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.