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Circuits Over Sets Of Natural Numbers
Circuits over natural numbers are a mathematical model used in studying computational complexity theory. They are a special case of circuits. The object is a labeled directed acyclic graph the nodes of which evaluate to sets of natural numbers, the leaves are finite sets, and the gates are set operations or arithmetic operations. As an algorithmic problem, the problem is to find if a given natural number is an element of the output node or if two circuits compute the same set. Decidability is still an open question. Formal definition A natural number circuit is a circuit, i.e. a labelled directed acyclic graph of in-degree at most 2. The nodes of in-degree 0, the leaves, are finite sets of natural numbers, the labels of the nodes of in-degree 1 are −, where \overline=\ and the labels of the nodes of in-degree 2 are +, ×, ∪ and ∩, where A+B=\, A\times B=\ and ∪ and ∩ with the usual set meaning. The subset of circuits which do not use all of the possible lab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circuit (computer Theory)
In theoretical computer science, a circuit is a model of computation in which input values proceed through a sequence of gates, each of which computes a function. Circuits of this kind provide a generalization of Boolean circuits and a mathematical model for digital logic circuits. Circuits are defined by the gates they contain and the values the gates can produce. For example, the values in a Boolean circuit are boolean values, and the circuit includes conjunction, disjunction, and negation gates. The values in an integer circuit are sets of integers and the gates compute set union, set intersection, and set complement, as well as the arithmetic operations addition and multiplication. Formal definition A circuit is a triple (M, L, G), where * M is a set of values, * L is a set of gate labels, each of which is a function from M^ to M for some non-negative integer i (where i represents the number of inputs to the gate), and * G is a labelled directed acyclic graph with labels fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded-error Probabilistic Polynomial
In computational complexity theory, a branch of computer science, 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 by 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 (complexity) , 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. Informally, a problem is in BPP if there is an algorithm for it that has the following properties: *It is allowed to flip coins and make random decisions *It is guaranteed to run in polynomial time *On any given run of the algorithm, it has a probability of at most 1/3 of giving the wrong answer, whether the answer is YES or NO. Def ... [...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] |
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] |
NC (complexity)
In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors. In other words, a problem with input size ''n'' is in NC if there exist constants ''c'' and ''k'' such that it can be solved in time using parallel processors. Stephen Cook coined the name "Nick's class" after Nick Pippenger, who had done extensive research on circuits with polylogarithmic depth and polynomial size.Arora & Barak (2009) p.120 Just as the class P can be thought of as the tractable problems ( Cobham's thesis), so NC can be thought of as the problems that can be efficiently solved on a parallel computer.Arora & Barak (2009) p.118 NC is a subset of P because polylogarithmic parallel computations can be simulated by polynomial-time sequential ones. It is unknown whether NC = P, but most researchers suspect this to be false, meaning that there are probably some tractable pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NL (complexity)
In computational complexity theory, NL (Nondeterministic Logarithmic-space) is the complexity class containing decision problems that can be solved by a nondeterministic Turing machine using a logarithmic amount of memory space. NL is a generalization of L, the class for logspace problems on a deterministic Turing machine. Since any deterministic Turing machine is also a nondeterministic Turing machine, we have that L is contained in NL. NL can be formally defined in terms of the computational resource nondeterministic space (or NSPACE) as NL = NSPACE(log ''n''). Important results in complexity theory allow us to relate this complexity class with other classes, telling us about the relative power of the resources involved. Results in the field of algorithms, on the other hand, tell us which problems can be solved with this resource. Like much of complexity theory, many important questions about NL are still open (see Unsolved problems in computer science). Occasionally NL ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L (complexity)
In computational complexity theory, L (also known as LSPACE or DLOGSPACE) is the complexity class containing decision problems that can be solved by a deterministic Turing machine using a logarithmic amount of writable memory space., Definition 8.12, p. 295., p. 177. Formally, the Turing machine has two tapes, one of which encodes the input and can only be read, whereas the other tape has logarithmic size but can be read as well as written. Logarithmic space is sufficient to hold a constant number of pointers into the input and a logarithmic number of boolean flags, and many basic logspace algorithms use the memory in this way. Complete problems and logical characterization Every non-trivial problem in L is complete under log-space reductions, so weaker reductions are required to identify meaningful notions of L-completeness, the most common being first-order reductions. A 2004 result by Omer Reingold shows that USTCON, the problem of whether there exists a path ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CL (complexity)
CL or cl may refer to: Arts and entertainment * CL (rapper) or Lee Chae-rin (born 1991), singer and rapper, former leader of the K-pop girl group 2NE1 * Creative Loafing, a newspaper publisher Brands and enterprises * Colgate-Palmolive's NYSE stock symbol * Companhia das Lezírias, an agribusiness company in Portugal Computing and technology * , the command-line C/C++ compiler for Microsoft Visual C++ * .cl, Internet country code top-level domain for Chile * CL register, the low byte of an X86 16-bit CX register * CAS latency, a measure used in computer memory * Common Lisp, a programming language * Common Logic, a framework for a family of logic languages * Control Language, a scripting language for the IBM AS/400 midrange platform Industry and technology * CL, the prefix for Canadair manufactured aircraft model numbers * Caseless ammunition Organizations * Catholic League (U.S.), also known as The Catholic League for Religious and Civil Rights * Communion and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LOGCFL
In computational complexity theory, LOGCFL is the complexity class that contains all decision problems that can be reduced in logarithmic space to a context-free language. This class is situated between NL and AC1, in the sense that it contains the former and is contained in the latter. Problems that are complete for LOGCFL include many problems whose instances can be characterized by acyclic hypergraphs: * evaluating acyclic Boolean conjunctive queries * checking the existence of a homomorphism between two acyclic relational structures * checking the existence of solutions of acyclic constraint satisfaction problems See also * List of complexity classes This is a list of complexity classes in computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and rela ... External links * Complexity classes {{comp-sci-theory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RP (complexity)
In computational complexity theory, randomized polynomial time (RP) is the complexity class of problems for which a probabilistic Turing machine exists with these properties: * It always runs in polynomial time in the input size * If the correct answer is NO, it always returns NO * If the correct answer is YES, then it returns YES with probability at least 1/2 (otherwise, it returns NO). In other words, the algorithm is allowed to flip a truly random coin while it is running. The only case in which the algorithm can return YES is if the actual answer is YES; therefore if the algorithm terminates and produces YES, then the correct answer is definitely YES; however, the algorithm can terminate with NO ''regardless'' of the actual answer. That is, if the algorithm returns NO, it might be wrong. Some authors call this class R, although this name is more commonly used for the class of recursive languages. If the correct answer is YES and the algorithm is run ''n'' times with the r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P (complexity)
In computational complexity theory, P, also known as PTIME or DTIME(''n''O(1)), 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. Cobham's thesis holds that P is the class of computational problems that are "efficiently solvable" or " tractable". This is inexact: in practice, some problems not known to be in P have practical solutions, and some that are in P do not, but this is a useful rule of thumb. Definition A language ''L'' is in P if and only if there exists a deterministic Turing machine ''M'', such that * ''M'' runs for polynomial time on all inputs * For all ''x'' in ''L'', ''M'' outputs 1 * For all ''x'' not in ''L'', ''M'' outputs 0 P can also be viewed as a uniform family of boolean circuits. A language ''L'' is in P if and only if there exists a polynomial-time uniform family of boolean circuits \, such that * For all n \in \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
