A formulae-as-type notion of control

Abstract

The programming language Scheme contains the control construct call/cc that allows access to the current continuation (the current control context). This, in effect, provides Scheme with first-class labels and jumps.We show that the well-known formulae-astypes correspondence, which relates a constructive proof of a formula a to a program of type (Y, can be extended to a typed Idealized Scheme. What is surprising about this correspondence is that it relates classical proofs to typed programs. The existence of computationally interesting "classical programs" -programs of type (Y, where Q holds classically, but not constructively -is illustrated by the definition of conjunctive, disjunctive, and existential types using standard classical definitions. We also prove that all evaluations of typed terms in Idealized Scheme are finite.

A proof-theoretic foundation of abortive continuations

“…Next, we give the structural presentation. Section 4 reviews the basic control operators and calculi together with their (Curry-Howard) isomorphism to classical logic discovered by Griffin (1990). Section 5 and 6 introduce two isomorphic calculi which are the computational counterparts of classical natural deduction with one and multiple conclusions, respectively.…”

Strong Normalization of the Dual Classical Sequent Calculus

“…In its traditional form, terms in the λ-calculus encode proofs in intuitionistic natural deduction; from another perspective the proofs serve as typing derivations for the terms. Griffin extended the Curry-Howard correspondence to classical logic in his seminal 1990 POPL paper [16], by observing that classical tautologies suggest typings for certain control operators. This initiated a vigorous line of research: on the one hand classical calculi can be seen as pure programming languages with explicit representations of control, while at the same time terms can be tools for extracting the constructive content of classical proofs [21,3].…”

Playful, Streamlike Computation

“…The formula ((A → B) → A) → A is called Pierce's law and is a typical tautology of classical logic. The connection between control operators and classical logic -and in particular the fact that call-cc corresponds to Pierce's law-was first discovered in [21]. Here is is the sequential algorithm interpreting call-cc for A = bool: …”