Titlebook: Gems of Theoretical Computer Science; Uwe Sch?ning,Randall Pruim Book 1998 Springer-Verlag Berlin Heidelberg 1998 Kolmogorov complexity.Re

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https://doi.org/10.1007/978-1-4757-2696-1In their pioneering work of 1984, Furst, Saxe and Sipser introduced the method of “random restrictions” to achieve lower bounds for circuits: The parity function cannot be computed by an AND-OR circuit of polynomial size and constant depth.

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The Meaning of the Constitutive Equation,The lower bound theory for circuits received an additional boost through algebraic techniques (in combination with probabilistic techniques) that go back to Razborov and Smolensky.

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https://doi.org/10.1007/978-1-4757-2257-4If all NP-complete languages were P-isomorphic to each other, then it would follow that P ≠ NP. This “Isomorphism Conjecture” has been the starting point of much research, in particular into sparse sets and their potential to be NP-complete.

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https://doi.org/10.1007/978-1-4471-3774-0The following results suggest that the Graph Isomorphism problem is not NP-complete, since, unlike the known NP-complete problems, Graph Isomorphism belongs to a class that can be defined by means of the BPoperator, an operator that has proven useful in many other applications as well.

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,Hilbert’s Tenth Problem,Hilbert’s Tenth Problem goes back to the year 1900 and concerns a fundamental question, namely whether there is an algorithmic method for solving Diophantine equations. The ultimate solution to this problem was not achieved until 1970. The “solution” wets, however, a negative one: there is no such algorithm.

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