The Solar Corona Problem

It's all been a bit abstract a bit lately. So I thought I'd bring things down to Earth a bit. But only metaphorically speaking.

I've decided to volunteer some time at the Chabot Space and Science Center. I'll mainly be manning their 20" reflector pictured here:



Basically I'll be something like a tour guide for visitors, answering any questions they may have. Only during daylight hours for now. It's a good excuse to read up a little on astronomy and if I stick at it then one day I'll be qualified enough to work the telescope at night, and maybe even catch a few photos through it with my own camera.

Anyway, one of the visitors was asking about the temperature of the Sun. As any photographer can tell you, the surface of the Sun has at a temperature of 5800K. But observations during eclipses show that the corona has some pretty weird spectral emission lines that correspond to atoms at very high temperatures, of the order of a few million K. Strangely, this is a bit of a mystery. At first you might think what's the problem? There's a gigantic fusion reactor with something like 2×10³⁰kg of fuel burning right next to it. But the problem is this - how is that energy getting to the corona when the surface of the sun is at 5800K. As everyone knows, heat doesn't generally flow from a cooler region to a hotter one. To me this is much more interesting than the Solar neutrino problem. Neutrinos don't play a large part in anyone's lives and all of the evidence that there is even a problem rests on indirect evidence and theoretical work. But the solar corona problem is pretty immediate. You can literally see the temperature of the sun's surface and the anomalous high temperature emission lines require nothing more abstract than spectrography. When a visitor asked about this I had no answer so I decided to read up on the issue.

Turns out there are a few possible explanations, though none of the corresponding predicted physical proceses have been directly observed yet. My money is on Alfvén waves. These are essentially waves that form solutions to the combined Maxwell and Navier-Stokes equations. If you imagine wave travelling down a whip when it is cracked, all that enerfy gets to the end of the whip and gets concentrated in the tip resulting in it moving at supersonic speeds. Something similar may be happening with these waves. They proceed out from the sun but when they get to the corona there's nowhere for the energy to go except into heating it to incredible temperatures.

Anyway, next time someone asks about the corona I won't be telling them this. Most people don't want this much detail and even if they did, they have impatient kids with them. But it's good to have people prodding me to read up on interesting physics.

Zog, The Devil and Me.

Zog runs a casino on the planet of Vega Sals. Like his competitors on the galaxy-famous Landing Strip he's been using probability to estimate his earnings based on the odds that he offers in his games. It needs quite a bit of care - offer odds that are too good and he'll lose the games, offer stingy odds and his competitors undercut him.

Anyway, Zog decided to attend some adult education classes at the local university and learnt about the concept of falsifiable propositions. The effect was disastrous, he found that he couldn't attach a meaning to the statement "this coin has a 50% probability of turning up left-head". (Note that the inhabitants of Vega Sals have two heads.) Whether it turns up left-head or right-head it's consistent with the original statement which can never be falsified. Yet surely it has a meaning of some sort.

Zog wrestled and wrestled with this problem, suffering nights of insomnia, when suddenly in a flash of smoke the Devil appeared in front of him.

"I'll do a deal with you," he said. "You're struggling because you can't get a falsifiable proposition from a probabilistic one. So here's what I'll do for you. Whenever you want, I'll let you convert a probabilistic statement into a falsifiable one with probability one.". "

"Sounds great," replied Zog, "but what's the catch?"

"Hmmm...I see my reputation precedes me. Anyway, it's just a small catch. For every proposition that I convert you'll have to suffer the fires of Hell. If it's an unlikely proposition you might suffer for trillions of years. But if it's 90% likely you'll only have to suffer, oh, let's say a billion years. If you can make it 99.99% likely you might only suffer for a million years but if you make it something like 99.9999999% likely you'll only have to suffer a short while. I have a brochure here that explains the details."

"Sign me up right away," said Zog, and he slept well that night.

But the next day when Zog was doing the weekly profit and loss forecast he found he had a problem. If he had a thousand people play space-roulette that day then it'd cost him trillions of years of sufffering to make the outcomes of these games certain. In the cold light of day he seemed no better off than before. But then he had an idea. Why not batch together the days takings? Instead of predicting the outcome of one game he would predict the outcome for the entire day. By using a suitable variant of the central limit theorem he found that he could predict the day's winnings to within 10 galactic credits with 99.9999% certainty. "Great!" he thought looking this up in the brochure. "99.9999% - 3 days of Hellfire." As the Astro-Priest had told him that even one second in Hell was worse than a lifetime of suffering it didn't seem such a good idea after all.

