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Computable and Continuous Partial Homomorphisms on Metric Partial Algebras

Published online by Cambridge University Press:  15 January 2014

Viggo Stoltenberg-Hansen  and
John V. Tucker
Viggo Stoltenberg-Hansen
Affiliation:
Department of Mathematics, Uppsala University, Box 480, S-75106 Uppsala, Sweden.E-mail:[email protected]
John V. Tucker
Affiliation:
Department of Computer Science, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, Wales, E-mail:[email protected]
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Abstract

We analyse the connection between the computability and continuity of functions in the case of homomorphisms between topological algebraic structures. Inspired by the Pour-El and Richards equivalence theorem between computability and boundedness for closed linear operators on Banach spaces, we study the rather general situation of partial homomorphisms between metric partial universal algebras. First, we develop a set of basic notions and results that reveal some of the delicate algebraic, topological and effective properties of partial algebras. Our main computability concepts are based on numerations and include those of effective metric partial algebras and effective partial homomorphisms. We prove a general equivalence theorem that includes a version of the Pour-El and Richards Theorem, and has other applications. Finally, the Pour-El and Richards axioms for computable sequence structures on Banach spaces are generalised to computable partial sequence structures on metric algebras, and we prove their equivalence with our computability model based on numerations.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 9 , Issue 3 , September 2003 , pp. 299 - 334
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1] Aberth, O., Computable analysis, McGraw-Hill, New York, 1980.Google Scholar
[2] Banach, S. and Mazur, S., Sur les fonctions calculables, Ann. Soc. Pol. de Math., vol. 16 (1937), p. 223.Google Scholar
[3] Berberian, S. K., Lectures in functional analysis and operator theory, Springer-Verlag, New York, 1974.CrossRefGoogle Scholar
[4] Blanck, J., Domain representability of metric spaces, Annals of Pure and Applied Logic, vol. 83 (1997), pp. 225–247.CrossRefGoogle Scholar
[5] Bourbaki, N., Elements of mathematics: General topology, Parts 1 and 2, Addison-Wesley, 1966.Google Scholar
[6] Ceitin, G. S., Algorithmic operators in constructive complete separable metric spaces, Doklady Akademii Nauk SSSR, vol. 128 (1959), pp. 49–52, in Russian.Google Scholar
[7] Ceitin, G. S., Algorithmic operators in constructive metric spaces, Trudy Matematicheskogo Instituta Imeni V. A. Steklova, vol. 67 (1962), pp. 259–361, in Russian; English translation in American Mathematics Society Translations vol. 64 (1967), pp. 1–80.Google Scholar
[8] Comfort, W. W., Topological groups, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J. E., editors), North-Holland, 1984, pp. 1143–1263.CrossRefGoogle Scholar
[9] Edalat, A., Domains for computation in mathematics, physics and exact real arithmetic, this Bulletin, vol. 3 (1997), pp. 401–452.Google Scholar
[10] Ershov, Y. L., Computable functionals of finite type, Algebra and Logic, vol. 11 (1972), p. 203–242.CrossRefGoogle Scholar
[11] Ershov, Y. L., Theorie der Numerierungen I, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 19 (1973), pp. 289–388.CrossRefGoogle Scholar
[12] Ershov, Y. L., Theorie der Numerierungen II, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 473–584.CrossRefGoogle Scholar
