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"... They first noticed that Gold’s limiting recursive functions which was originally introduced to formulate the learning processes of machines, serve as approximation algorithms. Here, Gold’s limiting recursive function is of the form $f(x) $ such that $f(x)=y \Leftrightarrow\exists t_{0}\forall t>t ..."

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They first noticed that Gold’s limiting recursive functions which was originally introduced to formulate the learning processes of machines, serve as approximation algorithms. Here, Gold’s limiting recursive function is of the form $f(x) $ such that $f(x)=y \Leftrightarrow\exists t_{0}\forall t>t_{0}.g(t,x)=y\Leftrightarrow\lim_{t}g(t, x)=y$, $t $ where $g(t, x) $ is called a guessing function, and is a limit variable. Then, they proved that some limiting recursive functions approximate arealizer of a semi-classical principle $\neg\neg\exists y\forall x.g(x, y)=0arrow\exists y\forall x.g(x, y)=0$. Also, they showed impressive usages of the semi-classical principle for mathematics and for software synthesis. In this way, Nakata-Hayashi opened up the possibility that limiting operations provide readability interpretation of semi-classical logical systems. They formulated the set of the limiting recursive functions as a Basic Recursive hnction Theory(brft, for short. Wagner[19] and Strong[16]). Then Nakata-Hayashi carried out their readability interpretation using the BRFT.