Skip to main content Accessibility help
×
Home
  • Print publication year: 2016
  • Online publication date: March 2016

13 - Turing's ‘Oracle’: From Absolute to Relative Computability and Back

from Part Five - Oracles, Infinitary Computation, and the Physics of the Mind

Summary

Introduction

We offer here some historical notes on the conceptual routes taken in the development of recursion theory over the last 60 years, and their possible significance for computational practice. These illustrate, incidentally, the vagaries to which mathematical ideas may be susceptible on the one hand, and – once keyed into a research program – their endless exploitation on the other.

At the hands primarily of mathematical logicians, the subject of effective computability, or recursion theory as it has come to be called (for historical reasons to be explained in the next section), has developed along several interrelated but conceptually distinctive lines. While this began with what were offered as analyses of the absolute limits of effective computability, the immediate primary aim was to establish negative results of the effective unsolvability of various problems in logic and mathematics. From this the subject turned to refined classifications of unsolvability for which a myriad of techniques were developed. The germinal step, conceptually, was provided by Turing's notion of computability relative to an ‘oracle’. At the hands of Post, this provided the beginning of the subject of degrees of unsolvability, which became a massive research program of great technical difficulty and combinatorial complexity. Less directly provided by Turing's notion, but implicit in it, were notions of uniform relative computability, which led to various important theories of recursive functionals. Finally the idea of computability has been relativized by extension, in various ways, to more or less arbitrary structures, leading to what has come to be called generalized recursion theory. Marching in under the banner of degree theory, these strands were to some extent woven together by the recursion theorists, but the trend has been to pull the subject of effective computability even farther away from questions of actual computation. The rise in recent years of computation theory as a subject with that as its primary concern forces a reconsideration of notions of computability theory both in theory and practice. Following the historical sections, I shall make the case for the primary significance for practice of the various notions of relative (rather than absolute) computability, but not of most methods or results obtained thereto in recursion theory.

