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A realizability interpretation of Church's simple theory of types

Published online by Cambridge University Press:  22 July 2016

ULRICH BERGER  and
TIE HOU
ULRICH BERGER
Affiliation:
Department of Computer Science, Swansea University, Swansea, U.K. Email: u.berger@swan.ac.uk, cshouA@swansea.ac.uk
TIE HOU
Affiliation:
Department of Computer Science, Swansea University, Swansea, U.K. Email: u.berger@swan.ac.uk, cshouA@swansea.ac.uk
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Abstract

We give a realizability interpretation of an intuitionistic version of Church's Simple Theory of Types (CST) which can be viewed as a formalization of intuitionistic higher-order logic. Although definable in CST we include operators for monotone induction and coinduction and provide simple realizers for them. Realizers are formally represented in an untyped lambda–calculus with pairing and case-construct. The purpose of this interpretation is to provide a foundation for the extraction of verified programs from formal proofs as an alternative to type-theoretic systems. The advantages of our approach are that (a) induction and coinduction are not restricted to the strictly positive case, (b) abstract mathematical structures and results may be imported, (c) the formalization is technically simpler than in other systems, for example, regarding the definition of realizability, which is a simple syntactical substitution, and the treatment of nested and simultaneous (co)inductive definitions.

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Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Special Issue 8: Continuity, Computability, Constructivity: From Logic to Algorithms 2013 , December 2017 , pp. 1364 - 1385
Copyright
Copyright © Cambridge University Press 2016 

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