On bar recursive interpretations of analysis

This dissertation concerns the computational interpretation of analysis via proof interpretations, and examines the variants of bar recursion that have been used to interpret the axiom of choice. It consists of an applied and a theoretical component. The applied part contains a series of case studies which address the issue of understanding the meaning and behaviour of bar recursive programs extracted from proofs in analysis. Taking as a starting point recent work of Escardó and Oliva on the product of selection functions, solutions to Gödel’s functional interpretation of several well known theorems of mathematics are given, and the semantics of the extracted programs described. In particular, new game-theoretic computational interpretations are found for weak König’s lemma for Σ1-trees and for the minimal-bad-sequence argument. On the theoretical side several new definability results which relate various modes of bar recursion are established. First, a hierarchy of fragments of system T based on finite bar recursion are defined, and it is shown that these fragments are in one-to-one correspondence with the usual fragments based on primitive recursion. Secondly, it is shown that the so called ‘special’ variant of Spector’s bar recursion actually defines the general one. Finally, it is proved that modified bar recursion (in the form of the implicitly controlled product of selection functions), open recursion, update recursion and the Berardi-BezemCoquand realizer for countable choice are all primitive recursively equivalent in the model of continuous functionals.

[1]  Frank Plumpton Ramsey,et al.  On a Problem of Formal Logic , 1930 .

[2]  Ernst Specker,et al.  Nicht konstruktiv beweisbare Sätze der Analysis , 1949, Journal of Symbolic Logic.

[3]  S. Kuroda Intuitionistische Untersuchungen der formalistischen Logik , 1951, Nagoya Mathematical Journal.

[4]  Georg Kreisel,et al.  On the interpretation of non-finitist proofs—Part I , 1951, Journal of Symbolic Logic.

[5]  Graham Higman,et al.  Ordering by Divisibility in Abstract Algebras , 1952 .

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

[7]  J. Kruskal Well-quasi-ordering, the Tree Theorem, and Vazsonyi’s conjecture , 1960 .

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

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

[10]  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.

[11]  W. A. Howard,et al.  Functional interpretation of bar induction by bar recursion , 1968 .

[12]  C. Parsons On a Number Theoretic Choice Schema and its Relation to Induction , 1970 .

[13]  W. Tait Normal Form Theorem for Bar Recursive Functions of Finite Type , 1971 .

[14]  Charles D. Parsons,et al.  On n-quantifier induction , 1972, Journal of Symbolic Logic.

[15]  H. Luckhardt Extensional Gödel Functional Interpretation , 1973 .

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

[17]  R. Gandy,et al.  Computable and recursively countable functions of higher type , 1977 .

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

[19]  J. Paris,et al.  ∑n-Collection Schemas in Arithmetic , 1978 .

[20]  J. Hyland,et al.  Filter spaces and continuous functionals , 1979 .

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

[22]  William A. Howard,et al.  Ordinal analysis of simple cases of bar recursion , 1981, Journal of Symbolic Logic.

[23]  Marc Bezem,et al.  Strong normalization of barrecursive terms without using infinite terms , 1985, Arch. Math. Log..

[24]  Wilfried Sieg,et al.  Fragments of arithmetic , 1985, Ann. Pure Appl. Log..

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

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

[27]  Marc Bezem Equivalence of bar recursors in the theory of functionals of finite type , 1988, Arch. Math. Log..

[28]  G. Bellin Ramsey Interpreted: A Parametric Version of Ramsey's Theorem , 1990 .

[29]  Chetan R. Murthy Extracting Constructive Content From Classical Proofs , 1990 .

[30]  de Paiva,et al.  The Dialectica categories , 1991 .

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

[32]  Bezem,et al.  Ramsey's theorem and the pigeonhole principle in intuitionistic mathematics , 1993 .

[33]  T. Coquand,et al.  A proof of Higman's lemma by structural induction , 1993 .

[34]  Alberto Marcone On the logical strength of Nash-Williams' theorem on transfinite sequences , 1994, math/9408204.

[35]  Thierry Coquand,et al.  An Analysis of Ramsey's Theorem , 1994, Inf. Comput..

[36]  Thierry Coquand,et al.  A semantics of evidence for classical arithmetic , 1995, Journal of Symbolic Logic.

[37]  Samuel R. Buss,et al.  Chapter II - First-Order Proof Theory of Arithmetic , 1998 .

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

[39]  Elias Tahhan-Bittar,et al.  Ordinal Recursive Bounds for Higman's Theorem , 1998, Theor. Comput. Sci..

[40]  Dag Normann,et al.  The Continuous Functionals , 1999, Handbook of Computability Theory.

[41]  Stephen G. Simpson,et al.  Subsystems of second order arithmetic , 1999, Perspectives in mathematical logic.

