Geodesic
On the classification of Post automaton bases by the decidability of the $A$-completeness property for definite automata
Diskretnaya Matematika, Tome 22 (2010) no. 2, pp. 80-95.

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We consider systems of the form $M=F\cup\nu$, where $F$ is some Post class and $\nu$ is a finite system of definite automata. We divide the Post classes into those for which the problem of $A$-completeness of such systems of definite automata is algorithmically decidable and those for which the problem of $A$-completeness is algorithmically undecidable. 
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D. N. Zhuk. On the classification of Post automaton bases by the decidability of the $A$-completeness property for definite automata. Diskretnaya Matematika, Tome 22 (2010) no. 2, pp. 80-95. http://geodesic.mathdoc.fr/item/DM_2010_22_2_a6/

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