[13] Ershov, Y. L., Theorie der Numerierungen III, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 23 (1977), pp. 289–371.Google Scholar
[14] Ge, X. and Richards, J. I., Computability in unitary representations of compact groups, Logical methods, in honour of A. Nerode, Birkhauser, Basel, 1993, pp. 386–421.CrossRefGoogle Scholar
[15] Goodstein, R. L., Recursive analysis, Constructivity in mathematics (Heyting, A., editor), North-Holland, Amsterdam, 1959, pp. 37–42.Google Scholar
[16] Grätzer, G., Universal algebra, Springer-Verlag, Berlin, 1979.CrossRefGoogle Scholar
[17] Grzegorczyk, A., Computable functionals, Fundamenta Mathematicae, vol. 42 (1955), pp. 168–202.CrossRefGoogle Scholar
[18] Grzegorczyk, A., On the definitions of computable real continuous functions, Fundamenta Mathematicae, vol. 44 (1957), pp. 61–71.CrossRefGoogle Scholar
[19] Hertling, P., Effectivity and effective continuity of functions between computable metric spaces, Combinatorics, complexity and logic (Bridges, D. et al., editors), Springer, 1997, pp. 264–275.Google Scholar
[20] Husain, T., Introduction to topological groups, W.B. Saunders Co., Philadelphia, 1966.Google Scholar
[21] Klaua, D., Konstruktive Analysis, DeutscherVerlag der Wissenschaften, Berlin, 1961.Google Scholar
[22] Kleene, S. C., Countable functionals, Constructivity in mathematics (Heyting, A., editor), North-Holland, 1959, pp. 81–100.Google Scholar
[23] Kreisel, G., Interpretation of analysis by means of constructive functionals of finite types, Constructivity in mathematics (Heyting, A., editor), North-Holland, 1959, pp. 101–128.Google Scholar
[24] Kushner, B. A., Lectures on constructive mathematical analysis, Translations of mathematical monographs, vol. 60, American Mathematical Society, Providence, 1984.Google Scholar
[25] Lacombe, D., Extension de la notion de fonction récursive aux fonctions d'une ou plusieurs variables réelles I, II, III, Comptes Rendus de l'Académie des Sciences, Serie A, vol. 240, pp. 2478–2480 and vol. 241, pp. 13–14, 151–153 (1955).Google Scholar
[26] Mazur, S., Computable analysis, Razprawy Matematyczne Warsaw, vol. 33 (1963).Google Scholar
[27] Meinke, K. and Tucker, J. V., Universal algebra, Handbook of logic in computer science (Abramsky, S. et al., editors), vol. I, Oxford University Press, 1995, pp. 189–411.Google Scholar
[28] Mori, T., Tsujii, Y., and Yasugi, M., Computability structures on metric spaces, Combinatorics, complexity and logic, proceedings of DMTCS'96 (Bridges, et al., editors), Springer, 1996, pp. 352–362.Google Scholar
[29] Moschovakis, Y. N., Recursive metric spaces, Fundamenta Mathematicae, vol. 55 (1964), pp. 215–238.CrossRefGoogle Scholar
[30] Moschovakis, Y. N., Notation systems and recursive ordered fields, Composito Mathematica, vol. 17 (1966), pp. 40–71.Google Scholar
[31] Pour-El, M. B. and Richards, J. I., Computability and noncomputability in classical analysis, Transactions of the American Mathematical Society, vol. 275 (1983), pp. 539–560.CrossRefGoogle Scholar
[32] Pour-El, M. B. and Richards, J. I., Computability in analysis and physics, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1989.CrossRefGoogle Scholar
[33] Scott, D. S., Outline of a mathematical theory of computation, Technical monograph, PRG-2, Oxford University, Oxford, November 1970.Google Scholar
[34] Spreen, D., On effective topological spaces, The Journal of Symbolic Logic, vol. 63 (1998), pp. 185–221.CrossRefGoogle Scholar
[35] Spreen, D., Representatons versus numberings: on the relationship of two computability notions, Theoretical Computer Science, vol. 263 (2001), pp. 473–499.CrossRefGoogle Scholar
