Algebraic Synchronization Criterion and Computing Reset Words

Abstract

AbstractWe 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 \(O(n \log ^3 n)\). Also, we prove the Černý conjecture for n-state automata with a letter of rank at most \(\root 3 \of {6n-6}\). 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. It follows that the expected value of the reset threshold of an n-state random synchronizing binary automaton is at most \(n^{7/4+o(1)}\).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.KeywordsMarkov ChainPolynomial AlgorithmShort WordHuffman CodeDeterministic Finite AutomatonThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.