Theoretical computer science is a subfield of computer science and mathematics that focuses on the abstract and mathematical foundations of computation .
89-570: The EATCS–IPEC Nerode Prize is a theoretical computer science prize awarded for outstanding research in the area of multivariate algorithmics . It is awarded by the European Association for Theoretical Computer Science and the International Symposium on Parameterized and Exact Computation . The prize was offered for the first time in 2013. The prize winners so far have been: This theoretical computer science –related article
178-511: A dual pair to show the underlying duality . This is similar to the use of bra–ket notation in quantum mechanics. In logic and the theory of computation , the function notation of lambda calculus is used to explicitly express the basic notions of function abstraction and application . In category theory and homological algebra , networks of functions are described in terms of how they and their compositions commute with each other using commutative diagrams that extend and generalize
267-406: A function from a set X to a set Y assigns to each element of X exactly one element of Y . The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically , the concept
356-432: A map or a mapping , but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. maps of manifolds ). In particular map may be used in place of homomorphism for the sake of succinctness (e.g., linear map or map from G to H instead of group homomorphism from G to H ). Some authors reserve
445-405: A roman type is customarily used instead, such as " sin " for the sine function , in contrast to italic font for single-letter symbols. The functional notation is often used colloquially for referring to a function and simultaneously naming its argument, such as in "let f ( x ) {\displaystyle f(x)} be a function". This is an abuse of notation that is useful for
534-399: A user interface for the input/output of mathematical expressions, a large set of routines to perform usual operations, like simplification of expressions, differentiation using chain rule , polynomial factorization , indefinite integration , etc. Very-large-scale integration ( VLSI ) is the process of creating an integrated circuit (IC) by combining thousands of transistors into
623-696: A distributed system is called a distributed program , and distributed programming is the process of writing such programs. There are many alternatives for the message passing mechanism, including RPC-like connectors and message queues . An important goal and challenge of distributed systems is location transparency . Information-based complexity (IBC) studies optimal algorithms and computational complexity for continuous problems. IBC has studied continuous problems as path integration, partial differential equations, systems of ordinary differential equations, nonlinear equations, integral equations, fixed points, and very-high-dimensional integration. Formal methods are
712-429: A function defined by an integral with variable upper bound: x ↦ ∫ a x f ( u ) d u {\textstyle x\mapsto \int _{a}^{x}f(u)\,du} . There are other, specialized notations for functions in sub-disciplines of mathematics. For example, in linear algebra and functional analysis , linear forms and the vectors they act upon are denoted using
801-401: A function has evolved significantly over centuries, from its informal origins in ancient mathematics to its formalization in the 19th century. See History of the function concept for details. A function f from a set X to a set Y is an assignment of one element of Y to each element of X . The set X is called the domain of the function and the set Y is called the codomain of
890-523: A function is commonly written as f ( x , y ) = x 2 + y 2 {\displaystyle f(x,y)=x^{2}+y^{2}} and referred to as "a function of two variables". Likewise one can have a function of three or more variables, with notations such as f ( w , x , y ) {\displaystyle f(w,x,y)} , f ( w , x , y , z ) {\displaystyle f(w,x,y,z)} . A function may also be called
979-513: A function is defined. In particular, it is common that one might only know, without some (possibly difficult) computation, that the domain of a specific function is contained in a larger set. For example, if f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is a real function , the determination of the domain of the function x ↦ 1 / f ( x ) {\displaystyle x\mapsto 1/f(x)} requires knowing
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#17330860608651068-405: A function is then called a partial function . The range or image of a function is the set of the images of all elements in the domain. A function f on a set S means a function from the domain S , without specifying a codomain. However, some authors use it as shorthand for saying that the function is f : S → S . The above definition of a function is essentially that of
1157-410: A landmark result in computational complexity theory . Modern theoretical computer science research is based on these basic developments, but includes many other mathematical and interdisciplinary problems that have been posed, as shown below: An algorithm is a step-by-step procedure for calculations. Algorithms are used for calculation , data processing , and automated reasoning . An algorithm
