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A Combinatory Algebra for Sequential Functionals of Finite Type
 University of Utrecht
, 1997
"... It is shown that the type structure of finitetype functionals associated to a combinatory algebra of partial functions from IN to IN (in the same way as the type structure of the countable functionals is associated to the partial combinatory algebra of total functions from IN to IN), is isomorphic ..."
Abstract

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It is shown that the type structure of finitetype functionals associated to a combinatory algebra of partial functions from IN to IN (in the same way as the type structure of the countable functionals is associated to the partial combinatory algebra of total functions from IN to IN), is isomorphic to the type structure generated by object N (the flat domain on the natural numbers) in Ehrhard's category of "dIdomains with coherence", or his "hypercoherences". AMS Subject Classification: Primary 03D65, 68Q55 Secondary 03B40, 03B70, 03D45, 06B35 Introduction PCF , "Godel's T with unlimited recursion", was defined in Plotkin's paper [16]. It is a simply typed calculus with a type o for integers and constants for basic arithmetical operations, definition by cases and fixed point recursion. More importantly, there is a special reduction relation attached to it which ensures (by Plotkin's "Activity Lemma") that all PCF definable highertype functionals have a sequential, i.e. nonparal...