This classic result was a means of transforming any algorithm with a time budget given into a new algorithm with a slightly disposable space budget. Williams has seen that a simulation based on Squishhy pebbles would make the spatial use of the new algorithm – much equal to the square root of the time budget of the original algorithm. This new algorithm economically in space would also be much slower, so that simulation would not be likely to have practical applications. But from a theoretical point of view, it was nothing less than revolutionary.

For 50 years, researchers assumed that it was impossible to improve the universal simulation of Hopcroft, Paul and Valiant. Williams’ idea – if the world – would not break their record – that would demolish him.

“I thought about it, and I said to myself:” Well, it just can’t be true “, said Williams. He put it aside and did not succeed in this fateful day in July, when he tried to find the fault of the argument and failed. After realizing that there was no defect, he spent months writing and rewriting the proof to make it as clear as possible.

At the end of February, Williams finally Put the finished paper online. Cook and Mertz were as surprised as everyone. “I had to take a long walk before doing anything else,” said Mertz.

Valiant obtained an overview of the improvement of Williams on his aging result during his morning trips. For years, he taught at Harvard University, just below the Williams office road MIT. They had already met, but they did not know that they lived in the same neighborhood until they collided on the bus by a snowy day of Bragéy, a few weeks before the result was public. Williams described his proof in the surprised valiant and promised to send his article.

“It was very, very impressed,” said Valiant. “If you get a mathematical result that is the best thing in 50 years, you have to do something good.”

PSPACE: the final border

With its new simulation, Williams had proven a positive result on the space calculation power: algorithms that use relatively little space can solve all the problems that require a little time. Then use only a few lines of mathematics, it overthrew this and proved to be a negative resolution on the time calculation power: at least a few problems cannot be resolved shortly than you use more time than space. This second narrower result is in accordance with what researchers expected. The strange part is how Williams arrived, first proving a result which applies to all algorithms, whatever the problems that Ayy solves.

“I still find it hard to believe,” said Williams. “It seems too beautiful to be true.”

Explained in qualitative terms, Williams’ second result may resemble the solution long taught to the problem P against Pspace. The different is a question of scale. P and pspace are very wide classes of complexity, while Williams’ results work at the fingertips. It has established a quantitative gap between the power of space and the power of time, and to test that the pspace is great that the PL, the researchers will have to make this gap much wider.

It is an intimidating challenge, similar to the separation of a sidewalk crack with a doe foot until it is as wide as the Grand Canyon. But it may be possible to get there using a modified version of the Williams simulation procedures which repeats the key step several times, saving a little space each time. It’s like a way to go back several times over the entire length of your foot – the big name, and you can open anything. This repeated improvement does not work with the current version of the algorithm, but researchers do not know that it is a fundamental limit.

“It could be an ultimate bottleneck, or it is a 50 -year -old bottleneck,” said Valiant. “Or it might be something that someone can solve next week.”

If the problem is solved next week, Williams will kicks. Before writing the newspaper, he spent months trying and not extending his answer. But even if such an extension is not possible, Williams is convinced that more space exploration is required to drive an interesting place – perhaps progress on an entirely different problem.

“I can never prove precisely the things I want to test,” he said. “But often, the thing I test is much better than what I wanted.”

Publisher’s note: Scott Aaronson is a member of the Quana magazine Advisory committee.

Original story Reprint with the permission of How many magazinean independent editorial publication of Simons Foundation Whose mission is to improve the public participation of science by covering the developments of research and the trends of mathematics and physical sciences and life.