Буледік қанағаттандыру және NP-толықтық туралы теоремалар

Boolean satisfiability is NP complete and therefore that NP complete problems exist
In computational complexity theory, the Cook–Levin theorem, also known as Cook's theorem, states that the Boolean satisfiability problem is NP complete. That is, it is in NP, and any problem in NP can be reduced in polynomial time by a deterministic Turing machine to the Boolean satisfiability problem. The theorem is named after Stephen Cook and Leonid Levin. The proof is due to Richard Karp, based on an earlier proof (using a different notion of reducibility) by Cook. in conference proceedings of the newly founded ACM Symposium on Theory of Computing. Richard Karp's subsequent paper, "Reducibility among
combinatorial problems",

>
The theoretical interest in NP completeness was also enhanced by the work of Theodore P. Baker, John Gill, and Robert Solovay who showed, in 1975, that solving NP problems in certain oracle machine models requires exponential time. That is, there exists an oracle A such that, for all subexponential deterministic time complexity classes T, the relativized complexity class NPA is not a subset of TA. In particular, for this oracle, PA ≠ NPA. In the USSR, a result equivalent to Baker, Gill, and Solovay's was published in 1969 by M. Dekhtiar. Later Leonid Levin's paper, "Universal search problems", was published in 1973, although it was mentioned in talks and submitted for publication a few years earlier. Levin's approach was slightly different from Cook's and Karp's in that he considered search problems, which require finding solutions rather than simply determining existence. He provided six such NP complete search problems, or universal problems. Additionally he found for each of these problems an algorithm that solves it in optimal time (in particular, these algorithms run in polynomial time if and only if P = NP).

A decision problem is in NP if it can be decided by a non deterministic Turing machine in polynomial time. An instance of the Boolean satisfiability problem is a Boolean expression that combines Boolean variables using Boolean operators. Such an expression is satisfiable if there is some assignment of truth values to the variables that makes the entire expression true.

Given any decision problem in NP, construct a non deterministic machine that solves it in polynomial time. Then for each input to that machine, build a Boolean expression that computes whether when that specific input is passed to the machine, the machine runs correctly, and the machine halts and answers "yes". Then the expression can be satisfied if and only if there is a way for the machine to run correctly and answer "yes", so the satisfiability of the constructed expression is equivalent to asking whether or not the machine will answer "yes".

While the above method encodes a non deterministic Turing machine in complexity , the literature describes more sophisticated approaches in complexity The quasilinear result first appeared seven years after Cook's original publication. The use of SAT to prove the existence of an NP complete problem can be extended to other computational problems in logic, and to completeness for other complexity classes. The quantified Boolean formula problem (QBF) involves Boolean formulas extended to include nested universal quantifiers and existential quantifiers for its variables. The QBF problem can be used to encode computation with a Turing machine limited to polynomial space complexity, proving that there exists a problem (the recognition of true quantified Boolean formulas) that is PSPACE complete. Analogously, dependency quantified boolean formulas encode computation with a Turing machine limited to logarithmic space complexity, proving that there exists a problem that is NL complete.

The proof shows that every problem in NP can be reduced in polynomial time (in fact, logarithmic space suffices) to an instance of the Boolean satisfiability problem. This means that if the Boolean satisfiability problem could be solved in polynomial time by a deterministic Turing machine, then all problems in NP could be solved in polynomial time, and so the complexity class NP would be equal to the complexity class P.

The significance of NP completeness was made clear by the publication in 1972 of Richard Karp's landmark paper, "Reducibility among combinatorial problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its intractability, are NP complete. Karp showed each of his problems to be NP complete by reducing another problem (already shown to be NP complete) to that problem. For example, he showed the problem 3SAT (the Boolean satisfiability problem for expressions in conjunctive normal form (CNF) with exactly three variables or negations of variables per clause) to be NP complete by showing how to reduce (in polynomial time) any instance of SAT to an equivalent instance of 3SAT. Garey and Johnson presented more than 300 NP complete problems in their book Computers and Intractability: A Guide to the Theory of NP Completeness, and new problems are still being discovered to be within that complexity class. Although many practical instances of SAT can be solved by heuristic methods, the question of whether there is a deterministic polynomial time algorithm for SAT (and consequently all other NP complete problems) is still a famous unsolved problem, despite decades of intense effort by complexity theorists, mathematical logicians, and others. For more details, see the article P versus NP problem.