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Co-NP
In computational complexity theory, co-NP is a complexity class. A decision problem X is a member of co-NP if and only if its complement is in the complexity class NP. The class can be defined as follows: a decision problem is in co-NP precisely if only ''no''-instances have a polynomial-length "certificate" and there is a polynomial-time algorithm that can be used to verify any purported certificate. That is, co-NP is the set of decision problems where there exists a polynomial ''p(n)'' and a polynomial-time bounded Turing machine ''M'' such that for every instance ''x'', ''x'' is a ''no''-instance if and only if: for some possible certificate ''c'' of length bounded by ''p(n)'', the Turing machine ''M'' accepts the pair (''x'', ''c''). Complementary Problems While an NP problem asks whether a given instance is a ''yes''-instance, its ''complement'' asks whether an instance is a ''no''-instance, which means the complement is in co-NP. Any ''yes''-instance for the original NP ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Co-NP-complete
In complexity theory, computational problems that are co-NP-complete are those that are the hardest problems in co-NP, in the sense that any problem in co-NP can be reformulated as a special case of any co-NP-complete problem with only polynomial overhead. If P is different from co-NP, then all of the co-NP-complete problems are not solvable in polynomial time. If there exists a way to solve a co-NP-complete problem quickly, then that algorithm can be used to solve all co-NP problems quickly. Each co-NP-complete problem is the complement of an NP-complete problem. There are some problems in both NP and co-NP, for example all problems in P or integer factorization. However, it is not known if the sets are equal, although inequality is thought more likely. See co-NP and NP-complete for more details. Fortune showed in 1979 that if any sparse language is co-NP-complete (or even just co-NP-hard), then , a critical foundation for Mahaney's theorem. Formal definition A decision prob ... [...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 ab ... [...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] |
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 det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty of this problem is important for the algorithms used in cryptography such as RSA public-key encryption and the RSA digital signature. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number ( RSA-240) utilizing approximately 900 core-years of computing power. The researchers estimated that a 1024- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexity Class
In computational complexity theory, a complexity class is a set of computational problems of related resource-based complexity. The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time or memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements. For instance, the class P is the set of decision problems solvable by a deterministic Turing machine in polynomial time. There are, however, many complexity classes defined in terms of other types of problems (e.g. counting problems and function problems) and using other models of computation (e.g. probabilistic Turing machines, interactive proof systems, Boolean circuits, and quantum computers). The study of the relationships between complexity class ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whether a given natural number is prime. Another is the problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?". The answer is either 'yes' or 'no' depending upon the values of ''x'' and ''y''. A method for solving a decision problem, given in the form of an algorithm, is called a decision procedure for that problem. A decision procedure for the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" would give the steps for determining whether ''x'' evenly divides ''y''. One such algorithm is long division. If the remainder is zero the answer is 'yes', otherwise it is 'no'. A decision problem which can be solved by an algorithm is called ''decidable''. Decision problems typically appear in ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Certificate (complexity)
In computational complexity theory, a certificate (also called a witness) is a string that certifies the answer to a computation, or certifies the membership of some string in a language. A certificate is often thought of as a solution path within a verification process, which is used to check whether a problem gives the answer "Yes" or "No". In the decision tree model of computation, certificate complexity is the minimum number of the n input variables of a decision tree that need to be assigned a value in order to definitely establish the value of the Boolean function f. Use in definitions The notion of certificate is used to define semi-decidability: a formal language ''L'' is semi-decidable if there is a two-place predicate relation R ⊆ Σ∗ × Σ∗ such that R is computable, and such that for all x ∈ Σ∗: x ∈ L ⇔ there exists y such that R(x, y) Certificates also give definitions for some complexity classes which can alternatively be characterised in te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Satisfiability Problem
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called ''satisfiable''. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is ''unsatisfiable''. For example, the formula "''a'' AND NOT ''b''" is satisfiable because one can find the values ''a'' = TRUE and ''b'' = FALSE, which make (''a'' AND NOT ''b'') = TRUE. In contrast, "''a'' AND NOT ''a''" is unsatisfiable. SAT is the first problem that was proved to be NP-complete; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautology (logic)
In mathematical logic, a tautology (from el, ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always true, regardless of the colour of the ball. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false. It cannot be untrue. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically contingent. Such a formula can be made either true or false based on the values assigned to its propositional variables. The doub ... [...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] |
