6

Talking of category theory...I can't remember if I previously mentioned the bizarre functorial property of the number six I came across in Designs, Codes and their Links

This is the actual theorem:

consider the category whose objects are the n element sets and whose arrows are the bijections between the sets. This category has a non-trivial functor to itself only for n=6.

By smart use of Google Print you should be able view the proof. It's the first five pages of Chapter 6. (Don't make the obvious mistake with Google Print and end up with only three pages of that chapter.)

Anyway, it's not too hard to give a bit of insight into what this means. Consider the set with n elements. You can build all kinds of combinatorial objects which have some underlying set. Permuting the original n-element set induces a permutation of the combinatorial object and hence its underlying set. If the underlying set also has n elements you usually just end up with the original permutation. For example consider n=3 and the combinatorial object that is the even permutations on the 3 element set. This also has three elements. But the induced permutations on this new set are equivalent to the original permutations (via a bijection from the 3 element set to the set of its even permutations). On the other hand, if n=6 then you can construct another 6 element combinatorial object where the induced action of S₆ is quite diffferent to the original one. In fact, it gives an outer automorphism of S₆, another bizarre thing that only exists for n=6. To see the actual details of the construction look here.

I should also mention that Todd wrote a paper on this subject: The Odd Number 6, JA Todd, Math. Proc. Camb. Phil. Soc. 41 (1945) 66--68.

It's also mentioned in the Wikipedia but that's only because yours truly wrote that bit.

Anyway, I'm vaguely interested in how this connects to other exceptional objects in mathematics such as S(5,6,12), the Mathieu groups, the Golay codes, the Leech lattice, Modular Forms, String Theory, as well as Life, the Universe and Everything.

Sets, Classes and Voodoo

After much anticipation I started reading Barwise and Moss's "Vicious Circles". Unfortunately by chapter 2 I've reached the point where the statements don't just seem incorrect, they don't even seem to be propositions. I raised my question on USENET but I may as well mention it here too.

Here's how I understand the concept of a class in ZF Set Theory: talk about classes is really just talk about predicates. We enrich the language of Set Theory with a bunch of new terms ('class', 'subclass') and overload other terms ('is an element of', 'is a subset of') to give a new language that reifies classes, but instead of adding new axioms to deal with classes we provide a translation back to ZF without these extra terms. For example if P and Q are classes then "x is in P" means "P(x)" and "P is contained in Q means" "for all x, P(x) implies Q(x)" or even "a subclass of a set is a set" which translates to the axiom of separation. (We could alternatively add new axioms, instead of the translation, and then we'd get NBG Set Theory.)

Am I right so far? (By the way, nobody ever seems to say what I've just said explicitly. In particular, it seems to me that once you add the term 'class' you need to start proving metatheorems about classes to show what kind of deductions about them are valid, but nobody ever seems to do this.)

I understand that this is a sensible thing to do because of the overloading - in the enriched language sets and classes look similar and that allows us to do category theory, for example, in a much wider context, without having to define everything twice. (And also, maybe, because talk about sets is really a substitute for talk about classes...but that's a philosophical point for another day...)

So what does "If a class is a member of a class then it is a set" mean in the context of ZF? A class is really just a predicate, so it doesn't make sense to me that there could be a predicate about predicates. So at this point the book is looking like Voodoo.

Can anyone out there clarify this for me?

(Hmmm...I wonder if the translation to ZF is an adjoint functor...ow...MUST STOP OBSESSING ABOUT ADJUNCTIONS...)

The Representation of Integers by Quadratic Forms

There's a nice article at Science News on the work of Bhargava on the representation of integers by quadratic forms. Rather than just restate what's written there (and in a few other blogs) let me quote the main theorem which is really quite amazing:

If Q is a quadratic form: Zⁿ->Z then if the image of Q contains {1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290}, then it contains every positiver integer.

Closely related is Conway's 15-theorem which originally inspired Bhargava and which (I think) I first read about in the excellent book The Sensual Quadratic Form Check out the section on topographs where Conway works his usual magic and makes some parts of the theory seem so clear and obvious that even your cat could start proving theorems about quadratic forms.

