Focalisation and Classical Realisability

We develop a polarised variant of Curien and Herbelin’s λ̄μμ̃ calculus suitable for sequent calculi that admit a focalising cut elimination (i.e. whose proofs are focalised when cut-free), such as Girard’s classical logic LC or linear logic. This gives a setting in which Krivine’s classical realisability extends naturally (in particular to callby-value), with a presentation in terms of orthogonality. We give examples of applications to the theory of programming languages. In this version extended with appendices, we in particular give the two-sided formulation of classical logic with the involutive classical negation. We also show that there is, in classical realisability, a notion of internal completeness similar to the one of Ludics.