[36] Stoltenberg-Hansen, V., Lindström, I., and Griffor, E. R., Mathematical theory of domains, Cambridge University Press, 1994.CrossRefGoogle Scholar
[37] Stoltenberg-Hansen, V. and Tucker, J. V., Complete local rings as domains, CTCS Report 1.85, University of Leeds, 1985.Google Scholar
[38] Stoltenberg-Hansen, V. and Tucker, J. V., Complete local rings as domains, The Journal of Symbolic Logic, vol. 53 (1988), pp. 603–624.CrossRefGoogle Scholar
[39] Stoltenberg-Hansen, V. and Tucker, J. V., Effective algebra, Handbook of logic in computer science (Abramsky, S. et al., editors), vol. IV, Oxford University Press, 1995, pp. 357–526.CrossRefGoogle Scholar
[40] Stoltenberg-Hansen, V. and Tucker, J. V., Computable rings and fields, Handbook of computability theory (Griffor, E.R., editor), Elsevier, 1999, pp. 363–447.CrossRefGoogle Scholar
[41] Stoltenberg-Hansen, V. and Tucker, J. V., Concrete models of computation for topological algebras, Theoretical Computer Science, vol. 219 (1999), pp. 347–378.CrossRefGoogle Scholar
[42] Tucker, J. V. and Zucker, J., Computation by while programs on topological partial algebras, Theoretical Computer Science, vol. 219 (1999), pp. 379–421.CrossRefGoogle Scholar
[43] Tucker, J. V. and Zucker, J., Computable functions and semicomputable sets on many sorted algebras, Handbook of logic for computer science (Abramsky, S. et al., editors), vol. V, Oxford University Press, 2000, pp. 317–523.Google Scholar
[44] Tucker, J. V. and Zucker, J., Abstract versus concrete computation on metric partial algebras, ACM Transactions on Computational Logic, (in press).Google Scholar
[45] Turing, A. M., On computable numbers, with an application to the entscheidungsproblem, Proceedings of the London Mathematical Society, Ser. 2, vol. 42, 1936, pp. 230–265.Google Scholar
[46] van Dalen, D., Mystic, geometer, and intuitionist. The life of L. E. J. Brouwer. Volume 1: The dawning revolution, Oxford University Press, 1999.Google Scholar
[47] Washihara, M., Computability and Frechet spaces, Mathematica Japonica, vol. 42 (1995), pp. 1–13.Google Scholar
[48] Wechler, W., Universal algebra for computer scientists, EATCS Monographs on Theoretical Computer Science 25, Springer-Verlag, Berlin, 1991.Google Scholar
[49] Weihrauch, K., Computability, EATCS Monographs on Theoretical Computer Science 9, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar
[50] Weihrauch, K., Computable analysis, An introduction, Springer-Verlag, Berlin, 2000.CrossRefGoogle Scholar
[51] Weihrauch, K. and Schreiber, U., Embedding metric spaces into cpo's, Theoretical Computer Science, vol. 16 (1981), pp. 5–24.CrossRefGoogle Scholar
[52] Weihrauch, K. and Zhong, N., The wave propagator is Turing computable, Automata, languages and programming, 26th colloquium, 1999 (Wiedermann, J., Boas, Peter van Emde, and Nielsen, M., editors), Springer Lecture Notes in Computer Science, vol. 1644, Springer-Verlag, Berlin, 1999, pp. 697–706.Google Scholar
[53] Weihrauch, K. and Zhong, N., Is the linear Schrödinger propagator Turing computable?, Computability and complexity in analysis, 2000 (Blanck, J. E., Brattka, V., and Hertling, P., editors), Springer Lecture Notes in Computer Science, vol. 2064, Springer-Verlag, Berlin, 2001, pp. 369–377.CrossRefGoogle Scholar
[54] Weihrauch, K. and Zhong, N., Is wave propogation computable or can wave computers beat the Turing machine?, Proceedings of London Mathematical Society, vol. 85 (2002), pp. 312–332.CrossRefGoogle Scholar
[55] Yasugi, M., Mori, T., and Tsujii, Y., Effective properties of sets and functions in metric spaces with computability structure, Theoretical Computer Science, vol. 219 (1999), pp. 467–486.CrossRefGoogle Scholar