On the Minimal Reset Words of Synchronizing Automata

Cerny’s conjecture is a 50 year old conjecture which concerns the combinatoral field of synchronizing automata. In particular, it postulates that the maximal length of the minimal reset word among all n-state automata is (n − 1). We present a proof for Pin’s Theorem, which applies Cerny’s conjecture to p-state automata consisting of a cycle and a nonpermutation, where p ≥ 3 is an odd prime. We also introduce families of the form F (p, k) of automata which consist of a cycle and a group of k simple merging arcs, and we define C(p, k) to be the maximal length of minimal reset words within these families. We provide a lower bound of C(p, k) for general k, and we find with proof the exact value of C(p, 2). Summary Suppose that you are given a group of identically shaped puzzle pieces, each of which is in one of finitely many possible orientations. Accompanying these pieces, you are given a set of machines, each of which can perform some operation which alters the orientation of each piece. Then this project examines the process of selecting the shortest possible series of machines to put in succession such that any puzzle piece entering the series leaves it with a fixed orientation.