[42]  Ulrich Kohlenbach,et al.  On the No-Counterexample Interpretation , 1997, Journal of Symbolic Logic.

[43]  J. Avigad Update Procedures and the 1-Consistency of Arithmetic , 2002, Math. Log. Q..

[44]  M. Seisenberger On the Constructive Content of Proofs , 2003 .

[45]  Ulrich Berger A computational interpretation of open induction , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

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

[47]  Wim Veldman,et al.  An intuitionistic proof of Kruskal’s theorem , 2004, Arch. Math. Log..

[48]  Seisenberger Monika,et al.  Applications of inductive definitions and choice principles to program synthesis , 2005 .

[49]  Paulo Oliva,et al.  Unifying Functional Interpretations , 2006, Notre Dame J. Formal Log..

[50]  Ulrich Berger,et al.  Modified bar recursion , 2006, Mathematical Structures in Computer Science.

[51]  Paulo Oliva Understanding and Using Spector's Bar Recursive Interpretation of Classical Analysis , 2006, CiE.

[52]  Helmut Schwichtenberg Dialectica interpretation of well-founded induction , 2008, Math. Log. Q..

[53]  Ulrich Kohlenbach,et al.  Applied Proof Theory - Proof Interpretations and their Use in Mathematics , 2008, Springer Monographs in Mathematics.

[54]  Martín Hötzel Escardó,et al.  Exhaustible Sets in Higher-type Computation , 2008, Log. Methods Comput. Sci..

[55]  Ulrich Kohlenbach,et al.  Ramsey's Theorem for Pairs and Provably Recursive Functions , 2009, Notre Dame J. Formal Log..

[56]  H. Towsner,et al.  LOCAL STABILITY OF ERGODIC AVERAGES , 2007, 0706.1512.

[57]  Stefano Berardi,et al.  Interactive Learning-Based Realizability for Heyting Arithmetic with EM1 , 2010, Log. Methods Comput. Sci..

[58]  Martín Hötzel Escardó,et al.  Computational Interpretations of Analysis via Products of Selection Functions , 2010, CiE.

[59]  Martín Hötzel Escardó,et al.  Selection functions, bar recursion and backward induction , 2010, Mathematical Structures in Computer Science.

[60]  Federico Aschieri,et al.  Learning, realizability and games in classical arithmetic , 2010, 1012.4992.

[61]  Ulrich Kohlenbach,et al.  On Tao's “finitary” infinite pigeonhole principle , 2010, The Journal of Symbolic Logic.

[62]  Paulo Oliva,et al.  Sequential games and optimal strategies , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[63]  Federico Aschieri Transfinite Update Procedures for Predicative Systems of Analysis , 2011, CSL.

[64]  Philippe Schnoebelen,et al.  Multiply-Recursive Upper Bounds with Higman's Lemma , 2011, ICALP.

[65]  Martín Hötzel Escardó,et al.  System T and the Product of Selection Functions , 2011, CSL.

[66]  Paulo Oliva,et al.  A Game-Theoretic Computational Interpretation of Proofs in Classical Analysis , 2012, ArXiv.

[67]  P. Erdös,et al.  Combinatorial Set Theory: Partition Relations for Cardinals , 2012 .

[68]  Paulo Oliva,et al.  On the Relation Between Various Negative Trans- lations , 2012 .

[69]  A. Kreuzer Proof mining and combinatorics : Program extraction for Ramsey's theorem for pairs , 2012 .

[70]  Ulrich Kohlenbach,et al.  Term extraction and Ramsey's theorem for pairs , 2012, The Journal of Symbolic Logic.

[71]  U. Kohlenbach A UNIFORM QUANTITATIVE FORM OF SEQUENTIAL WEAK COMPACTNESS AND BAILLON'S NONLINEAR ERGODIC THEOREM , 2012 .

[72]  Thomas Powell,et al.  Applying Gödel's Dialectica Interpretation to Obtain a Constructive Proof of Higman's Lemma , 2012, CL&C.

[73]  Danko Ilik,et al.  A Direct Constructive Proof of Open Induction on Cantor Space , 2012 .

[74]  Ulrich Kohlenbach Gödel functional interpretation and weak compactness , 2012, Ann. Pure Appl. Log..

[75]  Paulo Oliva,et al.  On Spector's bar recursion , 2012, Math. Log. Q..

[76]  Thomas Powell,et al.  The equivalence of bar recursion and open recursion , 2014, Ann. Pure Appl. Log..

[77]  R. Gorenflo,et al.  Multi-index Mittag-Leffler Functions , 2014 .

[78]  Thomas Powell,et al.  A constructive interpretation of Ramsey's theorem via the product of selection functions , 2015, Math. Struct. Comput. Sci..

[79]  Martín Hötzel Escardó,et al.  Bar Recursion and Products of Selection Functions , 2015, J. Symb. Log..