Probably many of the mathematicians reading this know about the Newton Institute in Cambridge, a mathematics institute run by Sir Michael Atiyah. It's a cozy little building, in a quiet neighborhood a certain distance from the center of town, which one can reach by taking a nice walk or bike ride over the bridge near Trinity College, across Grange Road, and down Clarkson Road. Inside it's one big space, with stairways slightly reminiscent of a certain picture by Escher, with a nice little library on the first floor, tea and coffee on the 3rd floor, blackboards in the bathrooms... everything a mathematician could want. This is where Wiles first announced his proof of Fermat's last theorem, and they sell T-shirts there commemorating that fact, which are unfortunately too small to contain the proof itself... as they do not refrain from pointing out.
I just got back from a conference there on New Connections between Mathematics and Computer Science. It was organized by Jeremy Gunawardena, who was eager to expose computer scientists and mathematicians to a wide gamut of new interactions between the two subjects. I spoke about n-categories in logic, topology and physics. Since I don't know anything about computer science, when I first got the invitation I thought it was a mistake: a wrong email address or something! But Gunawardena assured me otherwise. I assumed the idea was that n-categories, being so abstract, must have some application to just about everything, even computer science. Luckily, some other speakers at the conference gave some very nice applications of n-category theory to computer science, so now I know they really exist.
Unfortunately I had to miss the beginning of the conference, and therefore missed some interesting talks of a geometrical nature by Smale, Gromov, Shub and others. Let me say a bit about some of the talks I did catch. You can find a list of all the speakers and abstracts of their talks at
1) Basic Research Institute in the Mathematical Sciences, New Connections web page, http://www-uk.hpl.hp.com/brims/
Richard Jozsa gave an interesting talk on quantum computers, in part outlining Peter Shor's work (see "week34") on efficient factoring via quantum computation, but also presenting some new results on "counterfactual quantum computation". It turns out that - in principle - in some cases you can get a quantum computer to help you answer a question, even without running it, just as long as you COULD HAVE run it! (I should add that in practice a lot of things make this quite impractical.) This is a new twist on the Elitzur-Vaidman bomb-testing paradox about how if you have a bunch of bombs and half of them are duds, and the only way you can test a bomb is by lighting the fuse and seeing if it goes off, you can still get a bomb you're sure will work, if you use quantum mechanics. The trick involves getting a fuse that's so sensitive that even one photon will make the bomb go off, and then setting up a beam-splitter, and using the bomb to measure which path the photon followed, before recombining the beams. Check out:
2) A. C. Elitzur and L. Vaidman, Quantum mechanical interaction-free measurements, Foundations of Phys. 23 (1993), 987-997.
Graeme Mitchison and Richard Jozsa, Counterfactual quantum computation, Proc. Roy. Soc. Lond. A457 (2001) 1175-1194. Also available as quant-ph/9907007.
Jean-Yves Girard gave an overview of linear logic. Linear logic is a new version of logic that he invented, which has some new operations besides the good old ones like "and", "or", and "not". For example, there are things like "par" (written as an upside-down ampersand), "!" (usually pronounced "bang") and "?". Ever since I started going to conferences on category theory and computer science I have been hearing a lot about it, and I keep trying to get people to explain these weird new logical operations to me. Unfortunately, I keep getting very different answers, so it has remained rather mysterious to me, even though it seems like a lot of fun (see "week40"). Thus I was eager to hear it from the horse's mouth.
Indeed, Girard gave a fascinating talk on it which almost made me feel I understood it. I think the big thing I'd been missing was a good appreciation of topics in proof theory like "cut elimination". He noted that this subject usually appears to be all about the precise manipulation of formulas according to purely syntactic rules: "Very bureaucratic" he joked, "one parenthesis missing and you've had it!" (For full effect, one must imagine this being said in a French accent by someone stylishly dressed entirely in black.) He wanted to get a more geometrical way to think about proofs, but to do this it turned out to be important to refine ordinary logic in certain ways.... leading to linear logic. However, I still don't feel up to explaining it, so let me turn you to:
3) Jean-Yves Girard, Linear logic, Theoretical Computer Science 50, 1-102, 1987.