Category Theory Screws You Up!

Well it does. Since I started a burst of intense category theory reading a couple of weeks ago (not that intense as I have a full time job) I've been showing unpleasant symptoms. These include insomnia, lack of concentration and grumpiness. We're not just talking correlation here, I have causal mechanisms too: how can I sleep when an example of an adjunction might pop into my mind at any moment, how can I concentrate when my brain is already fully occupied in finding those examples, and of course I'm grumpy with all this effort to understand difficult theorems that always turn out to be trivial and content-free. Fortunately I find that drugs help with the insomnia, but there's no cure for the other symptoms.

At least I haven't reached the stage where I sit down to dinner wondering whether or not my eating it is an operation with a left or right adjoint. (But I thought it didn't I, so I must be pretty far gone.) And I'm not dreaming commutative diagrams yet.

So here's my advice: if someone comes up to you in a shady bar or alleyway and offers you a monad, or an adjunction, or even an innocent little natural transformation, just say "no!".

What can I do with adjoints? And Lemma 28.

By 'do' I mean 'compute'.

In order to teach myself about F-(co)algebras I wrote some literate Haskell that actually made use of them. The neat thing was that it gave me something for nothing. I wrote a generic unfold function (mostly not by thinking about it but by juggling all the functions that were available to me until I found one of the correct type) and I was amazed to find that it did something that I found useful, or at least interesting.

Can I do the same for adjunctions? Unfortunately, all of the examples I tried were basically uninteresting. They were quintessential category theory, you spend ages unpacking the definitions only to find that what you ended up with was trivial. Unlike the fold/unfold case I didn't find myself ever defining a new function that I could use for something.

I did just find this link with some intuition that I'd never read before: 'To get some understanding of adjunctions, it's best to think of adjunction as a relationship between a "path" to an element and a "data structure" or "space" of elements.' Hmmm...none of the category theory books talks about that. I think I can see it: a forgetful functor rubs out the path and just leaves you with the destination. (Don't take my word for it, I'm just guessing at this point.)

Maybe I'm now on target to be able to understand Lambek & Scott which I bought 20 years ago (I'm older than you thought), not really knowing what it was actually about. BTW Check out the review on Amazon: "I was looking for a book for my girlfriend this Christmas and stumbled upon this one..."

And I thought I'd mention Newton's Lemma 28. But after I asked a question about it on Usenet, some other people said far more interesting things that I could ever say.

Answers and questions

The answer to that cellular automaton puzzle was that it generates the derivatives of the tan function at zero. It uses the fact that if f=tan x, then f'=1+f^2. We can now repeatedly use this to compute higher derivatives, eg.

(f^n)' = nf^(n-1)f'=nf^(n-1)(1+f^2) = nf^(n-1)+nf^(n+1).

You should now see why the CA rule arises.

(Why n,n instead of (n-1),(n+1) as in the rule I gave? Because this is the 'push' version rather than the 'pull' version of the rule that I gave. If you imagine differentiation acting as a linear operation on the vectors with respect to the basis (1,f,f^2,f^3,...) then the 'push' and 'pull' versions are described by matrices that are transposes of each other.)

I was petty excited by this and tried applying it to all sorts of other functions. But actually it was hard to make it work. Maybe the fact that there is a nice formula for the tangent function is just an accident. Anyway, Jan Rutten, whose paper I got this from, has lots of nice papers on coalgebras and infinite streams. I think I've now learned enough about coalgebras to move onto the next section of my category theory book: Adjoints.

Homotopies between proofs and between programs

Many years ago, while out drinking after an algebraic topology seminar, someone mentioned the idea of using algebraic topology to analyse computer programs. I never did get the details but I'm guessing it must have been something like the material presented at this conference.

