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?"

Young babies count

At least according to an article in Scientific American.

Of course it's not clear that this is "counting" in any conventional sense. Additionally, some monkeys can count this well (and I think also many other mammals). So I'm not sure what conclusions can be drawn from this experiment.

What's the most implausible but true mathematical proposition?

Some would say the Banach-Tarski paradox but my attitude has always been that all bets are off when dealing with unmeasurable sets. I think the existence of a number satisfying the main property of Khinchin's constant is much less plasuble.

Addendum: According to this paper, even though the Khinchin property is true for almost all real numbers, nobody has explicitly found a number for which it is true! You can artificially construct a number for which it is true but that doesn't give you an explicit result, and anyway, that's cheating. Experimentally, π gives something near the Khinchin constant, but there's no proof of what happens in the limit.

Is Mathematics too Easy?

Over the years many people have felt that mathematicians have made life too easy for themselves by using axioms that seem much like the proverbial sledgehammer cracking open a nut. Look at the way analysts and algebraists will wield the Axiom of Choice when they can prove what they need with much weaker axioms. Wouldn't it be more enlightening to carry out these proofs using the weakest axiom system possible so we can see exactly what is needed to make them provable?

It turns out that this is more or less what reverse mathematicians do. They take theorems such as the well known Bolzano-Weierstrass theorem and try to figure out precisely how much mathematical machinery is required to prove them. There is no need to use all of ZFC, or even ZF, to prove such a result.

So reverse mathematicians sometimes start with something simpler like second order arithmetic. In its universe are integers and sets of integers, and nothing else. It can still talk about real numbers by encoding them as Cauchy sequences in the usual way but there is no way to encode subsets of the reals in this way. Maybe surprisingly, it is possible to represent continuous functions because the rationals are dense in the reals so continuous functions are defined by the values they take at rational numbers. It turns out that almost all classical mathematics can be encoded in this system even though it's much weaker than ZF.

But usually reverse mathematicians like to go even weaker still and work with recursive comprehension which is essentially the Peano axioms combined with induction and a restricted comprehension axiom. Using these weak tools it's still possible to prove things like the Intermediate Value Theorem, the uncountability of the reals and even results about metric spaces such as the Baire category theorem (but only for separable spaces).

One thing that has always intrigued me about number theory is how many theorems are proved by making excursions into analysis. For example, think about results in analytic number theory that can easily be stated using the language of Peano's axioms, and yet whose proofs require techniques like contour integration that make reference to infinite sized objects and limits. In 1988 Stephen G Simpson proved that such theorems can in fact be reduced to "primitive recursive arithmetic" and hence turned into "finitistic" proofs. On the other hand, I'm not sure that these proofs are necessarily going to give any insight into why the theorems are true. They probably end up being as unwieldy as Borbaki's definition of one.

An End to Coding Theory

Error correcting codes have inspired some beautiful mathematics. One of my favourite mathematical constructions, the binary Golay code, is an error correcting code.

The idea behind the theory is simple. Suppose there is a set of M messages, one of which you want to send over a noisy communication channel. If you simply send one of the messages it might be corrupted and interpreted as another. So instead you embed the set M inside a larger set C and send an element of C as a message. If the message is corrupted then they might receive a message in C but not in M. They'll know right away that the message has been corrupted. But even better, if the embedding of M in C is designed cunningly enough they'll be able to make a good guess as to what the original message was assuming some probability model for the corruption. You pay a price of course, the elements of C typically take longer to transmit than the elements of M, reducing the rate of data transmission.

There have been lots of proposed schemes over the years since Shannon started the field. They use a variety of methods from discrete mathematics and algebraic geometry. There are nice connections between the sporadic groups and coding theory. The Mathieu groups arise directly as symmetries of the Golay code and the Golay code can itself be derived from the Leech lattice whose symmetries are described by the Monster. A common feature of these codes is that they are linear. The set C is a vector space over F₂, the field with two elements. M is a subspace of this space. M is the kernel of an F₂-linear map represented by a matrix called the syndrome. If the syndrome, when applied to a message, gives zero, then it's likely that the message has arrived uncorrupted. Much of the work in coding theory has been about generating suitable syndrome matrices.

