A computational interpretation of open induction
Ulrich Berger · Logic in Computer Science · 2004
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 A-translation, and therefore proves the same /spl pi//sub 2//sup 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 /spl pi//sub 2//sup 0/-formulas using open induction. We also show that the computational interpretation of classical countable choice given by S. Berardi et al. (1998) can be derived from our results.