Section: New Results

Focused Linear Logic and the $\lambda$-calculus

Participants : Taus Brock-Nannestad, Nicolas Guenot.

Linear Logic enjoys strong symmetries inherited from classical logic while providing a constructive framework comparable to intuitionistic logic. However, the computational interpretation of sequent calculus presentations of linear logic remains problematic, mostly because of the many rule permutations allowed in the sequent calculus.

In focused variants of Linear Logic, most of these rule permutations are eliminated by the focusing restriction — during focusing, a single formula is decomposed eagerly, and the focus is passed down to its subformulas. Conversely, during inversion, all invertible connectives are decomposed. Moreover, this decomposition is made fully determinstic by keeping the connectives in question in a list, and only decomposing the first connective of this list.

The end result of this is that a focused proof in Linear Logic almost always has one particular formula singled out as the one that will be decomposed. Thus, somewhat curiously, focused Linear Logic behaves much more like an intuitionistic sequent calculus (where at all times there is a single “special” formula on the right hand side of the sequent) than a classical calculus.

In [26] (MFPS'15), we study a term assignment for a focused version of Multiplicative Exponential Linear Logic (MELL), and show how the focusing technique gives rise to a calculus that straightforwardly embeds both a linear variant of the $\lambda$-calculus, and a sequent-based formulation of Parigot's $\lambda \mu$-calculus.