Abstract
Starting with D. Scott's work on the mathematical foundations of programming language semantics, interest in topology has grown up in theoretical computer science, under the slogan "open sets are semidecidable properties". But whereas on Scott domains all such properties are also open, this is no longer true in general. In this paper we present a characterization of effectively given topological spaces that says which semidecidable sets are open. We consider countable topological T o-spaces that satisfy certain additional topological and computational requirements which can be verified for a general class of Scott domains and metric spaces, and we show that the given topology is the recursively finest topology generated by semidecidable sets which is compatible with it. From this general result we derive the above mentioned theorem about the correspondence of the semidecidable properties with the Scott open sets. This theorem, in its turn, is a generalization of the Rice/Shapiro theorem on index sets of classes of recursively enumerable sets. Moreover, characterizations of the canonical topology of a recursively separable recursive metric space are derived. It is shown that it is the recursively finest effective T 3-topology that can be generated by semidecidable sets the topological complement of which is also semidecidable.
Keywords
- Topological Space
- Directed Subset
- Partial Recursive Function
- Scott Topology
- Computable Indexing
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
America, P. and J. Rutten: Solving reflexive domain equations in a category of complete metric spaces. Mathematical Foundations of Programming Language Semantics, 3rd Workshop (Main, M. et al., eds.), 252–288, Lec. Notes Comp. Sci. 298. Berlin: Springer (1988).
De Bakker, J.W. and J.I. Zucker: Processes and the denotational semantics of concurrency. Inform. and Control 54, 70–120 (1982).
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).
Czászár, A.: Foundations of General Topology. New York: Pergamon (1963).
Dyment, E.Z.: Recursive metrizability of numbered topological spaces and bases of effective linear topological spaces. Izv. Vyssh. Uchebn. Zaved, Mat. (Kazan), no. 8, 59–61 (1984); English transl., Sov. Math. (Iz. VUZ) 28, 74–78 (1984).
Egli, H. and R.L. Constable: Computability concepts for programming language semantics. Theoret. Comp. Sci. 2, 133–145 (1976).
Eršov, Ju.L.: Computable functionals of finite type. Algebra i Logika 11, 367–437 (1972); English transl., Algebra and Logic 11, 203–242 (1972).
Eršov, Ju.L.: The theory of A-spaces. Algebra i Logika 12, 369–416 (1973); English transl., Algebra and Logic 12, 209–232 (1973).
Eršov, Ju.L.: Model ℝ of partial continuous functionals. Logic Colloquium 76 (Gandy, R. et al., eds.), 455–467. Amsterdam: North-Holland (1977).
Fletcher, P. and W.F. Lindgren: Quasi-Uniform Spaces. New York: Dekker (1982).
Gierz, G. et al.: A Compendium of Continuous Lattices. Berlin: Springer (1980).
Golson, W.G. and W.C. Rounds: Connections between two theories of concurrency: Metric spaces and synchronisation trees. Inform. and Control 57, 102–124 (1983).
Hauck, J.: Konstruktive Darstellungen in topologischen Räumen mit rekursiver Basis. Zeitschr. f. math. Logik Grundl. d. Math. 26, 565–576 (1980).
Hauck, J.: Berechenbarkeit in topologischen Räumen mit rekursiver Basis. Zeitschr. f. math. Logik Grundl. d. Math. 27, 473–480 (1981).
Hennessy, M. and G. Plotkin: A term model for CCS. Mathematical Foundations of Computer Science (Dembiński, E.P., ed.), 261–274. Lec. Notes Comp. Sci. 88. Berlin: Springer (1980).
Hingston, Ph.: Non-complemented open sets in effective topology. J. Austral. Math. Soc. (Series A) 44, 129–137 (1988).
Kalantari, I.: Major subsets in effective topology. Patras Logic Symposium (Metakides, G., ed.), 77–94. Amsterdam: North-Holland (1982).
Kalantari, I. and A. Leggett: Simplicity in effective topology. J. Symbolic Logic 47, 169–183 (1982).
Kalantari, I. and A. Leggett: Maximality in effective topology. J. Symbolic Logic 48, 100–112 (1983).
Kalantari, I. and J.B. Remmel: Degrees of recursively enumerable topological spaces. J. Symbolic Logic 48, 610–622 (1983).
