Computational complexity theoryIn 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.
Complexity classIn 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.
Computational complexityIn computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) and memory storage requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem. The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory.
Parameterized complexityIn computer science, parameterized complexity is a branch of computational complexity theory that focuses on classifying computational problems according to their inherent difficulty with respect to multiple parameters of the input or output. The complexity of a problem is then measured as a function of those parameters. This allows the classification of NP-hard problems on a finer scale than in the classical setting, where the complexity of a problem is only measured as a function of the number of bits in the input.
ComplexityComplexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, leading to non-linearity, randomness, collective dynamics, hierarchy, and emergence. The term is generally used to characterize something with many parts where those parts interact with each other in multiple ways, culminating in a higher order of emergence greater than the sum of its parts. The study of these complex linkages at various scales is the main goal of complex systems theory.
Pattern recognitionPattern recognition is the automated recognition of patterns and regularities in data. While similar, pattern recognition (PR) is not to be confused with pattern machines (PM) which may possess (PR) capabilities but their primary function is to distinguish and create emergent pattern. PR has applications in statistical data analysis, signal processing, , information retrieval, bioinformatics, data compression, computer graphics and machine learning.
Pseudo-polynomial timeIn computational complexity theory, a numeric algorithm runs in pseudo-polynomial time if its running time is a polynomial in the numeric value of the input (the largest integer present in the input)—but not necessarily in the length of the input (the number of bits required to represent it), which is the case for polynomial time algorithms. In general, the numeric value of the input is exponential in the input length, which is why a pseudo-polynomial time algorithm does not necessarily run in polynomial time with respect to the input length.
Time complexityIn computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor.
Quantum complexity theoryQuantum complexity theory is the subfield of computational complexity theory that deals with complexity classes defined using quantum computers, a computational model based on quantum mechanics. It studies the hardness of computational problems in relation to these complexity classes, as well as the relationship between quantum complexity classes and classical (i.e., non-quantum) complexity classes. Two important quantum complexity classes are BQP and QMA.
Polynomial-time reductionIn computational complexity theory, a polynomial-time reduction is a method for solving one problem using another. One shows that if a hypothetical subroutine solving the second problem exists, then the first problem can be solved by transforming or reducing it to inputs for the second problem and calling the subroutine one or more times. If both the time required to transform the first problem to the second, and the number of times the subroutine is called is polynomial, then the first problem is polynomial-time reducible to the second.
Space complexityThe space complexity of an algorithm or a computer program is the amount of memory space required to solve an instance of the computational problem as a function of characteristics of the input. It is the memory required by an algorithm until it executes completely. This includes the memory space used by its inputs, called input space, and any other (auxiliary) memory it uses during execution, which is called auxiliary space. Similar to time complexity, space complexity is often expressed asymptotically in big O notation, such as etc.
Polynomial-time approximation schemeIn computer science (particularly algorithmics), a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems (most often, NP-hard optimization problems). A PTAS is an algorithm which takes an instance of an optimization problem and a parameter ε > 0 and produces a solution that is within a factor 1 + ε of being optimal (or 1 – ε for maximization problems). For example, for the Euclidean traveling salesman problem, a PTAS would produce a tour with length at most (1 + ε)L, with L being the length of the shortest tour.
Multilayer perceptronA multilayer perceptron (MLP) is a misnomer for a modern feedforward artificial neural network, consisting of fully connected neurons with a nonlinear kind of activation function, organized in at least three layers, notable for being able to distinguish data that is not linearly separable. It is a misnomer because the original perceptron used a Heaviside step function, instead of a nonlinear kind of activation function (used by modern networks).
Handwriting recognitionHandwriting recognition (HWR), also known as handwritten text recognition (HTR), is the ability of a computer to receive and interpret intelligible handwritten input from sources such as paper documents, photographs, touch-screens and other devices. The image of the written text may be sensed "off line" from a piece of paper by optical scanning (optical character recognition) or intelligent word recognition. Alternatively, the movements of the pen tip may be sensed "on line", for example by a pen-based computer screen surface, a generally easier task as there are more clues available.
Speech recognitionSpeech recognition is an interdisciplinary subfield of computer science and computational linguistics that develops methodologies and technologies that enable the recognition and translation of spoken language into text by computers. It is also known as automatic speech recognition (ASR), computer speech recognition or speech to text (STT). It incorporates knowledge and research in the computer science, linguistics and computer engineering fields. The reverse process is speech synthesis.
PerceptronIn machine learning, the perceptron (or McCulloch-Pitts neuron) is an algorithm for supervised learning of binary classifiers. A binary classifier is a function which can decide whether or not an input, represented by a vector of numbers, belongs to some specific class. It is a type of linear classifier, i.e. a classification algorithm that makes its predictions based on a linear predictor function combining a set of weights with the feature vector.
Social classA social class or social stratum is a grouping of people into a set of hierarchical social categories, the most common being the upper, middle and lower classes. Membership in a social class can for example be dependent on education, wealth, occupation, income, and belonging to a particular subculture or social network. "Class" is a subject of analysis for sociologists, political scientists, anthropologists and social historians. The term has a wide range of sometimes conflicting meanings, and there is no broad consensus on a definition of "class".
Kolmogorov complexityIn algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produces the object as output. It is a measure of the computational resources needed to specify the object, and is also known as algorithmic complexity, Solomonoff–Kolmogorov–Chaitin complexity, program-size complexity, descriptive complexity, or algorithmic entropy.
Perceptrons (book)Perceptrons: an introduction to computational geometry is a book written by Marvin Minsky and Seymour Papert and published in 1969. An edition with handwritten corrections and additions was released in the early 1970s. An expanded edition was further published in 1987, containing a chapter dedicated to counter the criticisms made of it in the 1980s. The main subject of the book is the perceptron, a type of artificial neural network developed in the late 1950s and early 1960s.
Euclidean spaceEuclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes.
