# 15-816 Linear Logic Lecture 9: Focusing

We discuss further restrictions that can be imposed on sequent calculus proof search without losing completeness. A first step are the inversion properties of the left and right rules for the connectives in the sequent calculus. Weak invertibility means that the premises of a left or right rule are provable if the conclusion is provable. Strong invertibility means that the corresponding left or right rule can always be applied immediately without losing completness for proof search.

Once all strongly invertible rules have been applied, we can then focus, either on the goal or a particular hypothesis and apply a sequence of non-invertible rules to this focus proposition. When we encounter an invertible connective, we switch back to using invertible rules.

Remarkably, this strategy is both sound and complete. It is also robust in that related focusing strategies are sound and complete for many logics of interest, including classical logics. The idea originates in Andreoli's system for classical linear logic.

We formalize focusing as a deductive system with several related judgments.