|
BPLP
In computational complexity theory, BPL (Bounded-error Probabilistic Logarithmic-space), sometimes called BPLP (Bounded-error Probabilistic Logarithmic-space Polynomial-time), is the complexity class of problems solvable in logarithmic space and polynomial time with probabilistic Turing machines with two-sided error. It is named in analogy with BPP, which is similar but has no logarithmic space restriction. Error model The probabilistic Turing machines in the definition of BPL may only accept or reject incorrectly less than 1/3 of the time; this is called ''two-sided error''. The constant 1/3 is arbitrary; any ''x'' with 0 ≤ ''x'' < 1/2 would suffice. This error can be made 2−''p''(''x'') times smaller for any polynomial ''p''(''x'') without using more than polynomial time or logarithmic space by running the algorithm repeatedly. Related classes Since two-sided error is more general than one-sided error,[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probabilistic Turing Machine
In theoretical computer science, a probabilistic Turing machine is a non-deterministic Turing machine that chooses between the available transitions at each point according to some probability distribution. As a consequence, a probabilistic Turing machine can—unlike a deterministic Turing Machine—have stochastic results; that is, on a given input and instruction state machine, it may have different run times, or it may not halt at all; furthermore, it may accept an input in one execution and reject the same input in another execution. In the case of equal probabilities for the transitions, probabilistic Turing machines can be defined as deterministic Turing machines having an additional "write" instruction where the value of the write is uniform distribution (discrete), uniformly distributed in the Turing Machine's alphabet (generally, an equal likelihood of writing a "1" or a "0" on to the tape). Another common reformulation is simply a deterministic Turing machine with an ad ... [...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 compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PL (complexity)
PL, or probabilistic L, is the class of languages recognizable by a polynomial time logarithmic space randomized machine with probability > (this is called unbounded error). Equivalently, as shown below, PL is the class of languages recognized by unbounded time unbounded error logspace randomized machine. An example of PL complete problem (under logspace reduction) is finding whether the determinant of a matrix (with integral coefficients) is positive. Given a matrix and a number , testing with , M, >n is also PL complete. By contrast, testing whether matrix permanent is positive is PP complete. PLPL=PL in the sense that for every f in PL, PL is unchanged if it is extended to allow as a subroutine, where A is the input string. PL contains NL and BPL and is contained in NC2. Approximating determinant in PL Determinant of an integral matrix can be reduced to finding the difference between the number of accepting and rejecting paths on a polynomially sized directe ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symposium On Theory Of Computing
The Annual ACM Symposium on Theory of Computing (STOC) is an academic conference in the field of theoretical computer science. STOC has been organized annually since 1969, typically in May or June; the conference is sponsored by the Association for Computing Machinery special interest group SIGACT. Acceptance rate of STOC, averaged from 1970 to 2012, is 31%, with the rate of 29% in 2012. As writes, STOC and its annual IEEE counterpart FOCS (the Symposium on Foundations of Computer Science) 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 STOC and FOCS as one of several defining characteristics of theoretical computer scientists. Awards The Gödel Prize for outstanding papers in theoretical computer science is presented alternately ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SC (complexity)
In computational complexity theory, SC (Steve's Class, named after Stephen Cook) is the complexity class of problems solvable by a deterministic Turing machine in polynomial time (class P) and polylogarithmic space (class PolyL) (that is, O((log ''n'')k) space for some constant ''k''). It may also be called DTISP(poly, polylog), where DTISP stands for ''deterministic time and space''. Note that the definition of SC differs from P ∩ PolyL, since for the former, it is required that a single algorithm runs in both polynomial time and polylogarithmic space; while for the latter, two separate algorithms will suffice: one that runs in polynomial time, and another that runs in polylogarithmic space. (It is unknown whether SC and P ∩ PolyL are equivalent). DCFL, the strict subset of context-free languages recognized by deterministic pushdown automata, is contained in SC, as shown by Cook in 1979. It is open if all context-free languages can be recognized in SC, although they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derandomization
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are random variables. One has to distinguish between algorithms that use the random input so that they always terminate with the correct answer, but where the expected running time is finite (Las Vegas algorithms, for example Quicksort), and algorithms which have a chance of producing an incorrect result (Monte Carlo algorithms, for example the Monte Carlo algorithm for the MFAS problem) or fail to produce a result either by signaling a failure or failing to terminate. In some cases, probabilistic algorithms are the only practical means of solving a problem. In common practice, randomized algor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PP (complexity)
In complexity theory, PP 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 for it that is allowed to flip coins and make random decisions. It is guaranteed to run in polynomial time. If the answer is YES, the algorithm will answer YES with probability more than 1/2. If the answer is NO, the algorithm will answer YES with probability less 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 characterizatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RL (complexity)
Randomized Logarithmic-space (RL), sometimes called RLP (Randomized Logarithmic-space Polynomial-time), is the complexity class of computational complexity theory problems solvable in logarithmic space and polynomial time with probabilistic Turing machines with one-sided error. It is named in analogy with RP, which is similar but has no logarithmic space restriction. Definition The probabilistic Turing machines in the definition of RL never accept incorrectly but are allowed to reject incorrectly less than 1/3 of the time; this is called ''one-sided error''. The constant 1/3 is arbitrary; any ''x'' with 0 < ''x'' < 1 would suffice. This error can be made 2−''p''(''x'') times smaller for any polynomial ''p''(''x'') without using more than polynomial time or logarithmic space by running the algorithm repeatedly. Relation to other complexity classes Sometimes the name RL is reserved for the class of problems solvable by logarithmic-space probabilisti ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (complexity)
In computational complexity theory, the complement of a decision problem is the decision problem resulting from reversing the ''yes'' and ''no'' answers. Equivalently, if we define decision problems as sets of finite strings, then the complement of this set over some fixed domain is its complement problem. For example, one important problem is whether a number is a prime number. Its complement is to determine whether a number is a composite number (a number which is not prime). Here the domain of the complement is the set of all integers exceeding one. There is a Turing reduction from every problem to its complement problem. The complement operation is an involution, meaning it "undoes itself", or the complement of the complement is the original problem. One can generalize this to the complement of a complexity class, called the complement class, which is the set of complements of every problem in the class. If a class is called C, its complement is conventionally labelled co-C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Computing
The ''SIAM Journal on Computing'' is a scientific journal focusing on the mathematical and formal aspects of computer science. It is published by the Society for Industrial and Applied Mathematics (SIAM). Although its official ISO abbreviation is ''SIAM J. Comput.'', its publisher and contributors frequently use the shorter abbreviation ''SICOMP''. SICOMP typically hosts the special issues of the IEEE Annual Symposium on Foundations of Computer Science (FOCS) and the Annual ACM Symposium on Theory of Computing (STOC), where about 15% of papers published in FOCS and STOC each year are invited to these special issues. For example, Volume 48 contains 11 out of 85 papers published in FOCS 2016. References * External linksSIAM Journal on Computing on |
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, 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. Definition A langu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
