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In computational complexity theory, bounded-error probabilistic polynomial time (BPP) is the class of decision problems solvable by a probabilistic Turing machine in polynomial time with an error probability bounded away from 1/3 for all instances. BPP is one of the largest practical classes of problems, meaning most problems of interest in BPP have efficient probabilistic algorithms that can be run quickly on real modern machines. BPP also contains P, the class of problems solvable in polynomial time with a deterministic machine, since a deterministic machine is a special case of a probabilistic machine.
In logic and computer science, the Boolean satisfiability problem is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make = TRUE. In contrast, "a AND NOT a" is unsatisfiable.
The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified can also be solved quickly.
In computational complexity theory, co-NP is a complexity class. A decision problem X is a member of co-NP if and only if its complement X is in the complexity class NP. The class can be defined as follows: a decision problem is in co-NP precisely if for any no-instance x there is a "certificate" which a polynomial-time algorithm can use to verify that x is a no-instance.
Computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm.
In computational complexity theory, NP is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer is "yes", have proofs verifiable in polynomial time by a non deterministic Turing machine.
In computational complexity theory, the complexity class #P is the set of the counting problems associated with the decision problems in the set NP. More formally, #P is the class of function problems of the form "compute f(x)", where f is the number of accepting paths of a nondeterministic Turing machine running in polynomial time. Unlike most well-known complexity classes, it is not a class of decision problems but a class of function problems. The most difficult, representative problems of this class are #P-complete.
In computational complexity theory, NP-hardness is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard problem is the subset sum problem.
In computational complexity theory, a decision problem is P-complete if it is in P and every problem in P can be reduced to it by an appropriate reduction.
In computational complexity theory, a decision problem is PSPACE-complete if it can be solved using an amount of memory that is polynomial in the input length and if every other problem that can be solved in polynomial space can be transformed to it in polynomial time. The problems that are PSPACE-complete can be thought of as the hardest problems in PSPACE, because a solution to any one such problem could easily be used to solve any other problem in PSPACE.
In complexity theory, UP is the complexity class of decision problems solvable in polynomial time on an unambiguous Turing machine with at most one accepting path for each input. UP contains P and is contained in NP.
In computer science, 2-satisfiability, 2-SAT or just 2SAT is a computational problem of assigning values to variables, each of which has two possible values, in order to satisfy a system of constraints on pairs of variables. It is a special case of the general Boolean satisfiability problem, which can involve constraints on more than two variables, and of constraint satisfaction problems, which can allow more than two choices for the value of each variable. But in contrast to those more general problems, which are NP-complete, 2-satisfiability can be solved in polynomial time.
In computational complexity theory, a complexity class is a set of computational problems of related resource-based complexity. The two most commonly analyzed resources are time and memory.
In computational complexity theory, P, also known as PTIME or DTIME(nO ), is a fundamental complexity class. It contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time.
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.
In computational complexity theory, a function problem is a computational problem where a single output is expected for every input, but the output is more complex than that of a decision problem. For function problems, the output is not simply 'yes' or 'no'.
In complexity theory, the Karp–Lipton theorem states that if the Boolean satisfiability problem (SAT) can be solved by Boolean circuits with a polynomial number of logic gates, then
In computational complexity theory, the language TQBF is a formal language consisting of the true quantified Boolean formulas. A (fully) quantified Boolean formula is a formula in quantified propositional logic where every variable is quantified, using either existential or universal quantifiers, at the beginning of the sentence. Such a formula is equivalent to either true or false. If such a formula evaluates to true, then that formula is in the language TQBF. It is also known as QSAT.
In computational complexity theory, a problem is NP-complete when:
- a brute-force search algorithm can solve it, and the correctness of each solution can be verified quickly, and
- the problem can be used to simulate any other problem with similar solvability.
In computational complexity theory, the exponential time hypothesis is an unproven computational hardness assumption that was formulated by Impagliazzo & Paturi (1999). The hypothesis states that 3-SAT cannot be solved in subexponential time in the worst case. The exponential time hypothesis, if true, would imply that P ≠ NP, but it is a stronger statement. It can be used to show that many computational problems are equivalent in complexity, in the sense that if one of them has a subexponential time algorithm then they all do.
References
- ↑ Fortune, S. (1979). "A Note on Sparse Complete Sets" (PDF). SIAM Journal on Computing. 8 (3): 431–433. doi:10.1137/0208034. hdl: 1813/7473 .
- 1 2 Arora, Sanjeev; Barak, Boaz (2009). Complexity Theory: A Modern Approach. Cambridge University Press. ISBN 978-0-521-42426-4.