So Zog decided to consult a mathematician to buy a theorem. "I can construct just the thing you need," the mathematician said, "for only 1000 galactic credits." So the mathematician sold him a theorem a bit like this: "if, over the course of your lifetime, you make your decisions based on probability theory, then you have a 99.999999999% chance that you won't make a total loss at your casino tables and that nobody else will undercut you." The cunning mathematician had taken Zog's batching trick and batched together the entirety of Zog's future transactions to make this prediction. It was a perfectly valid theorem and Zog thought it was money well spent. He also no longer needed to worry about the meaning of probabilistic statements. He knew that if he just used probability theory, instead of thinking about its meaning, he would be guaranteed to succeed in the casino business.

And Zog lived happily ever after. When he died, the hundredth of a second that he suffered in Hell was pretty bad, but the everlasting happiness that came afterwards more than made up for it.

Of course you're probably getting suspicious by now. There isn't really such a place as Vega Sals. And I just made up Zog. This is of course a fable. The truth of the matter is that it was me who made the deal with the Devil, not Zog. And it wasn't exactly this deal. The deal I made is based on the theorem that I got for free from here. The Devil said that any time there was a system whose wavefunction was ψ he could make it φ where the amount of suffering I would have to endure was computed from |ψ-φ|. Using that theorem I was able to get the devil to remove from the universal wavefunction that part in which the probabilities of the Copenhagen interpretation didn't work. It was a pretty small part so I should suffer even less than Zog did. And the Devil offered me a 50% discount if I blogged aobut how good my deal was.

But I'm not the only person who's made such a deal with the Devil. Another person is David Deutsch in this paper. But I feel a little sorry for David. You see he claims "we have proved that a rational decision maker will maximise the expectation value of his utility." Well, yes, this is true. But only because of how he's defined 'rational'. But by using sleight of hand he's made it appear as if this statement is true for people who are rational in the usual sense of the word and of course people who read this paper will be less inclined to deal with the Devil. The Devil really doesn't like it when people deny they have had dealings with him, it makes him feel sordid and shady. So even as we speak the Devil is warming a very special place in Hell for David.

Oh...and maybe you can now see why I've been thinking about "almost certainty" a bit recently. I should also add that S4 doesn't quite do the job I want. I think ◊¬(◊p∧◊¬p) really ought to be a theorem of my logic but it's provably not a theorem of S4. (Where I now use ◊ to mean "almost certainly".) I think I need some more axioms in addition to S4...

S4 and Partial Evaluation

I never cease to be amazed by the ubiquity of mathematical ideas. I'm still not entirely certain that S4 captures what I mean by "almost certain" but it's neat that S4 does a half-decent job of representing what we mean when we say "I know that X". It also has another application that relates back to partial evaluation.

Under the Curry-Howard isomorphism we have an isomorphism between (intuitionistic) propositional calculus and 'types' in a programming language. If A and B are types then A∧B is the cartesian product type, A∨B is the disjoint union type and A→B is the type of functions that take an A as input and return a B. That's all standard stuff. But now we can augment these types with a new type □A which means a piece of data whose value can be worked out by the compiler. For example we normally think of 1.0 as an object of type Float, but it's really of type □Float because the compiler can do stuff with 1.0 during compilation. The meaning of that should become apparent in the next paragraph.

The function sin is of type □(Float->Float) because, like 1.0, sin is just a constant so the compiler knows its value. If the compiler knows the value of something that is a function, and it knows the value of its argument, then the compiler is in a position to perform the computation at compile time. In this case, the compiler could compute sin 1.0 during compilation rather than defer the computation to the final executable. In other words we have a function □(a→b)->(□a→□b). But this is just the axiom K of S4. It's also, essentially, the same as partial evaluation as a described here.

If the compiler knows the value of something, it's still at liberty to delay the evaluation until runtime. In other words, we have a map □a→a. This is the axiom of S4 known as T. Notice that we don't have the reverse. A value of type a might be computed from an input from the user. As the compiler is presumed to have run before the user gets to input anything to the program there's no way the compiler can have a general way to promote an object of type a to □a. This axiom might be used if we chose to write code to add a number of type □Float to one of type Float. The compiler could cast the one of type □Float to type Float in order to allow the addition be performed in the executable at runtime.

Though we don't have a→□a we do have □a→□□a. This is more or less by fiat. It just says the trivial thing that if you say it's known at compile time twice over then that's the same as knowing it once. So we have axiom 4 of S4.

So intuitionistic S4 can be viewed as the logic we need to annotate types with information that tells the compiler what it can compute at compile time and what should be left until runtime. In a sense it's a form of macro evaluation providing some of the features that the C/C++ preprocessor provides when it runs just before compilation. But what's nice is that it's expressed in the language itself by extending the type system rather than as a separate language (which the C/C++ preprocessor essentially is.)

