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- Claire Kouwenhoven-Gentil, Jaap van Oosten
- J. Symb. Log.
- 2005

- Jaap van Oosten
- Ann. Pure Appl. Logic
- 1997

- Jaap van Oosten
- 1997

It is shown that the type structure of nite-type 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 at domain on… (More)

- Jaap van Oosten
- Mathematical Structures in Computer Science
- 2002

The purpose of this short paper is to sketch the development of a few basic topics in the history of Realizability The number of topics is quite limited and re ects very much my own personal taste biases and prejudices Realizability has over the past years developed into a subject of such dimensions that a comprehensive overview would require a fat book… (More)

- Jaap van Oosten
- Notre Dame Journal of Formal Logic
- 2006

The purpose of this note is to observe a generalization of the concept “computable in. . . ” to arbitrary partial combinatory algebras. For every partial combinatory algebra (pca) A and every partial endofunction on A, a pca A[f ] is constructed such that in A[f ], the function f is representable by an element; a universal property of the construction is… (More)

- Jaap van Oosten
- 2015

b) (3 points) Define a new function F ′ by: F ′(0,m, ~y, x) = G(~y,Km(x)) F ′(n+ 1,m, ~y, x) = H(n, F ′(n,m, ~y, x), ~y,Km−̇(n+1)(x)) Recall that Km−̇(n+1) means: the function K applied m−̇(n+1) times. Prove: if n ≤ m then ∀k[F ′(n,m+ k, ~y, x) = F ′(n,m, ~y,Kk(x))] c) (3 points) Prove by induction: F (z, ~y, x) = F ′(z, z, ~y, x) and conclude that F is… (More)

- Jaap van Oosten
- Mathematical Structures in Computer Science
- 2015

We define a notion of homotopy in the effective topos. AMS Subject Classification (2000): 18B25 (Topos Theory),55U35 (Abstract and axiomatic homotopy theory)

- Jaap van Oosten
- Arch. Math. Log.
- 1991

- Lars Birkedal, Jaap van Oosten
- Ann. Pure Appl. Logic
- 2002