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\begin{center}{\huge  (Partial) Combinatory Algebras 
for Sequential Computation}\end{center}
\begin{center}{\large Jaap van Oosten}\end{center}
\begin{center}{\large University of Utrecht}\end{center}
%Provisional title 
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 The language PCF (a typed $\lambda$ calculus
    with constants for arithmetic operations and fixed
    points) is regarded by many as a suitable formalisation
    of the concept ``sequential functional of higher type''.
    (The exact relation with Kleene's formalism appears to be
    not fully understood, however)
    The quest for ``full abstraction'' (Milner) is the search
    for mathematically attractive models for PCF. Research has
    resulted in many, rather complex, categories of domains
    (Scott domains with extra properties, or extra structure).
    In my lecture I shall show that some of these categories
    (at least, the part of them which is generated by the
    natural numbers)
    come about in a natural way, as the type structure of
    (partial) combinatory algebras: defined in the Kleene style,
    with associates.
    Apart from a closer analysis of the domains, this gives us
    these categories as full sub-ccc's of a topos, which is then
    a model of the statement ``all functionals are sequential''.
    I hope to arrive at a similar decomposition of the categories
    of games, of Hyland-Ong and Abramsky-Jagadeesan-Malacaria.

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