Complexity classes P and NP - Complexity classes P and NP Computational complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are time (how many steps does it take to solve a problem) and space (how much memory does it take to solve a problem). In this theory, the class P consists of all those decision problems that can be solved on a deterministic sequential machine in an amount of time that is polynomial in the size of the input; the class NP consists of all those decision problems whose positive solutions can be verified in polynomial time given the right information, or equivalently, whose solution can be found in polynomial time on a non-deterministic machine..
NP (complexity) - NP (complexity) In computational complexity theory, NP ("non-deterministic polynomial-time") is the set of decision problems solvable in polynomial time on a non-deterministic Turing machine. Or, equivalently, YES answers are checkable in polynomial time on a deterministic Turing machine given the right information. A language L belongs to NP if there exists a two input polynomial time algorithm A and a constant c such that L = {x in {0,1}* ∃ certificate, y with y = O(xc) such that A(x,y) = 1} Algorithm A verifies L in polynomial time. See also: Complexity classes P and NP and NP-Complete..
Computational complexity theory - Computational complexity theory Complexity theory is part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are time (how many steps does it take to solve a problem) and space (how much memory does it take to solve a problem). Other resources can also be considered, such as how many parallel processors are needed to solve a problem in parallel. Complexity theory differs from computability theory, which deals with whether a problem can be solved at all, regardless of the resources required. Overview A single "problem" is an entire set of related questions, where each question is a finite-length string. For example, the problem FACTORIZE is: given an integer written in binary, return all of.
NP-complete - NP-complete In complexity theory, the NP-complete problems are the hardest problems in NP, in the sense that they are the ones most likely not to be in P. The reason is that if you could find a way to solve an NP-complete problem quickly, then you could use that algorithm to solve all NP problems quickly. (A more formal definition is given below. See also Complexity classes P and NP). One example of an NP-complete problem is the subset sum problem which is: given a finite set of integers, determine whether any non-empty subset of them adds up to zero. A supposed answer is very easy to verify for correctness, but no-one knows a faster way to solve the problem than to try every single possible.
NP-hard - NP-hard In computational complexity theory, NP-hard (Non-deterministic Polynomial-time hard) refers to the class of decision problems that contains all problems H such that for all decision problems L in NP there is a polynomial-time many-one reduction to H. Informally this class can be described as containing the decision problems that are at least as hard as any problem in NP. This intuition is supported by the fact that if we can find an algorithm A that solves one of these problems H in polynomial time then we can construct a polynomial time algorithm for every problem in NP by first executing the reduction from this problem to H and then executing the algorithm A. Assuming language L to be NP-complete, 1. L is in NP 2..
List of computability and complexity topics - List of computability and complexity topics This is a list of computability and complexity topics, by Wikipedia page. Computability theory is the part of the theory of computation that deals with what can be computed, in principle. Computational complexity theory deals with how hard computations are, in quantitative terms, both with upper bounds (algorithms whose complexity in the worst cases, as use of computing resources, can be estimated), and from below (proofs that no procedure to carry out some task can be very fast). For more abstract foundational matters, see the list of mathematical logic topics. Table of contents showTocToggle("show","hide") 1 Calculation 2 Computability theory: models of computation 3 Decision problems 4 Definability questions 5 Complexity theory 6 Complexity classes 7 Named problems 8 Extensions Calculation Mathematical expression.
Integer factorization - 45, the prime factorization would be 32·5. The factorization is always unique, according to the fundamental theorem of arithmetic. This problem is of significance in mathematics, cryptography, complexity theory, and quantum computers. The complete list of factors can be derived from the prime factorization by incrementing the exponents from zero until the number is reached. For example, since 45 = 32·5, 45 is divisible by 30·50, 30·51, 31·50, 31·51, 32·50, and 32·51, or 1, 5, 3, 15, 9, and 45. In contrast, the prime factorization only includes prime factors. Given two large prime numbers, it is easy to multiply them together. However, given their product, it appears to be difficult to find the factors. This is relevant for many modern systems in cryptography. If a fast method were found for solving.
