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Choice and uniformity in weak applicative theories

from ARTICLES

Published online by Cambridge University Press:  31 March 2017

By
Andrea Cantini
Edited by
Matthias Baaz ,
Sy-David Friedman  and
Jan Krajíček
Matthias Baaz
Affiliation:
Technische Universität Wien, Austria
Sy-David Friedman
Affiliation:
Universität Wien, Austria
Jan Krajíček
Affiliation:
Academy of Sciences of the Czech Republic, Prague
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Summary

Abstract. We are concerned with first order theories of operations, based on combinatory logic and extended with the type W of binary words. The theories include forms of “positive” and “bounded” induction on W and naturally characterize primitive recursive and polytime functions (respectively).

We prove that the recursive content of the theories under investigation (i.e., the associated class of provably total functions onW) is invariant under addition of

1. an axiomof choice for operations and a uniformity principle, restricted to positive conditions;

2. a (form of) self-referential truth, providing a fixed point theorem for predicates. As to the proof methods, we apply a kind of internal forcing semantics, non-standard variants of realizability and cut-elimination.

§1. Introduction. In this paper, we deal with theories of abstract computable operations, underlying the so-called explicit mathematics, introduced by Feferman in the midseventies as a logical frame to formalize Bishop's style constructive mathematics ([17], [18]). Following a common usage, these theories are termed applicative, since they primarily axiomatize structures, which are closed under a general binary operation of application (so that all objects represent abstract programs and self-application is allowed).

Themost important feature of applicative systems is that they include forms of combinatory logic or untyped lambda calculus, and hence they are farmore general than bounded arithmetical systems in the sense of Buss [8]. In particular, applicative theories have the strongest expressive power, as they comprise a Turing complete (functional) language, and they can justify suitably controlled recursion principles, without having to add them as primitive.

Although applicative systems are definitionally very strong, it has recently turned out that typical results from bounded arithmetic and the so called intrinsic (or implicit) approach to computational complexity,1 can be lifted to these systems (see [28], [29], [11], [12]).

Our starting point is given by two natural applicative theories PR and PT, which are considered in [29]. There it is shownthat the recursive content of PR coincideswith the class of primitive recursive functions, whilePTcharacterizes the class of polytime operations.

We strengthen Strahm's results in two directions: (i)we include principles of choice and uniformity in weak applicative systems; (ii) we study intuitionistic applicative theories and the relations with their classical counterparts.

Type
Chapter
Information
Logic Colloquium '01 , pp. 108 - 138
Publisher: Cambridge University Press
Print publication year: 2005

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References

[1] J., Avigad, Interpreting classical theories in constructive ones, The Journal of Symbolic Logic, vol. 65 (2000), pp. 1785–1812.Google Scholar
[2] J., Avigad and S., Feferman, Gödel's functional (Dialectica) interpretation, In Buss [9], pp. 337–405.
[3] H., Barendregt, The Lambda Calculus. Its Syntax and Semantics, Studies in Logic and the Foundations of Mathematics, Elsevier, Amsterdam, 1984.
[4] M. J., Beeson, Foundations of Constructive Mathematics, Springer, Berlin, 1985.
[5] S., Bellantoni, Predicative Recursion and Computational Complexity, Ph.D. thesis, University of Toronto, 1992.
[6] S., Bellantoni, K. H., Niggl, and H., Schwichtenberg, Ramification, modality and linearity in higher types, Annals of Pure and Applied Logic, vol. 104 (2000), pp. 17–30.Google Scholar
[7] M., Boffa, D., van Dalen, and K., McAloon (editors), Logic Colloquium –78, Studies in Logic and the Foundations of Mathematics, North Holland, Amsterdam, 1979.
[8] S. J., Buss, Bounded Arithmetic, Bibliopolis, Napoli, 1986.
[9] S. J., Buss (editor), Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics, Elsevier, Amsterdam, 1998.
[10] A., Cantini, Proof theoretical aspects of self-referential truth, In Dalla, Chiara et al. [16], pp. 7–27.
[11] A., Cantini, Feasible operations and applicative theories based on Mathematical Logic Quarterly, vol. 46 (2000), pp. 291–312.
[12] A., Cantini, Polytime, combinatory logic and positive safe induction, Archive forMathematical Logic, vol. 41 (2002), no. 2, pp. 169–189.Google Scholar
[13] S. A, Cook and Urquhart, A., Functional interpretations of feasibly constructive arithmetic, Annals of Pure and Applied Logic, vol. 63 (1993), pp. 103–200.Google Scholar
[14] T., Coquand and M., Hofmann, A new method for establishing conservativity of classical systems over their intuitionistic version, Mathematical Structures in Computer Science, vol. 9 (1999), pp. 323–333.Google Scholar
[15] J. N., Crossley (editor), Algebra and logic, Lecture Notes in Mathematics 450, Springer, Berlin, 1975.
[16] M. L., Dalla Chiara, H., Doets, D., Mundici, and J., Van Benthem (editors), Logic and Scientific Methods. LMPS –95, Kluwer, Dordrecht, 1996.
[17] S., Feferman, A language and axioms for explicit mathematics, In Crossley [15], pp. 87–139.
[18] S., Feferman, Constructive theories of functions and classes, In Boffa et al. [7], pp. 159–225.
[19] S., Feferman, Iterated inductive fixed point theories: application to Hancock's conjecture, In Metakides [24], pp. 171–195.
[20] F., Ferreira, Polynomial time computable arithmetic, In Sieg [26], pp. 147–156.
[21] D., Leivant, A foundational delineation of polytime, Information andComputation, vol. 110 (1994), pp. 391–420.Google Scholar
[22] D., Leivant, Intrinsic theories and computational complexity, In Logic and Computational Complexity [23], pp. 177–194.
[23] D., Leivant (editor), Logic and Computational Complexity, Lecture Notes in Computer Science 960, Springer, Berlin-Heidelberg, 1995.
[24] G., Metakides (editor), Patras Logic Colloquium, Studies in Logic and the Foundations of Mathematics, North Holland, Amsterdam, 1982.
[25] P., Minari, Theories with types and names with positive stratified comprehension, Studia Logica, vol. 62 (1999), pp. 215–242.Google Scholar
[26] W., Sieg (editor), Logic and Computation, Contemporary Mathematics, American Mathematical Society, Providence R. I., 1990.
[27] S. G., Simpson, Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic, Springer, Berlin-Heidelberg, 1998.
[28] T., Strahm, Polynomial time operations in explicit mathematics, The Journal of Symbolic Logic, vol. 62 (1997), pp. 575–594.Google Scholar
[29] T., Strahm, Theories with self-application and computational complexity, Information and Computation, vol. 185 (2003), pp. 263–297.Google Scholar
[30] A. S., Troelstra and H., Schwichtenberg, Basic Proof Theory, Cambridge University Press, 1997.
[31] A. S., Troelstra and D., van Dalen, Constructivism in Mathematics, Studies in Logic and the Foundations of Mathematics, Elsevier-North Holland, Amsterdam, 1988.

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