Matching typed and untyped realizability

John Longley
1999 Electronical Notes in Theoretical Computer Science  
Realizability i n terpretations of logics are given by s a ying what it means for computational objects of some kind to realize logical formulae. The computational objects in question might b e d r a wn from an untyped universe of computation, such a s a partial combinatory algebra, or they might b e t yped objects such as terms of a PCF-style programming language. In some instances, one can show that a particular untyped realizability i n terpretation matches a particular typed one, in the
more » ... e that they give the same set of realizable formulae. In this case, we h a ve a v ery good t indeed between the typed language and the untyped realizability m o d e l | w e r e f e r to this condition as (constructive) logical full abstraction. We g i v e some examples of this situation for a variety of extensions of PCF. Of particular interest are some models that are logically fully abstract for typed languages including non-functional features. Our results establish connections between what is computable in various programming languages and what is true inside various realizability toposes. We consider some examples of logical formulae to illustrate these ideas, in particular their application to exact real-number computability.
doi:10.1016/s1571-0661(04)00105-7 fatcat:6zp5v23rfjhkdow3vy7yyhaojy