Abstract
It is the aim of this paper to present a uniform generalization of both the Myhill/Shepherdson and the Kreisel/Lacombe/Shoenfield theorems on effective operators. To this end we consider countable topological T o-spaces that satisfy certain effectivity requirements which can be verified for both the set of all partial recursive functions and for the set of all total recursive functions, and we show that under appropriate further conditions all effective operators between such spaces are effectively continuous. From this general result we derive the above mentioned theorems and also the generalizations of these theorems which are due to Weihrauch, Moschovakis and Ceitin. There is a long history of interest in continuity theorems in recursive function theory, and such continuity results are basic in computer science when studying continuous partial orders introduced by Scott and by Eršov for studies of the semantics of the λ-calculus.
Supported by NSF Research Grant MCS 7609212A, University of Washington.
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References
Ceitin, G.S.: Algorithmic operators in constructive metric spaces. Trudy Mat. Inst. Steklov 67, 295–361 (1962); English transl., Amer. Math. Soc. Transl. (2) 64, 1–80 (1967).
Egli, H., Constable, R.L.: Computability concepts for programming language semantics. Theoret. Comp. Sci. 2, 133–145 (1976).
Eršov, Ju.L.: Computable functionals of finite types. Algebra i Logika 11, 367–437 (1972); English transl., Algebra and Logic 11, 203–242 (1972).
—: Model C of partial continuous functionals. Logic Colloquium 76 (Gandy, R., Hyland, M., eds.), 455–467. Amsterdam: North-Holland (1977).
Friedberg, R.: Un contre-exemple relatif aux fonctionelles récursives. Compt. Rend. Acad. Sci. Paris 247, 852–854 (1958).
Helm, J.: On effectively computable operators. Zeitschr. f. Math. Logik Grundl. d. Math. 17, 231–244 (1971).
Kreisel, G., Lacombe, D., Shoenfield, J.: Partial recursive functionals and effective operations. Constructivity in Mathematics (Heyting, A., ed.), 290–297. Amsterdam: North-Holland (1959).
Lachlan, A.: Effective operations in a general setting. J. Symbolic Logic 29, 163–178 (1964).
Myhill, J., Shepherdson, J.C.: Effective operations on partial recursive functions. Zeitschr. f. math. Logik Grundl. d. Math. 1, 310–317 (1955).
Moschovakis, Y.N.: Recursive analysis. Ph.D. Thesis, Univ. of Wisconsin, Madison, Wis. (1963).
—: Recursive metric spaces. Fund. Math. 55, 215–238 (1964).
Pour-El, M.B.: A comparison of five “computable” operators. Zeitschr. j. math. Logik Grundl. d. Math. 6, 325–340 (1960).
Rogers, H., Jr.: Theory of Recursive Functions and Effective Computability. New York: McGraw-Hill (1967).
Scott, D.: Outline of a mathematical theory of computation. Techn. Monograph PRG-2, Oxford Univ. Comp. Lab. (1970).
Weihrauch, K., Deil, Th.: Berechenbarkeit auf cpo-s. Schriften zur Angew. Math. u. Informatik Nr. 63, RWTH Aachen (1980).
Young, P.: An effective operator, continuous but not partial recursive. Proc. Amer. Math. Soc. 19, 103–108 (1968).
Young, P., Collins W.: Discontinuities of provably correct operators on the provably recursive real numbers. J. Symbolic Logic 48, 913–920 (1983).
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Spreen, D., Young, P. (1984). Effective operators in a topological setting. In: Börger, E., Oberschelp, W., Richter, M.M., Schinzel, B., Thomas, W. (eds) Computation and Proof Theory. Lecture Notes in Mathematics, vol 1104. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0099496
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DOI: https://doi.org/10.1007/BFb0099496
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