But in the sixties Robert Gallager looked at generating random sparse syndrome matrices and found that the resulting codes, called Low Density Parity Check (LDPC) codes, were good in the sense that they allowed messages to be transmitted at rates near the optimum rate found by Shannon - the so-called Shannon limit. Unfortunately the computers of the day weren't up to the task of finding the most likely element of M from a given element of C. But now they are. We now have near-optimal error correcting codes and the design of these codes is ridiculously simply. There was no need to use exotic mathematics, random matrices are as good as almost anything else. The past forty years of coding theory has been, more or less, a useless excursion. Any further research in the area can only yield tiny improvements.

A good place to learn about these codes is in the book by one of the 'rediscoverers' of Gallager's codes, David McKay. There's also the Wikipedia article. I originally heard of these codes from a friend of mine and was sceptical at first. Then I read this Science News article and I'm now a bit embarassed about my scepticism.

The Flyspeck Project

I just came across the Flyspeck Project. The problem they are dealing with is the verification of the Kepler conjecture. Although there is a potential proof nobody is 100% sure of its validity and it seems that the group asigned to verify it have run out of steam. So the Flyspeck Project seeks to build a complete formal proof using HOL Light, a computational framework for proving things. Unfortunately the guys behind the project expect it might take as long as twenty years to verify the potential proof. Let's hope that not too many interesting conjectures require this treatment.

Note that this is different from the four colour problem. For that problem, the proof required an enumeration that was too long and complicated to carry out by hand, so a machine carried it out. For this problem we have a slightly different situation: we already have a potential proof, but we need to reduce it to a completely formal form so that the entire thing can be verified to follow from axioms.

Biggest Test of Benford's Law?

Earlier this week a swapped a couple of emails with someone at Google involved with looking at web page statistics on the Google database. They probably have the one of the largestest corpus of textual data in the world and it's a mixture of data from a wide variety of sources. This makes it an ideal candidate for testing Benford's law and that's what I'm suggesting they do. The published tests I've seen so far are pretty small but it'd be fun to see the results from, say, a billion web pages. Coincidentally Boing Boing just had a linky to a New York Times story on it form 1998. Anyway...I asked the guy to email me if they carry out the test. It's probably pretty low down on their list of priorities. So of course I'll report back.

De Rham Cohomology and Computer Algebra

Computers are now good at a number of branches of mathematics. They've been performing symbolic integration, differentiation, summation and differential equation solving for decades. In recent years, with the advent of Grobner basis methods, they've become pretty competent at algebraic geometry. But one area has stood out in my mind as being unsuitable for computational methods and that's computing de Rham cohomology. In order to compute this we have to study spaces of differential forms on manifolds and these are large spaces. Given lots of pieces of differential form you can glue them to other bits of differential form using a partition of unity so in a sense you have a great freedom to mix and match forms. This is in stark contrast to algebraic geometry where spaces of functions frequently tend to be finite dimensional.

So I was surprised to find that in recent years computers have been calculating de Rham cohomology, and doing so using methods from algebraic geometry. In fact, functions to compute the de Rham cohomology of the complement of various algebraic varieties have been available in Macaulay2 for a while. There's a chapter of the Macaulay2 book available on the subject. The methods used are based on D-modules and discussed in the book Grobner Deformations of Hypergeometric Differential Equations.

I have no idea where the connection with hypergeometric series is coming from but it's interesting to see the connection with algebraic geometry. D-modules are rings with elements x_i and d_i sith the proprty that x_ix_j=x_jx_i and d_id_j=d_jd_i but that d_ix_j-x_jd_i=δ_ij. We can form ideals of these things and study them using Grobner basis methods. Given a smooth function we can form the ideal of elements of a D-module that annihilate the function by considering the d_i to be the differential operators d/dx_i. This gives a connection between the methods of algebraic geometry and differential geometry. But that's as far as my knowledge extends for now.

Geometry is Innate

It seems that geometry is innate according to a MSNBC. The study worked with the Mundurukú people from a remote region of the Amazon who lack vocabulary for geometric terms. Nonetheless they scored well on a variety of nonverbal tests. Judging by the pictures they were testing for the ability to identify connectedness, similarity and congruence. They also displayed an ability to use maps even though they are not a map-making culture.

