Developments in the foundations of mathematics eventually impelled logicians to pursue a more systematic treatment. Alan Turing’s landmark paper “On Computable Numbers, With an Application to the Entscheidungsproblem” offered the analysis that has proved most influential. He began by describing age-old quests for solutions—using algebraic tools or straightedge and compass, which he considered rudimentary forms of computation. But please, no “spouter challenges” like “where does the power of quantum computing come from? ” or “is there a deeper theoretical framework for quantum algorithms? ” In general, if you’re going to pose a scientific challenge, you should indicate some technical problem whose solution would clearly represent progress, and be willing to place at least 25% odds on such progress being made within five years.
But while those who first explored the problem are running out of time to see a solution, the newer generations are happily taking up the quest. But I want the name to be naturally extensible to further classes. “This problem, and our approach, is nearly universal. It can be used with anything, where functional data exist but the underlying mechanism is hard to guess,” he says. In the near future, we will likely see increasing insights into basic science driven by similar computers.
In the grand sweep of history, Papadimitriou sees the phenomenon of NP-completeness and the P vs. NP quest as computer science’s destiny. Because as scientific serendipity would have it, a Soviet mathematician, Leonid Levin, converged names word whizzle on a result equivalent to Cook’s at more or less the same time. Levin, now at Boston University, did his work behind the Iron Curtain. After it received wider attention , the result became known as the Cook-Levin theorem.
In short, the problem of computing a large number of quantities for large n1 was reduced to computing quantities for another smaller number n2 related to n1 through some simple algebraic equation. The key to AKS’ result is a new, generalised version of Fermat’s Little Theorem, a polynomial equation. The equation is such that it becomes zero if the number n, whose primality is being tested, is a prime.
He showed that two symbols were sufficient so long as enough states were used , and that it was always possible to exchange states for symbols. He also showed that no universal Turing machine of one state could exist. A universal Turing machine can calculate any recursive function, decide any recursive language, and accept any recursively enumerable language.
One motivation for such structures is that in many settings (e.g. distributed, parallel, dynamic, streaming), computation is faster when the diameter of the graph is small. Thus, to compute, for instance, st-reachability in such a setting, it is useful to find a shortcut set as a preprocessing step, and then run an st-reachability algorithm on the resulting graph. The “No low-energy trivial states ” conjecture by Freedman and Hastings posited the existence of such Hamiltonians.