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Arthur–Merlin Protocol
In computational complexity theory, an Arthur–Merlin protocol, introduced by , is an interactive proof system in which the verifier's coin tosses are constrained to be public (i.e. known to the prover too). proved that all (formal) languages with interactive proofs of arbitrary length with private coins also have interactive proofs with public coins. Given two participants in the protocol called Arthur and Merlin respectively, the basic assumption is that Arthur is a standard computer (or verifier) equipped with a random number generating device, while Merlin is effectively an oracle with infinite computational power (also known as a prover). However, Merlin is not necessarily honest, so Arthur must analyze the information provided by Merlin in response to Arthur's queries and decide the problem itself. A problem is considered to be solvable by this protocol if whenever the answer is "yes", Merlin has some series of responses which will cause Arthur to accept at least of t ... [...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 explores the relationships between these classifications. 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 logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Extended Riemann Hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these ''L''-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the algebraic function field case (not the number field case). Global ''L''-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothe ... [...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) that 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Knowledge Proof
In cryptography, a zero-knowledge proof (also known as a ZK proof or ZKP) is a protocol in which one party (the prover) can convince another party (the verifier) that some given statement is true, without conveying to the verifier any information ''beyond'' the mere fact of that statement's truth. The intuition underlying zero-knowledge proofs is that it is trivial to prove possession of the relevant information simply by revealing it; the hard part is to prove this possession without revealing this information (or any aspect of it whatsoever). In light of the fact that one should be able to generate a proof of some statement ''only'' when in possession of certain secret information connected to the statement, the verifier, even after having become convinced of the statement's truth, should nonetheless remain unable to prove the statement to further third parties. Zero-knowledge proofs can be interactive, meaning that the prover and verifier exchange messages according to some pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Isomorphism Problem
The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. The problem is not known to be solvable in polynomial time nor to be NP-complete, and therefore may be in the computational complexity class NP-intermediate. It is known that the graph isomorphism problem is in the low hierarchy of class NP, which implies that it is not NP-complete unless the polynomial time hierarchy collapses to its second level. At the same time, isomorphism for many special classes of graphs can be solved in polynomial time, and in practice graph isomorphism can often be solved efficiently. This problem is a special case of the subgraph isomorphism problem, which asks whether a given graph ''G'' contains a subgraph that is isomorphic to another given graph ''H''; this problem is known to be NP-complete. It is also known to be a special case of the non-abelian hidden subgroup problem over the symmetric group. In the area of image r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PP (complexity)
In complexity theory, PP, or PPT is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of less than 1/2 for all instances. The abbreviation PP refers to probabilistic polynomial time. The complexity class was defined by Gill in 1977. If a decision problem is in PP, then there is an algorithm running in polynomial time that is allowed to make random decisions, such that it returns the correct answer with chance higher than 1/2. In more practical terms, it is the class of problems that can be solved to any fixed degree of accuracy by running a randomized, polynomial-time algorithm a sufficient (but bounded) number of times. Turing machines that are polynomially-bound and probabilistic are characterized as PPT, which stands for probabilistic polynomial-time machines. This characterization of Turing machines does not require a bounded error probability. Hence, PP is the complexity class containing all problems s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P/poly
In computational complexity theory, P/poly is a complexity class that can be defined in both circuit complexity and non-uniform complexity. Since the two definitions are equivalent, this concept bridges the two areas. In the perspective of circuit complexity, P/poly is the class of problems that can be solved by small circuits. More precisely, it is the set of formal languages that have polynomial-size circuit families. In the perspective of non-uniform complexity, P/poly is defined 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. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advice (complexity)
In computational complexity theory, an advice string is an extra input to a Turing machine that is allowed to depend on the length ''n'' of the input, but not on the input itself. A decision problem is in the complexity class P/''f''(''n'') if there is a polynomial time Turing machine ''M'' with the following property: for any ''n'', there is an advice string ''A'' of length ''f''(''n'') such that, for any input ''x'' of length ''n'', the machine ''M'' correctly decides the problem on the input ''x'', given ''x'' and ''A''. The most common complexity class involving advice is P/poly where advice length ''f''(''n'') can be any polynomial in ''n''. P/poly is equal to the class of decision problems such that, for every ''n'', there exists a polynomial size Boolean circuit correctly deciding the problem on all inputs of length ''n''. One direction of the equivalence is easy to see. If, for every ''n'', there is a polynomial size Boolean circuit ''A''(''n'') deciding the problem, we ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP/poly
In computational complexity theory, NP/poly is a complexity class, a non-uniform analogue of the class NP of problems solvable in polynomial time by a non-deterministic Turing machine. It is the non-deterministic complexity class corresponding to the deterministic class P/poly. Definition NP/poly is defined as the class of problems solvable in polynomial time by a non-deterministic Turing machine that has access to a polynomial-bounded advice function. It may equivalently be defined as the class of problems such that, for each instance size n, there is a Boolean circuit of size polynomial in n that implements a verifier for the problem. That is, the circuit computes a function f(x,y) such that an input x of length n is a yes-instance for the problem if and only if there exists y for which f(x,y) is true. Applications NP/poly is used in a variation of Mahaney's theorem on the non-existence of sparse NP-complete languages. Mahaney's theorem itself states that the number of yes-ins ... [...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 time hierarchy, and more specifically Σ2 ∩ Π2. In 1983, Michael Sipser showed that BPP is contained in the 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, which is a much stronger statement than the Sipser–Lautemann theorem. Proof Here we present the proof by Lautemann. 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 ''x'' is in the language, ''L'', defined by ''M'' by us ... [...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) that 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
