Proof by self-checking software: the four colour problem

Colouring in maps

Computer scientists at Microsoft Research Cambridge (MSRC) have developed a self-checking software-based proof of the four colour theorem that they say sweeps away any remaining uncertainty surrounding earlier proofs.

MRSC researcher Georges Gonthier and Benjamin Werner, a researcher at INRIA, have devised a computer program that verifies the correctness of its own calculations, and applied it to the four colour problem. It constructs a precise mathematical proof and checks that it follows the strict rules of formal logic.

The theorem states that any geographical map may be drawn so that no two contiguous regions are filled in the same colour, provided a palette of four colours is available. It was proven, rather controversially, in 1976. As well as using formal logic, a pair of mathematicians from the University of Illinois, Kenneth Appel and Wolfgang Haken, used a computer to check the tenets of the theorem by brute force.

The controversy arose because the section of the proof that was worked on computer was too long for another mathematician to check. Although most people in the community were prepared to accept the proof, a lingering uncertainty bothered some - what if there had been an error in the software? Gonthier and Werner's work aims to remove that last doubt.

Andrew Herbert, managing director at MSRC said that the discovery had great implications for the future of computing. "Advances we make into self-checking software have the potential to be incorporated into software development tools to make computing in general more reliable and trustworthy," he said.

For more on the history of the Four Colour Theorem, click here. ®

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