MODIFIED BAR RECURSION AND CLASSICAL DEPENDENT CHOICE

We introduce a variant of Spector’s bar recursion in finite types (which we call “modified bar recursion”) to give a realizability interpretation of the classical axiom of dependent choice allowing for the extraction of witnesses from proofs of ∀∃-formulas in classical analysis. As another application, we show that the fan functional can be defined by modified bar recursion together with a version of bar recursion due to Kohlenbach. We also show that the type structure M of strongly majorizable functionals is a model for modified bar recursion. §

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

[2]  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.

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

[4]  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 .

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

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

[7]  Helmut Schwichtenberg,et al.  On bar recursion of types 0 and 1 , 1979, Journal of Symbolic Logic.

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

[9]  Marc Bezem,et al.  Strongly majorizable functionals of finite type: A model for barrecursion containing discontinuous functionals , 1985, Journal of Symbolic Logic.

[10]  Ulrich Berger,et al.  Program Extraction from Classical Proofs , 1994, LCC.

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

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

[13]  Alex K. Simpson,et al.  Lazy Functional Algorithms for Exact Real Functionals , 1998, MFCS.

[14]  S. Buss.  Handbook of proof theory , 1998 .

[15]  Paulo Oliva,et al.  Modified Bar Recursion , 2002 .

[16]  Ulrich Berger,et al.  REVIEWS-Refined program extraction from classical proofs , 2003 .