October 19, 1994

This Week's Finds in Mathematical Physics (Week 40)

John Baez

When I was an undergraduate I was quite interested in logic and the foundations of mathematics --- I was always looking for the most mind-blowing concepts I could get ahold of, and Goedel's theorem, the Loewenheim-Skolem theorem, and so on were right up there with quantum mechanics and general relativity as far as I was concerned. I did my undergrad thesis on computability and quantum mechanics, but then I sort of lost interest in logic and started thinking more and more about quantum gravity. The real reason was probably that my thesis didn't turn out as interesting as I'd hoped, but I remember feeling at the time that logic had become less revolutionary than in it was in the early part of the century. It seemed to me that logic had become a branch of mathematics like any other, studying obscure properties of models of the Zermelo-Fraenkel axioms, rather than questioning the basic presumptions implicit in those axioms and daring to pursue new, different approaches. I couldn't really get excited about the properties of super-huge cardinals. Of course, I knew a bit about intuitionistic logic and various forms of finitism, but these seemed to be the opposite of daring; instead, they seemed to appeal mainly to grumpy people who didn't trust abstractions and wanted to do everything as conservatively as possible. I was pretty interested in quantum logic, too, but I tended to think of this more as a branch of physics than "logic" proper.

Anyway, it's now quite clear to me that I just hadn't been reading the right stuff. I think Rota has said that the really interesting work in logic now goes under the name of "computer science', but for whatever reason, I didn't dig into the Journal of Philosophical Logic, other logic journals, or proceedings of conferences on category theory, computer science and the like and find the stuff that would have excited me. It goes to show that one really needs to keep digging! Anyway, I just went to a conference called the Lambda Calculus Jumelage up in Ottawa, thanks to a kind invitation by Prakash Panangaden and Phil Scott, who thought my ideas on category theory and physics might interest (or at least amuse) the folks who attend this annual bash. It became clear to me while up there that logic is alive and well!

Of course, I don't actually understand most of what these people are up to, so take what I say with a large grain of salt. My goal here is more to draw attention to some interesting-sounding ideas than to explain them.

One interesting subject, which I think I'm finally beginning to get an inkling of, is "linear logic". This was introduced in the following paper (which I haven't gotten around to looking at):

1) Linear Logic, by Jean-Yves Girard, Theoretical Computer Science 50 (1987) pp. 1-102.

When I first heard about linear logic, it made utterly no sense. It seemed to be a logic suitable for use in some completely different universe than the one I inhabited! For example, there were the familiar logical connectives "and" and "or", but they had weird alternate versions called "tensor" and "par", the latter written with an upside-down ampersand. There was also an alternate version of the material implication "→", and a strange operation called "!" (pronounced "bang") that somehow mediated between the logical connectives I knew and loved and their eerie alter egos.

I understand a wee bit about these things now; one can get a certain ways just by getting used to "tensor", since the rest of the weird connectives are defined in terms of this one and the familiar ones. (I won't worry about the "!" here.) One key idea, which finally penetrated my thick skull, is that there is a good reason why "tensor" does not satisfy the following deduction rule so characteristic of "and":

S |- p       S |- q
S |- p & q

meaning: if from the set of premisses S we can deduce p, and from S we can also deduce q, then from S we can deduce p&q. The point is that in linear logic one should not think of S as a set of premisses, but rather as a multiset, meaning that the same premiss can appear twice. The idea is that if we use one premiss in S to deduce something, we *use it up*, and we can only use it again if S has several copies of that premiss in it. As they say, linear logic is "resource-sensitive" (which is apparently why computer scientists like it). So the idea is that in linear logic,

S |- p&q

means something like "from the premisses S one can deduce p if one feels like it, or alternatively one can deduce q if one feels like it, but not necessarily both "at once", since there may not be enough copies of the premisses to do that." On the other hand,

S |- p tensor q

is stronger, since it means something like "from the premisses S one can deduce both p and q at once, since there are enough copies of all the premisses in S to do it." Thus "&" satisfies the above deduction rule in linear logic just as in classical logic, but "tensor" does not; instead, it satisfies

S |- p       T |- q
S U T |- p tensor q

where S U T denotes the union of the multisets S and T (so that if both S and T have one copy of a premiss, S U T has two copies of it).

Well, let me leave it at that. I should add that there is a paper available online,

2) Linear logic for generalized quantum mechanics, by Vaughan Pratt, available in LaTeX format (compressed) by anonymous ftp from boole.stanford.edu, as the file pub/ql.tex.Z,

which relates linear logic and quantum logic, and which is part of a body of work relating linear logic and category theory, with the key idea being that "linear logic is a logic of monoidal closed categories in much the same way that intuitionistic logic is a logic of Cartesian closed categories" --- here I quote

3) Hopf algebras and linear logic, by Richard Blute, to appear in Mathematical Structures in Computer Science.

I suppose to most people, explaining linear logic in terms of monoidal closed categories may seem like using mud to wipe ones windshield. However, to some of us monoidal closed categories are rather familiar things, and in fact anyone who knows about vector spaces, linear maps, and the vector spaces Hom(V,W) and V tensor W knows a really good example of a monoidal closed category. Thus monoidal closed categories can be viewed as an abstraction of linear algebra, and indeed this is how "linear logic" got its name.

It seems that I should read the following papers, too, before I really understand the connection between linear logic and category theory:

4) Linear logic, *-autonomous categories and cofree coalgebras, by R. A. G. Seely, in Categories in Computer Science and Logic, Contemp. Math. 92 (1989).

5) Quantales and (noncommutative) linear logic, by D. Yetter, Journal of Symbolic Logic 55 (1990), 41-64.

A terse summary of linear logic in terms a categorist might like can be found in Section 3.5 of Pratt's paper cited above. I should add that Pratt has lots of other interesting papers available online (try the file pub/README).

© 1994 John Baez