1246-552: A letter such as f , g or h . The value of a function f at an element x of its domain (that is, the element of the codomain that is associated with x ) is denoted by f ( x ) ; for example, the value of f at x = 4 is denoted by f (4) . Commonly, a specific function is defined by means of an expression depending on x , such as f ( x ) = x 2 + 1 ; {\displaystyle f(x)=x^{2}+1;} in this case, some computation, called function evaluation , may be needed for deducing
1335-537: A means to manage large amounts of data efficiently for uses such as large databases and internet indexing services . Usually, efficient data structures are key to designing efficient algorithms . Some formal design methods and programming languages emphasize data structures, rather than algorithms, as the key organizing factor in software design. Storing and retrieving can be carried out on data stored in both main memory and in secondary memory . Distributed computing studies distributed systems. A distributed system
1424-417: A multivariate function is a function that has a Cartesian product or a proper subset of a Cartesian product as a domain. where the domain U has the form If all the X i {\displaystyle X_{i}} are equal to the set R {\displaystyle \mathbb {R} } of the real numbers or to the set C {\displaystyle \mathbb {C} } of
1513-427: A particular kind of mathematics based techniques for the specification , development and verification of software and hardware systems. The use of formal methods for software and hardware design is motivated by the expectation that, as in other engineering disciplines, performing appropriate mathematical analysis can contribute to the reliability and robustness of a design. Formal methods are best described as
1602-450: A simpler formulation. Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. It uses the ↦ arrow symbol, pronounced " maps to ". For example, x ↦ x + 1 {\displaystyle x\mapsto x+1} is the function which takes a real number as input and outputs that number plus 1. Again, a domain and codomain of R {\displaystyle \mathbb {R} }
1691-469: A single chip. VLSI began in the 1970s when complex semiconductor and communication technologies were being developed. The microprocessor is a VLSI device. Before the introduction of VLSI technology most ICs had a limited set of functions they could perform. An electronic circuit might consist of a CPU , ROM , RAM and other glue logic . VLSI allows IC makers to add all of these circuits into one chip. Function (mathematics) In mathematics ,
1780-403: A specific programming language, showing the computation involved. In such a case that the evaluation would be of syntactically illegal strings, the result would be non-computation. Semantics describes the processes a computer follows when executing a program in that specific language. This can be shown by describing the relationship between the input and output of a program, or an explanation of how
1869-437: A type of inductive learning called supervised learning. In supervised learning, an algorithm is given samples that are labeled in some useful way. For example, the samples might be descriptions of mushrooms, and the labels could be whether or not the mushrooms are edible. The algorithm takes these previously labeled samples and uses them to induce a classifier. This classifier is a function that assigns labels to samples including
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#17330860608651958-503: Is a scientific discipline that deals with the construction and study of algorithms that can learn from data. Such algorithms operate by building a model based on inputs and using that to make predictions or decisions, rather than following only explicitly programmed instructions. Machine learning can be considered a subfield of computer science and statistics . It has strong ties to artificial intelligence and optimization , which deliver methods, theory and application domains to
2047-858: Is a stub . You can help Misplaced Pages by expanding it . This science awards article is a stub . You can help Misplaced Pages by expanding it . Theoretical computer science It is difficult to circumscribe the theoretical areas precisely. The ACM 's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: TCS covers a wide variety of topics including algorithms , data structures , computational complexity , parallel and distributed computation, probabilistic computation , quantum computation , automata theory , information theory , cryptography , program semantics and verification , algorithmic game theory , machine learning , computational biology , computational economics , computational geometry , and computational number theory and algebra . Work in this field
2136-1002: Is a branch of computer science devoted to the study of algorithms that can be stated in terms of geometry . Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational geometry. The main impetus for the development of computational geometry as a discipline was progress in computer graphics and computer-aided design and manufacturing ( CAD / CAM ), but many problems in computational geometry are classical in nature, and may come from mathematical visualization . Other important applications of computational geometry include robotics (motion planning and visibility problems), geographic information systems (GIS) (geometrical location and search, route planning), integrated circuit design (IC geometry design and verification), computer-aided engineering (CAE) (mesh generation), computer vision (3D reconstruction). Theoretical results in machine learning mainly deal with
