A computational interpretation of open induction

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.

[1]  Von Kurt Gödel,et al.  ÜBER EINE BISHER NOCH NICHT BENÜTZTE ERWEITERUNG DES FINITEN STANDPUNKTES , 1958 .

[2]  C. Spector.  Provably recursive functionals of analysis: a consistency proof of analysis by an extension of princ , 1962 .

[3]  C. Nash-Williams.  On well-quasi-ordering infinite trees , 1963, Mathematical Proceedings of the Cambridge Philosophical Society.

[4]  Georg Kreisel,et al.  Transfinite induction and bar induction of types zero and one, and the role of continuity in intuitionistic analysis , 1966, Journal of Symbolic Logic.

[5]  Dana S. Scott,et al.  Outline of a Mathematical Theory of Computation , 1970 .

[6]  D. Dalen.  Review: Georg Kreisel, Godel's Intepretation of Heyting's Arithmetic; G. Kreisel, Relations Between Classes of Constructive Functionals; Georg Kreisel, A. Heyting, Interpretation of Analysis by Means of Constructive Functionals of Finite Types , 1971 .

[7]  A. Troelstra.  Metamathematical investigation of intuitionistic arithmetic and analysis , 1973 .

[8]  H. Luckhardt.  Extensional Godel functional interpretation;: A consistency proof of classical analysis , 1973 .

[9]  J. Diller.  Eine Variante zur Dialectica-Interpretation der Heyting-Arithmetik endlicher Typen , 1974 .

[10]  G.D. Plotkin,et al.  LCF Considered as a Programming Language , 1977, Theor. Comput. Sci..

[11]  Y. Ershov.  Model of Partial Continuous Functionals , 1977 .

[12]  Harvey M. Friedman,et al.  Classically and intuitionistically provably recursive functions , 1978 .

[13]  D. Normann.  Recursion on the countable functionals , 1980 .

[14]  D. Normann.  The countable functionals , 1980 .

[15]  A. S. Troelstra,et al.  Extended Bar Induction of Type Zero , 1980 .

[16]  Jean-Claude Raoult,et al.  Proving Open Properties by Induction , 1988, Inf. Process. Lett..

[17]  S. Shelah,et al.  Annals of Pure and Applied Logic , 1991 .

[18]  Thierry Coquand,et al.  Constructive Topology and Combinatorics , 1992, Constructivity in Computer Science.

[19]  Solomon Feferman,et al.  Gödel's Dialectica Interpretation and Its Two-Way Stretch , 1993, Kurt Gödel Colloquium.

[20]  Helmut Schwichtenberg,et al.  Ordinal Bounds for Programs , 1995 .

[21]  T. Coquand.  A Note on the Open Induction Principle , 1997 .

[22]  Jeremy Avigad,et al.  Chapter V – Gödel’s Functional (“Dialectica”) Interpretation , 1998 .

[23]  Thierry Coquand,et al.  On the computational content of the axiom of choice , 1994, The Journal of Symbolic Logic.

[24]  P. Howard,et al.  Consequences of the axiom of choice , 1998 .

[25]  Ulrich Kohlenbach,et al.  On uniform weak König's lemma , 2002, Ann. Pure Appl. Log..

[26]  Helmut Schwichtenberg,et al.  Refined program extraction form classical proofs , 2002, Ann. Pure Appl. Log..

[27]  Paulo Oliva,et al.  MODIFIED BAR RECURSION AND CLASSICAL DEPENDENT CHOICE , 2004 .