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Lost Innocence
IF YOUR daughter starts puberty early, you might want to check her shampoo. Unbeknown to many parents, a few hair products-especially some marketed to black people-contain small amounts of hormones that could cause premature sexual development in girls.

The evidence that hair products containing oestrogens cause premature puberty is largely circumstantial, and the case is still unproven. But Ella Toombs, acting director for the Office of Cosmetics and Colors at the US Food and Drug Administration, told New Scientist: "No amount [of oestrogen] is considered safe and can be included in an over-the-counter product."

Under FDA regulations, over-the-counter products containing hormones are drugs, and thus require specific approval. However, there appears to be a grey area regarding products marketed before 1994. The FDA failed to respond to requests to clarify the position. At least five companies are still making hormone-containing hair products, a source within the industry-who preferred not to be named-told New Scientist.

Throughout the West, girls are tending to reach puberty earlier. This has been blamed on everything from improved diet to environmental contaminants. But African-American girls are developing even earlier than their white counterparts. About half of black girls in the US begin developing breasts or pubic hair by age 8, compared with just 15 per cent of white girls, one study has found. In Africa, girls enter puberty much later, regardless of their socioeconomic status.

That big discrepancy may be explained, at least in part, by the more frequent use of hormone-containing hair products among African Americans, says Chandra Tiwary, former chief of paediatric endocrinology at Brooke Army Medical Center in Texas. "I believe that the frequency of sexual precocity can be reduced simply if children do not use those hair products," he says.

The products are sold as shampoos or treatments to deep-condition dry, brittle hair. The labels usually state that they contain placenta, hormones or "estrogen", although not all products that make such claims contain active hormones. While New Scientist's inquiries suggest such products are no longer sold in Europe, many are still available worldwide over the Internet.

And they remain popular among African Americans. A small study published earlier this year by Su-Ting Li of the Child Health Institute in Seattle suggests that nearly half of African-American parents use such products, and that most also use them on their children. For other ethnic groups the figure is under 10 per cent. Tiwary told New Scientist that he has carried out a bigger, as yet unpublished, survey of 2000 households that confirms these findings.

In 1998 Tiwary, now retired, published a study of four girls-including a 14-month-old who developed breasts or pubic hair months after beginning to use such products. The symptoms started to disappear when they stopped using them. The year before, he published a study showing that some of the products used by his patients contained up to 4 milligrams of oestradiol per 100 grams. Others contained up to 2 grams of oestriol per 100 grams.

B&B Super Gro, for example, which was marketed before 1994 and is still on sale in the US, and claims to be "rich in hormones", was found to contain 1.6 grams of oestriol per 100 grams. While the levels of oestriol in the products were much higher, oestradiol is a far more potent form of oestrogen.

Stuff of miracles A replacement gene is giving boys a new lease of life

GENE therapy has cured Welsh baby Rhys Evans of the fatal "bubble boy" disease. "His progress seems nothing short of a miracle," says his mother Marie. Another boy treated more recently continues to improve.

The treatment, carried out at London's Great Ormond Street Hospital, is one of only a handful of successful gene therapy trials in people. lt is also only the third trial of gene therapy for severe combined immunodeficiency or SCID. Alain Fischer's team at the Necker Hospital in Paris reported the first-ever treatment in 2000, of two boys, while an Italian-Israeli team recently reported promising initial results with two people who have another form of the disease. Nine people in total have had gene therapy at the Necker Hospital, and seven of them are doing well. But the researchers at Great Ormond Street think they have developed a better way of delivering the gene to correct the fault that causes the disease.

The type of SCID that the Welsh baby had is "X-linked", caused by a fault in a gene on the X chromosome that makes an immune protein called interleukin-2. The disease affects boys because they only have one X chromosome. The faulty gene stops the development of T cells, a key part of the immune system. Children must be kept in isolation to protect them from catching infections and usually die young. A bone marrow transplant can cure the disease, but suitable donors are only found in a third of cases.

To treat the boys, the Great Ormond Street team took the stem cells that give rise to immune cells from the two boys' bone marrow. Then they used a modified form of a retrovirus found in gibbons to add a normal copy of the faulty gene to the stem cells. The virus has altered spikes on its surface which may mean it binds better to stem cells and transfers the gene to them more efficiently, team leader Adrian Thrasher told New Scientist.