It's curious to note that the people of this culture weave baskets that appear to involve interesting geometry and whose design look anything but 'innate'. And though 'remote' they also wear glasses and Western style clothing. So clearly they do transmit 'geometric' culture, even if it's nonverbal, and they have had non-trivial contact with cultures that do have a strong geometric vocabulary.

Eleven Reasons to use Haskell as a Mathematician

As a mathematician, what programming language should you use when you need to do anything algorithmic? For some specific domains there are languages tailored for the task, for example Macaulay2 is great for some types of algebra. For general problems many turn to Mathematica or Maple. But although they are universal (in the sense of Turing) they're not great platforms for general purpose algorithms and can perform very poorly unless the feature set of these languages just happens to marry well with the problem you're trying to solve. These tools are also like the proverbial sledgehammer - you're using a giant engine to solve what can sometimes be a simple problem and you'll pay the price in terms of memory and CPU usage. If you want a general purpose programming language the answer is simple. There are eleven reasons why you should use Haskell. So if you're a mathematician who occasionally needs to write code, get started on a tutorial right away!

(1) Haskell is Functional

Functions are the bread and butter of mathematics so it makes sense to use a programming language in which functions are first class objects. It's not at all unusual for a mathematician to take a function of two arguments, say f(.,.), and construct from it a function of a single variable defined by f_x(y) = f(x,y). Computer scientists call such a thing a closure - it's a function constructed by supplying values for some, but not all, of the arguments of another functions. It's one of the most natural operations in the world and yet it's actually quite horrible to construct such a thing in languages like Fortran, C or C++. It's slightly cleaner in C++ because you can use objects and templates to simulate closures, but the syntax is still pretty hideous. (I know there are libraries that make this even easier - but they do so by implementing a functional language as a 'domain specific language' within C++, essentially making my point for me.) Here's how you do this in Haskell: you write "f x". The way it works is quite neat. Given sets A, B and C there is a natural bijection between the set of functions AxB->C and the set of functions A->C^B. So rather than write functions of multiple arguments Haskell function always take one argument and return the function that operates on the rest of the arguments. At first this might seem like a bad idea because functions of multiple arguments are now, in some sense, a second class object. But it's not a problem at all. To apply the function f to the arguments 1 and 2 we simply write "f 1 2".

To get an idea of how horrible it can be to work with functions without functional programming consider the code in Numerical Recipes in C. Most of their code is 'functional' in character in the sense that it is intended to perform operations on functions. But look at how you have to make use of horrible void * pointers to pass in those extra arguments to simulate closures.

(2) Haskell Insulates You

As a mathematician you're more interested in solving the problem than dealing with the idiosyncracies of the hardware and issues like memory management. Haskell does a great job of insulating you from the underlying hardware so that you don't have to worry about these things. Languages like Java aren't too bad at this either. C++ is terrible, if you decide you don't want to manage memory yourself you can use shared_ptr<> objects and turn your code into a horribly unreadable mess.

(3) Haskell Performs Well

It's surprising how well Haskell peforms. You might think "if every time I add a to b it has to construct a temporary intermediate object that represents the function that adds a to a number and then apply that function to b it's going to be slow". But that is not the way to look at things. Haskell doesn't simply execute code by working top to bottom, left to right through your source code. The compiler (itself written in Haskell) is pretty sophisticated and frequently performs comparably to C. It's generally not as fast as C, but it's usually in the same league. The Haskell compiler rips apart your code and reconstitutes it as an executable in ways that are quite surprising. In Haskell you can often write code using idioms that feel natural, but are generally considered slow in other languges, without performance impact. One example is in list handling. Often you want to return a temporary list of objects and perform some operation on them. In many languages there is an overhead in making such a list, only to take it apart again when you want to operate on the actual elements so you instead have to restructure your code so that no list is constructed. In Haskell the compiler will often figure out that your list is temporary and do the restructuring behind the scenes for you.

