Groups of Automorphisms and Sets of Equivalence Classes of Input for Automata
TL;DR: An investigation is presented which continues the work of Fleck and Weeg concerning the relationships between the equivalence classes of inputs and the group of automorphisms of a finite automaton and the principal result is that if for each state of a strongly connected automaton there exists a subset of the set of equiwfience classes of the input semigroup which constitute t~ group.
Abstract: An investigation is presented which continues the work of Fleck and Weeg concerning the relationships between the equivalence classes of inputs and the group of automorphisms of a finite automaton. The principal result is that if for each state of a strongly connected automaton there exists a subset of the set of equiwfience classes of the input semigroup which constitute t~ group, then this group is isomorphic to a group of automorphisms of the automaton. The relationship between subautomata and subgroups of the group of autoraorphisms is also studied.
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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
26 Oct 1966
TL;DR: Applications to decomposition theory, in particular to the problem of factoring out identical parallel front components, are given and a generalization of the major parts of the theory to infinite strongly connected monadic algebras is obtained.
Abstract: The class of total automata is characterized. Relationships between the structure of the automorphism group G(A) of a finite automaton A and G(A/H), where A/H is a quotient [9] of A, are exhibited. It is shown that the poset PA of isomorphism classes of quotients of A is an antihomomorphic, image of the poset PG(A) of conjugacy classes of subgroups of G(A). Some results are obtained about natural series of quotient automata. Applications to decomposition theory, in particular to the problem of factoring out identical parallel front components, are given. A generalization of the major parts of the theory to infinite strongly connected monadic algebras is obtained.
18 citations
TL;DR: Some new results about automorphism groups of strongly connected automata are obtained by making use of a new type of representation of stronglyconnected automata introduced.
12 citations
01 Jun 1971
TL;DR: In this article, the relation between dominance and equivalence is considered and some properties are pointed out, such as equivalence and dominance relations, for the same class of faults, for which it is shown that there exist machines for which this property of being strongly connected is destroyed by every possible single fault.
Abstract: : The paper is concerned with the relationships among faults as they affect sequential machine behavior. Of particular interest are equivalence and dominance relations. It is shown that for output faults (i.e., faults that do not affect state behavior), fault equivalence is related to the existence of an automorphism of the state table. For the same class of faults, the relation between dominance and equivalence is considered and some properties are pointed out. Another class of possible faults is also considered, namely, memory faults (i.e., faults in the logic feedback lines). These clearly affect the state behavior of the machine, and their influence on machine properties, such as being strongly connected, is discussed. It is proven that there exist classes of machines for which this property of being strongly connected is destroyed by every possible single fault. Further results on both memory and output faults are also presented. (Author)
10 citations
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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: 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.
58 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