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Automaton semigroups: the two-state case. (English) Zbl 1345.20074
A Mealy automaton \((A,\Sigma,\delta=(\delta_i\colon A\to A)_{i\in\Sigma},\rho=(\rho_x\colon\Sigma\to\Sigma)_{x\in A})\) is invertible if the output functions \(\rho_x\) are permutations of the input-output alphabet \(\Sigma\) and reversible if the state transition functions \(\delta_i\) are permutations of the set of states \(A\). The output functions \(\rho_x\), \(x\in A\), can in natural way be extended to mappings \(\Sigma^*\to\Sigma^*\); it is proved, that the semigroup of a two-state reversible Mealy automaton generated by these mappings is either finite or free; for invertible automaton is presented also an effective decision procedure.

MSC:
20M35 Semigroups in automata theory, linguistics, etc.
68Q45 Formal languages and automata
20M05 Free semigroups, generators and relations, word problems
03B25 Decidability of theories and sets of sentences
03D05 Automata and formal grammars in connection with logical questions
Software:
FR; AutomGrp; GAP; automata
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