Light Dialectica Program Extraction from a Classical Fibonacci Proof

Abstract

We demonstrate program extraction by the Light Dialectica Interpretation (LDI) on a minimal logic proof of the classical existence of Fibonacci numbers. This semi-classical proof is available in MinLog's library of examples. The term of Gödel's T extracted by the LDI is, after strong normalization, exactly the usual recursive algorithm which defines the Fibonacci numbers (in pairs). This outcome of the Light Dialectica meta-algorithm is much better than the T-program extracted by means of the pure Gödel Dialectica Interpretation. It is also strictly less complex than the result obtained by means of the refined A-translation technique of Berger, Buchholz and Schwichtenberg on an artificially distorted variant of the input proof, but otherwise it is identical with the term yielded by Berger's Kripke-style refined A-translation. Although syntactically different, it also has the same computational complexity as the original program yielded by the refined A-translation from the undistorted input classical Fibonacci proof.

Modal Functional (Dialectica) Interpretation

“…Then it must be possible to adapt the proof of (4.3) to a proof in NA m or NA m l of ( ∀ x A ) → ∀ y B → ∀ z C . As noticed by Oliva in [Oli12], the Fibonacci example first treated with Dialectica in [Her07] falls into this category. Oliva also suggested an interesting example, which motivated the definition of our positively computational quantifier ∀ + (cf.…”

Modal Functional (Dialectica) Interpretation

“…Then it must be possible to adapt the proof of ( 18) to a proof in NA m or NA m l of ( ∀ x A ) → ∀ y B → ∀ z C . As noticed by Oliva in [28], the Fibonacci example first treated with Dialectica in [14] falls into this category.…”

“…In his PhD thesis (cf. Chapter 5 of [38] , in particular Section 5.6.2) the second author explains that three uniform quantifiers need to be inserted in order to remove the negative computational content from three universally quantified formulas inside the proof 14 . It turns out that this can be achieved by inserting a single in the formulation of the corollary he is proving (Unbounded Pigeonhole Principle).…”

Hybrid Functional Interpretations

“…As was noticed by the first author in [15], the negative universally quantified l 1 , l 2 and l 3 do not need to be witnessed in order to extract an algorithm for computing the Fibonacci numbers as a witness for m as function of n. The proof in [14] can thus be translated to a hybrid linear logic proof such that, in the pattern of (6), statement (8) becomes…”

“…The example was first used in [14] to illustrate the so-called "refined A-translation" and then in [15] to illustrate the light Dialectica (see also Section 4.3 of [12]). The semi-classical Fibonacci proof is a minimal-logic proof of ∀n∃ cl m G(n, m) , where…”