The engineered stem cells were then returned to the boys' bodies. Rhys Evans is now back at home, with normal T cell levels, seven months after treatment. The second child, treated just three months ago, continues to improve at home.

The Great Ormond Street researchers say they are planning to treat another four boys over the next two years. They are also treating a child with another immune deficiency, X-linked chronic granulomatous disease. In this disease T cells are present but don't work properly. To make room for new cells, the child's bone marrow will have to be partly destroyed by chemotherapy before the altered stem cells are put back.

While gene therapy is at last starting to live up to its promise, there's still a long way to go before many other inherited diseases can be treated. Getting genes into cells that can be removed from the body and putting them back is relatively easy. But diseases that affect organs such as the kidney can't be treated this way. Getting genes into enough cells inside the body, and controlling their expression, remains the big challenge. Sophie Petit-Zeman

You're my kind of curl
Love the shell babe, fancy starting a new species?

HOW does one species of animal split into two species without evolving in isolation for thousands of years? The problem has vexed biologists since Darwin, but now the curly shells of snails have suggested an answer.

Speciation is easy if two populations become separated and evolve in isolation for millions of years. But sometimes the split happens without a physical barrier. For instance, some crater lakes in East Africa are full of thousands of species of cichlid fish that evolved from a single ancestral species.

Now Mats Bjorklund and jonathan Stone at UpVsala U4iiversity in Sweden have found a quick trick for generating new species, at least in snails. Some snails have shells that coil in a right-handed or left-handed direction. "Righties" can't mate with 'lefties" because their reproductive organs won't meet.

The coil direction is controlled by a "maternal effect". A single gene in the mother codes for a protein that she deposits in her eggs, and this protein makes an embryo's early divisions happen in one of two orientations to make left or right-coiling shells.

Bjorklund says a righty snail population could generate a new lefty one when a rare mutation in a righty mother's egg switches the coiling gene to the left. The mother's egg protein still forces the embryo to coil to the right, resulting in a righty daughter who can mate with other righties-rather than a lone mutant who can't mate with anyone. But all her offspring are lefties who can only mate with each other, producing a distinct species.

Michael Doebeii, a biologist at the University of British Columbia in Vancouver, says the model is neat because it relies on the simple mechanism of enforced mate choice, and could work in snails. "But it is another thing to make it more general," he says.

Bjorklund thinks it could apply to other animals. For instance, an insect that normally mates in the morning could acquire a mutation that makes it mate only with its siblings in the afternoon. James Randerson

More at: Proceedings of the Royal Society 8 (DOI 10.1098/rspb.2001.1934)

India says yes to transgenic crops

But farmers and environmentalists are at odds over the decision

INDIA has become the latest developing country to embrace genetically modified crops. But while several farmers' groups are greeting the government's approval of Bt cotton with jubilation, environmentalists have condemned it. "This is like the fall of the Berlin Wall for Indian agriculture," Sharad joshi, founder of the Maharashtra-based farmers' organisation Shetkari Sanghatana, told New Scientist. "Farmers have been deprived of new technology for a long time but now they will haivttacces5 to it." "It's a recipe for environmental and social disaster," says Devinder Sharma of the Forum for Biotechnology and Food Security. "This will open the floodgates to genetically modified organisms." Bt cotton contains a gene from the common soil bacterium Bacillus thuringiensis. It codes for a protein that kills pests such as the bollworm. The bollworm has devastated cotton crops in states such as the Punjab for the past four years. Proponents of the technology say Bt cotton will help farmers cut down on pesticides, which are expensive, often damage famers' health and are also contaminating groundwater.

Official trials of Bt cotton have been going on in India for four years. "The field trials clearly established that Bt cotton grows in a healthy way in our conditions, and insects are killed by the toxin. There were no adverse effects on soil flora," says Achyut Madhav Gokhale, chair of the environment ministry's Genetic Engineering Approval Committee. Trials of transgenic mustard have already started. "We expect soya, corn and others to follow later," Gokhale says. "It's a technology we cannot stop."

