In computational complexity theory the compression theorem is an important theorem about the complexity of computable functions.
Computational complexity theory focuses on classifying computational problems according to their inherent difficulty, 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.
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithm, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output. Computable functions are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Any definition, however, must make reference to some specific model of computation but all valid definitions yield the same class of functions. Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the μ-recursive functions.
The theorem states that there exists no largest complexity class, with computable boundary, which contains all computable functions.
In computational complexity theory, a complexity class is a set of problems of related resource-based complexity. A typical complexity class has a definition of the form:
Given a Gödel numbering of the computable functions and a Blum complexity measure where a complexity class for a boundary function is defined as
In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was used by Kurt Gödel for the proof of his incompleteness theorems.
Then there exists a total computable function so that for all
and
In computability theory, Rice's theorem states that all non-trivial, semantic properties of programs are undecidable. A semantic property is one about the program's behavior, unlike a syntactic property. A property is non-trivial if it is neither true for every computable function, nor false for every computable function.
In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping f from U onto the open unit disk
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other.
Distributions are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.
In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first proved by Stephen Kleene in 1938 and appear in his 1952 book Introduction to Metamathematics. A related theorem which constructs fixed points of a computable function is known as Rogers's theorem and is due to Hartley Rogers, Jr. (1967).
In calculus, integration by substitution, also known as u-substitution, is a method for finding integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool in mathematics. It is the counterpart to the chain rule for differentiation.
In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm.
In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function, but can be every intersection of the graph itself with a hyperplane parallel to a fixed x-axis and to the y-axis.
In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.
In computational complexity theory the Blum axioms or Blum complexity axioms are axioms that specify desirable properties of complexity measures on the set of computable functions. The axioms were first defined by Manuel Blum in 1967.
In computability theory, productive sets and creative sets are types of sets of natural numbers that have important applications in mathematical logic. They are a standard topic in mathematical logic textbooks such as Soare (1987) and Rogers (1987).
In computational complexity theory, Blum's speedup theorem, first stated by Manuel Blum in 1967, is a fundamental theorem about the complexity of computable functions.
In computational complexity theory the Gap Theorem, also known as the Borodin-Trakhtenbrot Gap Theorem, is a major theorem about the complexity of computable functions.
In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.
In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.
In machine learning, kernel methods are a class of algorithms for pattern analysis, whose best known member is the support vector machine (SVM). The general task of pattern analysis is to find and study general types of relations in datasets. In its simplest form, the kernel trick means transforming data into another dimension that has a clear dividing margin between classes of data. For many algorithms that solve these tasks, the data in raw representation have to be explicitly transformed into feature vector representations via a user-specified feature map: in contrast, kernel methods require only a user-specified kernel, i.e., a similarity function over pairs of data points in raw representation.
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
In the mathematical analysis, and especially in real and harmonic analysis, a Birnbaum–Orlicz space is a type of function space which generalizes the Lp spaces. Like the Lp spaces, they are Banach spaces. The spaces are named for Władysław Orlicz and Zygmunt William Birnbaum, who first defined them in 1931.
In computational complexity theory, the language TQBF is a formal language consisting of the true quantified Boolean formulas. A (fully) quantified Boolean formula is a formula in quantified propositional logic where every variable is quantified, using either existential or universal quantifiers, at the beginning of the sentence. Such a formula is equivalent to either true or false. If such a formula evaluates to true, then that formula is in the language TQBF. It is also known as QSAT.
Computational anatomy is an interdisciplinary field of biology focused on quantitative investigation and modelling of anatomical shapes variability. It involves the development and application of mathematical, statistical and data-analytical methods for modelling and simulation of biological structures.
Arto K. Salomaa is a Finnish mathematician and computer scientist. His research career, which spans over forty years, is focused on formal languages and automata theory.
The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.
| P ≟ NP | This theoretical computer science–related article is a stub. You can help Wikipedia by expanding it. |