Anyway, this description is very rough and there are a few gotchas with this interpretation. So I suggest reading a real paper on the subject. My summary doesn't correspond exactly to any one paper I looked at so rather than reference one I can use the great universal qualifier known as Google to express them all in one go. Apparently there does exist an implementation of a subset of ML that supports these modal types.

By the way, I thought I'd revisit self-reproducing code armed with this modal logic to see if I could produce a typed version of the theorem that a language with partial evaluation always has a fixed point. But the axioms of S4 seem to rule out this kind of self-reference because I can't construct a function whose type allows it to accept inputs of its own type.

There seems to also be a connection with temporal logic with □ saying something about the time when a piece of data is available.

A Modal Logic of Near Certainty?

If this "□" isn't a square then the following will be unreadable for you. For some reason □ is less standard than ◊ and may be absent from a font when the latter is present, especially when printing.

So having read some stuff on modal logic (basically just what the Stanford Encyclopedia of Philosophy has to say) I was trying to make up a modal logic of my own. So what follows is just brainstorming. It might not make any sense at all if you try to work out the details. But as I was stuck in trying to figure out those details I thought I'd mention it here.

Usually □ is used to mean things like "it is necessarily the case that", "I know that" or "it ought to be the case that" but I thought I'd see if it could be used to mean "it is almost certainly the case that". Here's the intuition I'm trying to capture: □p would be roughly like saying p holds with probability 1-O(ε). So □□p would mean that we expect it holds with probability (1-O(ε))² which is the same as 1-O(ε). So we'd have □□p⇒□p. I intend "almost certainty" to mean so close to certainty that if A almost certainly tells the truth, and so do B, C, D up to Z, then if Z tells you that Y says that X claimed....that A said p then you could still be nearly certain that p is true.

Other examples of things that I'd expect to hold would be "continuity" conditions like

□A∧B⇒□(A∧B)⇒□A∧□B

and I'm nearly certain that

□A⇒◊A (where ◊A=¬□¬A by definition).

Anyway, it seems like a perfectly good modal logic to me. But when I tried to work out the possible world semantics I got nowhere, mainly because p⇒□p upsets things. I was also toying with the idea that "almost certain" is a bit like the strict mathematical sense of "almost always" and that there might be some kind of semantics that involves sets dense in some topology. But I couldn't work out the details of that either.

So can anyone who is knowledgable about these things tell me if what I'm trying to do is sensible? If I fill in more details can I use possible worlds to prove completeness and soundness?

The really wacky idea I had (and remember, I'm just brainstorming) is that you might be able to set up a differential calculus of logic. Just as you compute derivatives of a function w.r.t. x by perturbing x slightly and looking at how f(x) varies, □A might be seen as a slight deformation of A, hence my use of the term "continuity" above. I'd then expect to see a version of the Leibniz rule appear somewhere in the theorems of this modal logic.

Anyway, this might be going nowhere, so brainstorming over.

Update

Ordinary propositional calculus can be viewed as being set theory in disguise. Pick a set X. Now translate a logical proposition to a statement of set theory by 'atomic' propositions to subsets of X, ∧ to ∩, ∨ to ∪, ⊤ to X; ⊥ to ∅ and ¬ to the complementation in X operator. A proposition is true if and only if its translation always equals X. For example ¬p∨p translates to P∪(X\P) which is clearly X.

For my proposed modal logic put a topology on X. Then □ translates to the Cl operator where Cl(A) is the smallest closed set containing A. The things I want to be true again correspond to propositions whose traslations are all of X.

I think this modal logic captures exactly what I want. You can get a feel for it by thinking about propositions like x=3 on the real line. This is false "almost everywhere" (ie. on a dense subset of the real line, ie. a subset whose closure is the entire real line) so it makes some kind of sense to want to say □¬(x=3).

With some keywords to search on I was now able to find this paper. Page 29 (following the document's own page numbering) essentially describes what I have except that they have swapped □ and ◊. Essentially they're saying that this is the standard modal logic known as S4. This topological interpretation of S4 goes back to work by Tarski and McKinsey in 1944.

And now I can reason about almost-certainty with certainty.

Mongruences

Before saying anything I'd be interested in feedback on whether the mathematical symbols below are visible in your browser. Things like right arrow ⇒, infinity ∞ and union ∪.

Anyway, I've been trying to understand exactly what coindunction is all about. In particular I've been trying to figure out exactly what the duality is between induction and coinduction. The problem I was having was that induction and coinduction seemed not to be dual to each other at all, and weirder still, almost every discussion of coindunction that I looked at seemed to completely ignore the issue.

