A Fast Algorithm Finding the Shortest Reset Words

Abstract

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 $n$ 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 $n=100$ states. With our algorithm we are able to consider much larger sample of automata with up to $n=300$ states. In particular, we obtain a new more precise estimation of the expected length of the shortest reset word $\approx 2.5\sqrt{n-5}$.Comment: COCOON 2013. The final publication is available at http://link.springer.com/chapter/10.1007%2F978-3-642-38768-5_1

Parallel brute-force algorithm for deriving reset sequences from deterministic incomplete finite automata

“…Thus, a more scalable algorithm can process larger FA. Besides, although FA can model a wide range of systems, previous approaches (except the methods given in [20][21][22]) for deriving RSs have been developed for deriving short reset sequences from complete FA [23][24][25][26][27][28] and surprisingly, to our knowledge, there are no proposed works for effectively constructing short RSs from FA.…”

“…In this paper, we introduce a revolutionary strategy to derive RSs. Existing approaches to generate RSs from FA use successor trees [23][24][25][26][27][28]. However, due to the high memory demand of such trees, we abolish this strategy and introduce a vector-based approach by introducing a new and an efficient data structure that (1) is suitable for the memory architecture of GPUs and (2) demands less memory and effectively encapsulates data that are required to derive RSs.…”

Random Deterministic Automata

“…Hence, theČerný conjecture holds with high probability. There is still room for sharper results in this direction, as experimentations [32] seem to indicate that the expected length of the smallest synchronizing word grows in √ n.…”

Synchronizing Automata with Random Inputs

“…Let 2 ≤ ℓ ≤ n−3 2 . Applying equations (5) several times in successive order we get µ 1,ℓ (5) = µ n−ℓ,ℓ + n−ℓ−1 q (6) = pµ n−ℓ,ℓ+1 + qµ n−ℓ+1,ℓ + 1 + n−ℓ−1 q . Since µ n−ℓ,ℓ+1 = pµ 1,ℓ + qµ 1,ℓ+1 + 1 + ℓ q and µ n−ℓ+1,ℓ…”

“…It was shown in [9] that a random automaton with large number of letters satisfies the Černý conjecture with high probability. Furthermore, computational experiments performed in [10,5] suggest that expected reset threshold of a random synchronizing automaton is sublinear.…”