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When quoting this document, please refer to the following
DOI: 10.4230/OASIcs.CCA.2009.2254
URN: urn:nbn:de:0030-drops-22540
URL: http://drops.dagstuhl.de/opus/volltexte/2009/2254/
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Escardó, Martin
Tutorials

Theory and Practice of Higher-type Computation (Tutorial)

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Abstract

In higher-type computation, established by Kleene and Kreisel in the late 1950's (independently), one works with the data types obtained from the discrete natural numbers by closing under finite products and function spaces. For the theory of higher-type programming languages, it is natural to work with a corresponding hierarchy, or type structure, of domains, identified by Ershov and Scott in the late 1960's (again independently). The Kleene-Kreisel and Ershov-Scott hierarchies account for total and partial computation respectively. In this tutorial I'll explain the theory and practice of higher-type computation and programming languages, and develop old and new applications. From a theoretical point of view, I'll present Kleene-Kreisel spaces and Ershov-Scott domains, and relate the two. Moreover, I'll discuss common generalizations, chiefly QCB spaces and equilogical spaces, which admit further useful closure properties, and their relationship to TTE (Schroeder, Simpson. Scott, Bauer, Weihrauch and many others). I'll also present a natural higher-type model of computation/programming language, namely PCF (Platek, Scott, Plotkin). From a practical point of view, I'll introduce a fragment of the language Haskell as a faithful implementation of PCF. Moreover, I'll develop and run several examples (and prove theorems about them), pertaining to (i) exhaustive search of infinite sets in finite time in particular Ulrich Berger's algorithm and generalizations), and (ii) computation with real numbers (in particular Alex Simpson's integration algorithm and generalizations).

BibTeX - Entry

@InProceedings{escard:OASIcs:2009:2254,
  author =	{Martin Escard{\'o}},
  title =	{{Theory and Practice of Higher-type Computation (Tutorial)}},
  booktitle =	{6th International Conference on Computability and Complexity in Analysis (CCA'09)},
  series =	{OpenAccess Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-12-5},
  ISSN =	{2190-6807},
  year =	{2009},
  volume =	{11},
  editor =	{Andrej Bauer and Peter Hertling and Ker-I Ko},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2009/2254},
  URN =		{urn:nbn:de:0030-drops-22540},
  doi =		{http://dx.doi.org/10.4230/OASIcs.CCA.2009.2254},
  annote =	{Keywords: Higher-type computation, domain theory, Kleene-Kreisel spaces, Ershov-Scott domains, QCB spaces, equilogical spaces, PCF}
}

Keywords: Higher-type computation, domain theory, Kleene-Kreisel spaces, Ershov-Scott domains, QCB spaces, equilogical spaces, PCF
Seminar: 6th International Conference on Computability and Complexity in Analysis (CCA'09)
Issue Date: 2009
Date of publication: 25.11.2009


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