A proof-theoretic foundation of abortive continuations

Abstract

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 λµ corresponds to minimal classical logic. A continuation constant must be added to λµ to get full classical logic. The extended calculus is isomorphic to a syntactical restriction of Felleisen's theory of control that offers a more expressive reduction semantics. This isomorphic calculus is in correspondence with a refined version of Prawitz's natural deduction.

A Context-based Approach to Proving Termination of Evaluation

“…We can see that these rules agree with the standard typing for first-class continuations both from the semantics and logic viewpoints [1,19,27]. In particular, if we interpret them through the Curry-Howard correspondence, we obtain a natural deduction system for minimal classical logic, i.e., minimal logic + Peirce's law.…”

“…Here again we can use another representative of the class of programs equal to v, E t 1 , such as t 1 , v E . Now we can apply the induction hypothesis again, this time for t 1 , provided that v E is well typed and C cont C (v E) holds.…”

“…In effect, we obtain direct, simple proofs of termination that take advantage of the context-based formulation of the reduction semantics. In contrast, many of the existing proofs of normalization properties for typed lambda calculi with control operators are indirect and they use a translation to another language already known to be normalizable [1,18,24]. This line of work on proof-theoretic properties of typed control operators was originated by Griffin who gave a type assignment to Felleisen's C operator, abort and callcc, and who proved termination of evaluation for his language using a translation to the simply typed lambda calculus akin to Plotkin's colon translation [18].…”

Classical Call-by-Need and Duality

“…The study has also suggested how to provide a call-by-need version of Parigot's λµ-calculus, and in the minimal case, has led to a new notion of standard reduction, which applies the lift and assoc rule eagerly. In the minimal case, the single context variable, called , could be seen as the constant tp discussed in [6,5]. In the cited work, it is also presented how delimited control can be captured by extending tp to a dynamic variable named tp.…”

Control reduction theories: the benefit of structural substitution

“…As a consequence, no rule artificially breaks strong normalization (see e.g. (Ariola & Herbelin, 2003;Ariola et al, 2005) for a proof of strong normalization in the simply-typed case).…”