Kalantari, I. and A. Retzlaff: Recursive constructions in topological spaces. J. Symbolic Logic 44, 609–625 (1979).
Kalantari, I. and G. Weitkamp: Effective topological spaces I: A definability theory. Ann. Pure Appl. Logic 29, 1–27 (1985)
Kalantari, I. and G. Weitkamp: Effective topological spaces II: A hierarchy. Ann. Pure Appl. Logic 29, 207–224 (1985).
Kalantari, I. and G. Weitkamp: Effective topological spaces III: Forcing and definability. Ann. Pure Appl. Logic 36, 17–27 (1987).
Kanda, A.: Gödel numbering of domain theoretic computable functions. Report no. 138, Dept. of Comp. Sci., Univ. of Leeds (1980).
Kanda, A. and D. Park: When are two effectively given domains identical? Theoretical Computer Science, 4th GI Conference (Weihrauch, K., ed.), 170–181. Lec. Notes Comp. Sci. 67. Berlin: Springer (1979).
Kreisel, G., D. Lacombe, and J. Shoenfield: Partial recursive functionals and effective operations. Constructivity in Mathematics (Heyting, A., ed.), 290–297. Amsterdam: North-Holland (1959).
Kreitz, Ch.: Zulässige cpo-s, ein Entwurf für ein allgemeines Berechenbarkeitskonzept. Schriften zur Angew. Math. u. Informatik Nr. 76, Rheinisch-Westfälische Technische Hochschule Aachen (1982).
Mal'cev, A.I.: The Metamathematics of Algebraic Systems. Collected Papers: 1936–1967 (Wells III, B.F., ed.). Amsterdam: North-Holland (1971).
Milner, R.: A Calculus of Communicating Sequences. Lec. Notes Comp. Sci. 92. Berlin: Springer (1980).
Moschovakis, Y.N.: Recursive analysis. Ph.D. Thesis, Univ. of Wisconsin, Madison, Wis. (1963).
Moschovakis, Y.N.: Recursive metric spaces. Fund. Math. 55, 215–238 (1964).
Myhill, J. and J.C. Shepherdson: Effective operators on partial recursive functions. Zeitschr. f. math. Logik Grundl. d. Math. 1, 310–317 (1955).
Nivat, M.: Infinite words, infinite trees, infinite computations. Foundations of Computer Science III, Part 2 (de Bakker, J.W. et al., eds.), 1–52. Math. Centre Tracts 109. Amsterdam (1979).
Nogina, E.Ju.: On effectively topological spaces. Dokl. Akad. Nauk SSSR 169, 28–31 (Russian) (1966); English transl., Soviet Math. Dokl. 7, 865–868 (1966).
Nogina, E.Ju.: Relations between certain classes of effectively topological spaces. Mat. Zametki 5, 483–495 (Russian) (1969); English transl., Math. Notes 5, 288–294 (1969).
Nogina, E.Ju.: Enumerable topological spaces. Zeitschr. f. math. Logik Grundl. d. Math. 24, 141–176 (Russian) (1978).
Nogina, E. Ju.: On completely enumerable subsets of direct products of numbered sets. Mathematical Linguistics and Theory of Algorithms, 130–132 (Russian). Interuniv. thematic Collect., Kalinin Univ. (1978).
Nogina, E.Ju.: The relation between separability and tracebility of sets. Mathematical Logic and Mathematical Linguistics, 135–144 (Russian). Kalinin Univ. (1981).
Reed, G.M. and A.W. Roscoe: Metric spaces as models for real-time concurrency. Mathematical Foundations of Programming Language Semantics, 3rd Workshop (Main, M. et al., eds.), 330–343. Lec. Notes Comp. Sci. 298. Berlin: Springer (1988).
Roscoe, A.W.: A mathematical theory of communicating processes. D.Phil. Thesis, Oxford Univ. (1982).
Rogers, H., Jr.: Theory of Recursive Functions and Effective Computability. New York: McGraw-Hill (1967).
Rounds, W.C.: Applications of topology to the semantics of communicating processes. Seminar on Concurrency (Brookes, S.D. et al., eds.), 360–372. Lec. Notes Comp. Sci. 197. Berlin: Springer (1985).
