# Correspondence of Linear Logic & Geometric Algebra

• To: <types@cis.upenn.edu>
• Subject: Correspondence of Linear Logic & Geometric Algebra
• From: "Tim Sweeney" <tim@epicgames.com>
• Date: Fri, 27 Oct 2000 06:13:30 -0400
• Delivery-Date: Fri Oct 27 09:14:25 2000
• References: <200010131211.e9DCBDn28813@saul.cis.upenn.edu>

```While prototyping a compiler using Girard's Linear Logic as a type system
and implementing, in it, a math library supporting the Geometric (Clifford)
Algebra, I noticed a striking resemblance.  Are these two systems known to
coincide?

My conjecture: Linear Logic and Geometric Algebra are, in an axiomatic and
semantic view, equivalant; as:

- The linear logic's "times" operator corresponds to the Geometric Algebra
outer product.
- Linear "with" corresponds to the geometric product, with semantics
analogous to the set-theoretic "exclusive or".
- "Par" matches the meet operator (the dual of the "times" and therefore of
the outer product).
- "Plus" is the dual of the geometric product.
- "Nil", A_|_ corresponds to multiplication by the unit pseudoscalar.
- The "!" operator in linear logic appear to roughly correspond to the
reverse of a multivector, and "?" to the conjugate of a multivector.
- The grade of a homogeneous multivector is analogous to the number of
linear terms that appear in each conjunctive subterm of an expression.  GA
operations which raise, lower, and retain the grade, are analogous to linear
logic with respect to the number of linear assumptions.
- The GA interpretation of linear implication "A -o B" is (surprisingly) not
an exponential, but in fact the inner product, seeing that it is the
algebras' sole grade- (uniqueness count-) lowering operation.
- The entire duality of linear logic (between conjunction and disjunction,
and so on) is carried out identically to the duality of Clifford Algebra,
between outer products and the meet operation.
- The "1", "T", "_|_", and "0" of linear logic correspond to 1, the unit
pseudoscalar, the unit pseudoscalar again, and zero.

Going through Hestenes' book on Geometric Algebra and Girard et al's
"Advances in Linear Logic", the correspondence runs so deep that I find it
hard to believe the systems do not correspond.

Interestingly, the development of Geometric Algebra was developed to treat
spaces and their subspaces as first-class constructs in a single algebra,
with the new geometric product bringing together terms of dimensionality;
while Linear Logic was invented to treat uniqueness (in a certain view, a
form of dimensionality) as a first-class construct, using the "with"
(do-only-once) operator to formalize the uniqueness dimension of values.

It would thus be delightful to see that these two operators, and perhaps the
entire formalisms of GA and LL, contain the same substance, developed
independently, as they were, almost a hundred years apart.

I would be interested in a counterexample, or references on this
correspondence if it is known.

-Tim Sweeney
```