151 research outputs found

    Algebraic synchronization criterion and computing reset words

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    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 nn-state synchronizing decoder has a reset word of length at most O(nlog3n)O(n \log^3 n). In addition to that, we prove that the expected reset threshold of a uniformly random synchronizing binary nn-state decoder is at most O(nlogn)O(n \log n). We also show that for any non-unary alphabet there exist decoders whose reset threshold is in Θ(n)\varTheta(n). We prove the \v{C}ern\'{y} conjecture for nn-state automata with a letter of rank at most 6n63\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 nn-state random synchronizing binary automaton is at most n3/2+o(1)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

    Synchronizing Strongly Connected Partial DFAs

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    We study synchronizing partial DFAs, which extend the classical concept of synchronizing complete DFAs and are a special case of synchronizing unambiguous NFAs. A partial DFA is called synchronizing if it has a word (called a reset word) whose action brings a non-empty subset of states to a unique state and is undefined for all other states. While in the general case the problem of checking whether a partial DFA is synchronizing is PSPACE-complete, we show that in the strongly connected case this problem can be efficiently reduced to the same problem for a complete DFA. Using combinatorial, algebraic, and formal languages methods, we develop techniques that relate main synchronization problems for strongly connected partial DFAs with the same problems for complete DFAs. In particular, this includes the \v{C}ern\'{y} and the rank conjectures, the problem of finding a reset word, and upper bounds on the length of the shortest reset words of literal automata of finite prefix codes. We conclude that solving fundamental synchronization problems is equally hard in both models, as an essential improvement of the results for one model implies an improvement for the other.Comment: Full version of the paper at STACS 202

    Algebraic synchronization criterion and computing reset words

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    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

    Algebraic synchronization criterion and computing reset words

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    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

    A Fast Algorithm Finding the Shortest Reset Words

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    In this paper we present a new fast algorithm finding minimal reset words for finite synchronizing automata. The problem is know to be computationally hard, and our algorithm is exponential. Yet, it is faster than the algorithms used so far and it works well in practice. The main idea is to use a bidirectional BFS and radix (Patricia) tries to store and compare resulted subsets. We give both theoretical and practical arguments showing that the branching factor is reduced efficiently. As a practical test we perform an experimental study of the length of the shortest reset word for random automata with nn states and 2 input letters. We follow Skvorsov and Tipikin, who have performed such a study using a SAT solver and considering automata up to n=100n=100 states. With our algorithm we are able to consider much larger sample of automata with up to n=300n=300 states. In particular, we obtain a new more precise estimation of the expected length of the shortest reset word 2.5n5\approx 2.5\sqrt{n-5}.Comment: COCOON 2013. The final publication is available at http://link.springer.com/chapter/10.1007%2F978-3-642-38768-5_1

    Preimage problems for deterministic finite automata

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    Given a subset of states SS of a deterministic finite automaton and a word ww, the preimage is the subset of all states mapped to a state in SS by the action of ww. We study three natural problems concerning words giving certain preimages. The first problem is whether, for a given subset, there exists a word \emph{extending} the subset (giving a larger preimage). The second problem is whether there exists a \emph{totally extending} word (giving the whole set of states as a preimage)---equivalently, whether there exists an \emph{avoiding} word for the complementary subset. The third problem is whether there exists a \emph{resizing} word. We also consider variants where the length of the word is upper bounded, where the size of the given subset is restricted, and where the automaton is strongly connected, synchronizing, or binary. We conclude with a summary of the complexities in all combinations of the cases

    On the interplay between Babai and Cerny's conjectures

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    Motivated by the Babai conjecture and the Cerny conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with nn states in this class, we prove that the reset thresholds are upper-bounded by 2n26n+52n^2-6n+5 and can attain the value n(n1)2\tfrac{n(n-1)}{2}. In addition, we study diameters of the pair digraphs of permutation automata and construct nn-state permutation automata with diameter n24+o(n2)\tfrac{n^2}{4} + o(n^2).Comment: 21 pages version with full proof

    Acta Cybernetica : Volume 19. Number 2.

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