Anyway, the comment got me thinking. I spend much of my time refactoring code. This is making small modifications to software intended to restructure it to a better form without changing its behaviour (eg. I've just been dumped with a single C++ source file 0.25MB long most of which is one function and is in considerable need of some tidying up!). If you think of a program as a path to get you from an input to an output then a change to that program, that keeps the behaviour the same, is a kind of homotopy between paths. But I couldn't really get anywhere with that idea.

In Baez's This Week's Finds this week he talks about homotopies between proofs. These are ways to convert one proof of a proposition into another proof of the same proposition. Cut elimination is such an operation but there are surely others.

According to the Curry-Howard isomorphism, a computer program that maps type A to type B is essentially the same thing as a proof of B assuming A. So a homotopy between programs is the same thing as a homotopy between proofs. For example, Baez's example of converting between two proofs P -> R corresponds exactly to converting the piece of Haskell code f . ( g . h) to (f . g) . h using the associativity of . (. is Haskell for function composition.)

So, is there some kind of interesting topology to be extracted from this? Does the space of programs that map input x to output f(x) have interesting fundamental groups or homology groups? I guess a good place to start would be with two simple programs to perform the same operation that can't be transformed into each other by simple steps. And do I really need omega-categories to make sense of this?

Anyway, besides Baez's stuff there's also theo's comments with a real world example of what looks like a homotopy between proofs.

Cellular automaton puzzle

Here are the cells of a 1D cellular automaton:


+--+--+--+--+
|a0|a1|a2|a3| ...
+--+--+--+--+


Instead of a finite number of states the ai are integers.

Here's the update rule:

a0 <- a1
ai <- (i-1)*left+(i+1)*right, i>0

where left and right refer to the cells to the left or right of ai.

We start it in this state:

+--+--+--+--+--+--+--+
| 1| 0| 1| 0| 0| 0| 0| ... all zeroes
+--+--+--+--+--+--+--+


What is the significance of the numbers that appear in the leftmost cell?

If you can't figure it out the answer is here. (Though it might be easier to work it out for yourself than read that paper!)

I think this may actually be useful. I've spent quite a bit of time recently writing code to do the sorts of things that these automata do - but these automata may do it better.

Coalgebras and Automata

fold is a really useful operator in a number of programming languages, including Haskell. It also hs a dual partner, unfold. But more importantly, both can be generalised, using a little category theory, to F-algebras and F-coalgebras. Amazingly, unfold, in the right category (as I discovered after a weekend of frenzied category theory) gives a really nice way to convert descriptions of automata into runnable automata. Anyway, I wrote the whole thing up as a piece of literate Haskell.

UPDATE: That came out formatted ugly so I've moved it to my web site where I'll eventually format it better. Here's the link.

Hyperfun!

Smullyan's favourite game is called the Hypergame, invented by Zwicker. I hadn't heard of it until today when I read about it in Barwise and Moss's book Vicious Circles.

Consider games where the players take turns and the game is guaranteed to terminate in a finite number of moves. These are call well-founded games.

The Hypergame is quite simple. The first player chooses a well-founded game. The game now turns into that game and the second player starts by opening in that game with play continuing in that game. Eg. if the first player says "Chess (with whatever house rules are required to make it terminate)" then they start playing Chess with the second player moving first. Because of the stipulation of well-foundedness the Hypergame lasts precisely one move more than some well-founded game. Therefore the Hypergame is itself well-founded and always terminates.

Anyway, Alice and Bob decide to play:

Alice: Let's play the Hypergame!
Bob: Cool! I'll go first. My move is to select the Hypergame.
Alice: OK, now we're playing the hypergame. So my first move is to select the Hypergame.
Bob: The Hypergame is cool, I pick that.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
Alice: I pick the Hypergame.
Bob: I pick the Hypergame.
etc.

It's a square, square world!

I was browsing for books on elliptic functions on Ebay when I came across this link to a book by Oscar S Adams called "Elliptic Functions Applied To World Maps". This is pretty obscure, I thought to myself, so I googled Oscar. Sure enough, there really was an Oscar S Adams who worked on cartographic projections and his speciality was conformal (ie. angle preserving) projections including this one:


Anyway, there are lots of interesting conformal projections out there. Some of them are quite pretty so I thought I'd share. I like the use of the Schwarz-Christoffel transformation, better known (to me) from 2D fluid dynamics, to map the world into a variety of polygons. I was also surprised to see that Charles Sanders Peirce worked in this area.

