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dc.contributor.advisorDyckhoff, Roy
dc.contributor.advisorKesner, Delia
dc.contributor.authorLengrand, Stéphane J. E.
dc.coverage.spatial378en
dc.date.accessioned2007-04-20T15:04:31Z
dc.date.available2007-04-20T15:04:31Z
dc.date.issued2006-12-08
dc.identifieruk.bl.ethos.551994
dc.identifier.urihttp://hdl.handle.net/10023/319
dc.description.abstractAt the heart of the connections between Proof Theory and Type Theory, the Curry-Howard correspondence provides proof-terms with computational features and equational theories, i.e. notions of normalisation and equivalence. This dissertation contributes to extend its framework in the directions of proof-theoretic formalisms (such as sequent calculus) that are appealing for logical purposes like proof-search, powerful systems beyond propositional logic such as type theories, and classical (rather than intuitionistic) reasoning. Part I is entitled Proof-terms for Intuitionistic Implicational Logic. Its contributions use rewriting techniques on proof-terms for natural deduction (Lambda-calculus) and sequent calculus, and investigate normalisation and cut-elimination, with call-by-name and call-by-value semantics. In particular, it introduces proof-term calculi for multiplicative natural deduction and for the depth-bounded sequent calculus G4. The former gives rise to the calculus Lambdalxr with explicit substitutions, weakenings and contractions that refines the Lambda-calculus and Beta-reduction, and preserves strong normalisation with a full notion of composition of substitutions. The latter gives a new insight to cut-elimination in G4. Part II, entitled Type Theory in Sequent Calculus develops a theory of Pure Type Sequent Calculi (PTSC), which are sequent calculi that are equivalent (with respect to provability and normalisation) to Pure Type Systems but better suited for proof-search, in connection with proof-assistant tactics and proof-term enumeration algorithms. Part III, entitled Towards Classical Logic, presents some approaches to classical type theory. In particular it develops a sequent calculus for a classical version of System F_omega. Beyond such a type theory, the notion of equivalence of classical proofs becomes crucial and, with such a notion based on parallel rewriting in the Calculus of Structures, we compute canonical representatives of equivalent proofs.en
dc.format.extent2673747 bytes
dc.format.mimetypeapplication/pdf
dc.language.isoenen
dc.publisherUniversity of St Andrews
dc.subjectLogicen
dc.subjectProof theoryen
dc.subjectType theoryen
dc.subjectLambda-calculusen
dc.subjectRewritingen
dc.subject.lccQA9.54L4
dc.subject.lcshProof theoryen
dc.subject.lcshType theoryen
dc.subject.lcshCurry-Howard isomorphismen
dc.subject.lcshLambda calculusen
dc.titleNormalisation & equivalence in proof theory & type theoryen
dc.title.alternativeNormalisation & equivalence en théorie de la démonstration & théorie des typesen
dc.typeThesisen
dc.type.qualificationlevelDoctoralen
dc.type.qualificationnamePhD Doctor of Philosophyen
dc.publisher.institutionThe University of St Andrewsen
dc.publisher.departmentUniversité Paris 7 - Denis Diderot, Franceen


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