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Title: A continuous computational interpretation of type theories
Author: Xu, Chuangjie
ISNI:       0000 0004 5354 5784
Awarding Body: University of Birmingham
Current Institution: University of Birmingham
Date of Award: 2015
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This thesis provides a computational interpretation of type theory validating Brouwer’s uniform-continuity principle that all functions from the Cantor space to natural numbers are uniformly continuous, so that type-theoretic proofs with the principle as an assumption have computational content. For this, we develop a variation of Johnstone’s topological topos, which consists of sheaves on a certain uniform-continuity site that is suitable for predicative, constructive reasoning. Our concrete sheaves can be described as sets equipped with a suitable continuity structure, which we call C-spaces, and their natural transformations can be regarded as continuous maps. The Kleene-Kreisel continuous functional can be calculated within the category of C-spaces. Our C-spaces form a locally cartesian closed category with a natural numbers object, and hence give models of Gödel’s system T and of dependent type theory. Moreover, the category has a fan functional that continuously compute moduli of uniform continuity, which validates the uniform-continuity principle formulated as a skolemized formula in system T and as a type via the Curry-Howard interpretation in dependent type theory. We emphasize that the construction of C-spaces and the verification of the uniform-continuity principles have been formalized in intensional Martin-Löf type theory in Agda notation.
Supervisor: Not available Sponsor: University of Birmingham
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA75 Electronic computers. Computer science