Mathematics and Computation » Synthetic computability

Synthetic Computability (MFPS XXIII Tutorial)

Andrej Bauer — Thu, 24 May 2007 10:19:31 +0000

A tutorial presented at the Mathematical Foundations of Programming Semantics XXIII Tutorial Day.

Abstract: In this tutorial we show how to elegantly develop the basics of computability theory with simple set-theoretic and domain-theoretic ideas and constructions. Computability is never mentioned explicitly, instead we work in an intuitionistic set theory extended with suitable (classically inconsistent) axioms. The usual theorems of computability theory are expressed as statements of set-theoretic, domain-theoretic and topological nature. Classical theorems of computability theory are then just interpretations of our theorems in an appropriate realizability model (which will be presented in a separate tutorial).

Download slides: syncomp-mfps23.pdf

First Steps in Synthetic Computability Theory (Fischbachau)

Andrej Bauer — Sat, 17 Sep 2005 22:08:24 +0000

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

First Steps in Synthetic Computability Theory (MFPS XXI)

Andrej Bauer — Sat, 07 May 2005 23:47:09 +0000

Abstract: Computability theory, which investigates computable functions and computable sets, lies at the foundation of computer science. Its classical presentations usually involve a fair amount of Gödel encodings which sometime obscure ingenious arguments. 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 Gödel encodings, but rather use familiar concepts from set theory and topology.

Presented at: Mathematical Foundations of Programming Semantics XXI, Birmingham, 2004 (invited talk).

Download paper: synthetic.pdf, synthetic.ps.gz

Download slides: synthetic-slides.pdf