Focalisation and Classical Realisability [chapter]

Guillaume Munch-Maccagnoni
2009 Lecture Notes in Computer Science  
We develop a polarised variant of Curien and Herbelin'sλµμ 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
more » ... ith 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. Focalisation Here we tackle this problem from the point of view of focalisation [And92, Gir91]. In the realm of logic programming, Andreoli's focalisation [And92] divides the binary connectives of linear logic (LL) among two groups we shall call the positives and the negatives. The distinction is motivated by the fact that they can be subject to different assumptions during proof-search. Not long after Andreoli's work, Girard [Gir91] considered focalisation as a way to determinise classical sequent calculus with the classical logic LC, which gives an operational status to these polarities. In the first part of the paper (Section 2) we give a syntax for LC and LL derived from Curien-Herbelin's calculus, the focalising system L (L foc ). Despite the age of LC and the proximity of this logic with programming languages, it is the first time that such a Focalising System L Here we define the syntax and the reduction rules of L foc . Syntax Positive and negative variables are respectively written x, y, z . . . and α, β, γ . . . One defines the sets T + and T − of the positive and negative terms t + and t − , as well as the set C of commands c:
doi:10.1007/978-3-642-04027-6_30 fatcat:lfiecqfw3bezfc7yv2mwy2xmte