Just for the hell of it I might place a bid on the book.

UPDATE: Come on! 'fess up if you're the person bidding against me!

Blind Games

I've always been fascinated by games that involve bluffing and I've always been more fascinated by 'blind' games. I define a 'blind' game to be one where there is some state in the game that a player can choose not to reveal. Instead the player claims what that state is. The other player can choose to accept or reject this state of affairs, but if it is rejected then the actual state is revealed and one or other player is 'punished' as appropriate. (Eg. blind chess isn't 'blind'.)

Consider this trivial game: you deal a pack of cards to N players. The players take turns and the winner is the first one to get rid of all their cards. A turn consists of discarding a set of cards that all have the same number or rank. This game isn't very interesting.

But now apply the 'blind transform' to this game with the 'punishment' of picking up all of the discards. The players play cards face down and make some claim about what cards they are laying down. If someone challenges this claim you have a look and mete out the punishment. A trivial game has now been turned into a mildly entertaining game. It can be turned into an excellent game by means of the 'ethyl transform' - playing whle drunk. But moving on...

One of my favourite games years ago, when I could talk people into playing it, was "Blind Le Truc". It's Le Truc but with all of the cards played face down but players claiming what the cards are. In tis case the punishment is losing a trick. Even played 'sighted' this game is full of bluff. Playing it blind brings it to a whole new level. Players can play against each other for extended periods of time playing what is practically an imaginary game going through all the motions of Le Truc without ever seeing a card. It's a lot of fun. Well, I think so anyway.

Even the most trivial of games can become interesting. Eg. each player is given an entire suit of a deck of cards. Each round a player places one card in the middle. Whoever places the highest card in the centre wins that round. Play continues until nobody has any cards left and the winner is whoever won the most rounds. After applying the blind transform, with round losing as punishment, this game is turned into something fun, at least for a few minutes.

Poker is also a blind version of a trivial game.

How amenable are these games to analysis? That last game is pretty simple. Suppose two players each have only 3 cards. What is the optimal way to play?

You can also apply the blind transform multiple times. It's more interesting if the nth application has a tougher punishment than the (n-1)th. In that case it corresponds to raising the stakes for being caught, or accusing someone of, 'cheating'.

When is one thing equal to some other thing?

That's the title of a rough draft of a paper by Barry Mazur (courtesy of Lambda the Ultimate). It's basically an introduction to the ideas of category theory but it has some nice comments along the way on what we mean by equivalence - often a tricky subject in category theory.

And here's a coincidence. I'd just done a web search to find out what properties the number 691 has. (It pops up in the theory of modular forms.) And then I see a link to this paper (completely independently) which uses 691 as an example on the first page. Weird!

An Actual Application of Fractal Dimension

According to this story, and others, a trove of 32 paintings, purported to be Jackson Pollocks, was discovered last year. But many people had doubts about whether or not they were really by the master of paint pouring or not. So a physicist, Richard Taylor, used some software to compute the fractal dimension of Pollock's paintings and compare them to these new paintings. It turned out that the fractal dimension didn't fit in with the trend of Pollock's work and hence they look like fakes.

Of course, me mentioning this fact isn't intended to be an endorsement of the reliability of these methods. I put as much trust in them as I'd put in using catastrophe theory to navigate my way through a prison riot.

A Cautionary Tale for Would-Be Generalisers

You may or may not know this story already as it's been floating around for a while. But I'm going to retell it anyway:

Define sinc(x) = sin(x)/x and sinc(0) = 1

Let 'I' mean integration from 0 to infinity.

