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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
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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

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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
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