He adds that only conditional clearance has been given to the Maharashtra Hybrid Seed Company (Mahyco), which is partowned by Monsanto, holder of the patent on Bt cotton. Farmers will have to fulfil terms such as planting 80 per cent Bt cotton and 20 per traditional cotton to provide "refuges" to stop pests developing resistance to the Bt toxin. For the moment, however, Mahyco itself will monitor the transgenic crops. "How is this going to work in a landholding of less than half an acre where a small farmer is supposed to waste 20 per cent of the crop?" asks Suman Sahai, president of Gene Campaign in Delhi. She says the conditions are ludicrous for Indian farms, and the trials scientifically flawed and not transparent. Sanjay Kumar, New Delhi

Crops can coexist ... as long as they're kept apart

GENETICALLY modified and conventional crops can safely coexist in the European countryside, although they might sometimes need to be kept further apart than normal to prevent unwanted cross-pollination, says a scientific review issued last week by the European Environment Agency.

Plants that need extra isolation from GM crops include those grown solely to supply high-quality seeds. Also at risk are the highyielding "male-sterile" varieties of oilseed rape, which are pollinated almost exclusively by other plants. But for most crops, bxisting isolation barriers of up to 100 metres should be enough to keep the level of contamination below the accepted limit of I per cent. "it just needs careful farm management," says Jeremy Sweet of the National Institute of Agricultural Botany in Cambridge, who wrote the review with his colleague Katie Eastham.

They rank each of Europe's major GM trial crops according to their "promiscuity"-the likelihood that they will cross-pollinate with other crops or with wild relatives. Oilseed rape topped the league because it spreads its pollen far and wide, both in the air and via insects.

Sugar beet and maize pose "medium to high" risks.

Peter Melchett, the Soil Association's policy director, points out that these three crops are the ones grown in the British government's field trials to discover whether GM crops are a greater risk to farmland wildlife than their conventional counterparts. Sweet says the risk to organic farming is exaggerated. But there are circumstances in which the 100-metre barrier could justifiably be extended to around 400 metres, he says. "For seed production crops, there may have to be some zoning if you want very high levels of non-GM purity," says Sweet. Andy Coghlan

Ready to croak A fungal disease is wiping out New Zealand's ancient frogs

ALL of New Zealand's native frog species could be extinct within two years, warns one of the country's leading frog experts. His gloomy prediction follows the discovery of a fatal fungal disease in one of the country's "living fossil" frogs. Bruce Waldman of the University of Canterbury in Christchurch fears that the disease could quickly spread to other native species. "This is really scary. There's a real possibility that these unique frogs will disappear," Waldman told New Scientist.

New Zealand'sfour species of native frog belong to the ancient genus Leiopelma and differ'lli4le from the earliest frogs, which -lived 200 million years ago. All four are nocturnal, lack ears and don't croak. They hatch from the egg as fully developed froglets, missing out the tadpole stage. The chytrid fungus, Batrachochytrium dendrobatidis, was discovered in the late 1990s in the skin of dead and dying frogs in Australia and Central America. It has recently reached the US and Europe. Infection is usually fatal and the disease is implicated in a rash of recent extinctions.

Waldman found the first knoivn case of chytrid disease in New Zealand late in 1999 in the southern bell frog, a spqcies introduced a century ago from Australia. Bell frogs on both South and North Island are now infected. Waldman suspects the fungus came to New Zealand in frogs imported for the pet trade.

Now the disease has struck the endangered native Archey's frog. Dead and dying frogs from three separate populations have tested positive for the fungus. Biologists have noticed a decline in these terrestrial frogs since 1996, and at some sites 80 per cent of the frogs have gone. "Once one population is gone, extinction is just over the horizon," says Waldman. The fungus seems to be spreading at an alarming rate, and clearly not just via waterways. Ben Bell, a herpetologist at Victoria University in Wellington, thinks hikers and pig hunters carry the disease deep into the bush on their boots and clothes.

"It's possible that every species of native frog has the fungus," says Waldman. Hochstetter's frog, the most widespread native species, shows signs of decline but so far no infection. More worrying is the threat to the two species that live on islands in the Cook Strait. The Maud Island frog is down to a few thousand individuals. Hamilton's frog lives on a small rocky patch at the summit of Stephen's island, and there are fewer than 300 left. Stephanie Pain, Christchurch

Forlorn farewell to frogs

AMPHIBIANS the world over are in trouble. Habitat destruction, pesticides and other pollutants have sent their numbers crashing in recent decades. Then, in 1989, scientists identified a fungus that killed frogs in Panama and Australia, and which has since spread all over the world. Even knowing this, it is a shock to hear that the fungus could see off all New Zealand's native frogs (see p 17). These rare endangered species have changed little over the past 200 million years, and for such living fossils to go extinct would be a huge loss to evolutionary biology. But averting the tragedy will be tough. There is no good way of identifying the fungus in the field. Antibody and DNA tests are being developed, but for now the only sure test is when frogs start to die. Where this happens, the area needs to be cordoned off to stop the infection spreading. The fungus seems to travel via the pet trade in infected water or animals, and on the boots and equipment of hikers, hunters and even conservationists. Until "clean" and infected areas can be identified, it seems prudent to limit the trade in frogs and warn walkers to disinfect their kit.

