Primitive recursive selection functions for existential assertions over abstract algebras

Journal of Logic and Algebraic Programming

Volumn 76, Issue 2, 2008, Pages 175-197

  Strahm, Thomas ^a   Zucker, Jeffery ^b  

^a UNIVERSITY OF BERN   (Switzerland)

^b MCMASTER UNIVERSITY   (Canada)

Author keywords

Generalized computability; Realizability; Selection function

Indexed keywords

ABSTRACTING; RECURSIVE FUNCTIONS;

(MIN ,MAX ,+) FUNCTIONS; PARSONS (CO);

ALGEBRA;

EID: 47049116063     PISSN: 15678326     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jlap.2008.02.002     Document Type: Article

References (36)

1. Avigad J., and Feferman S. Gödel's functional ("Dialectica) interpretation. In: Buss S. (Ed). Handbook of Proof Theory (1998), Elsevier 337-405
2. Beeson M. Foundations of Constructive Mathematics (1985), Springer-Verlag
3. Buss S. The witness function method and fragments of Peano arithmetic. In: Prawitz D., Skyrms B., and Westerståhl D. (Eds). Proceedings of the Ninth International Congress on Logic, Methodology and Philosophy of Science, Uppsala, August 1991 (1994), Elsevier
4. Buss S. First-order proof theory of arithmetic. In: Buss S. (Ed). Handbook of Proof Theory (1998), Elsevier 79-147
5. Buss S. An introduction to proof theory. In: Buss S. (Ed). Handbook of Proof Theory (1998), Elsevier 1-78
6. S.R. Buss, Bounded Arithmetic, Bibliopolis, Napoli, 1986.
7. Cantini A. Polytime, combinatory logic and safe induction. Arch. Math. Logic 41 2 (2002) 169-189
8. A. Cantini, Choice and uniformity in weak applicative theories, in: M. Baaz, S. Friedman, J. Kraji{dotless}́ček (Ed.), Logic Colloquium '01, vol. 20 of Lecture Notes in Logic, Association for Symbolic Logic, 2005, pp. 108-138.
9. Feferman S. A language and axioms for explicit mathematics. Algebra and Logic. In: Crossley J. (Ed). Lecture Notes in Mathematics vol. 450 (1975), Springer, Berlin 87-139
10. Feferman S. Constructive theories of functions and classes. In: Boffa M., van Dalen D., and McAloon K. (Eds). Logic Colloquium '78 (1979), North Holland, Amsterdam 159-224
11. Feferman S. Definedness. Erkenntnis 43 (1995) 295-320
12. Gentzen G. Investigations into logical deduction. In: Szabo M. (Ed). The Collected Papers of Gerhard Gentzen (1969), North Holland 68-131
13. Kleene S. On the interpretation of intuitionistic number theory. J. Symbolic Logic 10 (1945) 109-124
14. Kleene S. Introduction to Metamathematics (1952), North Holland
15. Kohlenbach U. Remarks on Herbrand normal forms and Herbrand realizations. Arch. Math. Logic 301 (1992) 305-317
16. Kreisel G. Some reasons for generalizing recursion theory. In: Gandy R., and Yates C. (Eds). Logic Colloquium '69 (1971), North Holland 139-198
17. Leivant D. A foundational delineation of poly-time. Inform. and Comput. 110 (1994) 391-420
18. Meinke K., and Tucker J. Universal algebra. In: Abramsky S., Gabbay D., and Maibaum T. (Eds). Handbook of Logic in Computer Science vol. 1 (1992), Oxford University Press 189-411
19. Mints G. Quantifier-free and one-quantifier systems. J. Soviet Math. 1 (1973) 71-84
20. Parsons C. On a number theoretic choice scheme II. J. Symbolic Logic 36 (1971) 587 (Abstract)
21. Parsons C. On n-quantifier induction. J. Symbolic Logic 37 (1972) 466-482
22. A. Schlüter, An extension of Leivant's characterization of polytime by predicative arithmetic, Stanford University, 1995 (preprint).
23. Sieg W. Herband analyses. Arch. Math. Logic 30 (1991) 409-441
24. Strahm T. Theories with self-application and computational complexity. Inform. and Comput. 185 (2003) 263-297
25. Strahm T. A proof-theoretic characterization of the basic feasible functionals. Theoret. Comput. Sci. 329 (2004) 159-176
26. Takeuti G. Proof Theory. first ed. (1975), North Holland (second ed., 1987)
27. A. Troelstra (Ed.), Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, second corrected ed., Technical Notes X-93-05, Institute for Logic, Language and Computation, Amsterdam, 1993, Lecture Notes in Mathematics (originally published), vol. 344, Springer-Verlag, 1973.
28. Troelstra A. Realizability. In: Buss S. (Ed). Handbook of Proof Theory (1998), Elsevier 407-473
29. Tucker J., and Zucker J. Program Correctness over Abstract Data Types, with Error-State Semantics. CWI Monographs vol. 6 (1988), North Holland
30. Tucker J., and Zucker J. Provable computable selection functions on abstract structures. In: Aczel P., Simmons H., and Wainer S. (Eds). Proof Theory (1993), Cambridge University Press 277-306
31. Tucker J., and Zucker J. Computation by 'while' programs on topological partial algebras. Theoret. Comput. Sci. 219 (1999) 379-420
32. Tucker J., and Zucker J. Computable functions and semicomputable sets on many-sorted algebras. In: Abramsky S., Gabbay D., and Maibaum T. (Eds). Handbook of Logic in Computer Science vol. 5 (2000), Oxford University Press 317-523
33. Tucker J., and Zucker J. Abstract computability and algebraic specification. ACM Trans. Comput. Logic 3 (2002) 279-333
34. Tucker J., and Zucker J. Abstract versus concrete computation on metric partial algebras. ACM Trans. Comput. Logic 5 (2004) 611-668
35. Tucker J., and Zucker J. Computable total functions, algebraic specifications and dynamical systems. J. Log. Algebr. Program. 62 (2005) 71-108
36. Zucker J. Primitive recursive selection functions over abstract algebras. In: Beckmann A., Berger U., Löwe B., and Tucker J. (Eds). Logical Approaches to Computational Barriers: Second Conference on Computability in Europe, CiE 2006, Swansea, UK, June/July 2006: Proceedings. Lecture Notes in Computer Science vol. 3988 (2006), Springer-Verlag