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10/30/07

20-year UnderGraduate Proved Simplest Universal Turing Machine


Stephen Wolfram is the author of the book "A New Kind Of Science", and the founder of the software "Mathematica". He postulated in his book, that the 2,3 Turing Machine is universal.
This has now been proved by 20-year Alex Smith, an undergraduate from Birmingham university. He thereby won a prize of $ 25,000 set by Wolfram.

The idea of a universal machine originated from Alan Turing in 1936. He defined a machine now called Turing machine, capable of computing anything that is computable at all. The machine is only conceptual, not a hardware. It has an infinite 1-dimensional tape, and a read-write head which can also move left or right.

Digital computers are also universal in a sense that it can be programmed to compute any computable function. The difference with a Turing machine is that a Turing machine has infinite storage (tape), whereas digital computers are finite, so digital computers are not really universal.

People have asked what is the simplest possible Turing machine. A 7,4 machine is found to be universal. The numbers 7,4 refer to the number of states of the head (7) and the colors of the tape (4).
Wolfram later discovered a smaller 2,5 machine, and postulated that a 2,3 machine is universal, which according a 40-page proof by Alex Smith, indeed is.

This machine has only 2 states and 3 colors, and it is amazing that such a simple machine can compute (in theory) complex differential equations, make weather predictions and financial calculations. In theory of course, since the Turing machine is not known to be (time) efficient. Still it is more powerful than computers with highly dense electronics.

So, what is the practical application of the new discovery? Wolfram said "Perhaps one day there'll even be practical molecular computers built from this very 2,3 Turing machine.
With tapes a bit like RNA strands, and heads moving up and down like ribosomes.
When we think of nanoscale computers, we usually imagine carefully engineering them to mimic the architecture of the computers we know today."

Wolfram thinks the philosophical implications are also important, "that above a very low threshold, all systems will be exactly equivalent in their computational capabilities."

Background information:

1 komentar:

MD said...

"It might be wrong". See "Simplest Turing machine proof takes a Fermat's-last-theorem turn"
http://science-community.sciam.com/thread.jspa?threadID=300004284