of the words in the code.In [10,11] Biskup and Plandowski gave an O ( nh log n ) upper bound for the reset thresholds ofbinary decoders, where h is the maximum length of a word from the code. Since h can be linear interms of n , this is an O ( n log n ) general bound. Later, it was improved to O ( n ) in [3]. However, inthe worst case, only decoders with a reset threshold in Θ ( n ) are known [11], and it was conjecturedthat every synchronizing decoder possess a synchronizing word of length O ( n ) . Thus, there was abig gap between the upper and lower bounds for the worst case.The following lemma is a simple generalization of [11, Lemma 14] to k -ary decoders. Lemma 2.
Let A T = ( Q, Σ, δ ) be the n -state k -ary synchronizing decoder of a finite prefix code T . There is a word w of rank r ≤ ⌈ log k n ⌉ and length r .Proof. For a word w , we define Q ( w ) = { q v .w | q v ∈ Q such that no prefix of w maps q v to q ε } . Consider r > . Observe that for two distinct words w , w of the same length r the sets Q ( w ) and Q ( w ) are disjoint. Also the states in Q ( w ) are of level at least r + 1 . If for all words of length r the sets Q ( w ) are non-empty, then there are at least k r states in Q of level at least r + 1 , becausethere are k r different words of length r . Then k r + r + 1 ≤ n and r < log k n . Hence, if r = ⌈ log k n ⌉ then there exists a word w with the empty Q ( w ) . Since any state is mapped to q ε by a prefix of w ,the rank of w is at most | w | = r . ⊓⊔ Since there exists a short word of small rank r , we can apply Theorem 2 to improve the generalupper bounds for the reset threshold of decoders. Corollary 3.
Let A T = ( Q, Σ, δ ) be the n -state k -ary synchronizing decoder of a finite prefix code T , and let r = ⌈ log k n ⌉ . Then1. rt( A T ) ≤ ( r + n − r − r − if r ≥ r + n − r − if r ≤ .
2. Any pair of states from Q is compressible by a word of length at most r + ( r + n − r − r . Proof.
For Claim 1 we apply Theorem 2 with w being the word of rank at most r and length atmost r from Lemma 2, and d = n − . This gives ( r + n − r − r ) − ( n −
1) = r +( r + n − r − r − for r ≥ .We can slightly refine the bound by Pin’s result [27, Proposition 5], which states that if we cancompress Q.w , then a shortest compressing word for
Q.w has length at most | w | + n − | Q.w | + 1 .Thus if | Q.w | = r this is n + 1 , and we end up with r − ( r + n −
1) + ( n + 1) + ( r + n − (cid:18) r − r − (cid:19) = 2 + ( r + n − (cid:18) r − r − (cid:19) . Similar calculation applies when r ≤ .Claim 2 follows directly from Theorem 2. ⊓⊔ If the size k of the alphabet is fixed, Corollary 3 yields O ( n log n ) upper bound for the resetthreshold, and O ( n log n ) upper bound for the length of a word compressing a pair of states of adecoder.Note that the word w from Lemma 2 can be easily computed in O ( n ) time, since there are O ( n ) words of length at most ⌈ log k n ⌉ . Then a reset word within the bound of Corollary 3 can becomputed in polynomial time by the algorithm discussed in Section 3. lgebraic synchronization criterion and computing reset words 15 By a uniformly random n -state decoder we understand a decoder chosen uniformly at random fromthe set of all n -state decoders. We consider here random binary decoders. In [17] it was proved thata uniformly random binary n -state decoder is synchronizing with a probability that tends to as n goes to infinity. Since every binary n -state decoder correspond to a binary tree with n + 1 leaves,the number of all such decoders is the n -th Catalan number. Note that it is known that the averageheight of a binary n -state decoder is asymptotically Θ ( √ n ) [24]. Theorem 6.