(3) Haskell is Declarative

Mathematicians like to spend their time saying what is the case rather than thinking in terms of sequences of operations. If b = f(a) and c = f(b) then it doesn't matter if you say c = f(b) and then b = f(a). Order is irrelevant. In an imperative language like C it matters to a large extent what order your code is written in because a statement like b = f(a) is an instruction. In Haskell you aren't giving a list of instructions so much as stating what is true. This fits much better with the way that mathematicians work. Haskell can take care of some of the details of precisely what order the operations are going to be performed in and hide those details from you.

(4) Haskell is Lazy

This goes hand-in-hand with Haskell being declarative. Haskell only computes what it needs to compute in order to get the answer you require. It tries not to do unnecessary work. Languages that aren't lazy like this are called 'strict' languages. In a strict language, say, if you required the first component of the cross product of a vector, you would typically compute all three components of the vector and throw away the second and third components. If your code needed to be efficient you'd probably end up writing special purpose code to compute just the first component of the cross product. You'd be unable to reuse your cross product code. Mathematicians don't want to do this. They want to say that x is the first component of the cross product because it's true statement and not worry about what exactly gets implemented when. This laziness is ubiquitous throughout all of Haskell.

One place that laziness really pays off is in working with lists. Haskell has absolutely no problem working with infinite lists as long as you don't actually try to look at all of the elements. Instead of evaluating all of the elements it simply retains enough information to construct as many elements as are needed. Computing the elementwise sum, say, of two infinite lists is completely straightforward. And performing computations with power series, say, where the later elements are functions of the earlier elements, is absolutely straightforward. You don't have to check to see if you've computed enough of the earlier elements to compute the later one. Haskell does all that for you. If you want to see the first ten elements of the final infinite list you can just ask for them. You don't need to decide in advance how many elements you want and build that into the code from the beginning.

(5) Haskell is good at Datastructures

Haskell is good at dealing with datastructures. Often mathematicians don't realise that they are using datastructures all the time. But they are. For example a group is a set equipped with functions to perform multiplication and inversion as well as a singled out element, the identitiy. Haskell makes it very easy to work with structures that are tuples of objects like this. It's also very easy to build one datastructure out of another and even specify rules about datastructures. For example a matrix is a two-dimensional array of objects. But if the objects form a ring then so does the matrix. This kind of statement is easy to express in Haskell making it well suited to working with algebraic structures. If you've defined formal power series over a ring and you've defined the ring of polynomials over another ring it's trivial to build the ring of power series over polynomials over the integers, say. What's more, the compiler can tell you when the structure you're trying to create doesn't make sense, for example if you were to try to multiply two matrices whose underlying elements can't be multiplied. A programming language like Mathematica is very poor at this as it doesn't really distinguish between datatypes. You'll only discover your error when the program is actually running and it tries to multiply the underlying objects. Haskell also uses type inference. You don't have to explicitly tell it the type of something because it can often figure it out for itself. If you see the number '1' in a textbook what does it mean? Is it an element of N, Z, R, Q, R[sqrt(2)], C or some other completely different algebraic structure? It could be any of these things depending on the context. The same is true in Haskell, it'll figure out what you mean by '1' using completely unambiguous rules. This means that you don't have to use a different symbol for '1' in different contexts, and yet you don't have to laboriously state what you mean every time you use it.

And rather than having an ad-hoc system of datastructure types Haskell's is algebraic.

(6) Haskell is Compact

Not in the topological sense. Haskell code tends to be much shorter than its counterpart written in other languages. There is little extraneous verbiage and the verbiage that does exist is very powerful. This is more than a space saving exercise. Consider the history of calculus in n-dimensions. When Maxwell first wrote down his equations of electrodynamics they took up most of a page. WIth the introduction of vector notation they were simplified. By using the notation of exterior calculus they were simplified to dF = 0, d*F = *J. This isn't just typographically simpler but conceptually simpler. You can see immediately how you might generalise to n-dimensions instead of three, you can see trivially that d*J=0 from d^2 = 0 and you can now perform countless computations in a few lines that previously would have taken dozens of pages. What's more the new notation makes it harder to accidentally write equations of motion that don't make physical sense because they automatically obey symmetries that are difficult to see in more complex notation. Haskell is just like this too. You can do a lot with a tiny amount of code. And if two blocks of code appear to do similar things you can often factor out the common part and make it a separate block of code in a way that's much harder than with just about any other language. This makes it much easier to generalise your code and have insights about it. This works well with the way mathematicians like to think about things.