Jean-Yves Girard, Y. Lafont and P. Taylor, Proofs and Types, Cambridge Tracts in Theoretical Computer Science 7, Cambridge U. Press, 1989. Also available at http://www.cs.man.ac.uk/~pt/stable/Proofs+Types.html
Eric Goubault and Vaughan Pratt talked, in somewhat different ways, about a formalism for treating concurrency using "higher-dimensional automata". The basic idea is simple: say we have two jobs to do, one of which gets us from some starting-point A to some result B, and the other of which gets us from A' to B'. We can represent each task by an arrow, as follows:
A ----> B A' ---> B'
We can think of this arrow as a "morphism", that is, a completely abstract sort of operation taking A to B. Or, we can think of it more concretely as an interval of time, where our computer is doing something at each moment. Alternatively, we can think of it more discretely as a sequence of steps, starting with A and winding up with B.
If we now consider doing both these tasks concurrently, we can represent the situation by a square:
AA' -----> BA' | | | | | | | | v v AB' -----> BB'
Going first across and then down corresponds to completing one task before starting the other, while going first down and then across corresponds to doing the other one first. However, we can also imagine various roughly diagonal paths through the square, corresponding to doing both tasks at the same time. We might go horizontally for a while, then vertically, then diagonally, and so on. Of course, if the two tasks were not completely independent - for example, if some steps of one could only occur after some steps of the other were finished - we would have some constraints on what paths from AA' to BB' were allowed. The idea is then to model these constaints as "holes" in the square, forbidden regions where the path cannot go. There may then be several "essentially distinct" ways of getting from AA' to BB', that is, classes of paths that cannot be deformed into each other.
To anyone who knows homotopy theory, this will seem very familiar, homotopy theory being all about spaces with holes in them, and how those holes prevent you from continuously deforming one path into another. Goubault's title, "Scheduling problems and homotopy theory", emphasized the relationships. But there are also some big differences. Unlike homotopy theory, here the paths are typically required to be "monotonic": they can't double back and go backwards in time. And, as I mentioned, the tasks can be thought of more abstractly than as paths in some space. So we are really talking about 2-categories here: they give a general framework for studying situations with "dots" or "objects", "arrows between dots" or "morphisms", and "arrows between arrows between dots" or "2-morphisms". Similarly, when we study concurrency with more than 2 tasks at a time we can think of it in terms of n-categories.
By the way, since I don't know much about parallel processing, I'm not sure how much the above formalism actually helps the "working man". Probably not much, yet. I get the impression, however, that parallel processing is a complicated problem, and that people are busily dreaming up new formalisms for talking about it, hoping they will eventually be useful for inventing and analyzing parallel programming languages.
Some references for this are:
4) Eric Goubault, Schedulers as abstract interpretations of higher-dimensional automata, in Proc. PEPM '95 (La Jolla), ACM Press, 1995. Also available at http://www.di.ens.fr/%7Egoubault/GOUBAULTpapers.html
Eric Goubault and Thomas Jensen, Homology of higher-dimensional automata, in Proc. CONCUR '92 (New York), Lecture Notes in Computer Science 630, Springer, 1992. Also available at http://www.di.ens.fr/%7Egoubault/GOUBAULTpapers.html
5) Vaughan Pratt, Time and information in sequential and concurrent computation, in Proc. Theory and Practice of Parallel Programming, Sendai, Japan, 1994.
Yves Lafont also gave a talk with strong connections to n-category theory. Recall that a monoid is a set with an associative product having a unit element. One way to describe a monoid is by giving a presentation with "generators", say
a, b, c, d,
and "relations", say
ab = a, da = ac.
We get a monoid out of this in an obvious sort of way, namely by taking all strings built from the generators a,b,c, and d, and then identifying two strings if you can get from one to the other by repeated use of the relations. In math jargon, we form the free monoid on the generators and then mod out by the relations.
Suppose our monoid is finitely presented, that is, there are finitely many generators and finitely many relations. How can we tell whether two elements of it are equal? For example, does
dacb = acc
in the above monoid? Well, if the two are equal, we will always eventually find that out by an exhaustive search, applying the relations mechanicallly in all possible ways. But if they are not, we may never find out! (For the above example, the answer appears at the end of this article in case anyone wants to puzzle over it. Personally, I can't stand this sort of puzzle.) In fact, there is no general algorithm for solving this "word problem for monoids", and in fact one can even write down a specific finitely presented monoid for which no algorithm works.
However, sometimes things are nice. Suppose you write the relations as "rewrite rules", that go only one way:
ab → a
da → ac
Then if you have an equation you are trying to check, you can try to repeatedly apply the rewrite rules to each side, reducing it to "normal form", and see if the normal forms are equal. This will only work, however, if some good things happen! First of all, your rewrite rules had better terminate: it had better be that you can only apply them finitely many times to a given string. This happens to be true for the above pair of rewrite rules, because both rules decrease the number of b's and c's. Second of all, your rewrite rules had better be confluent: it had better be that if I use the rules one way until I can't go any further, and you use them some other way, that we both wind up with the same thing! If both these hold, then we can reduce any string to a unique normal form by applying the rules until we can't do it any more.
