New Proof Dramatically Compresses Space Needed for Computation
New proof considerably compresses the space necessary for the calculation
Surprising new work 50 years of assumptions on compromises between the calculation space and the time
Once upon a time, computers filled whole parts, reading the strip spinning numbers and by baring them through wires to make basic arithmetic chains. Today, they slip into our pockets, occurring in a small fraction of a second which was taking hours. But even if the chips shrink and gain speed, theorists return the question of the quantity of calculation space that we can wrap in a machine so that few things are sufficient to do the work.
This survey is at the heart of computer complexity, a measure of the limits of the problems that can be resolved and at what cost in time and space. For almost 50 years, theorists thought that if the resolution of a problem takes t steps, it should also roughly require t Memory bits – 0 and 1 that a machine uses to save information. (Technically, this equation was t /save(t), but for the Journal of numbers involved (t) is generally negligible.) If a task involves 100 steps, for example, you expect to need at least 100 bits, enough to register with diligence each step. The use of fewer bits was supposed to require more steps, like literacy by exchanging them by exchanging them by one on the shelf instead of removing them all and going up them. But in a surprising conclusion described this week at the ACM symposium on computer theory in Prague, Massachusetts Institute of Technology Ryan Williams IT found that any problem solved over time t only needs about √t Memory bits: A 100 -step calculation can be compressed and resolved with something around 10 bits. “This result shows that anterior intuition is completely false,” says Williams. “I thought there must be something wrong [with the proof] Because it is extremely unexpected.”
The breakthrough is based on a “reduction”, a means of transforming a problem into another which may seem unrelated but which is mathematically equivalent. With discounts, the packaging of a cartographic suitcase on the determination of a monthly budget: the size of your suitcase represents your total budget, the clothes correspond to potential expenses and carefully decide which clothes can adapt, it is like allocating your budget. The resolution of a problem would then directly solve the other. This idea is at the heart of Williams’ result: any problem can be transformed into the one you can solve by intelligently reusing the space, skillfully entering the necessary information into a number of square bits. Thus, the original problem must be resolved with this compact container.
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“This progress is incredible,” says Mahdi Cheraghchi, a computer scientist at the University of Michigan. “Before this result, there were problems that you could solve in a certain time, but many thought that you could not do it with such a small space.” Williams’ observation, he adds, is “a step in the right direction that we did not know how to take”.
While computers have continued to shrink, our theoretical understanding of their effectiveness has exploded, suggesting that real constraint is not the amount of memory we have, but how we use it wisely.
