Mathematics and Computation

At the EST training workshop in Fischbachau, Germany, I gave two lectures on syntehtic computability theory. This version of the talk contains material on recursive analysis which is not found in the MFPS XXI version of a similar talk.

Abstract:
Computability theory, which investigates computable functions and computable sets, lies at the foundation of logic and computer science. Its classical presentations usually involve a fair amount of Goedel encodings. Consequently, there have been a number of presentations of computability theory that aimed to present the subject in an abstract and conceptually pleasing way. We build on two such approaches, Hyland’s effective topos and Richman’s formulation in Bishop-style constructive mathematics, and develop basic computability theory, starting from a few simple axioms. Because we want a theory that resembles ordinary mathematics as much as possible, we never speak of Turing machines and Goedel encodings, but rather use familiar concepts from set theory and
topology.

Download slides: est.pdf

I recently lectured at an EST training workshop in Fischbachau, Germany. There were also a number of student talks, one of which was given by Luca Chiarabini from Munich. He talked about extraction of programs from proofs, using (a variant of) Curry-Howard isomorphism, also known as propositions-as-types. He had some very interesting ideas which were obviously related to old programming tricks, but he approached them from the logical point of view, rather than the programmer’s point of view. It got me thinking about how to write certain recursive programs as proofs. Since it is a nice application of program extraction, I want to share it with you here.

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