Machines That Think by Toby Walsh

Author:Toby Walsh
Language: eng
Format: mobi, epub
ISBN: 9781633883765
Publisher: Prometheus Books
Published: 2018-01-23T07:00:00+00:00

THE “COMPUTATIONAL COMPLEXITY” ARGUMENT

Another argument against the singularity comes from computational complexity. Humans are very poor at understanding exponentials. Many of us underestimate the effects of compound growth. But, equally, more of us overestimate the power of exponential growth. There is an idea that exponential improvements are adequate to crack any problem. This is a misconception.

Computational complexity is the branch of computer science that looks at how fast we can compute answers to problems. Some computational problems are easy. We can, for instance, sort even a long list of names into alphabetical order quickly. In fact, the time for an optimal algorithm to sort a list of n names grows faster than n, the size of the list, but slower than n2, the square of the size of the list. What does this mean in practice? It means that if we double the size of the list of names we are sorting, it takes more than twice as long but less than four times as long to compute a sorted list (since 22 or 2×2 is 4). On the other hand, if we triple the size of the list of names we are sorting, it takes more than three times as long but less than nine times as long to compute a sorted list (since 32 or 3×3 is 9). There are other computational problems that are more challenging. For example, we can multiply two n by n matrices of numbers in a time that grows faster than n2 but slower than n3. In practice, this means that if we quadruple the size of the matrices we are multiplying, the time to multiply them together increases by more than by a factor of 16 (which is 4×4 or 42), but by less than a factor of 64 (which is 4×4×4 or 43).

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