research-article

Semantics vs Syntax vs Computations

Author:

James S RoyerAuthors Info & Claims

Journal of Computer and System Sciences, Volume 54, Issue 3

Pages 424 - 436

https://doi.org/10.1006/jcss.1997.1487

Published: 01 June 1997 Publication History

Abstract

This paper investigates analogs of the Kreisel Lacombe Shoenfield Theorem in the context of the type-2 basic feasible functionals. We develop a direct, polynomial-time analog of effective operation in which the time bounding on computations is modeled after Kapron and Cook's scheme for their basic polynomial-time functionals. We show that if P=NP, these polynomial-time effective operations are strictly more powerful on R (the class of recursive functions) than the basic feasible functions. We also consider a weaker notion of polynomial-time effective operation where the machines computing these functionals have access to the computations of their procedural parameter, but not to its program text. For this version of polynomial-time effective operations, the analog of the Kreisel Lacombe Shoenfield is shown to hold their power matches that of the basic feasible functionals on R.

References

[1]

J. Balcázar, J. Díaz, J. Gabarró, Springer-Verlag, New York/Berlin, 1990.

[2]

R. Book, Tally languages and complexity classes, Inform. and Control, 26 (1974) 186-193.

[3]

S. Buss, The polynomial hierarchy and intuitionistic bounded arithmetic, Springer-Verlag, New York/Berlin, 1986.

[4]

S. Cook, B. Kapron, Characterizations of the basic feasible functions of finite type, in: Feasible Mathematics: A Mathematical Sciences Institute Workshop, Birkhäuser, Basel, 1990, pp. 71-95.

[5]

A. Cobham, 1965, The intrinsic computational difficulty of functions, Proc. International Conference on Logic, Methodology, and Philosophy, Y. Bar Hillel, 24, 30, North-Holland, Amsterdam

[6]

S. Cook, Computability and complexity of higher type functions, in: Logic from Computer Science, Springer-Verlag, New York/Berlin, 1991, pp. 51-72.

[7]

S. Cook, A. Urquhart, Functional interpretations of feasibly constructive arithmetic, Proc. 21st Annual ACM Symposium on Theory of Computing, 1989, 107, 112

[8]

S. Cook, A. Urquhart, Functional interpretations of feasibly constructive arithmetic, Ann. Pure Appl. Logic, 63 (1993) 103-200.

[9]

M. Davis, R. Sigal, E. Weyuker, Academic Press, New York, 1994.

[10]

M. Davis, E. Weyuker, Academic Press, New York, 1983.

[11]

R. Friedberg, Un contre-example relatif aux fonctionnelles récursives, C. Hebdomadaires Séances Acad. Sci., 247 (1958) 852-854.

[12]

J. Hopcroft, J. Ullman, Addison¿Wesley, Reading, 1979.

[13]

B. Kapron, S. Cook, A new characterization of Mehlhorn's polynomial time functionals, Proc. 32nd Annual IEEE Symposium on Foundations of Computer Science, 1991, 342, 347

Digital Library

[14]

B. Kapron, S. Cook, A new characterization of type 2 feasibility, SIAM J. Comput., 25 (1996) 117-132.

Digital Library

[15]

G. Kreisel, D. Lacombe, J. Shoenfield, 1957, Partial recursive functionals and effective operations, Constructivity in Mathematics: Proceedings of the Colloquium Held at Amsterdam, A. Heyting, 195, 207, North-Holland, Amsterdam

[16]

K. Mehlhorn, Polynomial and abstract subrecursive classes, J. Comput. System Sci., 12 (1976) 147-178.

Digital Library

[17]

J. Myhill, J. Shepherdson, Effective operations on partial recursive functions, Z. Math. Logik und Grundl. Math., 1 (1955) 310-317.

[18]

M. Machtey, P. Young, North-Holland, Amsterdam, 1978.

[19]

P. Odifreddi, North-Holland, Amsterdam, 1989.

[20]

G. Plotkin, LCF considered as a programming language, Theoret. Comput. Sci., 5 (1977) 223-255.

[21]

J. Royer, J. Case, Birkhäuser, Basel, 1994.

[22]

H. Rogers, McGraw¿Hill, New York, 1967.

[23]

1987, MIT Press, Cambridge, MA

[24]

D. Scott, 1975, Lambda calculus and recursion theory, Proceedings of the Third Scandinavian Logic Symposium, S. Kanger, 154, 193, North-Holland, Amsterdam

[25]

A. Seth, There is no recursive axiomatization for feasible functionals of type 2, Seventh Annual IEEE Symposium on Logic in Computer Science, 1992, 286, 295

[26]

A. Seth, 1994, Complexity Theory of Higher Type Functionals, University of Bombay

[27]

A. Seth, Turing machine characterizations of feasible functionals of all finite types, in: Feasible Mathematics II, Birkhäuser, Basel, 1995, pp. 407-428.

Cited By

Kapron BSteinberg F(2020)Type-two polynomial-time and restricted lookaheadTheoretical Computer Science10.1016/j.tcs.2019.07.003813:C(1-19)Online publication date: 12-Apr-2020
https://dl.acm.org/doi/10.1016/j.tcs.2019.07.003
Kapron BSteinberg F(2018)Type-two polynomial-time and restricted lookaheadProceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3209108.3209124(579-588)Online publication date: 9-Jul-2018
https://dl.acm.org/doi/10.1145/3209108.3209124
Li C(2007)Speed-Up Theorems in Type-2 ComputationProceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World10.1007/978-3-540-73001-9_50(478-487)Online publication date: 18-Jun-2007
https://dl.acm.org/doi/10.1007/978-3-540-73001-9_50
Show More Cited By

Index Terms

Semantics vs Syntax vs Computations
1. Theory of computation

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Published In

cover image Journal of Computer and System Sciences

Journal of Computer and System Sciences Volume 54, Issue 3

special issue on complexity theory

June 1997

141 pages

ISSN:0022-0000

Editor:
E. K. Blum
Univ. of Southern California, Los Angeles

Issue’s Table of Contents

Copyright © Academic Press.

Publisher

Academic Press, Inc.

United States

Publication History

Published: 01 June 1997

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Cited By

Kapron BSteinberg F(2020)Type-two polynomial-time and restricted lookaheadTheoretical Computer Science10.1016/j.tcs.2019.07.003813:C(1-19)Online publication date: 12-Apr-2020
https://dl.acm.org/doi/10.1016/j.tcs.2019.07.003
Kapron BSteinberg F(2018)Type-two polynomial-time and restricted lookaheadProceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3209108.3209124(579-588)Online publication date: 9-Jul-2018
https://dl.acm.org/doi/10.1145/3209108.3209124
Li C(2007)Speed-Up Theorems in Type-2 ComputationProceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World10.1007/978-3-540-73001-9_50(478-487)Online publication date: 18-Jun-2007
https://dl.acm.org/doi/10.1007/978-3-540-73001-9_50
Irwin RRoyer JKapron B(2001)On characterizations of the basic feasible functionals, Part IJournal of Functional Programming10.1017/S095679680000384111:1(117-153)Online publication date: 1-Jan-2001
https://dl.acm.org/doi/10.1017/S0956796800003841

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