Extensions of Scott’s Graph Model and Kleene’s Second Algebra

Abstract

We use a way to extend partial combinatory algebras (pcas) by forcing them to represent certain functions. In the case of Scott's Graph model, equality is computable relative to the complement function. However, the converse is not true. This creates a hierarchy of pcas which relates to similar structures of extensions on other pcas. We study one such structure on Kleene's second model and one on a pca equivalent but not isomorphic to it. For the recursively enumerable sub pca of the Graph model, results differ as we can compute the (partial) complement function using the equality.

Computability in Partial Combinatory Algebras

“…sets in pca's. The following definition is taken from van Oosten and Voorneveld [31], which in turn is based on Longley [19]. Definition 3.1.…”

“…We also have an analog of Turing reductions in any pca A. This is somewhat harder to define, and was carried out in van Oosten [29], see also [31]. This gives for any pca A and any partial function f : A → A a new pca A[ f] in which f is represented, in such a way that A[ f] contains A in a natural way.…”

Computability in partial combinatory algebras

“…The following definition is taken from van Oosten and Voorneveld [22], which in turn is based on Longley [17]. Definition 3.1.…”

“…For example, we can define the diagonal halting problem We also have an analog of Turing reductions in any pca A. This is somewhat harder to define, and was carried out in van Oosten [20], see also [22]. This gives for any pca A and any partial function f : A → A a new pca A[f ] in which f is represented, in such a way that A[f ] contains A in a natural way.…”

Impact of iron coordination isomerism on pyoverdine recognition by the FpvA membrane transporter of Pseudomonas aeruginosa

“…The all-atom starting structure of the Pseudomonas aeruginosa outer membrane was taken from the work of Kirschner et al 32 . The LPS outer leaflet was converted to a coarse-grained representation using the grain definitions of Van Oosten et al, 38 and the inner DPPE leaflet was replaced with more realistic POPE. The coarsegrained membrane was then solvated, energy-minimized and thoroughly equilibrated.…”