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In 1971, Stephen Cook published his paper "The complexity of theorem proving procedures" [2] in conference proceedings of the newly founded ACM Symposium on Theory of Computing. Richard Karp's subsequent paper, "Reducibility among combinatorial problems", [1] generated renewed interest in Cook's paper by providing a list of 21 NP-complete problems.
Kodaira vanishing theorem (complex manifold) Koebe 1/4 theorem (complex analysis) Kolmogorov extension theorem (stochastic processes) Kolmogorov's three-series theorem (mathematical series) Kolmogorov–Arnold representation theorem (real analysis, approximation theory) Kolmogorov–Arnold–Moser theorem (dynamical systems) Kőnig's theorem ...
The concept of NP-completeness was introduced in 1971 (see Cook–Levin theorem), though the term NP-complete was introduced later. At the 1971 STOC conference, there was a fierce debate between the computer scientists about whether NP-complete problems could be solved in polynomial time on a deterministic Turing machine .
Cook proved that Rule 110 was universal (or Turing complete) by showing it was possible to use the rule to emulate another computational model, the cyclic tag system, which is known to be universal. He first isolated a number of spaceships , self-perpetuating localized patterns, that could be constructed on an infinitely repeating pattern in a ...
This theorem was proven independently by Leonid Levin in the Soviet Union, and has thus been given the name the Cook–Levin theorem. The paper also formulated the most famous problem in computer science, the P vs. NP problem. Informally, the "P vs. NP" question asks whether every optimization problem whose answers can be efficiently verified ...
This NP-completeness theorem, often called the Cook–Levin theorem, was a basis for one of the seven Millennium Prize Problems declared by the Clay Mathematics Institute with a $1,000,000 prize offered. The Cook–Levin theorem was a breakthrough in computer science and an important step in the development of the theory of computational ...
For example, just as counting cannot be done by an circuit family of subexponential size, many tautologies relating to the pigeonhole principle cannot have subexponential proofs in a proof system based on bounded-depth formulas (and in particular, not by resolution-based systems, since they rely solely on depth 1 formulas).
Construction of the Malfatti circles: [3] For a given triangle determine three circles, which touch each other and two sides of the triangle each. Spherical version of Malfatti's problem: [4] The triangle is a spherical one. Essential tools for investigations on circles are the radical axis of two circles and the radical center of three circles.