19th Annual IEEE Symposium on Logic in Computer Science (LICS'04)
A Computational Interpretation of Open Induction
Turku, Finland
July 13-July 17
ISBN: 0-7695-2192-4
We study the proof-theoretic and computational properties of open induction, a principle which is classically equivalent to Nash-Williams' minimal-bad-sequence argument and also to (countable) dependent choice (and hence contains full classical analysis). We show that, intuitionistically, open induction and dependent choice are quite different: Unlike dependent choice, open induction is closed under negative- and -translation, and therefore proves the same II_2^0-formulas (over not necessarily decidable, basic predicates) with classical or intuitionistic arithmetic. Via modified realizability we obtain a new direct method for extracting programs from classical proofs of II_2^0-formulas using open induction. We also show that the computational interpretation of classical countable choice given by Berardi, Bezen and Coquand [On the computational content of the axiom of choice] can be derived from our results.