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S2P (complexity)
In computational complexity theory, S is a complexity class, intermediate between the first and second levels of the polynomial hierarchy. A language is in \mathsf S_2^P if there exists a polynomial-time predicate ''P'' such that * If x \in L, then there exists a ''y'' such that for all ''z'', P(x,y,z)=1, * If x \notin L, then there exists a ''z'' such that for all ''y'', P(x,y,z)=0, where size of ''y'' and ''z'' must be polynomial of ''x''. Relationship to other complexity classes It is immediate from the definition that S is closed under unions, intersections, and complements. Comparing the definition with that of \Sigma_^P and \Pi_^P, it also follows immediately that S is contained in \Sigma_^P \cap \Pi_^P. This inclusion can in fact be strengthened to ZPPNP.. A preliminary version of this paper appeared earlier, in FOCS 2001, , , . Every language in NP also belongs to For by definition, a language ''L'' is in NP, if and only if there exists a polynomial-time verifier ''V'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, 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. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of computationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexity Class
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 general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time or memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements. For instance, the class P is the set of decision problems solvable by a deterministic Turing machine in polynomial time. There are, however, many complexity classes defined in terms of other types of problems (e.g. counting problems and function problems) and using other models of computation (e.g. probabilistic Turing machines, interactive proof systems, Boolean circuits, and quantum computers). The study of the relationships between complexity classes is a ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Hierarchy
In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines. It is a resource-bounded counterpart to the arithmetical hierarchy and analytical hierarchy from mathematical logic. The union of the classes in the hierarchy is denoted PH. Classes within the hierarchy have complete problems (with respect to polynomial-time reductions) which ask if quantified Boolean formulae hold, for formulae with restrictions on the quantifier order. It is known that equality between classes on the same level or consecutive levels in the hierarchy would imply a "collapse" of the hierarchy to that level. Definitions There are multiple equivalent definitions of the classes of the polynomial hierarchy. Oracle definition For the oracle def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ZPP (complexity)
In complexity theory, ZPP (zero-error probabilistic polynomial time) is the complexity class of problems for which a probabilistic Turing machine exists with these properties: * It always returns the correct YES or NO answer. * The running time is polynomial in expectation for every input. In other words, if the algorithm is allowed to flip a truly-random coin while it is running, it will always return the correct answer and, for a problem of size ''n'', there is some polynomial ''p''(''n'') such that the average running time will be less than ''p''(''n''), even though it might occasionally be much longer. Such an algorithm is called a Las Vegas algorithm. Alternatively, ZPP can be defined as the class of problems for which a probabilistic Turing machine exists with these properties: * It always runs in polynomial time. * It returns an answer YES, NO or DO NOT KNOW. * The answer is always either DO NOT KNOW or the correct answer. * It returns DO NOT KNOW with probability at mos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symposium On Foundations Of Computer Science
The IEEE Annual Symposium on Foundations of Computer Science (FOCS) is an academic conference in the field of theoretical computer science. FOCS is sponsored by the IEEE Computer Society. As writes, FOCS and its annual Association for Computing Machinery counterpart STOC (the Symposium on Theory of Computing) are considered the two top conferences in theoretical computer science, considered broadly: they “are forums for some of the best work throughout theory of computing that promote breadth among theory of computing researchers and help to keep the community together.” includes regular attendance at FOCS and STOC as one of several defining characteristics of theoretical computer scientists. Awards The Knuth Prize for outstanding contributions to theoretical computer science is presented alternately at FOCS and STOC. Works of the highest quality presented at the conference are awarded the Best Paper Award. In addition, the Machtey Award is presented to the best student- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP (complexity)
