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Relative and Modified Relative Realizability

By Lars Birkedal and Jaap Van Oosten

Abstract

Introduction The notion of Relative Realizability was defined in [2] (see also [1, 4]). The idea is, that instead of doing realizability with one partial combinatory algebra A one uses an inclusion of partial combinatory algebras A ] ` A (such that there are combinators k; s 2 A ] which also serve as combinators for A); the principal point being that "(A ] -) computable" functions may also act on data (in A) that need not be computable. Of course this is reminiscent of Turing's computability with oracles and Kleene's definition ([12] and later papers) of a recursive functional of higher type, which, for example in the case of type 2, has to act on any (possibly non-recursive) function. In itself, relative realizability was not new; Kleene's 1957--realizability ([11]), a precursor of his later function realizability, was probably of thi

Year: 2000
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