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Classical logic, storage operators and second-order lambda-calculus. (English) Zbl 0814.03009
Storage operators, introduced by the author [Arch. Math. Logic 30, 241- 267 (1990; Zbl 0712.03009)], are a device to simulate call-by-value in a lambda-calculus with head-reduction. In this paper this concept is used to characterize the lambda-terms encoding integers in classical logic. Let $$\text{Nat} [x]$$ be the formula expressing in second-order lambda- calculus (also with first-order variables and quantifiers) the data type of natural numbers. An important consequence of the normalization theorem is that if $$\vdash M: \text{Nat} [x]$$, then $$M$$ is beta-equivalent to $$\underline{n}$$, the Church numeral for $$n$$. When one extends the calculus with axioms (i.e., when the typing context is extended with typed constants $$c_ i: \sigma_ i$$), this in general is no longer true. The paper considers the especially relevant extension obtained with the axiom $${\mathbf c}$$: $$\forall X(\neg\neg X\to X)$$. This system, in view of Gödel’s double-negation translation, axiomatizes classical logic. M. Parigot [Lect. Notes Comput Sci. 592, 361-380 (1991)] has shown that closed elements of type NAT in this extended system can be characterized in terms of storage operators: If $${\mathbf c} \vdash M: \text{Nat} [n]$$ and if $$T$$ is a storage operator for integers, then, for any variable $$f$$, $$TfM$$ reduces by head reduction to $$fN$$, with $$N$$ beta- equivalent to $$\underline {n}$$, depending only on $$n$$ (and not on $$M$$). The nontrivial proof uses a realizability argument. All the preliminaries, including those on storage operators and the proof of Parigot’s result, are clearly explained in the first part (pp. 53-70) of the paper. The result, besides its direct technical interest, is of potential relevance also for the ongoing research on the extraction of programs from classical proofs.
Reviewer: S.Martini (Pisa)

##### MSC:
 03B40 Combinatory logic and lambda calculus 68N15 Theory of programming languages
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