Abstract.Černý's conjecture asserts the existence of a synchronizing word of length at most (n − 1) 2 for any synchronized n-state deterministic automaton. We prove a quadratic upper bound on the length of a synchronizing word for any synchronized n-state deterministic automaton satisfying the following additional property: there is a letter a such that for any pair of states p, q, one has p · a r = q · a s for some integers r, s (for a state p and a word w, we denote by p · w the state reached from p by the path labeled w). As a consequence, we show that for any finite synchronized prefix code with an n-state decoder, there is a synchronizing word of length O(n 2 ). This applies in particular to Huffman codes.

Abstract. We prove that a random automaton with n states and any fixed non-singleton alphabet is synchronizing with high probability. Moreover, we also prove that the convergence rate is exactly 1 − Θ( 1 n ) as conjectured by Cameron [3] for the most interesting binary alphabet case. Finally, we describe a deterministic algorithm which decides whether a given random automaton is synchronizing in linear expected time.

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Under the assumption P = N P, we prove that two natural problems from the theory of synchronizing automata cannot be solved in polynomial time. The first problem is to decide whether a given reachable partial automaton is synchronizing. The second one is, given an n-state binary complete synchronizing automaton, to compute its reset threshold within performance ratio less than d ln (n) for a specific constant d > 0.

Abstract. 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 Θ(n). 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 Černý 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.