I sinc(x) = pi/2
I sinc(x)sinc(x/3) = pi/2
I sinc(x)sinc(x/3)sinc(x/5) = pi/2
I sinc(x)sinc(x/3)sinc(x/5)sinc(x/7) = pi/2
I sinc(x)sinc(x/3)sinc(x/5)sinc(x/7)sinc(x/9) = pi/2

You see the pattern right? So the story is that some guy was evaluating these integrals for some reason or other getting pi/2 all the time. They happily chugged on getting...

I sinc(x)sinc(x/3)sinc(x/5)sinc(x/7)sinc(x/9)sinc(x/11) = pi/2
I sinc(x)sinc(x/3)sinc(x/5)sinc(x/7)sinc(x/9)sinc(x/11)sinc(x/13) = pi/2

So when the computer algebra package this person was using said:

I sinc(x)sinc(x/3)sinc(x/5)sinc(x/7)sinc(x/9)sinc(x/11)sinc(x/13)sinc(x/15) = 467807924713440738696537864469/935615849440640907310521750000*pi

they knew they'd hit a bug. They complained to the vendor who agreed that, yes, definitely, there was something screwy in their integration routine.

Except there wasn't. Weird eh?

I've known this story for a while but I've only just stumbled on Borwein's paper that explains the phenomenon. They're called Borwein Integrals now.

Update: Corrected a small error. It's pi/2 for the first 7 terms, not just the first 6. Even more amazing!

Counter-counterfactual Computation

There has been quite a bit of reporting about Counterfactual Computation in the popular press. This work is essentially a realisation of a technique first published in this paper. I think it's worth looking at the process a little more clearly. Let's see if I can first cram a course in quantum mechanaics and quantum computing into a couple of paragraphs. (Physicists like to obfuscate these subjects but the principles are very simple. But I'm not sure if they're simple enough for two paragraphs however...)

Firstly, let's review classical computing. In a classical computer you have a machine in a state, call it s. You let a bit of time pass and it ends up in a new state, call it t. We can write this using the notation

|s> => |t>

That's basically all there is to know about computing, the rest is details. Let's look at an example. Suppose we have a computer with two one bit registers. The first is an on-off switch and the second will contain the result of the computation. For technical reasons we'll make it so that the answer doesn't just go into the second register but is exclusived-ored with it. We initialise the second register to zero so it's effectively the same thing as storing the result there. Write the state as |s,x> where s is the state of the switch and x is the second register. Suppose the result of the computation is r - this is the thing we are actually interested in. Then we have

|s,x> => |s,x^(s&r)>

In other words, this is saying that we only exclusive-or x with r if s is 1. (I used '^' to mean boolean 'exclusive-or' and '&' to mean 'and'.) For example

|0,0> => |0,0> and |1,0> => |1,r>.

Obviously s must be 1 in order to get a useful value of x.

Now let's move onto quantum computing. Now we no longer have a discrete set of states but instead a vector space. The time evolution of a quantum computer is given by a unitary linear operator. If the states evolves to state t we write this as

|s> => |t>

where we now interpret |s> and |t> as vectors in a normed vector space. The first important fact to know about quantum computers are that if |a> and |b> are possible states then so is any complex linear combination of these states. The second (fact(2)) is that if system A has states in a vector space V(A), and
system B has states given by vector space V(B), then the combined system is given by V(A) (x) V(B), where (x) is the usual tensor product over C. The third important fact (fact (3)) is about the interpretation of the state |a>+|b>. This makes no sense classically, but in QM linear combinations (called superpositions) are allowed. The rule is that if you observe this state to see whether it's in state |a> or state |b> then it 'collapses' into state |a> or |b>. The probability of state |s> collapsing into state |a> is given by |<a|s>|^2/|<a|a>||<s|s>>| where <x|y> is fancy notation for the inner product of |x> and |y>. (There's another rule of QM, if |a> and |b> are distinct possible observable outcomes of an experiment, then <a|b>=0.)

