Arbitrary finite automata are decomposed into their major substructures, the primaries. Several characterizations of homomorphisms, endomorphisms, isomorphisms, and automorphisms of arbitrary finite automata are presented via reduction to the primaries of the automata. Various characterizations of these transition-preserving functions on singly generated automata are presented and are used as a basis for the reduction. Estimates on the number of functions of each type are given.

Structure and Transition-Preserving Functions of Finite Automata

This paper is a continuation of the studies of Fleck, Weeg, and others concerning the theory of automorphisms of ordinary automata and of the work of Gil pertaining to time varying automata. A certain restricted class of time-varying automata, namely the class of polyadic automata, is investigated in detail. The results of an investigation of relationships concerning the group of automorphisms, the polyadic group of defined polyadic automata and the structure of the polyadic automation and the ordinary automata associated and with the polyadic automaton is presented.

Automorphisms of Polyadic Automata

The algebraic properties of automata are investigated. The automorphism group of an automaton and a certain associated semigroup are the devices used ir~ the study. Some relationships among various structures of the automaton, its group and semigroup are noted.

On the Automorphism Group of an Automaton

There are several known ways to define a product automaton on the cartesian product of the state sets of two given automata. This paper introduces a new product called the cartesian composition and discusses how various properties of the product automaton depend on the corresponding properties of the factors. A main result is that any finite connected automaton has a unique representation as a cartesian composition of prime automata.

The cartesian composition of automata

The purpose is to investigate the input structure of automata which have a group-like character. The class of perfect automata investigated by Fleck in [2] and Weeg in [6] is a proper subclass of those considered here.

Group-Type Automata

This paper persues a discussion of certain algebraic properties of automata and their relationship to the structure (i.e., properties of the next state function) of automata. The device which is used for this study is the association of a group with each automaton. We introduce functions on automata and study the group of an automaton, a representation for the group elements and the direct product of automata. Finally, for a certain class of automata a necessary and sufficient condition, in terms of the group of the automaton, is given for insuring that an automaton can be represented as a direct product.

A. C. Fleck

Papers

Isomorphism Groups of Automata