So here's my problem: in order to carry out an induction argument over the naturals, say, we find a predicate on the naturals, P, such that

P(0) and P(n) ⇒ P(s(n))

We can then conclude that P holds for all naturals.

In order to carry out a coindunction argument over the set of streams over A, say, we find a binary relation R (called a bisimulation) such that

R(a,b) ⇒ head(a)=head(b) and R(tail(a),tail(b)).

I can point out a certain amount of duality: The naturals form an initial F-algebra which is a fancy way of saying that they form a least fixed point to some equation, in this case N≅1+N. Here '1' means the type with one element called '*' and '+' is the disjoint union. We have a map s:1+N→N where s(*) = 0 and s(n) = n+1. s can
be thought of as a 'constructor'. Given something of type 1+N it shows how to construct an object of type N from it. Loosely speaking we can now see that induction is about forming a predicate that still holds after we apply a constructor.

Now consider the coinduction over all streams. A stream is a final F-coalgebra which is a fancy way of saying it's a greatest fixed point of the equation S≅A×S. We have a map (head,tail):S→A×S which maps a stream to its head (an element of A) and its tail (another stream). We can think of these as 'destructors' which reduce a stream back down to its component parts. And now we can see that coinduction is essentially about finding a binary predicate that still holds after applying a destructor.

But to my eyes, despite the duality there is a glaring difference: induction is about a unary predicate and coindunction is about a binary predicate. And through no amount of messing about with categories and their opposites was I able to find a way to see a unary predicate to be dual to a binary predicate.

So I gave in and tried to find something on the web about this subject and eventually was led to Bart Jacobs' paper Mongruences and Cofree Coalgebras. Inductive predicates aren't dual to bisimulations at all, rather inductive predicates are dual to what Jacobs calls mongruences. Unsurprisingly, this horrible neologism was dropped in favour of the term τ-invariant or just invariant in later papers. But I'm going to use the term out of sheer perversity.

But this paper still doesn't explicitly talk about coinduction over the naturals leaving me to work out the details. After chasing round the diagrams, including a nice demonstration of left and right adjoint functors (about which I seem to have managed to develop a crude intuition, who said you can't teach an old dog new tricks?) I managed to figure out what I think is the fairly simple definition of a congruence predicate for the naturals.

So instead of working over the naturals we actually work over N'=N∪{∞}, the set of 'conaturals'. The reason is that we are now looking at the final F-coalgebra that is the greatest fixed point to the equation X≅X+1, not the least fixed point. N' is equipped with a map p:N'→1+N' where p(0)=*, p(∞)=&infin and p(n)=n-1 otherwise. With this in mind a coinductive predicate for the conaturals (ie. a mongruence or invariant) turns out to be a P such that

P(n) ⇒ n=0 or n=∞ or P(n-1).

At least I think so. (This could be embarassing.) And now I can see why coinduction talks about bisimulations - mongruences aren't very interesting for the conaturals. Bit of a letdown really!

On the other hand there are some mongruences that are of mild interest in a more general setting. For example consider a finite state automaton with some terminal states. You can define a state to be safe if at no future point it's possible to reach the terminal state. So a state is safe if it isn't terminal and if all of its successor states are safe. If you think of a finite state automaton as a function f:X→(X×O)^I, where X is the set of states, I is the set of inputs and O is the set of outputs, then f is a 'destructor' and you can see that this definition fits into the same general pattern for coinductive definitions. So safety is a mongruence.

The obvious next question is: what is the dual of a bisimulation? These are called congruences but I haven't yet worked through the details of what these are.

My next question is this: given an inductive predicate you get to apply "proof by induction". What is the corresponding "proof by coinduction" for the conaturals? I'm not sure there is one. I have a vague sense of what proof by coinduction might look like for a more general mongruence but haven't figured out a non-trivial example yet.

But despite having learned about a concept that was uninteresting, and a proof method that I can't figure out how to apply, I have picked up something useful. The standard notions of proof by induction and proof by coinduction aren't straightforwardly dual to each other. So now I can stop worrying about it and get on with reading Vicious Circles.

The Minority Game

The El Farol restaurant in Santa Fe has "one of the best bars on earth" according to the New York Times. Unfortunately, on the days when it becomes too crowded it's more fun to stay at home. This can be modelled by a game known as the El Farol Bar Game or its simpler cousin, the Minority Game.