Unfortunately, the rules above aren't confluent; if we start with the word dab , you can apply the rules like this
dab → acb
while I apply them like this
dab → da → ac
and we both terminate, but at different answers. We could try to cure this by adding a new rule to our list,
acb → ac.
This is certainly a valid rule, which cures the problem at hand... but if we foolishly keep adding new rules to our list this way we may only succeed in getting confluence and termination when we have an infinite list of rules:
ab → a
da → ac
acb → ac
accb → acc
acccb → accc
accccb → acccc...
and so on. I leave you to check that this is really terminating and confluent. Because it is, and because it's a very predictable list of rules, we can use it to write a computer program in this case to solve the word problem for the monoid at hand. But in fact, if we had been cleverer, we could have invented a finite list of rules that was terminating and confluent:
ab → a
ac → da
Lafont's went on to describe some work by Squier:
6) Craig C. Squier, Word problems and a homological finiteness condition for monoids, Jour. Pure Appl. Algebra 49 (1987), 201-217.
Craig C. Squier, A finiteness condition for rewriting systems, revision by F. Otto and Y. Kobayashi, to appear in Theoretical Computer Science.
Craig C. Squier and F. Otto, The word problem for finitely presented monoids and finite canonical rewriting systems, in Rewriting Techniques and Applications, ed. J. P. Jouannuad, Lecture Notes in Computer Science 256, Springer, Berlin, 1987, 74-82.
which gave general conditions which must hold for there to be a finite terminating and confluent set of rewrite rules for a monoid. The nice thing is that this relies heavily on ideas from n-category theory. Note: we started with a monoid in which the relations are equations, but we then started thinking of the relations as rewrite rules or morphisms, so what we really have is a monoidal category. We then started worrying about "confluences", or equations between these morphisms. This is typical of "categorification", in which equations are replaced by morphisms, which we then want to satisfy new equations (see "week38").
For the experts, let me say exactly how it all goes. Given any monoid M, we can cook up a topological space called its "classifying space" KM, as follows. We can think of KM as a simplicial complex. We start by sticking in one 0-simplex, which we can visualize as a dot like this:
Then we stick in one 1-simplex for each element of the monoid, which we can visualize as an arrow going from the dot to itself. Unrolled a bit, it looks like this:
Really we should draw an arrow going from left to right, but soon things will get too messy if I do that, so I won't. Then, whenever we have ab = c in the monoid, we stick in a 2-simplex, which we can visualize as a triangle like this:
O / \ a b / \ O---c---O
Then, whenever we have abc = d in the monoid, we stick in a 3-simplex, which we can visualize as a tetrahedron like this
O /|\ / | \ / b \ a | bc / _O_ \ / / \_ \ / _ab c_ \ /_/ \_\ O--------d--------O
And so on... This is a wonderful space whose homology groups depend only on the monoid, so we can call them Hk(M). If we have a presentation of M with only finitely many generators, we can build KM using 1-simplices only for those generators, and it follows that H1(M) is finitely generated. (More precisely, we can build a space with the same homotopy type as KM using only the generators in our presentation.) Similarly, if we have a presentation with only finitely many relations, we can build KM using only finitely many 2-simplices, so H2(M) is finitely generated. What Squier showed is that if we can find a finite list of rewrite rules for M which is terminating and confluent, then we can build KM using only finitely many 3-simplices, so H3(M) is finitely generated! What's nice about this is that homological algebra gives an easy way to compute Hk(M) given a presentation of M, so in some cases we can prove that a monoid has no finite list of rewrite rules for M which is terminating and confluent, just by showing that H3(M) is too big. Examples of this, and many further details, appear in Lafont's work:
7) Yves Lafont and Alain Proute, Church-Rosser property and homology of monoids, Mathematical Structures in Computer Science 1 (1991), 297-326. Also available at http://iml.univ-mrs.fr/~lafont/publications.html
Yves Lafont, A new finiteness condition for monoids presented by complete rewriting systems (after Craig C. Squier), Journal of Pure and Applied Algebra 98 (1995), 229-244. Also available at http://iml.univ-mrs.fr/~lafont/publications.html
There were many other interesting talks, but I think I will quit here. Next time I want to talk a bit about topological quantum field theory. (Of course, folks who read "week38" will know that Lafont's work is deeply related to topological quantum field theory... but I won't go into that now.)
© 1995 John Baez