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In my work arose the problem of classification CFL under rational functions images. In other terms, what class of languages form languages $T(L)$ , for fixed context free language $L$ and deterministic finite state transducer $T$. I've obtained some easy results like Dyck language with two braces corresponds to CFL and Dyck language with one brace is a strict subset of CFL, now there are a few problems that are still interested to me, but I can't believe that nobody have discovered it yet. Are there any papers on this subject? Googling classification of CFL or Rational Functions (Deterministic FST) + CFL gives bad result.

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    $\begingroup$ The Chomsky-Schützenberger Theorem would fall in this family of results. People (i.e. Ginsburg and others) moved from there to the notion of Abstract Families of Languages (AFL). $\endgroup$ – Sylvain Oct 1 '12 at 17:43
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In general the determinism of the transducer seems to be no real restrition. The choices that can be made by a transducer can be moved into the input language. If that is OK for you, the objects sometimes denoted ${\cal M}(L)$ are called principal trio's. Examples include families of linear, blind counter and one turn counter languages (fixed number of counters).
The Habilitation-thesis of Kluas Reinhardt (Counting as Method, Model and Task in Theoretical Computer Science, available on-line) contains a nice inclusion diagram of several families (page 64).

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