Lower Bound for the Length of Synchronizing Words in Partially-Synchronizing Automata

Abstract

We introduce the generalized notion of automata synchronization, so called partial synchronization, which holds for automata with partial transition function. We give a lower bound for the length of minimal synchronizing words for partial synchronizing automata. The difference, in comparison to the ’classical’ synchronization, lies in the initial conditions: let $\mathcal{A}=(Q, A, \delta)$ be an automaton representing the dynamics of a particular system. In case of partial synchronization we assume that initial conditions (initial state of the system) can be represented by some particular states, that is by some P ⊂ Q, not necessarily by all possible states from Q. At first glance the above assumption limits our room for manoeuvre for constructing possibly long minimal synchronizing words (because of the lower number of states at the beginning). Unexpectedly this assumption allows us to construct longer minimal synchronizing words than in a standard case. In our proof we use Sperner’s Theorem and some basic combinatorics.