We now move our classical computer into Quantumland by considering the states |s,x> to be the basis elements of a vector space. Using fact (2) above we can consider these to be |s> (*) |x>. The rule

|s,x> => |s,x^(s&r)> (rule A)

defines what happens to each basis element and hence defines a linear map. For basis elements it acts just like a quantum system. But with a quantum computer we have available all kinds of interesting linear operators that make no sense for a classical computer. In particular, given a register we can apply this operator to it:

|0> => a*(|0>+|1>)
|1> => a*(|0>-|1>)

where a = 1/sqrt(2). This is linear and unitary. Notice also that if we do it twice we get back to where we started from. To make things easy we'll drop the constant a and work with linear operators that are unitary up to multiplication by a constant. The state |0>+|1> is a kind of weird mixture of on and off-ness.

So the trick to counterfactual computation is that instead of switching on the computer you instead 'half' switch it on using the linear operator above. Note that if you 'half' switch it on twice it just ends up being off again. But instead of 'half' switching it on twice in a row, we half switch it on, allow the quantum computer to evolve by rules (A) and then half switch it on again. Let's work through the details. First note that we are only 'half' switching on the first register. This means the linear operator applies only to the first factor in the tensor product |s>(x)|x>. So we get:

|0,0> => |0,0> + |1,0>

Now consider the result of this followed by allowing the computer to evolve by (A)

|0,0> => |0,0> + |1,r>.

And now follow this by another 'half' switch on:

|0,0> => |0,0> + |1,0> - |1,r> + |0,r>.

If r = 0 then:

|0,0> => |0,0> (modulo a constant multiple)

In other words, the state is unchanged.

If r = 1 then:

|0,0> => |0,0>+|1,0>+|0,1>-|1,1>

We no longer have the cancellation. Following fact (3), because of the |0,1> term above we have a non-zero probability of finding that the register x contains the answer r and yet the on-off switch is off.

And that's all there is to it. With a bit more cunning you can actually get the probability of finding r as high as you like. There are some great potential applications here. We have effectively left x untouched and this translates into practical 'interaction-free' experiments that allow us to make certain kinds of measurement leaving the measure system unchanged. (You may see why I used 'exclusive-or' above - just copying the result into the second register wouldn't have given me a unitary operator.)

BUT, and this is a big but, I see no valid way to interpret this as 'counterfactual computation'. In other words, I completely disagree with the interpretation of this result. In particular, suppose r equals zero. Then the reason we end up with the result |0,0> is that we have had a fortuitious cancellation. But we have actually passed the register |0>+|1> through the system. We only get |0,0> at the end because our second 'half' switching operation has made the effects of this 'destructively' interfere. In other words, it's just like we put on noise-cancelling headphones and then claim that there was no sound at all. That's silly, there was lots of sound, we just arranged for it to cancel at our ears. In our quantum computer both |0> and |1> waves passed through the system, but the second 'half' switch operation made them cancel. In the r=1 case you can thing of the shapes of these waves being messed up so they no longer cancel. (By the way, the talk of waves is quite fair here, you can think of the elements of these state vector spaces in fact being wavefunctions and the linear operators being time evolution of differential equations remarkably similar to the wave equation.)

In other words, I'm calling these physicists' bluff. There is nothing counterfactual about this computation at all. Though we might be able to get the result of our computation with the switch in the off state, that's only because it was partly on during the computation and we arranged for the state of that register to be destructively cancelled out at the end. I claim that the descriptions given of the experiment are completely bogus.

In fact, if you read Jozsa and Mitchison's paper you'll see that in his discussion his language is quite guarded. I think they realise that this talk of counterfactuality is a little bogus.

Update: Scott Aaronson rants about this too over here. Make sure you read Kwiat's response.

The ambiguous operator, pt.2

In 1963 John McCarthy, the inventor of Lisp, published the paper A Basis for a Mathematical Theory of Computation in which he proposed the function (in the computer program sense of the word) amb(.,.). The idea is that amb(x,y) is first equal to x. But if later in the computation it is found that this leads to some sort of contradiction the value of x is retracted and replaced with y. This is a much more complex business than it may seem to be at first. Retracting a value essentially means winding back the entire state of the computation to where it was when amb returned the value x, and then slipping in the value of y. This means somehow freezing and copying the entire state when x was first returned. When a contradiction is found the entire state of the program is discarded and replaced with the frozen version which is reactivated. These frozen states are known as continuations. In many ways it's like a GOTO statement on acid. It can cause a jump to an arbitrary spot on your code. But continuations are nicer than GOTOs because they are more amenable to logical reasoning.