In the Minority Game, N players (with N odd) get to cast a secret vote either for A or for B. When all the votes have been cast the winners are the members of the minority group and they receive a payoff. But what is surprising about this game isn't just the game itself but the people who are studying it - physicists. The idea is that for large N the game looks like certain types of system from statistical thermodynamics. For example, consider the Ising model. In this we have an array of atoms, each of which can be spin up or spin down. Depending on the parameters of the model atoms either want to be aligned like their neighbours or aligned in the opposite direction. In the latter case we have a situation similar to the Minority Game. The Minority Game looks a lot like a 'frustrated' physical system with no obvious ground state. Just as a group start moving towards what seems like the optimum it ceases to be an optimum.

The game has been studied by many people using computer simulations. Typically the simulations result in the number of winners in each round hovering around one value. But the feature that many researchers have found interesting is the way that over multiple plays, independent players seem to spontaneously cooperate, despite the lack of communication. The result is that they reduce the variance, and hence risk, of the outcome below what you would expect from a purely randomised strategy.

MG can be used to model a wide variety of phenomena besides Saturday night drinking - including financial markets, traffic congestion, network congestion and even the ebb and flow of fashion.

The original paper by Mallet and Zhang may be found here but a web search will reveal that there are in fact hundreds of papers on the subject. It seems to be the in thing.

Anti-Foundation and Strong Extensionality

Barwise and Moss's Vicious Circles is about the Anti-Foundation axiom, an alternative to the usual axiom of foundation in set theory. I had a look at various online introductions to this subject and they all seem a bit long winded. So here's my own quick summary pared down to the bare minimum. (I sacrifice a bit of accuracy along the way but it should be enough to give a feel for the subject.)

Firstly: the whole point of the Axiom of Foundation is to eliminate infinite chains of membership. So the Axiom of Foundation says that there are no infinite descending chains:

...∈A₃∈A₂∈A₁∈A

(I hope you can read that, I'm using this list.)

The kind of thing that the Axiom of Foundation eliminates are sets x such that x∈x. It means we can define functions inductively over sets as we're always guaranteed to have a base case to start the induction. More generally imagine a system of equations such as

x = {x,y,z,4}
y = {x,z,2}
z = {{3},x,y}

The Axiom of Foundation rules out any solution to this. The Axiom of Anti-Foundation, on the other hand, says that any set of equations like this, with unknowns on the left, and sets containing those unknowns on the right, has a unique solution. (Note that the containing in that sentence is essential, x = x doesn't have a unique solution.)

And that's it, the Axiom of Anti-Foundation.

But there's a slight problem. Consider the equation x = {x} and the equation pair y = {z}, z = {y}. The solution to the first equation seems like it ought to be the solution to the second as well. But we have no way of telling this. In fact, there are issues just telling whether or not the sets x = {x} and y = {y} are equal. Normally in set theory we'd use the Axiom of Extension. Two sets are equal iff their elements are equal. So in this case we have x = y iff their elements are equal, ie. iff x = y. So the Axiom of Extension tells us nothing here. We need a new kind of Extension.

The idea is to find something like an isomorphism between two sets. For example, define f(x) = y for our last example above. Notice that for every element x' of x, there is an element y' of y such that f(x') = y'. f also has an inverse. It's a bit trivial in this case but it's a start. But suppose we compare the set x defined by x={x} with y defined by y={z} and z = {y}. We want some kind of isomorphism that maps x to both y and z. And with more complex sets of equations there might be any number of elements of one set that we need to map to any number of elements in the other. So what we need isn't a function but a relation connecting elements in one set to the other. The relation R is said to be a bisimulation between sets a and b if

aRb
for all a'∈a there exists a b'∈b such that a'Rb'
for all b'∈b there exists an a'∈a such that a'Rb'

If two sets are related by a bisimilarity then it can be shown that they satisfy the same equation and hence by the uniqueness of solutions they must be the same set. This property is known as Strong Extension. (I've paraphrased this from the original but I think it's essentially correct.)

And that's it - you can now go off and make your own non-well founded sets and compare them for equality.

I should add some interesting connections with computer science. Bisimulations are more familiar as a relation holding between automata. The usual Axiom of Extension gives a recursive definition of equality. Recursive definitions require a base case and with a base case we can get inductin proofs started. Without base cases we have corecursion instead and have to use a different proof technique, coindunction, whose main tool is bisimilarity.

Another interesting link with computer science is through F-algebras and F-coalgebras. I mentioned that F-algebras and F-coalgebras may have initial and final objects so that there is an isomorphism X≅F(X). The Axiom of Anti-Foundation often allows us to make the isomorphism equality. For example it's not hard to see that for a finite state automaton we can make a state simply equal to the set of transitions available from that state. In fact, that gives a nice way to answer the question I was originally asking in that earlier posting - how can I remove the labelling of the states and leave just the bare information about what transitions take me where?

