Constructive Completeness Proofs and Delimited Control

Danko Ilik 1, 2, 3
2 PI.R2 - Design, study and implementation of languages for proofs and programs
PPS - Preuves, Programmes et Systèmes, Inria Paris-Rocquencourt, UPD7 - Université Paris Diderot - Paris 7, CNRS - Centre National de la Recherche Scientifique : UMR7126
Abstract : Motivated by facilitating reasoning with logical meta-theory inside the Coq proof assistant, we investigate the constructive versions of some completeness theorems. We start by analysing the proofs of Krivine and Berardi-Valentini, that classical logic is constructively complete with respect to (relaxed) Boolean models, and the algorithm behind the proof. In an effort to make a more canonical proof of completeness for classical logic, inspired by the normalisation-by-evaluation (NBE) methodology of Berger and Schwichtenberg, we design a completeness proof for classical logic by introducing a notion of model in the style of Kripke models. We then turn our attention to NBE for full intuitionistic predicate logic, that is, to its completeness with respect to Kripke models. Inspired by the computer program of Danvy for normalising terms of $\lambda$-calculus with sums, which makes use of delimited control operators, we develop a notion of model, again similar to Kripke models, which is sound and complete for full intuitionistic predicate logic, and is coincidentally very similar to the notion of Kripke-style model introduced for classical logic. Finally, based on observations of Herbelin, we show that one can have an intuitionistic logic extended with delimited control operators which is equi-consistent with intuitionistic logic, preserves the disjunction and existence properties, and is able to derive the Double Negation Shift schema and Markov's Principle.
Document type :
Theses
Software Engineering [cs.SE]. Ecole Polytechnique X, 2010. English
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Danko Ilik. Constructive Completeness Proofs and Delimited Control. Software Engineering [cs.SE]. Ecole Polytechnique X, 2010. English. <tel-00529021>

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