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Opérateurs de mise en mémoire et traduction de Gödel. (Storage operators and Gödel translation). (French) Zbl 0712.03009
The leftmost reduction strategy S of $$\lambda$$-calculus (“call by name”) has many mathematical advantages: for example it always terminates when applied to a normalizable term. A practical advantage of S is that the argument of a function is evaluated only if it is really needed; on the negative side it will be evaluated each time it is used.
The author remedies to this drawback by means of “storage operators” (for each data type). A storage operator T, say for the integers, will be a term of pure $$\lambda$$-calculus, with the property that: for each $$\lambda$$-term $$\phi$$ and for each $$\nu$$ which is $$\beta$$-equivalent to an integer there will be a reduced form $$\nu_ 0$$ of $$\nu$$ (not far from its normal form) such that the left computation of $$T\nu$$ f is equivalent (same length and same result) to the computation of $$\nu$$ to $$\nu_ 0$$ followed by the (left) computation of $$\phi \nu_ 0$$. Thus, the call-by- value computation of $$\phi\nu$$ will be simulated by the call-by-name computation of $$T\nu\phi$$.
Let D[x] be a formula of second order functional arithmetic defining a data type D, and let $$D^*[x]$$ be a Gödel transform of D[x]. The main theorem of the paper states that any term T of $$\lambda$$-calculus which encodes a 2nd order intuitionistic proof of the theorem $$\forall x(D[x]^*\to \neg \neg D[x])$$ will be a storage operator. This solves the conjecture raised by the author [RAIRO, Inf. Théor. Appl. 25, No.1, 67-84 (1991)] (with a slightly different definition of storage operators).
The logical framework (and type system) is the second order functional arithmetic, $$AF_ 2$$, developed by the author in his book [Lambda- calcul, types et modèles (1990; Zbl 0697.03004)], however the paper is self-contained. The conceptual part of the proof, which is also highly technical, relies on a simple version of Gödel’s translation and a systematic use of realisability in standard models of $$AF_ 2$$.
Reviewer: C.Berline

##### MSC:
 03B40 Combinatory logic and lambda calculus 68Q65 Abstract data types; algebraic specification 03F35 Second- and higher-order arithmetic and fragments
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##### References:
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