Geodesic
On the properties of a special composition of automata
Diskretnaya Matematika, Tome 15 (2003) no. 3, pp. 66-75.

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We suggest algorithms for constructing models of a finite automaton which are new automata such that the equations describing the functioning of these models are consequences of the equations describing the functioning of the initial automaton. These new automata are constructed with the use of powers of the initial automaton and some additional functions on the output words. Applications of the results are related to the problem on determining the state of an automaton on the base of the input sequences and the corresponding output sequences. 
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A. V. Babash. On the properties of a special composition of automata. Diskretnaya Matematika, Tome 15 (2003) no. 3, pp. 66-75. http://geodesic.mathdoc.fr/item/DM_2003_15_3_a3/

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