EXPTIME - EXPTIME In complexity theory, EXPTIME is the set of all decision problems solvable by a deterministic Turing machine in O(2p(n)) time, where p(n) is a polynomial function of n. Some authors restrict p(n) to be a linear function, but a more common definition is to allow p(n) to be any polynomial. If p(n) is a linear function, the resulting class is often called E, and is obviously a subset of EXPTIME. EXPTIME is known to be a subset of EXPSPACE and a superset of PSPACE, NP-complete, NP, and P. That is significant because it is currently unknown which (if any) of those four sets are equal to each other. It is known however that P is a strict subset of EXPTIME. The complexity class EXPTIME-complete is also.
Algorithm - potatoes and end when the potato salad is ready to be eaten. Correctly performing an algorithm will not solve a problem if the algorithm is flawed, or not appropriate to the problem. For example, performing the potato salad algorithm will fail if there are no potatoes present, even if all the motions of preparing the salad are performed as if the potatoes were there. Table of contents showTocToggle("show","hide") 1 Formalized algorithms 2 Implementing algorithms 3 Example 4 History 5 Classes of algorithms 6 See also 7 References 8 Other Web Reference Formalized algorithms Algorithms are essential to the way computers process information, because a computer program is essentially an algorithm that tells the computer what specific steps to perform (in what specific order) in order to carry out a specified task,.
Computation - equivalent in power to the familiar electronic computer, if one pretends that electronic computers have infinite memory. Indeed, it is widely believed that all "proper" formalizations of the concept of algorithm will be equivalent in power to Turing machines; this is known as the Church-Turing thesis. In general, questions of what can be computed by various machines are investigated in computability theory. The theory of computation studies these models of general computation, along with the limits of computing: Which problems are (provably) unsolvable by a computer? (See the halting problem and the Post correspondence problem.) Which problems are solvable by a computer, but require such an enormously long time to compute that the solution is impractical? (See Presburger arithmetic.) Can it be harder to solve a problem than to check a.
Computational learning theory - learning. In COLT the main interest is the computational aspects of learning, e.g., the time complexity of learning problem. The computational aspects are considered in a learning framework, like the very common one of Probably approximately correct learning. Note that a computation is considered feasible if it can be done in polynomial time. There are two kind of results in COLT. Possitive results - Showing the a certain class of function is learnable in polynomial time. Negative results - Showing that certain classes cannot be learned in polynomial time. Negative results were proven only by assumption. The assumptions the are common in negative results are: Computational - P<>NP Cryptographic - One way functions exits. A list of important COLT papers Surveys [Angluin, 92] Angluin, D. 1992. Computational learning theory: Survey and.
Stephen Cook - a noted computer scientist. Cook formalised the notion of NP-completeness in a famous 1971 paper "The Complexity of Theorem Proving Procedures", which also contained a proof that the boolean satisfiability problem was NP-complete. The paper left open theoretical computer science's greatest unsolved question - whether complexity classes P and NP are equivalent, the answer to which has eluded researchers since. Cook received the Turing Award in 1982 for his discovery. His citation reads: For his advancement of our understanding of the complexity of computation in a significant and profound way. His seminal paper, The Complexity of Theorem Proving Procedures, presented at the 1971 ACM SIGACT Symposium on the Theory of Computing, Laid the foundations for the theory of NP-Completeness. The ensuing exploration of the boundaries and nature of NP-complete class of.
Subset sum problem - The subset sum problem is an important problem in complexity theory and cryptography. The problem is this: given a set of integers, does any subset sum to exactly zero? For example, given the set {-7, -3, -2, 5, 8}, the answer is YES because the subset {-3, -2, 5} sums to zero. The problem is NP-Complete, and is perhaps the simplest such problem to describe. An equivalent problem is this: given a set of integers and an integer s, does any subset sum to s? Subset sum can also be thought of as a special case of the knapsack problem. Although the problem was described above in terms of integers and addition, it can actually be defined using any group. For example, the problem could be: given an integer n and.
