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
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