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We consider various methodologically well motivated notions which appear to be
essentially different, but surprisingly capture the same class of relations.
These are: the notion of FM-representability (representability in Finite
Models), statistical representability, limiting recursion or algorithmic
learning with empty data, and decidability by accelerating Turing machines by
computations of length $\omega$.
Finally, we give a new result characterizing FM-representability in poor
finite arithmetics, weaker than divisibility arithmetic, but stronger than