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Cardinality problems of compositions of morphisms and inverse morphisms. (English) Zbl 0685.68065
Summary: We show that we cannot effectively determine whether, for morphisms $$\alpha_ i$$, $$\beta_ i$$, $$card(u\alpha_ 0^{-1}\alpha_ 1)=card(u\beta_ 0^{-1}\beta_ 1)$$ for all words u over the domain alphabets of the two given compositions. In contrast it is decidable for morphisms $$\alpha_ i$$, $$\beta_ i$$ and a regular set R whether $$card(u\alpha_ 0\alpha_ 1^{-1})=card(u\beta_ 0\beta_ 1^{-1})$$ for all words u in R. In order to prove the latter result we give a characterization of the multiplicity functions of simple finite automata by using cardinalities of compositions of the above form. Finally, we show that the above decidability result also holds when we consider rational functions rather than morphisms.

##### MSC:
 68Q45 Formal languages and automata 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 08A50 Word problems (aspects of algebraic structures) 03D40 Word problems, etc. in computability and recursion theory
##### Keywords:
compositions of morphisms; cardinality function
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##### References:
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