On the Probability of Being Synchronizable

Abstract

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.

Exact synchronization in partial deterministic automata

“…Recall the same effect was experimentally observed for CFAs and was then theoretically justified by Berlinkov [4] and Nicaud [27,28]: the probability P S (n) that a random binary CFA with n states is synchronizing tends to 1 as n tends to infinity. Moreover, it is shown in [4] is not difficult to extend the latter result to random almost complete PFAs. Since we have not found such an extension in the literature, we have included it here, without any originality claim.…”

“…, ℓ, add all the lines obtained from the lines that correspond to the clauses in T ′ by keeping the sign of every non-zero integer and adding (t − 1)(n + 1) to its absolute value. 4. In each line corresponding to a clause in S ′ , substitute every nonzero integer ±k by the integer ±(k + (ℓ − 1)(n + 1)).…”

Random Deterministic Automata

“…In a recent preprint [10], Berlinkov proved that a random automaton is synchronizing with probability 1 − Θ( 1 n ) on a two-letter alphabet. This is a deep and difficult result, which was expected for quite some time, since simulations clearly shows that automata are synchronizing with high probability.…”

Reduced checking sequences using unreliable reset

“…However, being able to use a synchronizing sequence for reset means that our method is in fact widely applicable. Berlinkov [12] claims that a random FSM with n states and q inputs has a synchronizing sequence with probability 1 − Θ(1/n 0.5×q ). This claim was experimentally supported by Kisielewicz et al [13].…”