2225-464: Is a form of computation in which many calculations are carried out simultaneously, operating on the principle that large problems can often be divided into smaller ones, which are then solved "in parallel" . There are several different forms of parallel computing: bit-level , instruction level , data , and task parallelism . Parallelism has been employed for many years, mainly in high-performance computing , but interest in it has grown lately due to
2314-423: Is a function in two variables, and we want to refer to a partially applied function X → Y {\displaystyle X\to Y} produced by fixing the second argument to the value t 0 without introducing a new function name. The map in question could be denoted x ↦ f ( x , t 0 ) {\displaystyle x\mapsto f(x,t_{0})} using
2403-400: Is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time travelled and its average speed. Formally, a function of n variables is a function whose domain is a set of n -tuples. For example, multiplication of integers is a function of two variables, or bivariate function , whose domain is
2492-604: Is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects . Although, properly speaking, computer algebra should be a subfield of scientific computing , they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers , while symbolic computation emphasizes exact computation with expressions containing variables that have not any given value and are thus manipulated as symbols (therefore
2581-577: Is a software system in which components located on networked computers communicate and coordinate their actions by passing messages . The components interact with each other in order to achieve a common goal. Three significant characteristics of distributed systems are: concurrency of components, lack of a global clock, and independent failure of components. Examples of distributed systems vary from SOA-based systems to massively multiplayer online games to peer-to-peer applications , and blockchain networks like Bitcoin . A computer program that runs in
2670-452: Is always in one of two definite states (0 or 1), quantum computation uses qubits (quantum bits), which can be in superpositions of states. A theoretical model is the quantum Turing machine , also known as the universal quantum computer. Quantum computers share theoretical similarities with non-deterministic and probabilistic computers ; one example is the ability to be in more than one state simultaneously. The field of quantum computing
2759-424: Is an effective method expressed as a finite list of well-defined instructions for calculating a function . Starting from an initial state and initial input (perhaps empty ), the instructions describe a computation that, when executed , proceeds through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state. The transition from one state to
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2848-765: Is at the intersection of mathematics , statistics , computer science , physics , neurobiology , and electrical engineering . Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones, the development of the Internet , the study of linguistics and of human perception, the understanding of black holes , and numerous other fields. Important sub-fields of information theory are source coding , channel coding , algorithmic complexity theory , algorithmic information theory , information-theoretic security , and measures of information. Machine learning
2937-479: Is called the Cartesian product of X and Y and denoted X × Y . {\displaystyle X\times Y.} Thus, the above definition may be formalized as follows. A function with domain X and codomain Y is a binary relation R between X and Y that satisfies the two following conditions: This definition may be rewritten more formally, without referring explicitly to
3026-438: Is implied. The domain and codomain can also be explicitly stated, for example: This defines a function sqr from the integers to the integers that returns the square of its input. As a common application of the arrow notation, suppose f : X × X → Y ; ( x , t ) ↦ f ( x , t ) {\displaystyle f:X\times X\to Y;\;(x,t)\mapsto f(x,t)}
3115-432: Is in Y , or it is undefined. The set of the elements of X such that f ( x ) {\displaystyle f(x)} is defined and belongs to Y is called the domain of definition of the function. A partial function from X to Y is thus a ordinary function that has as its domain a subset of X called the domain of definition of the function. If the domain of definition equals X , one often says that
3204-420: Is often distinguished by its emphasis on mathematical technique and rigor . While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with a 1948 mathematical theory of communication by Claude Shannon . In
3293-451: Is often the letter f . Then, the application of the function to an argument is denoted by its name followed by its argument (or, in the case of a multivariate functions, its arguments) enclosed between parentheses, such as in The argument between the parentheses may be a variable , often x , that represents an arbitrary element of the domain of the function, a specific element of the domain ( 3 in
