A Proof-Theoretic Foundation of Abortive Continuations (Extended version)

Abstract

We give an analysis of various classical axioms and characterize a notion of minimal classical logic that enforces Peirce's law without enforcing Ex Falso Quodlibet. We show that a natural implementation of this logic is Parigot's classical natural deduction. We then move on to the computational side and emphasize that Parigot's lambda-mu corresponds to minimal classical logic. A continuation constant must be added to lambda-mu to get full classical logic. We then map the extended lambda-mu to a new theory of control, lambda-C-minus-top, which extends Felleisen's reduction theory. The new theory lambda-C-minus-top distinguishes between aborting and throwing to a continuation and is in correspondence with a refined version of Prawitz's natural deduction system.