Journal ArticleDOI
Isomorphism Groups of Automata
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TLDR
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.read more
Citations
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Journal ArticleDOI
Note Concerning the Algebraic Theory of Automata
TL;DR: It is proved that the reduced semigroup of an automaton is a regular group of permutations iff the automorphism group of @@@@ is regular and @ @@@ is strongly connected.
Journal ArticleDOI
Externally induced operations
TL;DR: Using the compact-open topology and the standing hypotheses, it is seen that ~b' is a map and, indeed, X x is a semigroup (continuous associative multiplication) under functional composition.
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On structure and representations of cyclic automata
TL;DR: Three binary relations say L, R and H on cyclic automata are introduced, corresponding to Green's equivalences in semigroup theory, to provide representations of strict automata by representing the states as vectors and describing the state transitions in terms of matrix operations.
References
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Book
The theory of groups
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.
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Finite automata and their decision problems
Michael O. Rabin,Dana Scott +1 more
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.
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The Structure of an Automaton and Its Operation-Preserving Transformation Group
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.
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Some remarks on abstract machines
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.