Topology and Computability Thesis Proposal
نویسنده
چکیده
In this thesis I will relate standard computability to computability of reals|and related spaces|and link these notions with denotational semantics of computer languages. The framework for the proposed study was worked out in collaboration with Steve Awodey, Lars Birkedal and Dana Scott. The rst section gives a motivation for our setup and indicates how standard computability ts in it. The subsequent sections discuss applications of the ideas to examples in analysis, topology, and domain theory. They are meant to demonstrate the nature of the work that I intend to accomplish. 1 Motivation and Background We take as a canonical example of a space the powerset of natural numbers P = PN ordered by inclusion. It is an algebraic lattice and a countably based T 0-space under the so-called Scott topology, i.e., the weak topology rather than the compact Hausdorr topology on 2 N. The family RE of recur-sively enumerable sets of natural numbers is a subspace of P, and an element x 2 P is computable when x 2 RE. We next want to extend this well known idea of computability of elements to computability of functions.
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