15-816 Linear Logic

Lecture 27: Non-Commutative Linear Logic

We analyze the structural properties of various logic we have considered:

• linear logic only satisfies exchange,
• strict logic satisfies exchange and contraction, and
• intuitionistic logic satisfies exchange, contraction, and weakening
.

Thus there are 8 possibilities of allowing or denying weakening, exchange, and contraction, and we consider the various logics arising from that (not all of which may make sense). The pure versions of the weaker logics in this diagram are clearly not very useful, but by using modalities or multiple hypothesis judgments, one can recover the expressive power of stronger versions logics.

We discuss what is known about this constellation and give a formulation of the most expressive version: intuitionistic, non-commutative linear logic. We illustrate through a number of logic programming examples how a non-commutative context may be used to model stack and queues. I can also be employed to eliminate don't-care nondeterminism, thus improving robustness of linear logic programs with respect to failure.

• Next: Lecture 28: Stacks and Non-Commutativity
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