(7) Haskell is Mind Expanding

Haskell draws on ideas from interesting branches of mathematics, for example Category Theory and Logic. And the logic part doesn't appear the way you expect - it appears via the Curry-Howard 'isomorphism' in the type system used to define datastructures. As mentioned above, Haskell code lends itself well to generalisations. You'll find all sorts of commonality between things that previously didn't seem similar, and you'll find ways to express that similarity in your code. One of my favourite examples is by Ralph Hinze. He doesn't just illustrate the
similarity between binomial tree and binary arithmetic, he actually shows how merging binomials and adding binary numbers are actually instances of the very same thing. Haskell encourages this kind of abstraction. Though I've wandered into the realm of computer science with talk of trees this is still the type of abstraction that mathematicians find interesting.

(8) Haskell Has No Side Effects

When you add two numbers you get the sum. You don't find that something else in the universe has happened as well (except maybe the temperature in the room goes up if it's a really hard sum). So functions in a programming language shouldn't have any side effects. In Haskell a
function does nothing other than return the value of its result. This means that it's easy to bring your mathematical experience to bear when analysing programs. (Almost) all of the usual rules for manipulating mathematical expressions apply so that, for example, if b = f(a) and c =
g(b,b), then c = g(f(a),f(a)). In many other programming languages you'd have to make sure that calling f twice didn't have some side effect that made it behave differently to calling it once.

(9) Haskell Eats Circular Definitions for Breakfast

In most imperative languages a line of code like x=f(x) means replace the value of x with f applied to the old value of x. In Haskell it means x equals f(x). It has no problem with you defining a variable in terms of itself and x=f(x) means that x is a fixed point of the function f. In practice it's not so simple, you can't just write x = 1-x^2 and expect it to figure out that x is the golden ratio. Haskell is a programming language not an equation solving package. But in my formal power series library I'm able to use a line like "e = 1+integrate e" to define the power series for exp x. See here for another example of using circular definitions to solve differential
equations.

(10) Haskell is Well Founded

What I mean is that Haskell is built on fairly solid mathematical foundations in lambda calculus and proof theory. It's not just an ad hoc language that has accreted features over the years. Everything is there for a reason and people have thought long and hard about the theoretical underpinnings of what went into it.

(11) Haskell is Pretty

Well, that's a matter of taste. But I find it very elegant.

That's 11. I think there's a rule against making lists with prime numbers of elements but I'm ignoring it.

To be fair I ought to point out that it's not all a bowl of cherries. Haskell isn't very good at dealing with arrays. In most programming languages it takes time O(1) to access the ith element of an n element array. In Haskell you can get this kind of performace if you don't mind using the various painful to use array operators available, or you can sacrifice performance, keep the elegance, and get O(log n) performance. On the other hand, you'd be surprised just how many useful matrix algortihms don't require random access to the ith element.

Because Haskell functions don't have side effects it doesn't make sense to perform an operation more than once because it'll just do the same thing each time. So there's no notion of a 'for' loop (well, maybe there is) in Haskell and instead you must find other ways to express iterative operations - ie. using 'higher order functions' or recusrion. But sometimes you feel like you just want to write the loop and be done with it. There's always another way of expressing the problem but when the iterative approach is the first thing to pop into your head it can occasionally be frustrating. And sometimes you'd like side effects.

Haskell comes with a library of functions called the 'standard prelude'. Unfortunately, whoever designed it knew just enough mathematics to be dangerous and made a complete hash of it. Instead of using a well known hierarchy of algebraic structures such as group->ring->field they defined all kinds of bizarre structures starting with something like a ring with a norm or valuation. And while the library supports complex numbers it's not flexible enough to support the Gaussian integers (probably a consequence of forcing rings to have a norm). All things that could have been implemented correctly in a few seconds if a mathematician had been consulted early on.

