Universal asynchronous iterative arrays of Mealy automata

Summary

Asynchronous two-dimensional iterative arrays of automata will be introduced where the underlying automata are not of Moore-type but of Mealy-type. We will prove that there exists a Mealy automaton, $\mathfrak{U}_{ 0}$, with only two states and one input and output for each of its four distinguished directions, such that any given Mealy-automaton can be realized by an iterative array with only $\mathfrak{U}_{ 0}$ for its component-machines. It is known that loop-free nets cannot be as powerful as Mealy automata; however, we will show that any Mealy automaton can be realized by a network, N, with very restrictive component machines, where no signal may pass a loop in N. Using this fact asynchronous iterative arrays can be built up with one component machine, $\mathfrak{U}_{ 1}$ such that any given Mealy automaton can be realized under the restriction that no signal passes a loop more than once. $\mathfrak{U}_{ 1}$ contains only four states and one input and output for each direction.