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Title: Normalisation & equivalence in proof theory & type theory
Author: Lengrand, Stéphane J. E.
Awarding Body: University of St Andrews
Current Institution: University of St Andrews
Date of Award: 2006
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At 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.
Supervisor: Dyckhoff, Roy 1948-. ; Kesner, Delia Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Logic ; Proof theory ; Type theory ; Lambda-calculus ; Rewriting ; QA9.54L4 ; Proof theory ; Type theory ; Curry-Howard isomorphism ; Lambda calculus