3382-584: Is the one-time pad —but these schemes are more difficult to implement than the best theoretically breakable but computationally secure mechanisms. A data structure is a particular way of organizing data in a computer so that it can be used efficiently . Different kinds of data structures are suited to different kinds of applications, and some are highly specialized to specific tasks. For example, databases use B-tree indexes for small percentages of data retrieval and compilers and databases use dynamic hash tables as look up tables. Data structures provide
3471-440: Is the value of the function at x , or the image of x under the function. A function f , its domain X , and its codomain Y are often specified by the notation f : X → Y . {\displaystyle f:X\to Y.} One may write x ↦ y {\displaystyle x\mapsto y} instead of y = f ( x ) {\displaystyle y=f(x)} , where
3560-464: Is theoretically possible to break such a system, but it is infeasible to do so by any known practical means. These schemes are therefore termed computationally secure; theoretical advances, e.g., improvements in integer factorization algorithms, and faster computing technology require these solutions to be continually adapted. There exist information-theoretically secure schemes that provably cannot be broken even with unlimited computing power—an example
3649-469: Is typically the case for functions whose domain is the set of the natural numbers . Such a function is called a sequence , and, in this case the element f n {\displaystyle f_{n}} is called the n th element of the sequence. The index notation can also be used for distinguishing some variables called parameters from the "true variables". In fact, parameters are specific variables that are considered as being fixed during
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3738-479: Is understood to be a task that is in principle amenable to being solved by a computer, which is equivalent to stating that the problem may be solved 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
3827-442: The graph of the function , a popular means of illustrating the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane. Functions are widely used in science , engineering , and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics. The concept of
3916-687: The Riemann hypothesis . In computability theory , a general recursive function is a partial function from the integers to the integers whose values can be computed by an algorithm (roughly speaking). The domain of definition of such a function is the set of inputs for which the algorithm does not run forever. A fundamental theorem of computability theory is that there cannot exist an algorithm that takes an arbitrary general recursive function as input and tests whether 0 belongs to its domain of definition (see Halting problem ). A multivariate function , multivariable function , or function of several variables
4005-413: The complex numbers , one talks respectively of a function of several real variables or of a function of several complex variables . There are various standard ways for denoting functions. The most commonly used notation is functional notation, which is the first notation described below. The functional notation requires that a name is given to the function, which, in the case of a unspecified function
4094-412: The quantification of information . Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and communicating data. Since its inception it has broadened to find applications in many other areas, including statistical inference , natural language processing , cryptography , neurobiology ,
4183-420: The zeros of f. This is one of the reasons for which, in mathematical analysis , "a function from X to Y " may refer to a function having a proper subset of X as a domain. For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable whose domain is a proper subset of the real numbers , typically a subset that contains a non-empty open interval . Such
4272-550: The "total" condition removed. That is, a partial function from X to Y is a binary relation R between X and Y such that, for every x ∈ X , {\displaystyle x\in X,} there is at most one y in Y such that ( x , y ) ∈ R . {\displaystyle (x,y)\in R.} Using functional notation, this means that, given x ∈ X , {\displaystyle x\in X,} either f ( x ) {\displaystyle f(x)}
4361-688: The Greek word αὐτόματα meaning "self-acting". Automata Theory is the study of self-operating virtual machines to help in the logical understanding of input and output process, without or with intermediate stage(s) of computation (or any function /process). Coding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression , cryptography , error correction and more recently also for network coding . Codes are studied by various scientific disciplines – such as information theory , electrical engineering , mathematics , and computer science – for
4450-452: The above example), or an expression that can be evaluated to an element of the domain ( x 2 + 1 {\displaystyle x^{2}+1} in the above example). The use of a unspecified variable between parentheses is useful for defining a function explicitly such as in "let f ( x ) = sin ( x 2 + 1 ) {\displaystyle f(x)=\sin(x^{2}+1)} ". When
