A Computational Interpretation of Open Induction Ulrich Berger 1 University of Wales Swansea Department of Computer Science Singleton Park, Swansea SA2 0BD, UK u.berger@swansea.ac.uk Abstract We study the proof-theoretic and computational prop- erties of open induction, a principle which is classically equivalent to Nash-Williams’ minimal-bad-sequence argu- ment and also to (countable) dependent choice (and hence contains full classical analysis). We show that, intuitionisti- cally, open induction and dependent choice are quite differ- ent: Unlike dependent choice, open induction is closed un- der negative- and A-translation, and therefore proves the same Π 0 2 -formulas (over not necessarily decidable, basic predicates) with classical or intuitionistic arithmetic. Via modified realizability we obtain a new direct method for ex- tracting programs from classical proofs of Π 0 2 -formulas us- ing open induction. We also show that the computational in- terpretation of classical countable choice given by Berardi, Bezem and Coquand [2] can be derived from our results. 1. Introduction The combination of G¨ odel’s negative- and Friedman’s A-translation yields a simple and flexible method for prov- ing the Π 0 2 -conservativity of the classical versions of various formal systems over their intuitionistic counterparts. With Kleene/Kreisel (modified) realizabiliy added one obtains a method for extracting computational content from classical proofs which can be applied to, for example, various exten- sions of Peano arithmetic and Zermelo-Fraenkel set theory [11]. Unfortunately, the method does not work for classi- cal analysis formalized as an extension of Peano arithmetic with function variables plus the axiom scheme of countable choice AC ∀n ∃xA(n, x) →∃f ∀nA(n,fn) or dependent choice 1 Supported by the British Engineering and Physical Sciences Re- search Council, grant GR/R16020/01 DC ∀n ∀x ∃yA(n, x, y) →∃f ∀nA(n, f n, f (n + 1)). (the latter scheme is called ωAC in [19]; see e.g. [14], or [13] for other, equivalent forms of dependent choice). The method fails in this case because ACN, the negative trans- lation of AC, is not intutitionistically provable (from AC or any other intuitionistically valid principles). The same holds for DCN and DC. There are, however, other methods: Spector [27] ex- tended G¨ odel’s Dialectica interpretation [12] to classical analysis by interpreting ACN by bar recursion in finite types. Another solution was given by Berardi, Bezem and Coquand [2] who used a special form of realizability to in- terpret ACN. Oliva and the author [3] gave a modified re- alizability interpretation of the A-translation of ACN and DCN based on a variant of bar recursion in finite types. Re- cently, a rather different, more machine oriented interpreta- tion was proposed by Krivine [18]. The significance of Spector’s interpretation for reductive proof theory is widely regarded rather limited [10, 1] (but see [19]), and the other interpretations mentioned above do not seem to make new contributions in this respect. How- ever, there are improvements concerning the algorithmic behavior of the extracted realizers: In particular the realizer, let us call it Φ, of ACN given by Berardi, Bezem and Co- quand is very appealing as it implements a clever ‘demand driven’ algorithm [2] which seems to be superior over the other solutions (e.g. bar recursion) which rather perform a ‘blind’ (though terminating) search. The research presented in our paper was prompted by the desire to find a simple explanation and correctness argu- ment for Φ replacing the somewhat ad-hoc realizability in- terpretation and complicated proof in [2]. Indeed, we show that Φ is the computational content of the (negative- and A- translated) classical proof of AC using the principle of open induction (and a realizer thereof). The principle of open induction was formulated by Raoult [24] in a classical context and discussed by Co- quand [6] from an intuitionistic point of view. It is a classi-