Related content

Powered by UNSILO
A., Aho, J.E., Hopcroft, and J., Ullman 1974. The Design and Analysis of Computer Algroithms. Addison–Wesley.
S., Alagic and M.A., Arbib 1978. The Design of Well-Structured and Correct Programs. Springer Verlag.
T., Baker, J, Gill, and R., Solovay 1975. Relativisations of the P =? NP question. SIAM J. Comput., 4(4):431–442.
J.L., Balcázar, J, Díaz, and J, Gabouró 1988. Structural Complexity I. Springer Verlag.
J, Barwise 1975. Admissible Sets and Structures. Springer Verlag.
J., Barwise, editor 1977. Handbook of Mathematical Logic. North Holland.
A.K., Chandra and D., Harel 1980. Computable queries for relational data bases. J. Comput. System Sci., 21:156–178.
A., Church 1937. An unsolvable problem of elementary number theory. Amer. J. Math., 58:345–363. Reprinted in Davis (1965).
S.A., Cook 1971. The complexity of theorem-proving procedures. In Proceedings of the Third ACM Symposium on the Theory of Computing, pp. 151–158, Shaker Heights, Ohio.
S. B., Cooper and J., van Leeuwen, editors 2013. Alan Turing. His Work and Impact. Elsevier.
B.J., Copeland, C.J., Posy, and O., Shagrir, editors 2013. Computability. Turing, Gödel, Church, and Beyond. MIT Press.
M., Davis 1965. The Undecidable. Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions. Raven Press.
M., Davis 1982. Why Gödel didn't have Church's thesis. Information and Control, 54:3–24.
S., Feferman 1977. Inductive schemata and recursively continuous functionals. In Logic Colloquium ’76, pp. 373–392. North Holland.
S., Feferman 1998. Turing in the land of 0(z). In Herken (1988), pp. 113–147.
S., Feferman 1990. Polymorphic typed λ-calculi in a type-free axiomatic framework. In Logic and Computation, Contemporary Mathematics, volume 104, pp. 101–137.AMS.
S., Feferman 1991. Logics for termination and correctness of functional programs. In Logic from Computer Science, pp. 95–127. MSRI Publications, Springer Verlag.
S., Feferman 1992. Turing's ‘oracle’: from absolute to relative computability – and back. In The Space of Mathematics, J., Echeverria et al., editors, pp. 314–348. de Gruyter.
S., Feferman 2015. Theses for computation and recursion on concrete and abstract structures. To appear in Turing's Revolution. The Impact of his Ideas about Computability,G., Sommaruga and T., Strahm, editors.
J.E., Fenstad 1980. General Recursion Theory: An Axiomatic Approach. Springer Verlag.
J.E., Fenstad and P., Hinman, editors 1974. Generalized Recursion Theory. North Holland.
J.E., Fenstad, R., Gandy, and G., Sacks, editors 1978. Generalized Recursion Theory II. North Holland.
M., Fitting 1987. Computability Theory, Semantics and Logic Programming. Oxford University Press.
R.M., Friedberg 1957. Two recursively enumerable sets of incomparable degrees of unsolvability (solution of Post's problem 1944). Proc. Nat. Acad. Sci., 43:236–238.
H., Friedman 1971. Algorithmic procedures, generalized Turing algorithms, and elementary recursion theory. In Logic Colloquium ’69, pp. 361–389. North Holland.
R.O., Gandy 1980. Church's thesis and principles for mechanisms. In The Kleene Symposium, pp. 123–148. North Holland.
R.O., Gandy 1988. The confluence of ideas in 1936. In Herken (1988), pp. 55-111.
R.O., Gandy and C.E.M., Yates, editors 2001. Collected Works of A.M. Turing. Mathematical Logic. Elsevier.
M., Garey and D., Johnson 1979. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Co..
K., Gödel 1986. Collected Works Volume I: Publications1926–1936. Oxford University Press.
K., Gödel 1990. Collected Works Volume II: Publications1938–1974. Oxford University Press.
E.R., Griffor, editor 1999. Handbook of Computability Theory. Elsevier.
D., Harel 1987. Algorithmics: The Spirit of Computing. Addison–Wesley.
R., Herken, editor 1988. The Universal Turing Machine. A Half Century Survey. Oxford University Press.
P., Hinman 1978. Recursion-Theoretic Hierarchies. Springer Verlag.
A., Hodges 1983. Alan Turing: The Enigma. Simon and Shuster.
A., Kechris and Y., Moschovakis 1977. Recursion in higher types. In: Barwise (1977), pp. 681–737.
S.C., Kleene 1938. On notation for ordinal numbers. J. Symbolic Logic, 3:150–155.
S.C., Kleene 1952. Introduction to Metamathematics. North Holland.
S.C., Kleene 1959a. Countable functionals. In Constructivity in Mathematics, pp. 81–100. North Holland.
S.C., Kleene 1959b. Recursive functionals and quantifiers of finite types I. Trans. Amer. Math. Soc., 91:1–52.
S.C., Kleene 1981. Origins of recursive function theory. Ann. History Comput., 3:52–67.
S.C., Kleene and E.L., Post 1954. The upper semi-lattice of degrees of unsolvability. Ann. Math., 59:379–407.
G., Kreisel 1959. Interpretation of analysis by means of constructive functionals of finite types. In Constructivity in Mathematics, pp. 101–128. North Holland.
G., Kreisel 1971. Some reasons for generalizing recursion theory. In Logic Colloquium ’69, pp. 139–198. North Holland.
M., Lerman 1983. Degrees of Unsolvability. Springer Verlag.
W., Maas and T., Slaman 1989. Some problems and results in the theory of actually computable functions. In Logic Colloquium88, pp. 79–89. North Holland.
J., Mitchell and R., Harper 1988. The essence of ML. In Proc. 15th ACM/POPL, pp. 28–46.
J., Mitchell and G., Plotkin 1984. Abstract types have existential type. In Proc. 12th ACM/POPL, pp. 37–51.
Y., Moschovakis 1984. Abstract recursion as a foundation for the theory of algorithms. In Computation and Proof Theory, Lecture Notes in Maths. 1104, pp. 289–364. Springer Verlag.
P. G., Odifreddi 1989. Classical Recursion Theory. Elsevier.
P.G., Odifreddi, editor 1990. Logic and Computer Science. Academic Press.
R., Platek 1966. Foundations of Recursion Theory. PhD thesis, Stanford University.
E., Post 1944. Recursively enumerable sets of integers and their decision problems. Bull. Amer. Math. Soc., 50:284–316.
C., Reade 1989. Elements of Functional Programming. Addison–Wesley.
H., Rogers 1967. Theory of Recursive Functions and Effective Computability. McGraw– Hill.
G.E., Sacks 1963. Degrees of Unsolvability. Annals of Mathematics Studies volume 55. Princeton University Press.
G.E., Sacks 1990. Higher Recursion Theory. Perspectives in Mathematical Logic. Springer Verlag.
J., Shepherdson 1988. Mechanisms for computing over arbitrary structures. In Herken (1988), pp. 581–601.
J., Shepherdson and H., Sturgis 1963. Computability of recursive functions. J. ACM10:217– 255.
R., Shore 1977. a-recursion theory. In Barwise (1977), pp. 653–680.
W., Sieg 2008. Church without dogma: xxioms for computability. In New Computational Paradigms, B.L, öwe, A., Sorbi, S.B., Cooper, editors, pp. 139–152. Springer Verlag.
S., Simpson 1977. Degrees of unsolvability: a survey of results. In Barwise (1977), pp. 631–652.
R., Smullyan 1961. Theory of Formal Systems. Annals of Mathematics Studies volume 47. Princeton University Press.
R.I., Soare 1987. Recursively Enumerable Sets and Degrees. Springer Verlag.
G., Tamburrini 1987. Reflections on Mechanism. PhD thesis, Columbia University.
A.S., Troelstra 1977. Aspects of constructive mathematics. In Barwise (1977), pp. 973–1052.
J., Tucker and J., Zucker 1988. Program Correctness over Abstract Data Types with Error- State Semantics. CWI Monographs volume 6. Centre for Mathematics and Computer Science, Amsterdam.
A.M., Turing 1936. On computable numbers with an application to the Entscheidungs problem. Proc. London Math. Soc., 42:230–265. Reprinted in Davis (1965) and Gandy and Yates (2001).
A.M., Turing 1937. A correction. Proc. London Math. Soc., 43:544–546. Reprinted in Davis (1965) and Gandy and Yates (2001).
A.M., Turing 1939. Systems of logic based on ordinals. Proc. London Math. Soc., 45:161–228. Reprinted in Davis (1965) and Gandy and Yates (2001).