# 15-816 Linear Logic Lecture 4: Normal Deductions

We complete the example by showing how to represent regular expressions in linear logic. It is easy to show that for every string in the language of a regular expression, we can prove the corresponding proposition in linear logic. However, the other direction is intractable, unless we understand more about the structure of proofs and proof search.

We therefore turn our attention to developing complete strategies for proof search. We begin with so-called normal deductions where search proceeds only with introduction rules from the conclusion and elimination rules from the hypotheses. The fact that this strategy is complete has many important consequences. For example, the logic is consistent, and all the quantifiers and connectives are orthogonal to each other. Furthermore, our explanation of the connectives via introduction and elimination rules is a well-founded definition of truth.

Normal deductions are defined via two mutually recursive judgments. We consider some of their properties, and how they can help in the proof that the representation of regular expressions in linear logic is adequate.