4539-469: The amount of resources needed to solve them, such as time and storage. Other complexity measures 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 computational complexity theory is to determine the practical limits on what computers can and cannot do. Computational geometry
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#17330860608654628-423: The application of a fairly broad variety of theoretical computer science fundamentals, in particular logic calculi, formal languages , automata theory , and program semantics , but also type systems and algebraic data types to problems in software and hardware specification and verification. Information theory is a branch of applied mathematics , electrical engineering , and computer science involving
4717-432: The arrow notation for functions described above. In some cases the argument of a function may be an ordered pair of elements taken from some set or sets. For example, a function f can be defined as mapping any pair of real numbers ( x , y ) {\displaystyle (x,y)} to the sum of their squares, x 2 + y 2 {\displaystyle x^{2}+y^{2}} . Such
4806-590: The arrow notation. The expression x ↦ f ( x , t 0 ) {\displaystyle x\mapsto f(x,t_{0})} (read: "the map taking x to f of x comma t nought") represents this new function with just one argument, whereas the expression f ( x 0 , t 0 ) refers to the value of the function f at the point ( x 0 , t 0 ) . Index notation may be used instead of functional notation. That is, instead of writing f ( x ) , one writes f x . {\displaystyle f_{x}.} This
4895-434: The concept of a relation, but using more notation (including set-builder notation ): A function is formed by three sets, the domain X , {\displaystyle X,} the codomain Y , {\displaystyle Y,} and the graph R {\displaystyle R} that satisfy the three following conditions. Partial functions are defined similarly to ordinary functions, with
4984-443: The discipline of theoretical computer science, both depending on and affecting mathematics , software engineering, and linguistics . It is an active research area, with numerous dedicated academic journals. In programming language theory , semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages . It does so by evaluating the meaning of syntactically legal strings defined by
5073-428: The disciplines of mathematics , computer science , and electrical engineering . Applications of cryptography include ATM cards , computer passwords , and electronic commerce . Modern cryptography is heavily based on mathematical theory and computer science practice; cryptographic algorithms are designed around computational hardness assumptions , making such algorithms hard to break in practice by any adversary. It
5162-476: The domain and some (possibly all) elements of the codomain. Mathematically, a binary relation between two sets X and Y is a subset of the set of all ordered pairs ( x , y ) {\displaystyle (x,y)} such that x ∈ X {\displaystyle x\in X} and y ∈ Y . {\displaystyle y\in Y.} The set of all these pairs
5251-410: The domain of definition of a complex function is illustrated by the multiplicative inverse of the Riemann zeta function : the determination of the domain of definition of the function z ↦ 1 / ζ ( z ) {\displaystyle z\mapsto 1/\zeta (z)} is more or less equivalent to the proof or disproof of one of the major open problems in mathematics,
5340-448: The domain of definition of a multiplicative inverse of a (partial) function amounts to compute the zeros of the function, the values where the function is defined but not its multiplicative inverse. Similarly, a function of a complex variable is generally a partial function with a domain of definition included in the set C {\displaystyle \mathbb {C} } of the complex numbers . The difficulty of determining
5429-482: The evolution and function of molecular codes, model selection in statistics, thermal physics, quantum computing , linguistics , plagiarism detection, pattern recognition , anomaly detection and other forms of data analysis . Applications of fundamental topics of information theory include lossless data compression (e.g. ZIP files ), lossy data compression (e.g. MP3s and JPEGs ), and channel coding (e.g. for Digital Subscriber Line (DSL) ). The field
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#17330860608655518-480: The field is integer factorization . Cryptography is the practice and study of techniques for secure communication in the presence of third parties (called adversaries ). More generally, it is about constructing and analyzing protocols that overcome the influence of adversaries and that are related to various aspects in information security such as data confidentiality , data integrity , authentication , and non-repudiation . Modern cryptography intersects
5607-471: The field. Machine learning is employed in a range of computing tasks where designing and programming explicit, rule-based algorithms is infeasible. Example applications include spam filtering , optical character recognition (OCR), search engines and computer vision . Machine learning is sometimes conflated with data mining , although that focuses more on exploratory data analysis. Machine learning and pattern recognition "can be viewed as two facets of
