On the Synchronization of Planar Automata

Planar automata seems to be representative of the synchronizing behavior of deterministic finite state automata. We conjecture that Černy’s conjecture holds true, if and only if, it holds true for planar automata. In this paper we have gathered some evidence concerning this conjecture. This evidence amounts to show that the class of planar automata is representative of the algorithmic hardness of synchronization This work is related to the synchronization of deterministic finite state automata (DFAs, for short). Let M be a DFA, and let ΣM be its input alphabet, we use the symbol ΣM to denote the set of finite strings over the alphabet ΣM. The function δ̂M : Σ ∗ M ×QM → QM is defined by the equation: δ̂M (w1...wn, q) = δM ( wn, δ̂M (w1...wn−1, q) ) , where δM is the transition function of M. A synchronizing string (reset word) for M, is a string w ∈ ΣM such that for all p, q ∈ QM, the equality δ̂M (w, p) = δ̂M (w, q) holds We say that automaton M is synchronizing, if and only if, there exists a synchronizing string for M. Let M be a synchronizing automaton, its minimal reset length, denoted by rlM, is the length of its minimal synchronizing strings. It is easy to prove that rlM ∈ O ( |QM| 3 ) . Černy [5] conjectured that rlM ≤ (|QM| − 1) 2 . This conjecture is called Černy’s Conjecture, and it is considered the most important open problem in the combinatorial theory of finite state automata. The universality conjecture for planar automata. It is well known that if Černy’s conjecture holds true for strongly connected automata, then it holds true for all the deterministic finite state automata. Therefore, we say that