Sciore, E. and A. Tang: Admissible coherent c.p.o.-s. Automata, Languages and Programming (Ausiello, G. et al., eds.), 440–456. Lec. Notes Comp. Sci. 62. Berlin: Springer (1978).
Sciore, E. and A. Tang: Computability theory in admissible domains. 10th Annual ACM Symp. on Theory of Computing, 95–104. New York: Ass. Comp. Mach. (1978).
Scott, D.: Outline of a mathematical theory of computation. Techn. Monograph PRG-2, Oxford Univ. Comp. Lab. (1970).
Scott, D.: Continuous lattices. Toposes, Algebraic Geometry and Logic (Bucur, I. et al., eds.), 97–136. Lec. Notes Math. 274. Berlin: Springer (1971).
Scott, D.: Lattice theory, data types and semantics. Formal Semantics of Programming Languages (Rustin, R., ed.), 65–106. Englewood Cliffs, N.J.: Prentice-Hall (1972).
Scott, D.: Models for various type-free calculi. Logic, Methodology and Philosophy of Science IV (Suppes, P. et al., eds.), 157–187. Amsterdam: North-Holland (1973).
Scott, D.: Data types as lattices. SIAM J. on Computing 5, 522–587 (1976).
Scott, D.: Lectures on a mathematical theory of computation. Techn. Monograph PRG-19, Oxford Univ. Comp. Lab. (1981).
Scott, D.: Domains for denotational semantics. Automata, Languages and Programming (Nielsen, M. et al., eds.), 577–613. Lec. Notes Comp. Sci. 140. Berlin: Springer (1982).
Scott, D. and Ch. Strachey: Towards a mathematical semantics for computer languages. Computers and Automata (Fox, J., ed.), 19–46. Brooklyn, N.Y.: Polytechnic Press (1971).
Smyth, M.B.: Effectively given domains. Theoret. Comp. Sci. 5, 257–274 (1977).
Smyth, M.B.: Power domains and predicate transformers. Automata, Languages and Programming (Diaz, J., ed.), 662–675. Lec. Notes Comp. Sci. 154. Berlin: Springer (1983).
Smyth, M.B.: Completeness of quasi-uniform spaces in terms of filters. Manuscript (1987).
Smyth, M.B.: Quasi-uniformities: reconciling domains with metric spaces. Mathematical Foundations of Programming Language Semantics, 3rd Workshop (Main, M. et al., eds.), 236–253. Lec. Notes Comp. Sci. 298. Berlin: Springer (1988).
Spreen, D. and P. Young: Effective operators in a topological setting. Computation and Proof Theory, Proc. Logic Colloquium '83 (Richter, M.M. et al., eds.), 437–451. Lec. Notes Math. 1104. Berlin: Springer (1984).
Spreen, D.: Rekursionstheorie auf Teilmengen partieller Funktionen. Habilitationsschrift, Rheinisch-Westfälische Technische Hochschule Aachen (1985).
Vainberg, Ju.R. and E.Ju. Nogina: Categories of effectively topological spaces. Studies in Formalized Languages and Nonclassical Logics, 253–273 (Russian). Moscow: Izdat. "Nauka" (1974).
Vainberg, Ju.R. and E.Ju. Nogina: Two types of continuity of computable mappings of numerated topological spaces. Studies in the Theory of Algorithms and Mathematical Logic, Vol. 2, 84–99, 159 (Russian). Moscow: Vyčisl. Centr Akad. Nauk SSSR (1976).
Vickers, St.: Topology via Logic. Cambridge: Cambridge Univ. Press (1989).
Weihrauch, K. and Th. Deil: Berechenbarkeit auf cpo-s. Schriften zur Angew. Math. u. Informatik Nr. 63, Rheinisch-Westfälische Technische Hochschule Aachen (1980).
Weihrauch, K.: Computability. Berlin: Springer (1987).
Xiang, Li: Everywhere nonrecursive r.e. sets in recursively presented topological spaces. J. Austral. Math. Soc. (Series A) 44, 105–128 (1988).
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Spreen, D. (1990). A characterization of effective topological spaces. In: Ambos-Spies, K., Müller, G.H., Sacks, G.E. (eds) Recursion Theory Week. Lecture Notes in Mathematics, vol 1432. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086126
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DOI: https://doi.org/10.1007/BFb0086126
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