Thesis Open Access

Optimized programs from (non-constructive) proofs by the light (monotone) Dialectica interpretation

HERNEST MIRCEA DAN

Thesis supervisor(s)

Jean-Pierre JOUANNAUD; Helmut SCHWICHTENBERG

This thesis presents a new optimization of Gödel’s Dialectica interpretation
for the extraction of more efficient exact realizers from (classical) arithmeti-
cal proofs. The “light” variant of Dialectica also combines and even more
smoothly with Kohlenbach’s “monotone” optimization of Gödel’s functional
interpretation for the extraction of more efficient majorants and bounds from
(classical) monotonic arithmetical and even analytical proofs. “Light Dialec-
tica” is obtained by adapting Berger’s “uniform” or “non-computational”
quantifiers. Moreover, its presentation is given in Natural Deduction style,
as an improvement of Jørgensen’s recent adaptation of pure Gödel’s Dialec-
tica. A number of concrete examples are treated on the computer by means
of the novel technique. The machine comparison with the more established
program-synthesis technique of refined A-translation shows a very good per-
formance of Light Dialectica, which is outperformed only in the case of Dick-
son’s Lemma. Also the theory of synthesis of feasible, poly-time computable
programs is developed for the new “Light Monotone Dialectica” extraction
technique. Two pre-existent frameworks due to Cook-Urquhart-Ferreira and
respectively Kohlenbach are crossbreeded for this purpose into a “poly-time
bounded Analysis”. The theoretical result is promising, yet practical examples
are to be found for the difference with the pure Kohlenbach’s “polynomially
bounded Analysis”.

This is the unique revision of my PhD thesis, as accepted by the 5-strong Exam committee. Only the French parts were removed. Here is the link to the full thesis https://pastel.archives-ouvertes.fr/pastel-00002286
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