15-816 Linear Logic
Lecture 23: Ordered Logic

As a final topic of the course, we introduce a further refinement to linear logic by introducing an ordered hypothetical judgment. This further restrict the use of hypotheses. In ordered logic we have two forms of implication: one that adds a hypothesis at the right, and one that adds a hypothesis at the left.

Ordered logic allows a more natural representation of a number of examples, including parsing and various algorithms involving queues and stacks. It can also be used to remove unwanted don't care non-determinism as in the specification of focussed proofs by Andreoli.

We discuss several examples in some detail. We begin with a representation of finite automata, context-free grammars, and Turing machines. The last shows that even a small fragment of multiplicative exponential ordered logic is undecidable. Then we give an implementation of concurrent evaluation for a small functional language that does not require destination-passing style, and an implementation of merge sort that can be changed to insertion sort through the change of one connective.

[PP99] Jeff Polakow and Frank Pfenning.
Relating Natural Deduction and Sequent Calculus for Intuitionistic Non-Commutative Linear Logic.
Proceedings of the 15th Conference on Mathematical Foundations of Programming Semantics, A. Scedrov and A. Jung, editors, New Orleans, Louisiana, April 1999.
Available in Electronic Notes in Theoretical Computer Science, Vol. 20.

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Frank Pfenning