In computational complexity theory, NP (nondeterministic polynomial time) 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 deterministic Turing machine, or alternatively the set of problems that can be solved in polynomial time by a nondeterministic Turing machine.''Polynomial time'' refers to how quickly the number of operations needed by an algorithm, relative to the size of the problem, grows. It is therefore a measure of efficiency of an algorithm. An equivalent definition of NP is the set of decision problems ''solvable'' in polynomial time by a nondeterministic Turing machine. This definition is the basis for the abbreviation NP; " nondeterministic, polynomial time". These two definitions are equivalent because the algorithm based on the Turing machine consists of two phases, the first of which consists of a guess abou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Co-NP
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 is in the complexity class NP. The class can be defined as follows: a decision problem is in co-NP precisely if only ''no''-instances have a polynomial-length " certificate" and there is a polynomial-time algorithm that can be used to verify any purported certificate. That is, co-NP is the set of decision problems where there exists a polynomial ''p(n)'' and a polynomial-time bounded Turing machine ''M'' such that for every instance ''x'', ''x'' is a ''no''-instance if and only if: for some possible certificate ''c'' of length bounded by ''p(n)'', the Turing machine ''M'' accepts the pair (''x'', ''c''). Complementary Problems While an NP problem asks whether a given instance is a ''yes''-instance, its ''complement'' asks whether an instance is a ''no''-instance, which means the complement is in co-NP. Any ''yes''-instance for the original NP ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sipser–Lautemann Theorem
In computational complexity theory, the Sipser–Lautemann theorem or Sipser–Gács–Lautemann theorem states that bounded-error probabilistic polynomial (BPP) time is contained in the Polynomial hierarchy, polynomial time hierarchy, and more specifically Σ2 ∩ Π2. In 1983, Michael Sipser showed that BPP is contained in the Polynomial hierarchy, polynomial time hierarchy. Péter Gács showed that BPP is actually contained in Σ2 ∩ Π2. Clemens Lautemann contributed by giving a simple proof of BPP’s membership in Σ2 ∩ Π2, also in 1983. It is conjectured that in fact BPP=P (complexity), P, which is a much stronger statement than the Sipser–Lautemann theorem. Proof Here we present the Lautemann's proof. Without loss of generality, a machine ''M'' ∈ BPP with error ≤ 2−, ''x'', can be chosen. (All BPP problems can be amplified to reduce the error probability exponentially.) The basic idea of the proof is to define a Σ2 sentence that is equivalent to stating that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karp–Lipton Theorem
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 :\Pi_2 = \Sigma_2 \, and therefore \mathrm = \Sigma_2. \, That is, if we assume that NP, the class of nondeterministic polynomial time problems, can be contained in the non-uniform polynomial time complexity class P/poly, then this assumption implies the collapse of the polynomial hierarchy at its second level. Such a collapse is believed unlikely, so the theorem is generally viewed by complexity theorists as evidence for the nonexistence of polynomial size circuits for SAT or for other NP-complete problems. A proof that such circuits do not exist would imply that P ≠ NP. As P/poly contains all problems solvable in randomized polynomial time (Adleman's theorem), the theorem is also evidence that the use of randomization does not lead to polynomial time algorithms for NP-complete problems. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P/poly
In computational complexity theory, P/poly is a complexity class representing problems that can be solved by small circuits. More precisely, it is the set of formal languages that have polynomial-size circuit families. It can also be defined equivalently in terms of Turing machines with advice, extra information supplied to the Turing machine along with its input, that may depend on the input length but not on the input itself. In this formulation, P/poly is the class of decision problems that can be solved by a polynomial-time Turing machine with advice strings of length polynomial in the input size. These two different definitions make P/poly central to circuit complexity and non-uniform complexity. For example, the popular Miller–Rabin primality test can be formulated as a P/poly algorithm: the "advice" is a list of candidate values to test. It is possible to precompute a list of O(n) values such that every composite n-bit number will be certain to have a witness a in the list ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Time Hierarchy
In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines. It is a resource-bounded counterpart to the arithmetical hierarchy and analytical hierarchy from mathematical logic. The union of the classes in the hierarchy is denoted PH. Classes within the hierarchy have complete problems (with respect to polynomial-time reductions) which ask if quantified Boolean formulae hold, for formulae with restrictions on the quantifier order. It is known that equality between classes on the same level or consecutive levels in the hierarchy would imply a "collapse" of the hierarchy to that level. Definitions There are multiple equivalent definitions of the classes of the polynomial hierarchy. Oracle definition For the oracle def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