The expected reset threshold of a uniformly random synchronizing binary n -state de-coder is at most O ( n log n ) .Proof. Let T be the code (set of codewords) of a uniformly random synchronizing binary n -statedecoder. Let a be the first letter of the alphabet. First we show that the length of the left-mostbranch of the decoder is at most logarithmic with high probability. In other words, the unique wordfrom a ∗ ∩ T has length ℓ ∈ O (log n ) with high probability. Because a uniformly random binary n -state decoder is synchronizing with high probability, these probabilities transfer to a uniformlyrandom synchronizing binary n -state decoder. Note that if a ℓ is in the code, then the the letter a in the decoder is one-cluster with the cycle of length ℓ . Then we apply the upper bound O ( ℓn ) onthe reset thresholds of one-cluster automata whose the length of the cycle is ℓ [3].From the proof of [17, Lemma 5.9] we have that the fraction of decoders whose code contains a ℓ (for ≤ ℓ ≤ n − ) is equal to: ℓ ( n + 1)( n ) . . . ( n − ℓ + 1)(2 n − n − . . . (2 n − ℓ − . Let d be such that ≤ d ≤ n − . Then n − d + 12 n − d − − d − n − d − ≤ / Hence for ℓ ≥ , we have ℓ ( n + 1)( n ) . . . ( n − ℓ + 1)(2 n − n − . . . (2 n − ℓ − ≤ ℓ ℓ +1 . It follows that the probabilities that a ℓ is in the code for ℓ ≥ n is at most O (1 /n ) .For the case ℓ < n , we use the upper bound O ( ℓn ) from [3] and for ℓ ≥ n withprobability O (1 /n ) we use the general upper bound O ( n log n ) for decoders from Corollary 3.Summing up these cases yield our upper bound on the expected reset threshold of the decoder: O ( n log n ) + O ( n log n ) · O (1 /n ) = O ( n log n ) . ⊓⊔ As in the general case, a reset word of average length O ( n log n ) can be computed in polynomialtime. This can be done using the algorithm discussed in Subsection 3.2, which finds a reset wordwithin the bounds for (quasi-)one-cluster automata. Biskup and Plandowski [10,11] presented a series of binary n -state decoders with the reset threshold n − for even n and n − for odd n . However, only binary decoders were studied. Here we presenta series of k -ary decoders for every k ≥ with large reset thresholds. This shows that, in the worstcase, also for arbitrary large alphabets a decoder can have the reset threshold in Θ ( n ) .
01 2 . . . ka a a k . . .a ( k + 1) i ( k + 1) i + 1 ( k + 1) i + 2 . . . ( k + 1) i + ka a a a k . . .a ( k + 1) ℓa . . .n − a a Fig. 2.
The decoder X n,k with reset threshold ⌈ n/ ( k + 1) ⌉ . For k ≥ and n ≥ k + 2 , we define X n,k = ( Q, Σ, δ ) (shown in Fig. 2). Let Q = { , . . . , n − } and Σ = { a , . . . , a k } , and let ℓ = ⌈ n/ ( k + 1) ⌉ − (so ℓ ≥ ). We define δ as follows: For each i with ≤ i ≤ ℓ − and each ≤ j ≤ k , if ( k +1) i + j ≤ n − then we define: δ (( k +1) i +1 , a j ) = ( k +1) i + j .Also for i with ( k + 1) ℓ ≤ i ≤ n − we define δ ( i, a ) = i + 1 . For all the remaining states i andletters a j we set δ ( i, a j ) = 0 . Theorem 7.
The automaton X n,k is synchronizing and its reset threshold is ℓ + 2 = 2 ⌈ n/ ( k + 1) ⌉ .Proof. One verifies that the action of the word a k ( a ) ℓ a k synchronizes the automaton.Let w be a shortest reset word for the automaton. Consider the first two letters w , w of w .Observe that Q.w and Q.w w contains . So Q.w w also contains a state p from { , . . . , k } .State is at the level , and state p is at the level . For all states q < ( k + 1) ℓ the action of allthe letters alternates the parity of the level of the states. Thus two such states with an odd and aneven level cannot be compressed by the action of a single letter. So, to compress { , p } , one of thestates must be first mapped to a state q ≥ ( k + 1) ℓ . The shortest such a path is from to ( k + 1) ℓ lgebraic synchronization criterion and computing reset words 17 labeled by ( a ) ℓ − . Then we need one more letter ( a k ) to synchronize the pair. It follows that and p requires a word of length at least ℓ to be compressed. Hence, the length of w is at least ℓ . ⊓⊔ Using a more sophisticated construction, it is possible to modify our series and obtain decoderswith slightly larger reset thresholds, though still of order n/ ( k + 1) + O (1) . We suppose that thisorder of growth is tight for k ≥ up to the constant within O (1) . We have shown constructible upper bounds for the reset threshold, which turned out to be usefulin several important cases of automata. They are obtained using a uniform approach basing onMarkov chains. In all the cases, there exists a polynomial algorithm finding a reset word of lengthwithin the bounds. Also, note that if the Černý conjecture is true, then our bounds become reduced;in particular, we would get O ( n log n ) for the reset thresholds of decoders of finite prefix codes.The questions about tight bounds in the case of finite prefix codes and random automata remainopen. For finite prefix codes the bound O ( n ) was conjectured. Note that for some applications it canbe also important to get bounds in terms of the maximal length of the words in the code (e.g. [13]).There is also an interesting question about the expected reset threshold of the decoder of a randomfinite prefix code. So far for this case we have only the bound O ( n log n ) , which comes from thebounds for one-cluster automata.Also, there is the open problem of designing a polynomial algorithm finding reset words withinthe bound of [15] for circular automata. Acknowledgments
This work was supported by the Presidential Program “Leading Scientific Schools of the RussianFederation”, project no. 5161.2014.1, the Russian Foundation for Basic Research, project no. 13-01-00852, the Ministry of Education and Science of the Russian Federation, project no. 1.1999.2014/K,and the Competitiveness Program of Ural Federal University (Mikhail Berlinkov), and by the Na-tional Science Centre, Poland under project number 2014/15/B/ST6/00615 (Marek Szykuła).
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