Some scientists also want to see captive breeding programmes set up for different populations. This will mean screening captured creatures for the fungus, and treati ng any that turn out to be infected. New Zealand has an fine record of conserving endangered species i6ch as the kakapo, the flightless parrot that is intensively monitored. It's time to lavish similar attention on the nation's amphibians. They may not be as cute as the kakapo, but they are no less important.

Lightspeed Quantum Computing

WE HAVE a hardware problem. Every computer ever built is just a snazzier version of a blueprint drawn up by the mathematician Alan Turing in the 1930s. He created a hypothetical "universal Turing machine", a stripped-down processor which, he proved, could do anything that any digital computer would ever be able to do. By going back to the very roots of what maths can and cannot prove, Turing was also able to show that there are some things that none of our computers could ever doproblems no program could ever solve. This result places limits on all of science. When physicists use computers to investigate the workings of the Universe, there are certain 'things they just won't be able to find out.

But who says we're stuck with ordinary computers? Cristian Calude and Boris Pavlov of the University of Auckland in New Zealand have proved that Turing didn't have all the answers. They have found a trick that could sidestep Turing's computational barrier and give access to otherwise forbidden knowledge. While some things are uncomputable by Turing machines, that doesn't mean they're totally off-limits. If digital processing can't give you the answer, what about doing the number crunching with a totally different physical mechanism? One possibility, Calude says, is a quantum computer. These machines-if anyone ever manages to build one-perform computations using particles such as electrons and photons, which can exist in different quantum states. The states act just like digital ls and Os, with the crucial difference that the strange laws of quantum mechanics allow these particles ' to exist in a "superposition" of states. That means they can be in two or more states at once-such as spinning clockwise and anticlockwise simultaneously, or having several different energies at once. Quantum computers can exploit this counterintuitive property to carry out thousands of separate computations simultaneously. It's miles faster than classical computing, but by itself that's not enough to get you past the Turing barrier. After all, as the Caltech physicist and Nobel laureate Richard Feynman said 20 years ago, if you're going nowhere, then simply doing things more quickly will 4iist get you nowhere faster. Calude has something more subtle in mind. ,Ins suggestion is to think bigger: why not create'a superposition of every conceivable state at once? Something like a hydrogen atom has infinitely many possible energy levels. While the levels start out well-spaced, they get closer as the energies grow higher, until they become almost indistinguishable. In a paper to be published in the inaugural edition of MIT's new journal Quantum Information Processing, Calude and Pavlov have shown that a superposition of an infinite number of energy states would allow a quantum computer to do things no classical computer can ever manage-almost like running "forever" in a finite time. This leap means that a quantum computer can overcome Turing's most famous barrier to computing power: the 'halting problem". Given an y computer progr am and an input, can a Turing machine tell in advance whether that program will eventually halt or grind away forever? Turing himself proved the answer is no-the program might stop after a couple of days, or a billion years, but the only way to find out is to run it and wait. That might seem no more than a curiosity, but getting round this barrier would be a gigantic breakthrough for science, solving many important questions in maths and physics.

Stop and search

Goldbach's conjecture, for example, states that every even number is the sum of two primes, and it can be recast as a hatting problem. All you do is write a computer program that searches for a counterexamplethat is, for an even number that is not the sum of two primes. If it finds one, it stops, proving Goldbach's conjecture wrong. If it goes on forever, the conjecture is, apparently, right. A host of other mathematical questions can be written as halting problems-including the Riemann hypothesis, an unproven assumption upon which a great deal of physics and maths rests.

