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On the group of automorphisms of strongly connected automata

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Abstract

LetA = (S, I, M) be a strongly connected finite automaton withn states. Weeg has shown that ifA has a group of automorphisms of orderm, then there is a partitionπ of the setS inton/m blocks each withm states. Furthermore ifs i ands j are in the same block ofπ, thenT ii =T jj , whereT ii = {x|x ∈ Σ* and thenM(s i , x) =s i }. It will be shown that the partitionπ also must have the substitution property and that these two conditions are sufficient for ann state strongly connected automaton to have a group of automorphisms of orderm.

Necessary and sufficient conditions for twon-state strongly connected automata to have isomorphic automorphism groups are given. Also, it is demonstrated that forT ii to equalT jj it is necessary to check only a finite number of tapes and consequently provide an algorithm for determining whether or notA has a group of automorphisms of orderm.

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References

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    B. Barnes, Groups of automorphisms and sets of equivalence classes of input for automata,J. Assoc. Comput. Mach. 12 (1965), 561–565.

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    A. C. Fleck, Isomorphism groups of automata,J. Assoc. Comput. Mach. (1962), 469–476.

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    A. C. Fleck, On the automorphism group of an automaton,J. Assoc. Comput. Mach. 12 (1965), 566–569.

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    J. Hartmanis andR. E. Stearns,Algebraic Structure Theory of Sequential Machines, Prentice-Hall, Englewood Cliffs, N.J., 1966.

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    G. P. Weeg, The structure of an automaton and its operation-preserving transformation group,J. Assoc. Comput. Mach. 9 (1962), 345–349.

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    G. P. Weeg, The automorphism group of the direct product of strongly related automata.,J. Assoc. Comput. Mach. 12 (1965), 187–195.

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Additional information

This research was partially supported by the National Science Foundation under Grant No. G.P. 7077.

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Barnes, B.H. On the group of automorphisms of strongly connected automata. Math. Systems Theory 4, 289–294 (1970). https://doi.org/10.1007/BF01695770

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Keywords

  • Computational Mathematic
  • Finite Number
  • Automorphism Group
  • Finite Automaton
  • Substitution Property