Abstract. We define a constructive model for ∆ 0 2-maps, that is, maps recursively definable from a map deciding the halting problem. Our model refines existing constructive interpretation for classical reasoning over one-quantifier formulas: it is compositional (Modus Ponens is interpreted as an application) and semantical (rather than translating classical proofs into intuitionistic ones, we define a mathematical structure intuitionistically validating Excluded Middle for one-quantifier formulas). Iniziato a Torino (Italia), il 24 Gennaio 2001. Ultimo salvataggio: November 26, 2004. Acknowledgement. This paper comes out of a collection of notes taken in preparation of a short course in Kyoto, 7-19 January 2001. Such course would never have been possible without all the support and suggestions coming from Prof. Susumu Hayashi of Kobe University, and all people of the Proof Animation project. To them it goes my warmest gratitude. I thank also Prof. Yohji Akama, of Tohoku University, for proof-checking an earlier version of this paper, and for many valuable comments. 1

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.