Synchronization and Diversity of Solutions

A central computational problem in the realm of automata theory is the problem of determining whether a finite automaton A has a synchronizing word. This problem has found applications in a variety of subfields of artificial intelligence, such as robotics, game theory, and the theory of multi-agent systems. In this work, we study this problem within the framework of diversity of solutions, an up-and-coming trend in the field of artificial intelligence where the goal is to compute a set of solutions that are sufficiently distinct from one another.

In our work, the language L is specified by a finite automaton B given at the input. First, we define a notion of diversity of solutions that is suitable to the context of synchronization. Subsequently, we show that for each fixed r ∈ N, each fixed finite automaton A, and each finite automaton B given at the input, the problem of determining the existence of a diverse set {w_1 , w_2 , . . . , w_r } ⊆ L(B) of words that are synchronizing for A can be solved in polynomial time. This result yields new applications of the notion of synchronization to the subfields of artificial intelligence mentioned above.