So how would a quantum processor overcome Turing's barrier and see whether these programs halt? Any given program investigating another can do only one thing: run the program under test and then ask the question, "Does it halt within x amount of time?" Because the answer can only be yes or no, the processor might report that the program under test doesn't stop, whereas, in fact, it just hadn't stopped within that time. However long a time interval you allow, if the answer is always no you can still never believe it.

But if you assign different investigating questions to an infinite superposition of quantum states, something remarkable happens. While the program under test runs as normal, the questions in the infinite superposition ask every possible "does it stop" question at once. Calude has used this to uncover a subtle signal in the program under test. This signal is invisible to any classical analysis but shows up under the infinite superposition, and gives a clue as to whether or not the program will halt.

More precisely, it gives a measure of the proportion of misleading answers to the "does it stop" question. Although a non-halting program gives you an infinite number of worthless 'no" answers, Calude and Pavlov have shown that, as the quantum program runs, it can measure the error introduced by these worthless answers. And the longer you run the quantum algorithm, the less significant the error becomes. "This is our key result," Calude says. "It gives insight into the behaviour of non-halting problems no other mathematical result has been able to give."

Calude and Pavlov's algorithm doesn't provide a definite "yes" or "no" to the question of whether a program halts, but it does give an answer accompanied by a percentage certainty. If you want to get a more accurate result, you just run the quantum program for longer.

Calude is extremely proud of this result: he believes it could be implemented on a real-life quantum computer, laying much that is "unknowable" open to attack. "Using infinite superpositions is rather theoretical, but not necessarily non-practical or nontestable," he says.

He is bound to face a great deal of scepticism, of course. Most quantum computing researchers still think Feynman was right; quantum computers remain bound by Turing's barrier. "If you look at the theory of quantum mechanics, everything in there is computable," says Richard jozsa, a quantum algorithms researcher at Bristol University. And by computable, he means that a Turing machine could eventually do it. Jozsa does admit that there's more to the quantum world than we know about just yet, so we can't rule out finding ways past the Turing barrier in future. But he says he hasn't seen any way to do it so far.

Proof positive

Calude counters that such doubts are only a matter of opinion, and don't prove that quantum theory isn't up to the challenge. He thinks Feynman's 20-year-old pronouncement has closed people's minds. 'People were brainwashed by Feynman into thinking it was impossible," he says. Calude says his paper offers a proof that quantum computers can-in theory at least-breakthe Turing barrier.

In practice, however, this may prove enormously difficult. Gregory Chaitin, a mathematician based at IBM's Yorktown Heights laboratory, thinks the signal that hints in advance whether the program will ever halt will be too small for measurements to pick up. "I think they [email protected] being able to measure real numbers with infinite precision, which I don't think is possible," he says.

But Calude believes that such problems are often not insurmountable-and he has reason to be bullish. Last year he managed to do another calculation that Chaitin had thought beyond all hope (see "A glimpse of the impossible", opposite). Similarly, Calude is confident that reading the hidden quantum signal-and thus breaking the Turing barrier-will be possible. And he's not alone. In the past couple of years, several more researchers have begun to ask whether Turing only had half the picture. 'rien Kieu of Swinburne University of Technology in Australia, for example, has also come up with a way for quantum computers to surpass Turing's barrier. Like Calude's method, it exploits the possibility of infinite compu-

tations by encoding the problem in the energy states of, say, an atom or molecule. Others have suggested that black holes or DNA might provide a way to peek into the unknowable (see "To infinity and beyond", left). "It seems the problem's ripe for solving," Calude says.

Testing time

But it's not yet clear whether the theory will translate into hard knowledge. Although Calude is convinced his maths is right, only an experiment can reveal whether his idea is practical. And Kieu believes that the Universe might not be around for long enough to complete the execution of such a quantum algorithm. Even so, Calude thinks that some ftindamental aspect of maths and physics will have to change. "Already we've realised that classically uncomputable is not the same as quantum-mechanically uncomputable,' he says. And that alone might be enough to challenge the nature of mathematical thought about, for instance, what constitutes proof of a theorem. "Because of these new computational models, the idea of 'proof' might - and I personally believe that it will change," he says. "And if the nature of proof changes, the idea of mathematical knowledge will also change." This could mark the return to a more positive mathematical era. In 1930, the German mathematician David Hilbert showed his colleagues that maths would' meet all the challenges that faced it. Then, in the following year, Kurt G6del deflated the optimism with proofs that there were things in the mathematical universe that might be true, but were unprovable. It was Godel's work that enabled Turing to show that his computation machine would not be able to answer certain questions-such as the halting problem. But, thanks to Calude, that definition of "uncomputable' will have to go. Who knows what other 'insurmountable" barriers will crumble to the ground? An audacious assault on the limits of knowledge could reveal and unravel more than we ever thought possible. All we need is a machine that can break the laws of logic. 1-1