Polynomial time - Polynomial time In Complexity theory, Polynomial time refers to the computation time of a problem where the time, m(n), is no greater than a polynomial function of the problem size, n. Written mathematically, m(n) = O(nk) where k is a constant. Mathematicians sometimes use the notion of "polynomial time on the length of the input" as a definition of a "fast" computation, as opposed to "super-polynomial time", which is anything slower than that. Exponential time is one example of a super-polynomial time. The class of decision problems that can be solved on a deterministic sequential machine in polynomial time is known as P. The class of decision problems that can be verified in polynomial time is known as NP. Equivalently, NP is the class of decision problems that.
Polynomial-time many-one reduction - many-one reduction In computational complexity theory a polynomial-time many-one reduction (also known as polynomial transformation or Karp reduction) is a certain way to "reduce" one decision problem to another one in such a way that any algorithm solving the latter immediately yields an algorithm solving the former, with only a modest slow-down. Specifically, suppose L and M are formal languages over the alphabets Σ and Γ, respectively. A polynomial-time many-one reduction from L to M is a function f : Σ* → Γ* which can be computed in polynomial time in the input size and has the property that If such a function f exists, we also say that "L is polynomial-time many-one reducible to M". This notion of reducibility is used in the standard definitions of several complexity classes, for.
P - consonant. Those who speak Arabic usually have difficulty pronouncing this sound; they pronounce like b instead. A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z Papa represents the letter P in the NATO phonetic alphabet. P is also: In chemistry, P is the symbol for the element phosphorus. In complexity theory, P is the set of decision problems solvable in polynomial time on a deterministic Turing machine. See Complexity classes P and NP. In physics, p is the symbol for momentum, also for pressure; P for power and polarisation; p is also the symbol for the proton. As p, the symbol for the pond, a now obsolete unit of force, see kilopond. In units.
PTIME - PTIME In complexity theory PTIME is the class of decision problems that can be solved on a deterministic Turing machine in polynomial time, i.e. the number of computation steps is bounded by a polynomial function, where the length of the input is the argument. The length of the input is the length of the appropriate coding, e.g. for numbers this coding is usually of logarithmic length with respect to the number itself (a number n can be coded using log n symbols). PTIME is usually just named "P" and then forms part of one of the big open problems in mathematics: P =? NP (see Complexity classes P and NP). The problems in PTIME are often considered to be the problems that are effectively computable, since no.
Oracle machine - Oracle machine In complexity theory and computability theory, an oracle is a black box that computes a function in a single step. This could be a function solving an NP-complete problem such as the subset sum problem. It could even be an "uncomputable" (non-recursive) function like the halting problem. Table of contents showTocToggle("show","hide") 1 Oracle machines 2 Oracles and Complexity Theory 3 Oracles and Halting Problems 4 References in Popular Culture Oracle machines An oracle machine is a Turing machine connected to an oracle. The Turing machine can write on its own tape an input for the oracle, then tell the oracle to execute. In a single step, the oracle computes its function, erases its input, and writes its output to the tape. Sometimes the Turing machine is.
Non-deterministic Turing machine - NTM has a "computation tree". If any branch of the tree halts with an "accept" condition, we say that the NTM accepts the input. Intuitively it seems that the NTM is more powerful than the DTM, since it can execute many possible computations in parallel, requiring only that one of them succeeds. Any computation carried out by a NTM can be simulated by a DTM, although it may require significantly longer time. How much longer is not known in general - this is, in a nutshell, the definition of the "Is P = NP?" problem (see Complexity classes P and NP). It is a common misconception that quantum computers are NTMs. However, whereas NTMs can solve NP-complete problems in polynomial time, it is believed that quantum computers cannot..
Michael O. Rabin - joint paper "Finite Automata and Their Decision Problem," which introduced the idea of nondeterministic machines, which has proved to be an enormously valuable concept. Their (Scott & Rabin) classic paper has been a continuous source of inspiration for subsequent work in this field. The paper was written in 1959. Nondeterministic machines have become a key concept in computational complexity theory, particularly with the description of complexity classes P and NP, as the most well-known example. In 1975, Rabin also invented a randomized algorithm that could determine very quickly, but with a tiny probability of error, whether a number was a prime number. Fast primality testing is key in the successful implementation of most public-key cryptography. Rabin's more recent research has concentrated on computer security. He is currently the Thomas J. Watson.