5696-408: The founders of calculus , Leibniz , Newton and Euler . However, it cannot be formalized , since there is no mathematical definition of an "assignment". It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms of set theory . This set-theoretic definition is based on the fact that a function establishes a relation between the elements of
5785-474: The function f (⋅) from its value f ( x ) at x . For example, a ( ⋅ ) 2 {\displaystyle a(\cdot )^{2}} may stand for the function x ↦ a x 2 {\displaystyle x\mapsto ax^{2}} , and ∫ a ( ⋅ ) f ( u ) d u {\textstyle \int _{a}^{\,(\cdot )}f(u)\,du} may stand for
5874-407: The function. If the element y in Y is assigned to x in X by the function f , one says that f maps x to y , and this is commonly written y = f ( x ) . {\displaystyle y=f(x).} In this notation, x is the argument or variable of the function. A specific element x of X is a value of the variable , and the corresponding element of Y
5963-823: The functioning of the brain , Darwinian evolution , group behavior , the immune system , the defining properties of life forms, cell membranes , and morphogenesis . Besides traditional electronic hardware , these computational paradigms can be implemented on alternative physical media such as biomolecules (DNA, RNA), or trapped-ion quantum computing devices. Dually, one can view processes occurring in nature as information processing. Such processes include self-assembly , developmental processes , gene regulation networks, protein–protein interaction networks, biological transport ( active transport , passive transport ) networks, and gene assembly in unicellular organisms . Efforts to understand biological systems also include engineering of semi-synthetic organisms, and understanding
6052-520: The most common. Communication and synchronization between the different subtasks are typically some of the greatest obstacles to getting good parallel program performance. The maximum possible speed-up of a single program as a result of parallelization is known as Amdahl's law . Programming language theory is a branch of computer science that deals with the design, implementation, analysis, characterization, and classification of programming languages and their individual features . It falls within
6141-416: The name of symbolic computation ). Software applications that perform symbolic calculations are called computer algebra systems , with the term system alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the language used for the implementation), a dedicated memory manager,
6230-427: The next is not necessarily deterministic ; some algorithms, known as randomized algorithms , incorporate random input. Automata theory is the study of abstract machines and automata , as well as the computational problems that can be solved using them. It is a theory in theoretical computer science, under discrete mathematics (a section of mathematics and also of computer science ). Automata comes from
6319-436: The notation x ↦ f ( x ) , {\displaystyle x\mapsto f(x),} the symbol x does not represent any value; it is simply a placeholder , meaning that, if x is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, x may be replaced by any symbol, often an interpunct " ⋅ ". This may be useful for distinguishing
6408-404: The partial function is a total function . In several areas of mathematics the term "function" refers to partial functions rather than to ordinary functions. This is typically the case when functions may be specified in a way that makes difficult or even impossible to determine their domain. In calculus , a real-valued function of a real variable or real function is a partial function from
6497-487: The physical constraints preventing frequency scaling . As power consumption (and consequently heat generation) by computers has become a concern in recent years, parallel computing has become the dominant paradigm in computer architecture , mainly in the form of multi-core processors . Parallel computer programs are more difficult to write than sequential ones, because concurrency introduces several new classes of potential software bugs , of which race conditions are
6586-453: The program will execute on a certain platform , hence creating a model of computation . A quantum computer is a computation system that makes direct use of quantum-mechanical phenomena , such as superposition and entanglement , to perform operations on data . Quantum computers are different from digital computers based on transistors . Whereas digital computers require data to be encoded into binary digits ( bits ), each of which
6675-435: The purpose of designing efficient and reliable data transmission methods. This typically involves the removal of redundancy and the correction (or detection) of errors in the transmitted data. Computational complexity theory is a branch of the theory of computation that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other. A computational problem
6764-408: The same decade, Donald Hebb introduced a mathematical model of learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and parallel distributed processing were established. In 1971, Stephen Cook and, working independently , Leonid Levin , proved that there exist practically relevant problems that are NP-complete –