Further reading: "Coins, quantum measurements, and Turing's barrier" by Cristian Calude and Boris Pavlov (www.arxiv.org/abs/quant-ph/0112087) "Incompleteness, complexity, randomness and beyond" by Cristian Calude (www.arxiv.org/abs/quant-ph/0111118) "Hitbert's incompleteness, Chaitin's Omega number and quantum physics" by Tiend Kieu (www.arxiv.org/abs/quant-ph/0111062) Cristian Calude's website is at www.cs.auckland.ac.nz/-cristian

Dark forces at work

IS DARK energy blowing the Universe apart? The idea caught on in 1998 after astronomers noticed that light from distant supernovae seemed to be going astray. Critics said intergalactic dust or strange quantum effects might be responsible, but now there's evidence from an independent source.

Astronomers compared the distribution of10,000 nearby galaxies with wrinkles in the cosmic microwave background, which show the distribution of matter in the Universe when it was only 300,000 years old (Monthly Notices of the Royal Astronomical Society, vol 330, p 29). They say the two only match if you assume dark energy is speeding up the Universe's expansion. The next question is what dark energy is made of. "The ball is now firmly in the theorists' court," says ' team leader George Efstathiou of Cambridge University.

A glimpse of the impossible ~

Calude has little respect for "unbreakable" barriers. Last year,

New Scientist published a story about Omega, a bizarre number linked to Turing's proof that there are things computers can't do (10 March 2001, p 28). There was thought to be no way to even begin calculating the random sequence of digits that make up

Omega. But we're now able to publish the first 64 digits (below). Contrary to all expectations' calculating these bits wasn't that hard. The first n digits of Omega represent the probabiiity that a program that's less than n bits long (when translated into binary from whatever programming ianguage is used) will halt. Calude, together with Michael Dinneen and Chi-Kou Shu, also at Auckland University, ran all possible programs that are 1 bit long, then

2 bits, then 3 and so on up to 84 bits, to see whether they halted. They made their task more manageable by weeding out all the

programs that duplicated each other. This reduced the size of the computational task by a factor of more than 1017, but they still had to work with around 3 gigabytes of compressed data.

That didn't give the first 84 digits exactly, however. Calude couldn't ignore the potential that longer programs might affect Omega's first few digits: it's conceivabie that a large set of very long programs could contribute to the values of the first bits of Omega.

The breakthrough Calude's team made was to discover a structure that all halting programs longer than 84 bits must have. They found that, in the finite set of programs the team had computed, these programs all have to start with a particular sequence of bits. This limits the contribution that the remaining infinite set of halting programs can make to the first bits of Omega. "in our case, this set cannot influence the first 69 bits of Omega," says Calude. "However, for technical reasons, only the first 64 are exact." Calude may well have prised open the door to the Universe's deepest secrets.

00000010000001000001100010000110100011111100101110111010000100OO

"Omega's first few thousand digits contain the answers to more mathematical questions than can be written down in the entire Universe," says IBM's Charles Bennett, a pioneer in the field of quantum information. John Casti of the Santa Fe Institute echoes this. "Omega's digits encode the 'secret of the Universe'," he says. "Almost every unsolved problem in mathematics and many in physics and elsewhere could be settled by knowing enough digits of Omega." But Calude doesn't think their method for unraveiling Omega will take them much further. In fact, there's not much hope of significant progress at all, according to Omega's discoverer, Gregory Chaitin of IBM's Yorktown Heights laboratory. He says the fact that Calude could calculate these 64 digits simply shows that there's no classically uncomputable mathematical problem that can be tackled by a program just 64 bits long. Finding more of Omega's digits would take us closer to finding the threshold between computable and uncomputable, though, so it's worth pursuing, Calude says. But until we have a very different kind of computer, the secrets of the Universe will stay hidden. Further reading: "Computing a glimpse of randomness" by Cristian Calude Michael Dineen and Chi-Kou Shu (www.arxiv.org/abs/nlin.cd/0112022