Skip to main content
    • Aa
    • Aa

Computable and Continuous Partial Homomorphisms on Metric Partial Algebras

  • Viggo Stoltenberg-Hansen (a1) and John V. Tucker (a2)

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.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[3] S. K. Berberian , Lectures in functional analysis and operator theory, Springer-Verlag, New York, 1974.

[4] J. Blanck , Domain representability of metric spaces, Annals of Pure and Applied Logic, vol. 83 (1997), pp. 225247.

[8] W. W. Comfort , Topological groups, Handbook of set-theoretic topology ( K. Kunen and J. E. Vaughan , editors), North-Holland, 1984, pp. 11431263.

[10] Y. L. Ershov , Computable functionals of finite type, Algebra and Logic, vol. 11 (1972), p. 203242.

[11] Y. L. Ershov , Theorie der Numerierungen I, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 19 (1973), pp. 289388.

[12] Y. L. Ershov , Theorie der Numerierungen II, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 21 (1975), pp. 473584.

[14] X. Ge and J. I. Richards , Computability in unitary representations of compact groups, Logical methods, in honour of A. Nerode, Birkhauser, Basel, 1993, pp. 386421.

[16] G. Grätzer , Universal algebra, Springer-Verlag, Berlin, 1979.

[31] M. B. Pour-El and J. I. Richards , Computability and noncomputability in classical analysis, Transactions of the American Mathematical Society, vol. 275 (1983), pp. 539560.

[32] M. B. Pour-El and J. I. Richards , Computability in analysis and physics, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1989.

[36] V. Stoltenberg-Hansen , I. Lindström , and E. R. Griffor , Mathematical theory of domains, Cambridge University Press, 1994.

[38] V. Stoltenberg-Hansen and J. V. Tucker , Complete local rings as domains, The Journal of Symbolic Logic, vol. 53 (1988), pp. 603624.

[40] V. Stoltenberg-Hansen and J. V. Tucker , Computable rings and fields, Handbook of computability theory ( E.R. Griffor , editor), Elsevier, 1999, pp. 363447.

[41] V. Stoltenberg-Hansen and J. V. Tucker , Concrete models of computation for topological algebras, Theoretical Computer Science, vol. 219 (1999), pp. 347378.

[42] J. V. Tucker and J. Zucker , Computation by while programs on topological partial algebras, Theoretical Computer Science, vol. 219 (1999), pp. 379421.

[49] K. Weihrauch , Computability, EATCS Monographs on Theoretical Computer Science 9, Springer-Verlag, Berlin, 1987.

[50] K. Weihrauch , Computable analysis, An introduction, Springer-Verlag, Berlin, 2000.

[51] K. Weihrauch and U. Schreiber , Embedding metric spaces into cpo's, Theoretical Computer Science, vol. 16 (1981), pp. 524.

[53] K. Weihrauch and N. Zhong , Is the linear Schrödinger propagator Turing computable?, Computability and complexity in analysis, 2000 ( J. E. Blanck , V. Brattka , and P. Hertling , editors), Springer Lecture Notes in Computer Science, vol. 2064, Springer-Verlag, Berlin, 2001, pp. 369377.

[55] M. Yasugi , T. Mori , and Y. Tsujii , Effective properties of sets and functions in metric spaces with computability structure, Theoretical Computer Science, vol. 219 (1999), pp. 467486.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 2 *
Loading metrics...

Abstract views

Total abstract views: 44 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 27th May 2017. This data will be updated every 24 hours.