A computational interpretation of open induction

U. Berger
2004 Proceedings of the 19th Annual IEEE Symposium on 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 -translation, and therefore proves the same ¡ £ ¢ ¤ -formulas (over not necessarily
more » ... decidable, basic predicates) with classical or intuitionistic arithmetic. Via modified realizability we obtain a new direct method for extracting programs from classical proofs of ¡ £ ¢ ¤ -formulas using open induction. We also show that the computational interpretation of classical countable choice given by Berardi, Bezem and Coquand [2] can be derived from our results. P is the computational content of the (negative-andtranslated) classical proof of ¥ £ ¦ using the principle of open induction (and a realizer thereof). The principle of open induction was formulated by Raoult [21] in a classical context and discussed by Coquand [6] from an intuitionistic point of view. It is a classical reformulation of Nash-Williams minimal-bad-sequence argument which is used in classical proofs of Kruskal's theorem and related theorems [18, 6]. As we will see, open induction fits well into the extraction method based on neg-is closed under the following weak extensionality rule (see e.g. [14]): If z t § © where z is a set of quantifier free formulas, then
doi:10.1109/lics.2004.1319627 dblp:conf/lics/Berger04 fatcat:aihxty33a5evzetc57jlzwbiz4