There are a number of languages that have support for continuations built in, including Scheme with its call-with-current-continuation function. But it takes a little care to understand exactly what is meant by "wind back" in this context. Although the state of the current computation is wound back, including things like local variables and the state of the functions currently being executed, global variables remain unchanged. So even though we wind back time and throw away much of the current state some information can still "leak through" from the path that we are now pretending never happened.

So, to get back to the devilish story of part 1: suppose we have a C keywords called TRY and FAIL with the bizarre property that TRY functions just like the return statement except that if a later FAIL statement is met your program winds back to the last TRY statement, undoing any effects caused by the previous return value, and continues execution after the TRY statement. The ambiguous operator would be implemented like this:


int amb(int a,int b)
{
    TRY(a);
    return b;
}


amb(a,b) would return a. But if a later FAIL is met it would unwind the computer back to its previous state and then continue execution with the return b line. Global variables would remain unundone. Given such keywords we can now define:


int devilChoice()
{
    printf("I choose B\n");
    TRY(B);
    printf("Sorry I meant to say A\n");
    TRY(A);
}


void devilCheats()
{
    FAIL;
}


But C has no such keywords and there's no way to write such functions - unless you cheat a bit. The required code is on my web site here. The trick is to freeze the state of the program by literally reaching in and taking a copy of the C stack. Remarkably this code runs fine, without modification, when compiled with gcc, mipspro or MSVC on IA32, AMD64, PowerPC and mips under Windows, MacOS X, Irix and Linux for all legal combinations of those compilers, processors and operating systems.

But despite the implementation of TRY and FAIL being an ugly hack they really are nice operators to have. Papers like Wadler's lend a little legitimacy to continuation shenanigans by showing there is a nice theory underlying them, and that by allowing them you bridge the gap between intuitionistic and classical logic in the Curry-Howard isomorphism. They can also make for some surprisingly simple implementations of problems. For example, using TRY you can write an elegant non-brute force C sudoku solver in a couple of lines. In fact, you find yourself able to write code that is declarative rather than imperative, even though you are working in C. The backtracking is handled behind the scenes by TRY and FAIL which can be nested recursively (as suggested in McCarthy's original paper) without any difficulty. I originally wrote that code in order to implement a pattern matcher similar to Mathematica's over ten years ago. It took me several years to figure out how to rewrite it without using continuations - using continuation passing style instead. (I had no Computer Science training, I hadn't even heard of this stuff.) Nowadays I'd just use a backtracking monad in Haskell...

For an implementation in Ruby see randomhacks. For code similar to my C code see Aubrey Jaffer's implementation (in C) of Scheme. You can also do the same thing semilegally through the setcontext() and getcontext() functions if they are available for your operating system. You can probably do something similar using threads or fork().

The ambiguous operator, pt. 1

This was originally intended as my programming and mathematics blog but I never got around to any programming here, until today.

My recent interest in logic is partly motivated by a desire to understand the paper Call by Name is Dual to Call by Value. I suspect the paper is almost trivial, but that you have to be very familiar with the prerequisites before you can appreciate its triviality. (Ie. I'm using the mathematician's definition of trivial of course :-) )

Now I don't understand that paper yet, but on the seventh page there is a story about the Devil. Below I have a computational representation of the story written in C. The idea is, what must the functions devilChoice() and devilCheats() do in order to allow the devil to capture this soul? devilChoice() must return A or B.

