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Counting Problem (complexity)
In computational complexity theory and computability theory, a counting problem is a type of computational problem. If ''R'' is a search problem then :c_R(x)=\vert\\vert \, is the corresponding counting function and :\#R=\ denotes the corresponding decision problem. Note that ''cR'' is a search problem while #''R'' is a decision problem, however ''cR'' can be ''C'' Cook-reduced to #''R'' (for appropriate ''C'') using a binary search (the reason #''R'' is defined the way it is, rather than being the graph of ''cR'', is to make this binary search possible). Counting complexity class Just as NP has NP-complete In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any ... problems via many-one reductions, #P has #P-complete problems via parsimonious reductions, problem transformatio ... [...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] |
Computability Theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definable set, definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: * What does it mean for a function (mathematics), function on the natural numbers to be computable? * How can noncomputable functions be classified into a hierarchy based on their level of noncomputability? Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of computational complexity theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Problem
In theoretical computer science, a computational problem is one that asks for a solution in terms of an algorithm. For example, the problem of factoring :"Given a positive integer ''n'', find a nontrivial prime factor of ''n''." is a computational problem that has a solution, as there are many known integer factorization algorithms. A computational problem can be viewed as a set of ''instances'' or ''cases'' together with a, possibly empty, set of ''solutions'' for every instance/case. The question then is, whether there exists an algorithm that maps instances to solutions. For example, in the factoring problem, the instances are the integers ''n'', and solutions are prime numbers ''p'' that are the nontrivial prime factors of ''n''. An example of a computational problem without a solution is the Halting problem. Computational problems are one of the main objects of study in theoretical computer science. One is often interested not only in mere existence of an algorithm, b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Search Problem
In computational complexity theory and computability theory, a search problem is a computational problem of finding an ''admissible'' answer for a given input value, provided that such an answer exists. In fact, a search problem is specified by a binary relation where if and only if "'' is an admissible answer given ''". Search problems frequently occur in graph theory and combinatorial optimization, e.g. searching for matchings, optional cliques, and stable sets in a given undirected graph. An algorithm is said to solve a search problem if, for every input value , it returns an admissible answer for when such an answer exists; otherwise, it returns any appropriate output, e.g. "not found" for with no such answer. Definition PlanetMath defines the problem as follows: If R is a binary relation such that \operatorname(R)\subseteq\Gamma^ and T is a Turing machine, then T calculates f if: * If x is such that there is some y such that R(x,y) then T accepts x with output z suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counting Function
Counting is the process of determining the number of elements of a finite set of objects; that is, determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term ''enumeration'' refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element. Counting sometimes involves numbers other than one; for example, when counting money, counting out change, "counting by twos" (2, 4, 6, 8, 10, 12, ...), or "counting by fives" (5, 10, 15, 20, 25, ...). There is archaeological evidence suggesting that humans have been ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cook Reduction
In computability theory, a Turing reduction from a decision problem A to a decision problem B is an oracle machine that decides problem A given an oracle for B (Rogers 1967, Soare 1987) in finitely many steps. It can be understood as an algorithm that could be used to solve A if it had access to a subroutine for solving B. The concept can be analogously applied to function problems. If a Turing reduction from A to B exists, then every algorithm for B can be used to produce an algorithm for A, by inserting the algorithm for B at each place where the oracle machine computing A queries the oracle for B. However, because the oracle machine may query the oracle a large number of times, the resulting algorithm may require more time asymptotically than either the algorithm for B or the oracle machine computing A. A Turing reduction in which the oracle machine runs in polynomial time is known as a Cook reduction. The first formal definition of relative computability, then called relat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Search
In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the middle element of the array. If they are not equal, the half in which the target cannot lie is eliminated and the search continues on the remaining half, again taking the middle element to compare to the target value, and repeating this until the target value is found. If the search ends with the remaining half being empty, the target is not in the array. Binary search runs in Time complexity#Logarithmic time, logarithmic time in the Best, worst and average case, worst case, making O(\log n) comparisons, where n is the number of elements in the array. Binary search is faster than linear search except for small arrays. However, the array must be sorted first to be able to apply binary search. There are specialized data structures designed fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no". # When the answer is "yes", this can be demonstrated through the existence of a short (polynomial length) ''solution''. # 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. Hence, 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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Many-one Reduction
In computability theory and computational complexity theory, a many-one reduction (also called mapping reduction) is a reduction that converts instances of one decision problem (whether an instance is in L_1) to another decision problem (whether an instance is in L_2) using a computable function. The reduced instance is in the language L_2 if and only if the initial instance is in its language L_1. Thus if we can decide whether L_2 instances are in the language L_2, we can decide whether L_1 instances are in the language L_1 by applying the reduction and solving for L_2. Thus, reductions can be used to measure the relative computational difficulty of two problems. It is said that L_1 reduces to L_2 if, in layman's terms L_2 is at least as hard to solve as L_1. This means that any algorithm that solves L_2 can also be used as part of a (otherwise relatively simple) program that solves L_1. Many-one reductions are a special case and stronger form of Turing reductions. With many-one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parsimonious Reduction
In computational complexity theory and game complexity, a parsimonious reduction is a transformation from one problem to another (a reduction) that preserves the number of solutions. Informally, it is a bijection between the respective sets of solutions of two problems. A general reduction from problem A to problem B is a transformation that guarantees that whenever A has a solution B also has ''at least one'' solution and vice versa. A parsimonious reduction guarantees that for every solution of A, there exists ''a unique solution'' of B and vice versa. Parsimonious reductions are commonly used in computational complexity for proving the hardness of counting problems, for counting complexity classes such as #P. Additionally, they are used in game complexity, as a way to design hard puzzles that have a unique solution, as many types of puzzles require. Formal definition Let x be an instance of problem X. A ''Parsimonious reduction'' R from problem X to problem Y is a reduction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GapP
GapP is a counting complexity class, consisting of all of the functions ''f'' such that there exists a polynomial-time non-deterministic Turing machine ''M'' where, for any input ''x'', ''f(x)'' is equal to the number of accepting paths of ''M'' minus the number of rejecting paths of ''M''. GapP is exactly the closure of #P under subtraction. It also has all the other closure properties of #P, such as addition, multiplication, and binomial coefficients. The counting class AWPP is defined in terms of GapP functions. References * S. Fenner, L. Fortnow, and S. KurtzGap-definable counting classes ''Journal of Computer and System Sciences'' 48(1):116-148, 1994. * {{Comp-sci-theory-stub Complexity classes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
