Polynomial-time Algorithms from Ineffective Proofs

We present a constructive procedure for extracting polynomial-time realizers from ineffective proofs of Π2theorems in feasible analysis. By ineffective proof we mean a proof which involves the non-computational principle weak König’s lemma WKL, and by feasible analysis we mean Cook and Urquhart’s system CPV plus quantifierfree choice QF-AC. We shall also discuss the relation between the system CPV+QF-AC and Ferreira’s base theory for feasible analysis BTFA, for which Π2-conservation of WKL has been non-constructively proven. This paper treats the case of weak König’s lemma for trees defined by Π1formulas. Illustrating the applicability of CPV + QF-AC extended with this form of weak König’s lemma, we indicate how to formalize the proof of the Heine/Borel covering lemma in this system. The main techniques used in the paper are Gödel’s functional interpretation and a novel form of binary bar recursion.