6853-791: The same field." Natural computing , also called natural computation, is a terminology introduced to encompass three classes of methods: 1) those that take inspiration from nature for the development of novel problem-solving techniques; 2) those that are based on the use of computers to synthesize natural phenomena; and 3) those that employ natural materials (e.g., molecules) to compute. The main fields of research that compose these three branches are artificial neural networks , evolutionary algorithms , swarm intelligence , artificial immune systems , fractal geometry, artificial life , DNA computing , and quantum computing , among others. Computational paradigms studied by natural computing are abstracted from natural phenomena as diverse as self-replication ,
6942-399: The samples that have never been previously seen by the algorithm. The goal of the supervised learning algorithm is to optimize some measure of performance such as minimizing the number of mistakes made on new samples. Computational number theory , also known as algorithmic number theory , is the study of algorithms for performing number theoretic computations . The best known problem in
7031-408: The set R {\displaystyle \mathbb {R} } of the real numbers to itself. Given a real function f : x ↦ f ( x ) {\displaystyle f:x\mapsto f(x)} its multiplicative inverse x ↦ 1 / f ( x ) {\displaystyle x\mapsto 1/f(x)} is also a real function. The determination of
7120-795: The set of all n -tuples ( x 1 , … , x n ) {\displaystyle (x_{1},\ldots ,x_{n})} such that x 1 ∈ X 1 , … , x n ∈ X n {\displaystyle x_{1}\in X_{1},\ldots ,x_{n}\in X_{n}} is called the Cartesian product of X 1 , … , X n , {\displaystyle X_{1},\ldots ,X_{n},} and denoted X 1 × ⋯ × X n . {\displaystyle X_{1}\times \cdots \times X_{n}.} Therefore,
7209-865: The set of all ordered pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation . Commonly, an n -tuple is denoted enclosed between parentheses, such as in ( 1 , 2 , … , n ) . {\displaystyle (1,2,\ldots ,n).} When using functional notation , one usually omits the parentheses surrounding tuples, writing f ( x 1 , … , x n ) {\displaystyle f(x_{1},\ldots ,x_{n})} instead of f ( ( x 1 , … , x n ) ) . {\displaystyle f((x_{1},\ldots ,x_{n})).} Given n sets X 1 , … , X n , {\displaystyle X_{1},\ldots ,X_{n},}
7298-594: The study of a problem. For example, the map x ↦ f ( x , t ) {\displaystyle x\mapsto f(x,t)} (see above) would be denoted f t {\displaystyle f_{t}} using index notation, if we define the collection of maps f t {\displaystyle f_{t}} by the formula f t ( x ) = f ( x , t ) {\displaystyle f_{t}(x)=f(x,t)} for all x , t ∈ X {\displaystyle x,t\in X} . In
7387-443: The symbol ↦ {\displaystyle \mapsto } (read ' maps to ') is used to specify where a particular element x in the domain is mapped to by f . This allows the definition of a function without naming. For example, the square function is the function x ↦ x 2 . {\displaystyle x\mapsto x^{2}.} The domain and codomain are not always explicitly given when
7476-447: The symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write sin x instead of sin( x ) . Functional notation was first used by Leonhard Euler in 1734. Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). In this case,
7565-460: The universe itself from the point of view of information processing. Indeed, the idea was even advanced that information is more fundamental than matter or energy. The Zuse-Fredkin thesis, dating back to the 1960s, states that the entire universe is a huge cellular automaton which continuously updates its rules. Recently it has been suggested that the whole universe is a quantum computer that computes its own behaviour. Parallel computing
7654-432: The value of the function at a particular value; for example, if f ( x ) = x 2 + 1 , {\displaystyle f(x)=x^{2}+1,} then f ( 4 ) = 4 2 + 1 = 17. {\displaystyle f(4)=4^{2}+1=17.} Given its domain and its codomain, a function is uniquely represented by the set of all pairs ( x , f ( x )) , called
7743-504: The word mapping for the case where the structure of the codomain belongs explicitly to the definition of the function. Some authors, such as Serge Lang , use "function" only to refer to maps for which the codomain is a subset of the real or complex numbers, and use the term mapping for more general functions. In the theory of dynamical systems , a map denotes an evolution function used to create discrete dynamical systems . See also Poincaré map . Whichever definition of map
7832-415: Was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory , and this greatly increased the possible applications of the concept. A function is often denoted by
7921-648: Was first introduced by Yuri Manin in 1980 and Richard Feynman in 1982. A quantum computer with spins as quantum bits was also formulated for use as a quantum space–time in 1968. Experiments have been carried out in which quantum computational operations were executed on a very small number of qubits. Both practical and theoretical research continues, and many national governments and military funding agencies support quantum computing research to develop quantum computers for both civilian and national security purposes, such as cryptanalysis . Computer algebra , also called symbolic computation or algebraic computation
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