Proofs of strong normalisation for second order classical natural deduction

Abstract

We give two proofs of strong normalisation for second order classical natural deduction. The first one is an adaptation of the method of reducibility candidates introduced in [9] for second order intuitionistic natural deduction; the extension to the classical case requires in particular a simplification of the notion of reducibility candidate. The second one is a reduction to the intuitionistic case, using a Kolmogorov translation.

Strong Normalization of the Dual Classical Sequent Calculus

“…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]. In particular the λµ calculus of Parigot [23] has been the basis of a number of investigations [24,11,22,5,1] into the relationship between classical logic and theories of control in programming languages.…”

Relational Parametricity and Control

“…The second-order λµ-calculus, λµ2, is given as follows. We essentially follow Parigot's formulation [27] (with some flavour from Selinger's [36]). The types are the same as those of the second-order λ-calculus λ2:…”

“…The CPS translation. We present a call-by-name CPS translation which can be considered as an extension of that introduced by Streicher [16,39,36] (rather than the translations by Plotkin [29], Parigot [27] or Fujita [7] which introduce extra negations and do not respect extensionality).…”

“…The second-order λµ-calculus (λµ2), again due to Parigot [27], has been studied in depth as a calculus for second-order classical natural deduction. In particular, strong normalization results of λµ2 [27,23] and its extensions, e.g.…”

“…In particular, strong normalization results of λµ2 [27,23] and its extensions, e.g. with inductive types [21], have been a central research topic, because of the proof-theoretical importance of strong normalization.…”