Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-11T08:49:36.078Z Has data issue: false hasContentIssue false
  • English
  • Français

A logical presentation of the continuous functionals

Published online by Cambridge University Press:  12 March 2014

Erik Palmgren  and
Viggo Stoltenberg-Hansen
Erik Palmgren
Affiliation:
Department of Mathematics, Uppsala University, Po Box 480, S-751 06 Uppsala, Sweden, E-mail: palmgren@math.uu.se
Viggo Stoltenberg-Hansen
Affiliation:
Department of Mathematics, Uppsala University, Po Box 480, S-751 06 Uppsala, Sweden, E-mail: viggo@math.uu.se
Get access Rights & Permissions [Opens in a new window]

Extract

The Kleene-Kreisel continuous functionals [6, 7] have been given several alternative characterisations: using Kuratowski's limit spaces (Scarpellini [20]), using their generalisation, filter spaces (Hyland [4]), via hyperfinite functionals (Normann [11]) and perhaps most elegantly using Scott-Ershov domains (Ershov [3], later generalised by Berger [1]). We propose to add yet another characterisation to this list, which may be called model-theoretic in contrast to the others, but which is in fact closely related to Ershov's approach.

We use the notion of logically presented domains developed in Palmgren and Stoltenberg-Hansen [17] (originating in the work of [15]). Certain logical types, i.e., finitely consistent sets of formulas, over the full type structure built from ℕ correspond to the continuous functionals in the sense of Kreisel, or to Kleene's associates, while the elements realising the types correspond to functionals having associates (cf. [6]). To define these—the total types—we make a nonstandard extension of the full type structure. The set of nonstandard elements is sufficiently rich to single out the total types. The nonstandard extension also makes it possible to relate the logical presentation to Ershov's approach. The logical form of the domain constructions allows us to use a (generalised) Fréchet power as a nonstandard extension. This extension is constructive, in the sense that it avoids the axiom of choice, as distinguished from the one employed in [11].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 3 , September 1997 , pp. 1021 - 1034
Copyright
Copyright © Association for Symbolic Logic 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Berger, U., Total sets and objects in domain theory, Annals of Pure and Applied Logic, vol. 60 (1993), pp. 91–117.CrossRefGoogle Scholar
[2] Chang, C. C. and Keisler, H. J., Model theory, third ed., North-Holland, Amsterdam, 1990.Google Scholar
[3] Ershov, Yu. L., The model C of the continuous functionals, Logic colloquium '76 (Gandy, R. and Hyland, M., editors), North-Holland, Amsterdam, 1977.Google Scholar
[4] Hyland, J. M. E., Filter spaces and continuous functionals, Annals of Mathematical Logic, vol. 16 (1979), pp. 101–143.CrossRefGoogle Scholar
[5] Jónsson, B. and Olin, P., Almost direct products and saturation, Compositio Mathematica, vol. 20 (1968), pp. 125–132.Google Scholar
[6] Kleene, S. C., Countable functionals, Constructivity in mathematics (Heyting, A., editor), North-Holland, Amsterdam, 1959.Google Scholar
[7] Kreisel, G., Interpretation of analysis by means of functionals of finite types I, Constructivity in mathematics (Heyting, A., editor), North-Holland, Amsterdam, 1959.Google Scholar
[8] Lassez, J.-L. and McAloon, K., A constraint sequent calculus, Proceedings of the 5th IEEE symposium on logic in computer science (Washington D.C.), IEEE Computer Society Press, 1990.Google Scholar
[9] Normann, D., The continuous functionals, Handbook of recursion theory (Griffor, E., editor), North-Holland, forthcoming.Google Scholar
[10] Normann, D., Recursion on the continuous functionals, Lecture notes in mathematics, vol. 811, Springer-Verlag, Berlin, 1980.Google Scholar
[11] Normann, D., Characterizing the continuous functionals, this Journal, vol. 48 (1983), pp. 965–969.Google Scholar
[12] Normann, D., Formalizing the notion of total information, Mathematical logic (Petkov, P. P., editor), Plenum Press, 1990, pp. 67–94.CrossRefGoogle Scholar
[13] Normann, D., A hierarchy of domains with totality but without density, Computability, enumerability, unsolvability. Directions in recursion theory (Cooper, S. B.et al., editors), Cambridge University Press, 1996.Google Scholar
[14] Omarov, A. I., A syntactical description of filtering formulas, Soviet Math. Dokl., vol. 44 (1992), pp. 53–55.Google Scholar
[15] Palmgren, E., Denotational semantics of constraint logic programming—a nonstandard approach, Constraint programming (Mayoh, B., Tyugu, E., and Penjam, J., editors), NATO ASI Series F, Springer-Verlag, 1994, pp. 261–288.CrossRefGoogle Scholar
[16] Palmgren, E., A direct proof that certain reduced products are countably saturated, Department of Mathematics Report 27, Uppsala University, 1994.Google Scholar
[17] Palmgren, E. and Stoltenberg-Hansen, V., Logically presented domains, Proceedings of the 10th IEEE symposium on logic in computer science (Washington D.C.), IEEE Computer Society Press, 1995.Google Scholar
[18] Palyutin, E. A., Categorical Horn classes I, Algebra and Logic, vol. 19 (1980), pp. 377–400.CrossRefGoogle Scholar
[19] Richter, M. M. and Szabo, M. E., Towards a nonstandard analysis of programs, Nonstandard analysis—recent developments (Hurd, A. E., editor), Lecture Notes in Mathematics, vol. 983, Springer-Verlag, Berlin, 1983.Google Scholar
[20] Scarpellini, B., A model for barrecursion of higher types, Compositio Mathematica, vol. 23 (1971), pp. 123–153.Google Scholar
[21] Stoltenberg-Hansen, V., Lindström, I., and Griffor, E. R., Mathematical theory of domains, Cambridge University Press, 1994.CrossRefGoogle Scholar