An analysis of Gödel's dialectica Interpretation via linear logic

Dialectica

Volumn 62, Issue 2, 2008, Pages 269-290

  Oliva, Paulo ^a  

^a QUEEN MARY UNIVERSITY OF LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 45849128567     PISSN: 00122017     EISSN: None     Source Type: Journal    
DOI: 10.1111/j.1746-8361.2008.01135.x     Document Type: Article

References (34)

1. AVIGAD, J. and FEFERMAN, S. 1998, 'Gödel's Functional ("dialectica") Interpretation', in S. R. Buss, ed., Handbook of Proof Theory, vol. 137 of Studies in Logic and the Foundations of Mathematics, Amsterdam: North Holland, pp. 337-105
2. BERGER, U. and OLIVA, P. 2005, 'Modified Bar Recursion and Classical Dependent Choice', Lecture Notes in Logic 20, pp. 89-107
3. BERGER, U. and OLIVA, P. 2006, 'Modified Bar Recursion', Mathematical Structures in Computer Science 16, pp. 163-183
4. BEZEM, M. 1985, 'Strongly Majorizable Functionals of Finite Type: A Model for Bar Recursion Containing Discontinuous Functionals', The Journal of Symbolic Logic 50, pp. 652-660
5. BISHOP, E. 1970, 'Mathematics as a Numerical Language', in: A. Kino, J. Myhill, and R. E. Vesley, eds, Intuitionism and Proof Theory, Amsterdam: North Holland, pp. 53-71
6. DILLER, J. and NAHM, W. 1974, 'Eine Variante zur Dialectica-Interpretation der Heyting-Arithmetik endlicher Typen', Arch. Math. Logik Grundl. 16, pp. 49-66
7. FEFERMAN, S. 1993, 'Gödel's dialectica Interpretation and its Two-Way Stretch', in: G. Gottlob etal., Computational Logic and Proof Theory, Berlin: Springer, pp. 23-40
8. FERREIRA, F. and OLIVA, P. 2005,' Bounded Functional Interpretation', Annals of Pure and Applied Logic 135, pp. 73-112
9. FERREIRA, F. and OLIVA, P. 2007, 'Bounded Functional Interpretation in Feasible Analysis', Annals of Pure and Applied Logic 145, pp. 115-129
10. GERHARDY, P. 2006, 'Functional Interpretation and Modified Realizability Interpretation of the Doublenegation Shift', in: A. Beckmann, U. Berger, B. Löwe, and J. V. Tucker, eds, Logical Approaches to Computational Barriers, CiE 2006, Report CSR 7-2006, Swansea, pp. 109-118
11. GlRARD, J-Y. 1987, 'Linear Logic', Theoretical Computer Science 50, pp. 1-102
12. GIRARD, J-Y. 1989, 'Towards a Geometry of Interaction', Contemporary Mathematics 92, pp. 69-108
13. GÖDEL, K. 1933, 'Zur intuitionistischen Arithmetik und Zahlentheorie', Ergebnisse eines Mathema-tischen Kolloquiums 4, pp. 34-38
14. GÖDEL, K. 1958, 'Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes', dialectica 12, pp. 280-287
15. KOHLENBACH, U. 1992, 'Effective Bounds from Ineffective Proofs in Analysis: An Application of Functional Interpretation and Majorization', The Journal of Symbolic Logic 57, pp. 1239-1273
16. KOHLENBACH, U. 1996, 'Analysing Proofs in Analysis', in: W. Hodges, M. Hyland, C. Steinhorn, and J. Truss, eds, Logic: From Foundations to Applications, Oxford: Oxford University Press, pp. 225-260
17. KOHLENBACH, U. and OLIVA, P. 2003a, 'Proof Mining: A Systematic Way of Analysing Proofs in Mathematics', Proceedings of the Steklov Institute of Mathematics 242, pp. 136-164
18. KOHLENBACH, U. and OLIVA, P. 2003b, 'Proof Mining in Ll-approximation', Annals of Pure and Applied Logic 121, pp. 1-38
19. KREISEL, G. 1959, 'Interpretation of Analysis by Means of Constructive Functionals of Finite Types', in: A. Heyting, ed., Constructivity in Mathematics, Amsterdam: North Holland, pp. 101-128
20. KURODA, S. 1951, Intuitionistische Untersuchungen der formalistischen Logik', Nagoya Mathematical Journal 3, pp. 35-47
21. OLIVA, P. 2006a, 'Understanding and Using Spector' s Bar Recursive Interpretation of Classical Analysis, in: A. Beckmann, U. Berger, B. Löwe, and J. V. Tucker, eds, Proceedings of CiE'2006, LNCS 3988, Berlin: Springer, pp. 423-234
22. OLIVA, P. 2006b, 'Unifying Functional Interpretations', Notre Dame Journal of Formal Logic 47, pp. 263-290
23. OLIVA, P. 2007a, 'Computational Interpretations of Classical Linear Logic', in: Proceedings of WoLLIC '07, LNCS 4576, Berlin: Springer, pp. 285-296
24. OLIVA, P. 2007b, 'Modified Realizability Interpretation of Classical Linear Logic', in: Proceedings of the Eighteenth Annual IEEE Symposium on Logic in Computer Science L1CS '07, IEEE Press
25. DE PAIVA, V. C. V. 1989a, 'The dialectica categories', in: J. W. Gray and A. Scedrov, eds, Proceedings of Categories in Computer Science and Logic, Boulder, CO, 1987, Contemporary Mathematics, vol. 92, Providence, RI: American Mathematical Society, pp. 47-62
26. DE PAIVA, V. C. V. 1989b, 'A dialectica-like model of linear logic', in: D. Pitt, D. Rydeheard, P. Dybjer, A. Pitts, and A. Poigné, eds, Category Theory and Computer Science, Manchester, UK, Berlin: Springer-Verlag LNCS 389, pp. 341-356
27. DE PAIVA, V. C. V. 2007, 'Dialectica and Chu Constructions: Cousins?' Theory and Applications of Categories 17, pp. 127-152
28. SCHELLINX, H. 1991, 'Some Syntactical Observations on Linear Logic', Journal of Logic and Computation 1, pp. 537-559
29. SCHWICHTENBERG, H. to appear, 'Dialectica Interpretation of Well-founded Induction', Mathematical Logic Quarterly
30. SHIRAHATA, M. 2006, 'The dialectica Interpretation of First-Order Classical Linear Logic', Theory and Applications of Categories 17, pp. 49-79
31. SHOENFIELD, J. R. 1967, Mathematical Logic, Boston: Addison-Wesley Publishing Company
32. SPECTOR, C. 1962, Provably Recursive Functionals of Analysis: A Consistency Proof of Analysis by an Extension of Principles in Current Intuitionistic Mathematics, in: F. D. E. Dekker, ed., Recursive Function Theory: Proc. Symposia in Pure Mathematics, vol. 5, Providence, RI: American Mathematical Society, pp. 1-27
33. STRETCHER, T. and KOHLENBACH, U. 2007, Shoenfield is Gödel after Krivine, Mathematical Logic Quarterly 53, pp. 176-179
34. TROELSTRA, A. S. 1973, Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, vol. 344 of Lecture Notes in Mathematics, Berlin: Springer