And the last caveat. I found learning Haskell hard. Let me put it in perspective: I'm 39. I've been programming since I was 11. I've been writing recursive code since about 15 and I've been using whatever approximations to functional programming are avalable to me in any language I use for almost that long (including torturing myself with C++ template metaprogramming). I've even implemented a couple of declarative languages, Prolog and SASL, the latter being a functional lazy language a lot like Haskell that I thought I'd investigate before Haskell. And I still found Haskell tricky to learn. But it was worth it. And I believe that the trickiness of the language is the kind of trickiness that mathematicians should be able to handle. (And it probably helps if you don't know any C or C++...)

A friend of mine, and I, have been writing all kinds of mathematical code and finding Haskell well suited to the task. Logic (eg. building analytic tableau as in Smullyan's book on first order logic), Group Theory (eg. the Schreier-Sims algorithm), Calculus (automatic differentiation), Combinatorics (via formal power series), Algebra (eg. Grobner bases) and now we're looking into investigating relationships between elliptic functions and modular forms. (Hmmm...my friend takes most of the credit here.) Haskell is great at all of this stuff even though it's not in any way an algebra or mathematics package.

And one list thing about Haskell. If x is an integer (called an "Integer" in Haskell) then x+1>x. It's nice to work over the integers rather than a cheap and nasty 2-adic approximation (though the cheap and nasty approximation is there if you want it.)

You're needed!

As reported in Business Week, and cited on Slashdot, here's a story about why mathematicians aren't so useless after all.

General Relativity for Dummies

No, really, for complete dummies. John Baez has just posted a description of Einstein's field equations that just about anyone can understand. So simple, even, that it can be described in words. And yet the formulation is equivalent to the field equations. It goes like this:

Given a small ball of freely falling test particles initially at rest with respect to each other, the rate at which it begins to shrink is proportional to its volume times: the energy density at the center of the ball, plus the pressure in the $x$ direction at that point, plus the pressure in the $y$ direction, plus the pressure in the $z$ direction.

More information is here. He also has a derivation of Newton's law of gravition here.

As Baez points out - at first it looks implausible because the Einstein field equation is a tensor equation and this looks like a scalar equation. But the statement must hold true for any set of test particles that are at rest relative to each other. So at any point it doesn't just give one constraint.

This is pretty neat stuff and it ought to go into all GR textbooks...

Self-Reference in the Real World

A sign. From Boing Boing,

Theorem Proving and Code Generation

According to the Curry-Howard isomorphism a type in a programming language is essentially the same thing as a proposition and an object or program of the type is a(n intuitionist) proof of the proposition. So an automatic proof generator should also be usable to generate objects or programs of a given type. This is exactly what Djinn does. It seems pretty neat.

(I only learned about the Curry-Howard isomorphism in the last couple of years but it's probably of general interest. So I recommend reading the Wikipedia article I link to if you haven't met it before. It's also probably not what you're currently guessing it is. It's not about program correctness but type correctness and it doesn't prove programs to be correct but shows how programs themselves are proofs of type correctness. In the language of the article, Djinn solves the "Type Inhabitation Problem".)

Rant: Just for kids?

Search Google News for stories about mathematics and you'll find that they are dominated by a particular subsection of our society: kids. It seems that the only people in the world doing mathematics are children. Very occasionally
you'll see a story about an adult, but then it's usually about an adult teaching children. Do a search on history and you'll see some actual history. Search on physics and you'll get stories about kids mixed in with stories about people doing actual physics. So why are there so few stories about adults doing mathematics?

It's not just in news stories. Look at Texas Instruments' and Hewlett Packard's web sites and you'll see that calculators are largely marketed towards kids. TI are much worse on this front. They have powerful calculators that can solve all kinds of real world problems and yet they're not being marketed as devices for solving such problems, they're about getting kids through exams. HP at least have a section for professionals. Even so, in the old days if an HP calculator had exchangeable faceplates it was so that you could label the keys with new functions, but now it's to change the colour to make it look cool. The only people working with numbers are clearly children.

On closer inspection I see that I was wrong. I missed a story. It's about an adult. No mention of children or education. Ah...it's about an adult who can't do mathematics.

So clearly no adults use mathematics. So why do we bother teaching children a skill they'll never use in adulthood?

Bees can fly after all!

According to this article it seems that bees can fly after all.

PS I'm wondering about the field of 'human aviation' that the article mentions. Am I lacking a superpower that normally comes standard with being human?