And one important thing I should add - you can make a model in ZF for set theory with Anti-Foundation and Strong Extensionality. So it's consistent if ZF is. Additionally, Barwise and Moss also work with urelements, elements of sets that are not the empty set and yet contain no elements. I'm not sure, but I don't think these are essential to making sense of Anti-Foundation. Anti-Foundation, as described by Barwise and Moss, does make use of urelements in the definition of an equation (in particular to label the 'unknowns') but I think Anti-Foundation can be rewritten so as not to use urelements.

Update: I originally stated that Strong Extension was an axiom which it isn't. Just to add to the confusion, the Axiom of Anti-Foundation that I described above is called "The Solution Lemma" in Vicious Circles, even though it is an axiom.

SIGFPE's Law

I've decided it's time to stake my claim to the law that will ever be remembered and linked with my name. If Moore can get a law named after him, or even Murphy, then so can I.

SIGFPE's Law
This law is the prediction that the rate of growth of the number of bits on the best available quantum computers is going to grow linearly with time.

My reason for saying this is essentially the same as the reason that many people are sceptical of quantum computers - decoherence. It seems obvious to me that even with the cleverest of error correction schemes we're going to see the coherence of states decay exponentially fast. I'm bored of reading papers on quantum error correction and seeing that every one makes some assumption or other about the Hamiltonian that is not going to hold in practice. And any little deviation from that assumption becomes the coefficient in an exponential describing that decay.

On the other hand people are ingenious. As time goes by people will find ever more ingenious ways to fix the errors caused by decoherence. But the difficulty is going to grow exponentially and the solutions won't scale. So just as classical computing has seen an exponential growth in power I expect to see the logarithm of this and hence a linear growth. Maybe, just maybe, there'll be polynomial growth. But definitely not exponential growth. In fact, I expect it to be linear with a pretty shallow gradient, just a couple of qubits a year at most, and probably less. Humans will, one day in the distant future, crack large RSA keys using quantum computers - but when that day is reached humans will be well beyond caring about such trivial matters.

I also have nothing to lose by saying this. If I turn out to be wrong, and I sincerely hope I am, I can look forward to a world full of wonderful computing machines whose interest value will far outweigh the ignominy of being forever linked with an incorrect law.

BTW When I say n-bit quantum computer I mean a computer whose total memory is n qubits, not a computer with the qubits addressable by an n-bit word.

:-)

Quantum Probability

I took part in a brief discussion over at antimeta which reminded me that I ought to get back to a document I started writing on quantum mechanics for dummies. One of my pet peeves is that I believe there to be a little bit of a conspiracy to make quantum mechanics seem less accessible to people. Not a deliberate conspiracy - but people maintaining an aura of mystery about it that puts people off the subject. All of the fuzzy talk about quantum mechanics in the popular science press does nothing to help the situation. In particular, there is a core of quantum mechanics that I believe requires few prerequisites beyond elementary probability theory, vector spaces and complex numbers.

Anyway, I did some more digging on the web and found this course by Greg Kuperberg. The opening paragraphs almost take the words I wanted to say out of my mouth. In particular, despite the mystical mumbo-jumbo that is often written on the subject, the rules of quantum mechanics are "rigorous and clear" and "The precepts of quantum mechanics are neither a set of physical forces nor a geometrical model for physical objects. Rather, they are a variant, and ultimately a generalization, of classical probability theory." Most of all "...more mathematicians could and should learn quantum mechanics...". You don't even have to understand F=ma to get started with quantum mechanics and get to the point where you can really and truly get to grips, directly, with the so-called paradoxes of quantum mechanics such as the Bell Paradox. The strange thing is that you won't find words like this in most of the quantum mechanics textbooks. They throw you into physical situations that require finding tricky solutions to the Schrödinger equation while completely failing to give any insight into the real subject matter of quantum mechanics. Most QM books I know are really introductions to solving partial differential equations. (Remark to physicists: I bet you didn't know you could get the simultaneous eigenvalues for the energy and angular momentum operators for the hydrogen atom by a beautifully simple method that doesn't require even looking at a differential equation...) The best thing about the newly appearing field of quantum computing is that it's slowly forcing people to thing about quantum mechanics separately from the mechanics.

So even though I haven't read that course myself yet, I'm recommending that everyone read it :-) And some time I might get back to the even more elementary introduction I hope to put together.

A Neat Proof Technique

Last night I read this paper. There wasn't really anything in the paper I wanted, I was more interesting in flexing my newly developing computer science muscles. Anyway, like with all computer science papers I've managed to finish, I felt like I understood the paper but had no idea what the punch line was. Still, it was worthwhile because part way through there was a neat mathematical proof technique used that I think is worthy of a mention.

