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On the group of automorphisms of strongly connected automata

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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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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).

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