15-816 Linear Logic

Lecture 11: Weakly Focussed Derivations

Inversion principles provide us with one way to remove unnecessary disjunctive non-determinism in proof search. Inversion principles for right rules in the sequent calculus lead to the notion of a weakly goal-directed derivation.

In this lecture we extend this further by taking advantage of inversion principles for left rules and one additional observation which will lead us to the notion of a weakly focussed derivation. A derivation which is both weakly goal-directed and focussed is called weakly uniform, a class of derivations which is sound and complete for proof search in linear logic. When combined with unification and occurrence constraints, much of the non-determinism has been either eliminated or isolated into a few crucial choices.

We show how this can be turned into flexible basis for a theorem proving procedure combining interactive and automatic deduction using tactics and tacticals. In general, a tactic is a function which may reduce a goal to some subgoals which are sufficient for proving the original goal, or fail. The idea of possibly failing functions gives rise to a rich set of combinators for combining tactics called tacticals. Tactics and tacticals are the basis for some of the most successful theorem proving environments in use today, such as Coq, Isabelle, or NuPrl.

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Frank Pfenning