The paper is about the generalisations of 'fold' and 'unfold' that I played with recently. The usual fold function acts on a list like

[4,8,15,16,23,42].

Imagine it written as

cons(4,cons(8,cons(15,cons(16,cons(23,cons(42,[]))))))

where the cons function is a 'constructor' that constructs a list from a list element and another list. [] is the empty list. The fold function takes three arguments, a binary function f, a constant g, and a list. It then replaces cons by f and [] by g. For example

fold (+) 0 [4,8,15,16,23,42]

becomes

4+(8+(15+(16+(23+(42+0)))))

and so is 108, the sum of the numbers. An obvious theorem is that

fold cons [] x = x.

fold also generalises to structures such as trees. In this case there is another constructor that builds a tree from a node and its children. I'll stick with lists as the proof technique is much the same. The authors prove a bunch of theorems about the fold function. But now they want to prove something is true of all of the elements in a list (actually, a tree, but I use a list). Suppose the list is as above and you want to prove the property P holds for all elements in that list. Then you want to prove

P(4) && P(8) && P(15) && P(16) && P(23) && P(42) = True.

(I'm using Haskell's && to mean 'and'.) In other words you want to prove that

fold (&&) True [P(4),P(8),P(15),P(16),P(23),P(42)] = True.

The authors then proceed to show their theorem is true using the theorems about fold they just proved. Neat eh? But Maybe I haven't explained very well so I'll try to spell it out differently.

The authors want to prove theorems about all elements of a certain datastructure. So what they do is take the logical proposition that expresses what they want and rewrite the proposition so that the proposition itself in the same 'shape' as the datastructure. The statement of the truth of the proposition is an application of fold to the datastructure. So they can now use the theory of datastructures they develop to prove something about the structure of the logical proposition itself - namely that it's true. You need to read the actual paper (top left, page 5) to see the actual details of the proof technique.

In the particular case of lists the proof technique turns out to be standard induction. If you can prove P of an an element of the list, and you can prove "if P is true of an element it must be true of the next element", then P holds for every element of the list. Every time you generalise fold to a different datastructure there is a new induction principle that goes with it.

I suppose what they have actually done is prove a metatheorem about certain types of logical proposition and then used that to prove specific propositions.

Well, I hope that makes some kind of sense. I think it's quite a neat trick and that many of the paper's readers might miss the cool bit of self-reference that's going on here.

The General Theory of Self-Reproducing Programs

I'm guessing that anyone reading this is already familiar with the idea of programs that output themselves. If you're not there's a great list of such programs here. But what I was surprised to discover at the weekend was that there is in fact a bit of general theory about this. In particular, what do we need in a computer language to guarantee we can write a self-replicator?

Consider the following in Haskell:

let p x = x ++ show x in putStrLn $ p"let p x = x ++ show x in putStrLn $ p"

Evaluate this expression in an interactive Haskell session and it prints itself out. But there's a nice little cheat that made this easy: the Haskell 'show' function conveniently wraps a string in quotation marks. So we simply have two copies of once piece of code: one without quotes followed by one in quotes. In C, on the other hand, there is a bit of a gotcha. You need to explicitly write code to print those extra quotation marks. And of course, just like in Haskell, this code needs to appear twice, once out of quotes and once in. But the version in quotes needs the quotation marks to be 'escaped' using backslash so it's notactually the same as the first version. And that means we can't use exactly the same method as with Haskell. The standard workaround is not to represent the quotation marks directly in the strings, but instead to use the ASCII code for this character and use C's convenient %c mechanism to print at. For example:


main(a){printf(a="main(a){printf(a=%c%s%c,34,a,34);}",34,a,34);}


Again we were lucky, C provides this great %c mechanism. What do you need in a language to be sure you can write a self-replicator?

It turns out there is a very general approach to writing self-replicators that's described in Vicious Circles. What follows is essentially from there except that I've simplified the proofs by reducing generality.

We'll use capital letters to represent programs. Typically these mean 'inert' strings of characters. I'll use square brackets to indicate the function that the program evaluates. So if P is a program to compute the mathematical function p, we write [P](x) = p(x). P is a program and [P] is a function. We'll consider both programs that take arguments like the P I just mentioned, and also programs, R, that take no arguments, so [R] is simply the output or return value of the program R.

