COMPLEX NUMBERS

Dick Pountain/15 February 2001 14:52/Idealog 79

I'd been bracing myself for a hurricane of hype to follow the simultaneous publication of two first maps of the human genome, but after the initial reports things have gone eerily quiet. The great surprise contained in the maps -  published in Nature by the public sector teams, and separately by Celera genomics, the private US firm - is that they both pretty well agree that the human genome contains far fewer genes than anyone was predicting, somewhere around 30,000, which is not even twice the number possessed by a fruit fly or a humble weed. The newspapers that first reported this were clearly disappointed: their science correspondents concurred that with so few genes it is no longer plausible that there is a gene for every little human quirk and foible like Morris dancing, smoking or watching Thunderbirds. Indeed it becomes a puzzle, according to some, that there are even enough genes to account for human evolution, speech or intelligence. People seem to feel diminished by the knowledge, rather as if their bank statement had shown up with five hundred quid missing...

I have to smugly say that none of this fazes me in the slightest. That isn't quite the same as saying I predicted it, or I told you so, but merely to say that it does not contradict any of my cherished beliefs. I've always been sceptical of the new genetic hyperdeterminism that became fashionable in recent years (though I am still, and always have been, a rather orthodox Darwinian). This is partly through education - I was taught biochemistry at college by Steven Rose, that implacable foe of genetic determinists - but also because I have another set of beliefs that provide a much more satisfying answer to the puzzle of why so few genes are enough. These beliefs derive from Algorithmic Information Theory, the work of Chaitin, Kolmogorov and Solomonoff (CKS for short) whom I may have mentioned in these columns before.

The connection is not at first obvious as CKS's work started from the pure mathematics of probability, and in particular pseudo-random sequences or bit strings. Chaitin's theorem is an extension of (and includes) Goedel's Theorem and therefore has to do with limits on knowledge: very roughly summarised it says no consistent theorem can prove any result (that is, the existence of a bit string) much more complex than itself. From there we move on to consider programs (running on a PC, a Turing machine, a fruit fly, whatever) that can generate bit strings, and in particular the size of such programs and the time they take to execute. Some very long bit strings can be generated by very short programs, while others are so random that no program shorter than themself can generate them. For example the infinitely long string  0.1428571428571... is not at all complex because it can be generated to as many places as you want by merely dividing 1 by 7; on the other hand pi ( 3.14159265359...) is also infinite but requires a much longer program to generate as many places as you want, for example one that evaluates a Taylor series; finally a truly random sequence like 0.1636203524... (let's not worry about where it could come from) could only be printed out as an infinitely long literal value, because if any other way were available it wouldn't be random. You might be feeling uneasy that these are only numbers I'm talking about, but a reader of this magazine, of all people, ought to be happy with the notion that anything at all can be described by some suitable numeric encoding, for example an MP3 of Eminem's 'Stan' or a JPEG of Anna Kournikova. We end up with these three notions: the length of a bit string, its complexity (that is, the length of the shortest program that can generate it) and a third notion contributed by Chaitin's colleague Charles Bennett called its algorithmic depth, which is the number of computation steps (ie. the time) it takes to execute that program.

Now you can use these three concepts as tools to start talking some sense about the little subjects like the Universe As We Know It, or the human genome, or life in general. It is not impossible for example that there is (as Kabbalists, Alchemists and Douglas Adams fans have hoped for centuries) a short formula that describes the whole known universe: however it is equally possible that the depth of this formula is greater than (or perhaps equal to: God-as-a-computer anyone?) the age of the universe, in which case it would be of very little use even if you discovered it. Then again there is the nihilist's worst suspicion, that the universe is utterly random with perhaps infinite length and complexity but no depth. What is certain, as a consequence of Chaitin's theorem, is that we can never prove that either is or is not the case, given the finite number of brain-cells we have available for theorem handling. Bennett, who was a originally a chemist, has speculated that human DNA (another kind of bit string) has a depth considerably greater than its length - in other words, you can build a pretty complex being from not all that many genes if you run these 'programs' in parallel on millions of little Turing machines called ribosomes, and for somewhere around 80 years. As to whether these programs ever terminate, Alan Turing tells us that we can only wait and see, though the Pope may believe otherwise...