Isomorphism Groups of Automata
TL;DR: 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.
Abstract: 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.
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TL;DR: Arbitrary finite automata are decomposed into their major substructures, the primaries, and various characterizations of these transition-preserving functions on singly generated Automata are presented and are used as a basis for the reduction.
Abstract: 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.
36 citations
TL;DR: The results of an investigation of relationships concerning the group of automorphisms, the polyadic group of defined polyadic automata and the structure of thepolyadic automation and the ordinary automata associated and with thePolyadic automaton are presented.
Abstract: 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.
35 citations
TL;DR: The algebraic properties of automata are investigated and the automorphism group of an automaton and a certain associated semigroup are the devices used in the study.
Abstract: 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.
31 citations
TL;DR: This paper introduces a new product called the cartesian composition and discusses how various properties of the product automaton depend on the corresponding properties ofThe factors.
Abstract: 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.
22 citations
Cites background from "Isomorphism Groups of Automata"
...The first possibility has been particularly studied in several papers [5, 6, 8, 9, 11]....
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TL;DR: The purpose is to investigate the input structure of automata which have a group-like character and the class of perfect automata investigated by Fleck and Weeg is a proper subclass of those considered here.
Abstract: 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.
21 citations
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01 Jan 1959
TL;DR: The theory of normal subgroups and homomorphisms was introduced in this article, along with the theory of $p$-groups regular $p-groups and their relation to abelian groups.
Abstract: Introduction Normal subgroups and homomorphisms Elementary theory of abelian groups Sylow theorems Permutation groups Automorphisms Free groups Lattices and composition series A theorem of Frobenius solvable groups Supersolvable and nilpotent groups Basic commutators The theory of $p$-groups regular $p$-groups Further theory of abelian groups Monomial representations and the transfer Group extensions and cohomology of groups Group representation Free and amalgamated products The Burnside problem Lattices of subgroups Group theory and projective planes Bibliography Index Index of special symbols.
2,960 citations
TL;DR: Finite automata are considered as instruments for classifying finite tapes as well as generalizations of the notion of an automaton are introduced and their relation to the classical automata is determined.
Abstract: Finite automata are considered in this paper as instruments for classifying finite tapes. Each one-tape automaton defines a set of tapes, a two-tape automaton defines a set of pairs of tapes, et cetera. The structure of the defined sets is studied. Various generalizations of the notion of an automaton are introduced and their relation to the classical automata is determined. Some decision problems concerning automata are shown to be solvable by effective algorithms; others turn out to be unsolvable by algorithms.
1,930 citations
TL;DR: The main result shows the group of operation-preserving transformations of a strongly connected au tomaton onto itself is isomorphic to a group of subsets of input sequences under a certain operation.
Abstract: This paper is mot iva ted by Fleck's s tudy [1] on certain classes of structurepreserving, nontrivial t ransformations of au tomata . In tha t paper the class of those transformations which preserve \"strongly-connectedness\" is completely characterized. An interesting subclass, the class of operation-preserving functions (which are essentially homomorphisms) is introduced there. Fleck showed tha t the set of all operation-preserving functions of an au tomaton A onto itself constitutes a group G(A). In [2] some of the properties of G(A) when A is strongly connected were studied. I t was shown in the lat ter paper tha t corresponding to every finite group G of regular permutat ions there is a strongly connected automaton A for which G = G(A). Since, in fact, the group G(A) determines the structure of A, it would appear tha t the structure of G(A) and of A should be related. The present paper investigates tha t relationship. The main result shows tha t the group of operation-preserving transformations of a strongly connected au tomaton onto itself is isomorphic to a group of subsets of input sequences under a certain operation.
37 citations
TL;DR: The Moore-Mealy machine as discussed by the authors is defined as a nonempty set K (of states), D (of inputs), F (of outputs), and two functions a (the next state) function and X (the output) function.
Abstract: Introduction. In 1954 the mathematical entity called a (sequential) machine was found to be a valuable tool in designing sequential switching circuits [2; 8; 9]. Since then there has been considerable mathematical activity by mathematicians and nonmathematicians relating to the analysis and the synthesis of these machines. As was to be expected of a topic which arose because of an engineering need, most of these results have appeared in engineering and computing journals. Recently though, some of the papers have appeared in mathematical journals [3; 4; 5; 6; 10]. Also, much of the recent literature has dealt with questions almost exclusively of mathematical, as contrasted with engineering, interest [1; 3; 4; 5; 6; 10; 12]. The present paper is written in that spirit. The Moore-Mealy (complete, sequential) machine is defined [8; 9] as a nonempty set K (of "states"), a nonempty set D (of 'inputs"), a nonempty set F (of "outputs"), and two functions a (the "next state" function), and X (the "output" function), 5 mapping KXD into K and X mapping KXD into F. Then 5 and X are extended to sequences of inputs I1 ... Ik (written without commas) by
30 citations
"Isomorphism Groups of Automata" refers result in this paper
...This definition closely parallels that of Rabin and Scott [1] and more exactly that of Ginsburg [ 2 ] (except for outputs)....
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