Algebraic synchronization criterion and computing reset words

We refine a uniform algebraic approach for deriving upper bounds on reset thresholds of synchronizing automata. We express the condition that an automaton is synchronizing in terms of linear algebra, and obtain upper bounds for the reset thresholds of automata with a short word of a small rank. The results are applied to make several improvements in the area. We improve the best general upper bound for reset thresholds of finite prefix codes (Huffman codes): we show that an $n$-state synchronizing decoder has a reset word of length at most $O(n \log^3 n)$. In addition to that, we prove that the expected reset threshold of a uniformly random synchronizing binary $n$-state decoder is at most $O(n \log n)$. We also show that for any non-unary alphabet there exist decoders whose reset threshold is in $\varTheta(n)$. We prove the \v{C}ern\'{y} conjecture for $n$-state automata with a letter of rank at most $\sqrt[3]{6n-6}$. In another corollary, based on the recent results of Nicaud, we show that the probability that the \v{C}ern\'y conjecture does not hold for a random synchronizing binary automaton is exponentially small in terms of the number of states, and also that the expected value of the reset threshold of an $n$-state random synchronizing binary automaton is at most $n^{3/2+o(1)}$. Moreover, reset words of lengths within all of our bounds are computable in polynomial time. 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.Comment: 18 pages, 2 figure

On random primitive sets, directable NDFAs and the generation of slowly synchronizing DFAs

We tackle the problem of the randomized generation of slowly synchronizing deterministic automata (DFAs) by generating random primitive sets of matrices. We show that when the randomized procedure is too simple the exponent of the generated sets is O(n log n) with high probability, thus the procedure fails to return DFAs with large reset threshold. We extend this result to random nondeterministic automata (NDFAs) by showing, in particular, that a uniformly sampled NDFA has both a 2-directing word and a 3-directing word of length O(n log n) with high probability. We then present a more involved randomized algorithm that manages to generate DFAs with large reset threshold and we finally leverage this finding for exhibiting new families of DFAs with reset threshold of order $\Omega(n^2/4)$.Comment: 31 pages, 9 figures. arXiv admin note: text overlap with arXiv:1805.0672

On Randomized Generation of Slowly Synchronizing Automata

Motivated by the randomized generation of slowly synchronizing automata, we study automata made of permutation letters and a merging letter of rank n-1 . We present a constructive randomized procedure to generate synchronizing automata of that kind with (potentially) large alphabet size based on recent results on primitive sets of matrices. We report numerical results showing that our algorithm finds automata with much larger reset threshold than a mere uniform random generation and we present new families of automata with reset threshold of Omega(n^2/4) . We finally report theoretical results on randomized generation of primitive sets of matrices: a set of permutation matrices with a 0 entry changed into a 1 is primitive and has exponent of O(n log n) with high probability in case of uniform random distribution and the same holds for a random set of binary matrices where each entry is set, independently, equal to 1 with probability p and equal to 0 with probability 1-pwhen np-log n - > infty as n - > infty

On Synchronizing Colorings and the Eigenvectors of Digraphs

An automaton is synchronizing if there exists a word that sends all states of the automaton to a single state. A coloring of a digraph with a fixed out-degree k is a distribution of k labels over the edges resulting in a deterministic finite automaton. The famous road coloring theorem states that every primitive digraph has a synchronizing coloring. We study recent conjectures claiming that the number of synchronizing colorings is large in the worst and average cases. Our approach is based on the spectral properties of the adjacency matrix A(G) of a digraph G. Namely, we study the relation between the number of synchronizing colorings of G and the structure of the dominant eigenvector v of A(G). We show that a vector v has no partition of coordinates into blocks of equal sum if and only if all colorings of the digraphs associated with v are synchronizing. Furthermore, if for each b there exists at most one partition of the coordinates of v into blocks summing up to b, and the total number of partitions is equal to s, then the fraction of synchronizing colorings among all colorings of G is at least (k-s)/k. We also give a combinatorial interpretation of some known results concerning an upper bound on the minimal length of synchronizing words in terms of v

Algebraic synchronization criterion and computing reset words

We refine results about relations between Markov chains and synchronizing automata. We express the condition that an automaton is synchronizing in terms of linear algebra, and obtain upper bounds for the reset thresholds of automata with a short word of a small rank. The results are applied to make several improvements in the area. We improve the best general upper bound for reset thresholds of finite prefix codes (Huffman codes): we show that an n-state synchronizing decoder has a reset word of length at most O(n log3 n). Also, we prove the Černý conjecture for n-state automata with a letter of rank at most 3√6n-6. In another corollary, based on the recent results of Nicaud, we show that the probability that the Čern conjecture does not hold for a random synchronizing binary automaton is exponentially small in terms of the number of states. It follows that the expected value of the reset threshold of an n-state random synchronizing binary automaton is at most n7/4+o(1). Moreover, reset words of the lengths within our bounds are computable in polynomial time. We present suitable algorithms for this task for various classes of automata for which our results can be applied. These include (quasi-)one-cluster and (quasi-)Eulerian automata. © Springer-Verlag Berlin Heidelberg 2015

Algebraic synchronization criterion and computing reset words

We refine a uniform algebraic approach for deriving upper bounds on reset thresholds of synchronizing automata. We express the condition that an automaton is synchronizing in terms of linear algebra, and obtain new upper bounds for automata with a short word of small rank. The results are applied to make several improvements in the area. In particular, we improve the upper bound for reset thresholds of finite prefix codes (Huffman codes): we show that an n-state synchronizing decoder has a reset word of length at most O(nlog3n). In addition to that, we prove that the expected reset threshold of a uniformly random synchronizing binary n-state decoder is at most O(nlog n). We prove the Černý conjecture for n-state automata with a letter of rank ≤6n−63. In another corollary, we show that the probability that the Černý conjecture does not hold for a random synchronizing binary automaton is exponentially small in terms of the number of states, and that the expected value of the reset threshold is at most n3/2+o(1). Moreover, all of our bounds are constructible. We present suitable polynomial algorithms for the task of finding a reset word of length within our bounds. © 201

Parameterized Complexity of Synchronization and Road Coloring

First, we close the multivariate analysis of a canonical problem concerning short reset words (SYN), as it was started by Fernau et al. (2013). Namely, we prove that the problem, parameterized by the number of states, does not admit a polynomial kernel unless the polynomial hierarchy collapses. Second, we consider a related canonical problem concerning synchronizing road colorings (SRCP). Here we give a similar complete multivariate analysis. Namely, we show that the problem, parameterized by the number of states, admits a polynomial kernel and we close the previous research of restrictions to particular values of both the alphabet size and the maximum word length