Now we come to an important operation. We've defined [P](x) to be the result of running P with input x. Now we define P(x) to be the program P modified so that it no longer takes an argument or input but instead substitutes the 'hard-coded' value of x instead. In other words [P(x)] = [P](x). P(x) is, of course, another program. There are also many ways of implementing P(x). We could simply evaluate [P](x) and write a program to simply print this out or return it. On the other hand, we could do the absolute minimum and write a new piece of code that simply calls P and supplies it with a hard-coded argument. Whatever we choose is irrelevant to the following discussion. So here's the demand that we make of our programming language: that it's powerful enough for us to write a program that can compute P(x) from inputs P and x. This might not be a trivial program to write, but it's not conceptually hard either. It doesn't have gotchas like the quotation mark issue above. Typically we can compute P(x) by some kind of textual substitution on P.

With that assumption in mind, here's a theorem: any program P that takes one argument or input has a fixed point, X, in the sense that running P with input X gives the same result as just running X. Given an input X, P acts just like an interpreter for the programming language as it outputs the same thing as an
interpreter would given input X.

So here's a proof:
Define the function f(Q) = [P](Q(Q)). We've assumed that we can write a program that computes P(x) from P and x so we know we can write a program to compute Q(Q) for any Q. We can then feed this as an input to [P]. So f is obviously computable by some program which we call Q₀. So [Q₀](Q) = [P](Q(Q)).

Now the fun starts:

[P](Q0(Q0)) = [Q0](Q0) (by definition of Q0)
            = [Q0(Q0)] (by definition of P(x))

In other words Q₀(Q₀) is our fixed point.

So now take P to compute the identity function. Then [Q₀(Q₀)] = [P](Q₀(Q₀)) = Q₀(Q₀). So Q₀(Q₀) outputs itself when run! What's more, this also tells us how to do other fun stuff like write a program to print itself out backwards. And it tells us how to do this in any reasonably powerful programming language. We don't need to worry about having to work around problems like 'escaping' quotation marks - we can always find a way to replicate the escape mechanism too.

So does it work in practice? Well it does for Haskell - I derived the Haskell fragment above by applying this theorem directly, and then simplifying a bit. For C++, however, it might give you a piece of code that is longer than you want. In fact, you can go one step further and write a program that automatically generates a self-replicator. Check out Samuel Moelius's kpp. It is a preprocessor that converts an ordinary C++ program into one that can access its own source code by including the code to generate its own source within it.

Another example of an application of these methods is Futamura's theorem which states that there exists a program that can take as input an interpreter for a language and output a compiler. I personally think this is a little bogus.

Stanislaw Lem has Passed Away

He has an obituary in The Times. If you haven't read any of his science fiction you really should read some. For lovers of mathematical poetry out there, here is a translation of a passage from his Cyberiad:

Come, let us hasten to a higher plane,
Where dyads tread the fairy fields of Venn,
Their indices bedecked from one to n,
Comingled in an endless Markov chain!

Come, every frustrum longs to be a cone,
And every vector dreams of matrices.
Hark to the gentle gradient of the breeze
It whispers of a more ergodic zone.

In Riemann, Hilbert or in Banach space
Let superscripts and subscripts go their ways.
Our asymptotes no longer out of phase,
We shall encounter, counting, face to face.

I'll grant thee random access to my heart,
Thou'lt tell me all the constants of thy love;
And so we two shall all love's lemmas prove,
And in our bound partition never part.

For what did Cauchy know, or Christoffel,
Or Fourier, or any Boole or Euler,
Wielding their compasses, their pens and rulers,
Of thy supernal sinusoidal spell?

Cancel me not--for what then shall remain?
Abscissas, some mantissas, modules, modes,
A root or two, a torus and a node:
The inverse of my verse, a null domain.

Ellipse of bliss, converge, O lips divine!
The product of our scalars is defined!
Cyberiad draws nigh, and the skew mind
Cuts capers like a happy haversine.

I see the eigenvalue in thine eye,
I hear the tender tensor in thy sigh.
Bernoulli would have been content to die,
Had he known such a²cos(2φ) !

The Most Amazing and Mysterious Thing in All of Mathematics

I think that a good candidate is the table of the Homotopy Groups of Spheres.

For those not familiar with algebraic topology, π_m(Sⁿ) is the set of equivalence classes of continuous functions from the m-dimensional sphere to the n-dimensional sphere where two functions are considered equivalent if they are homotopic. An easy way to visualise this is that two functions are homotopic if you can interpolate a continuous animation between them. (Can you guess what industry I work in?) This set also has a group structure which is straightforward to define but which I won't go into here (unless someone requests it). That's all there is to it. How can such simplicity generate such complexity?

Monstrous Moonshine is pretty mysterious too - but it takes a lot of work to state it for a non-expert. So this wins by default.

Like John Baez I also wonder about the curious appearance of 24 on row 3.

And an aside I discovered on Anarchaia: foldr.com related to an earlier post.

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?"