An Approach to a Unified Theory of Automata

Abstract

A model of an automaton, called a balloon automaton is proposed, It consists of a finite control, which may be deterministic or nondeterministic, an input tape which may be one way or two way, and an abstract, infinite memory, called the balloon, which can enter any of a countable number of states. There is assumed to be a recursive function which manipulates the state of the balloon, and another which passes a finite amount of information from the balloon to the finite control. A subset of the balloon automata is considered a closed class if it obeys two very simple closure properties. Certain closed classes recognize exactly the languages recognized by such familiar automata as the pushdown automaton or stack automaton. Unfortunately, no closed class recognizes the sets accepted by linear bounded automata or the time and tape complexity classes of Turing machines. It is shown that many of the usual closure properties of languages accepted by the pushdown automaton, stack automaton, etc., hold for an arbitrary closed class of balloon automata. For example, the languages accepted by a closed class of one-way, nondeterministic balloon automata are closed under concatenation. Of special interest is the fact that a closed class of two-way deterministic balloon automata is closed under inverse g.s.m. mappings. This fact is not obvious, and was not known for all of the types of automata which form closed classes of balloon automata. It should be emphasized that the purpose of this paper is not to propose another “model of a computer.” Rather, we are proposing a method of proving the standard theorems about existing and future models. Hopefully, when a model is proposed in the future, one will simply show it equivalent to a closed class of balloon automata, and have many of the closure properties automatically proven.