Compositions over a Finite Domain: From Completeness to Synchronizable Automata

Abstract

We will consider functions whose domain is a fixed finite set N with n elements, n ≥ 2, and whose range is included in N. Such a setup occurs in many and very diverse situations. Depending on the interpretation, different questions will be asked. The two interpretations we have had mostly in mind are many-valued logic and finite deterministic automata. In the former, the set N consists of n truth values, and the functions are truth functions. In the latter, the set N consists of the states of an automaton, whereas each letter of the input alphabet induces a specific function: the next state when reading that letter. We will consider two specific issues concerning functions of the kind mentioned: completeness and complexity of compositions. While the former is fairly well understood, very little is known about the latter. Our starting point is an old conjecture, falling within the framework of the complexity of computations, concerning finite deterministic automata. Variants and generalizations of this conjecture are presented. It is also shown that the conjecture does not hold for functions of several variables