15-816 Linear Logic

Lecture 07: Proof Search and Inversion

In the search for proofs in intuitionistic linear logic, we can restrict ourselves to cut-free derivations. This is one important consequence of the cut elimination theorem. Proof search then proceeds via a purely bottom-up construction of a derivation, always reducing the goal of proving a sequent to the subgoal of proving the premisses of an applicable inference rule.

In this process, many non-deterministic choices remain: which left or right rule do we apply, and how do we split the resources in the multiplicative rules? To obtain feasible, yet complete theorem proving strategies these choices should be restricted further. For example, it is a complete strategy to always apply the right rule for alternative conjunction, when the proposition we want to prove has the right form.

Thus the simplest form of strategy arises from inversion principles: whenever the conclusion of an inference rule is provable, then the premisses of the rule are also derivable. Thus we will not miss a possibly true proposition, if we apply the rule in the backward direction. In this lecture we investigate which inference rules are invertible in this sense, and which critical choices remain in the search for a derivation.

In upcoming lectures we will develop further techniques to limit the non-determinism and postpone choices that are difficult to make.

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Frank Pfenning