Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T19:39:24.686Z Has data issue: false hasContentIssue false
  • English
  • Français

Definability via enumerations

Published online by Cambridge University Press:  12 March 2014

Ivan N. Soskov
Ivan N. Soskov*
Affiliation:
Laboratory for Applied Logic, Sofia University, Sofia 1126, Bulgaria
Get access Rights & Permissions [Opens in a new window]

Extract

The notion of ∀-recursiviness, introduced by Lacombe [1], is intended to describe the effectively definable functions and predicates in abstract structures with equality and denumerable domains. The fact that on every such structure ∀-recursiviness and search computability are equivalent is proved by Moschovakis in [2].

The definition of search computability [3] does not require the presence of the equality among the basic predicates of the structure. There exist abstract structures where the equality is not search-computable and even not semicomputable. On the other hand, in some structures the equality is not an “effective” predicate. Consider, for example, a structure whose domain consists of all partial recursive functions.

A notion of relative computability in abstract structures with denumerable domains, which we shall call here ∀-admissibility, was introduced by D. Skordev in 1977. The notion of ∀-admissibility is a generalization of Lacombe's ∀-recursiviness and does not require the presence of the equality among the basic predicates. In 1977 Skordev conjectured that, in every partial structure with denumerable domain, ∀-admissibility and search computability are equivalent.

Since 1977 some attempts have been made to establish Skordev's conjecture. It is proved in [4] for structures with total basic functions and without basic predicates, and in [5] for structures with finite domains. The proofs in [4] and [5] make use of the priority method and are very complicated.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 2 , June 1989 , pp. 428 - 440
Copyright
Copyright © Association for Symbolic Logic 1989

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]Lacombe, D., Deux généralisations de la notion de récursivité relative, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, vol. 258 (1964), pp. 3410–3413.Google Scholar
[2]Moschovakis, Y. N., Abstract computability and invariant definability, this Journal, vol. 34 (1969), pp. 605–633.Google Scholar
[3]Moschovakis, Y. N., Abstract first order computability. I, Transactions of the American Mathematical Society, vol. 138(1969), pp. 427–464.Google Scholar
[4]Ditchev, A. V., D. Skordev's conjecture, Algebra i Logika, vol. 24 (1985), pp. 379–391; English translation, Algebra and Logic, vol. 24 (1985), pp. 245–253.Google Scholar
[5]Ditchev, A. V., Search computability and computability with numberings are equivalent in the case of a finite set of objects, Proceedings of the summer school and conference on mathematical logic and its applications (Druzhba, Bulgaria, 1986), Plenum Press, New York (to appear).Google Scholar
[6]Friedman, H., Algorithmic procedures, generalized Turing algorithms and elementary recursion theory, Logic colloquium '69 (Gandy, R. O. and Yates, C. E. M., editors), North-Holland, Amsterdam, 1971, pp. 361–389.CrossRefGoogle Scholar
[7]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[8]Ditchev, A. V., Computability in the sense of Moschovakis and its connection with partial recursiveness via numerations, Serdica, vol. 7 (1981), pp. 117–130. (Russian)Google Scholar