#define A 0
#define B 1

int my_main() {
  int choice;

  static int devilMoney = 0;
  static int suckerMoney = 0;
  static int suckerKarma = 0;

  printf("Devil says:\n");
  printf("Here's my offer.\n");
  printf("I will choose either A or B\n");
  printf("If A I'll give you $1000000000\n");
  printf("If B I'll give you any wish for $1000000000\n");
  printf("Do you accept\n");

  printf("Sucker says: 'yes'.\n");
  choice = devilChoice();

  switch (choice)
  {
  case A:
        if (devilMoney<1000000000)
        {
            printf("Devil is unable to pay up.\n");
            printf("Devil is in big trouble...\n");
            exit(1);
        }
        printf("Devil gives 1000000000 to sucker\n");
        suckerMoney += 1000000000;
        devilMoney -= 1000000000;
        break;
    case B:
        printf("Sucker begs, borrows, steals $1000000000\n");
        suckerMoney += 1000000000;
        suckerKarma -= 1000000000;

        printf("Sucker gives money to Devil\n");
        devilMoney += 1000000000;
        suckerMoney -= 1000000000;

        printf("Sucker is about to make wish...\n");
        printf("'I wish to go to heaven'\n");
        devilCheats();
        printf("Sucker goes to heaven and devil loses a soul\n");
        exit(1);
        break;
    }

    if (suckerKarma<0)
    {
        printf("Sucker goes to hell with $%d\n",suckerMoney);
    }
}

(Hmmm...syntax colouring didn't work out quite as well as I wanted.)

In fact, I'm going to leave this as an exercise and give a possible solution another day. But the title of this post gives a clue - and of course the paper proposes a solution too. The code doesn't need to be 100% portable.

PS You don't need to ask. I have a solution in not quite 100% portable C. But of course I cheat. But is there an interesting cheat?

The cut rule and reading seminars

Despite being moderately good at mathematics, even managing to scrape together a PhD, there are certain topics that are always brick walls to me so that I find it hard to get started even at the most elementary level. In algebraic topology I always had problems with spectral sequences but they're not so elementary and are notoriously tricky. But in logic I can barely get off the ground. Here's an example of a sentence from an introduction to linear logic that baffles the hell out of me "One of the most important properties of the proof-rules for Classical Logic is that the cut-rule is redundant". This is one of the most ridiculous things I have read in mathematics writing. If it's redundant then don't study it. Excise it from the list of derivation rules and don't bother with it every again.

I'm sure than when set theorists first tried to write down the axioms that became ZF they found lots of redundant axioms. Over the years they were whittled them down to the list we have today so that I bet you can't even name the axioms that were jettisoned for redundancy. Not so in "Gentzen style" logic. Every document ever written on the subject seems to introduce this rule and then with a flourish they show how it can be eliminated. They develop the whole subject and then proceed to demolish the earlier work by showing how they can rewrite everything they did earlier without using this rule. The only explanation I can come up with is that authors of books on logic are paid by the word and that this allows them a few extra chapters for very little work.

Of course the problem here must be me. I'm sure there's a perfectly good reason for harping on about the cut rule, I just don't see it. And I think this points to a difficulty with trying to read mathematics texts outside of academia. When you're a student you're sharing extratextual material all the time. People tell you that such and such a result is interesting because it has an application somewhere and then you go and read the formal details knowing what the motivation was. Or someone will give an informal seminar where they write equations on the board, but speak much more informally between the equations. But most mathematics texts hide this material from you and present the bare facts. This is fine when you're in an academic environment, but I have to confess to finding it difficult when working on my own.

One thing I'd love to see online is the equivalent of the reading seminars we did during my PhD work. Each week we'd read a chapter and then have a seminar to discuss what we'd just read. Does anyone do this? Blogs seem like a great way to do this but I've seen no evidence of groups working like this.

Anyway, I shall try to persist until I see the light...

Mathematics has Competition

Mathematics is the lingua franca of science. Except that Bernard Chazelle now claims that there are two such languages - mathematics and computer science. It's a curious claim. He likens mathematics to "epithets" and computer science to "novels" and claims that a computer science background is imperative to studying biology. All he thinks we need are a great populariser of computer science (wasn't that what Wolfram tried to be?) and an Einstein.

I'm inclined to ask, with the article's author, "Isn't computer science really just a stepchild of mathematics?"