A linear bound on the k-rendezvous time for primitive sets of NZ matrices

A set of nonnegative matrices is called primitive if there exists a product of these matrices that is entrywise positive. Motivated by recent results relating synchronizing automata and primitive sets, we study the length of the shortest product of a primitive set having a column or a row with k positive entries, called its k-rendezvous time (k-RT}), in the case of sets of matrices having no zero rows and no zero columns. We prove that the k-RT is at most linear w.r.t. the matrix size n for small k, while the problem is still open for synchronizing automata. We provide two upper bounds on the k-RT: the second is an improvement of the first one, although the latter can be written in closed form. We then report numerical results comparing our upper bounds on the k-RT with heuristic approximation methods.Comment: 27 pages, 10 figur

Improving the Upper Bound on the Length of the Shortest Reset Word

We improve the best known upper bound on the length of the shortest reset words of synchronizing automata. The new bound is slightly better than 114 n^3 / 685 + O(n^2). The Cerny conjecture states that (n-1)^2 is an upper bound. So far, the best general upper bound was (n^3-n)/6-1 obtained by J.-E. Pin and P. Frankl in 1982. Despite a number of efforts, it remained unchanged for about 35 years. To obtain the new upper bound we utilize avoiding words. A word is avoiding for a state q if after reading the word the automaton cannot be in q. We obtain upper bounds on the length of the shortest avoiding words, and using the approach of Trahtman from 2011 combined with the well-known Frankl theorem from 1982, we improve the general upper bound on the length of the shortest reset words. For all the bounds, there exist polynomial algorithms finding a word of length not exceeding the bound

Complexity of Preimage Problems for Deterministic Finite Automata

Given a subset of states S of a deterministic finite automaton and a word w, the preimage is the subset of all states that are mapped to a state from S by the action of w. We study the computational complexity of three problems related to the existence of words yielding certain preimages, which are especially motivated by the theory of synchronizing automata. The first problem is whether, for a given subset, there exists a word extending the subset (giving a larger preimage). The second problem is whether there exists a word totally extending the subset (giving the whole set of states) - it is equivalent to the problem whether there exists an avoiding word for the complementary subset. The third problem is whether there exists a word resizing the subset (giving a preimage of a different size). We also consider the variants of the problem where an upper bound on the length of the word is given in the input. Because in most cases our problems are computationally hard, we additionally consider parametrized complexity by the size of the given subset. We focus on the most interesting cases that are the subclasses of strongly connected, synchronizing, and binary automata

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 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

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