AI Coding Interview Questions

AI Coding Interview Questions — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Application software

  Application software is software that is intended for end-user use – not operating, administering or programming a computer. It includes programs such as word processors, web browsers, media players, and mobile applications used in daily tasks. An application (app, application program, software application) is any program that can be categorized as application software. Application is a subjective classification that is often used to differentiate from system and utility software. Application software represents the user-facing layer of computing systems, designed to translate complex system capabilities into task-oriented, goal-driven workflows. Unlike system software, which focuses on hardware orchestration and resource management, application software is centered on problem abstraction, user interaction, and domain-specific functionality. The abbreviation app became popular with the 2008 introduction of the iOS App Store, to refer to applications for mobile devices such as smartphones and tablets. Later, with the release of the Mac App Store in 2010 and the Windows Store in 2011, it began to be used to refer to end-user software in general, regardless of platform. Applications may be bundled with the computer and its system software or published separately. Applications may be proprietary or open-source. == Terminology == === Meaning program and software === When used as an adjective, application can have a broader meaning than that described in this article. For example, concepts such as application programming interface (API), application server, application virtualization, application lifecycle management and portable application refer to programs and software in general. === Distinction between system and application software === The distinction between system and application software is subjective and has been the subject of controversy. For example, one of the key questions in the United States v. Microsoft Corp. antitrust trial was whether Microsoft's Internet Explorer web browser was part of its Windows operating system or a separate piece of application software. As another example, the GNU/Linux naming controversy is, in part, due to disagreement about the relationship between the Linux kernel and the operating systems built over this kernel. In some types of embedded systems, the application software and the operating system software may be indistinguishable by the user, as in the case of software used to control a VCR, DVD player, or microwave oven. The above definitions may exclude some applications that may exist on some computers in large organizations. For an alternative definition of an app: see Application Portfolio Management. === Killer application === A killer application (killer app, coined in the late 1980s) is an application that is so popular that it causes demand for its host platform to increase. For example, VisiCalc was the first modern spreadsheet software for the Apple II and helped sell the then-new personal computers into offices. For the BlackBerry, it was its email software. === Software suite === As software suite consists of multiple applications bundled together. They usually have related functions, features, and user interfaces, and may be able to interact with each other, e.g. open each other's files. Business applications often come in suites, e.g. Microsoft Office, LibreOffice and iWork, which bundle together a word processor, a spreadsheet, etc.; but suites exist for other purposes, e.g. graphics or music. == Ways to classify == As there so many applications and since their attributes vary so dramatically, there are many different ways to classify them. === By legal aspects === Proprietary software is protected under an exclusive copyright, and a software license grants limited usage rights. Such applications may allow add-ons from third parties. Free and open-source software (FOSS) can be run, distributed, sold, and extended for any purpose. FOSS software released under a free license may be perpetual and also royalty-free. Perhaps, the owner, the holder or third-party enforcer of any right (copyright, trademark, patent, or ius in re aliena) are entitled to add exceptions, limitations, time decays or expiring dates to the license terms of use. Public-domain software is a type of FOSS that is royalty-free and can be run, distributed, modified, reversed, republished, or created in derivative works without any copyright attribution and therefore revocation. It can even be sold, but without transferring the public domain property to other single subjects. Public-domain software can be released under a (un)licensing legal statement, which enforces those terms and conditions for an indefinite duration (for a lifetime, or forever). === By platform === An application can be categorized by the host platform on which it runs. Notable platforms include operating system (native), web browser, cloud computing and mobile. For example a web application runs in a web browser whereas a more traditional, native application runs in the environment of a computer's operating system. There has been a contentious debate regarding web applications replacing native applications for many purposes, especially on mobile devices such as smartphones and tablets. Web apps have indeed greatly increased in popularity for some uses, but the advantages of applications make them unlikely to disappear soon, if ever. Furthermore, the two can be complementary, and even integrated. === Horizontal vs. vertical === Application software can be seen as either horizontal or vertical. Horizontal applications are more popular and widespread, because they are general purpose, for example word processors or databases. Vertical applications are niche products, designed for a particular type of industry or business, or department within an organization. Integrated suites of software will try to handle every specific aspect possible of, for example, manufacturing or banking worker, accounting, or customer service. === By purpose === There are many types of application software: Enterprise Addresses the needs of an entire organization's processes and data flows, across several departments, often in a large distributed environment. Examples include enterprise resource planning systems, customer relationship management (CRM) systems, data replication engines, and supply chain management software. Departmental Software is a sub-type of enterprise software with a focus on smaller organizations or groups within a large organization. (Examples include travel expense management and IT Helpdesk.) Enterprise infrastructure Provides common capabilities needed to support enterprise software systems. (Examples include databases, email servers, and systems for managing networks and security.) Application platform as a service (aPaaS) A cloud computing service that offers development and deployment environments for application services. Knowledge worker Lets users create and manage information, often for and individual media editors may aid in multiple information worker tasks. Content access Used primarily to access content without editing, but may include software that allows for content editing. Such software addresses the needs of individuals and groups to consume digital entertainment and published digital content. (Examples include media players, web browsers, and help browsers.) Educational Related to content access software, but has the content or features adapted for use by educators or students. For example, it may deliver evaluations (tests), track progress through material, or include collaborative capabilities. Simulation Simulates physical or abstract systems for either research, training, or entertainment purposes. Media development Generates print and electronic media for others to consume, most often in a commercial or educational setting. This includes graphic-art software, desktop publishing software, multimedia development software, HTML editors, digital-animation editors, digital audio and video composition, and many others. Engineering Used in developing hardware and software products. This includes computer-aided design (CAD), computer-aided engineering (CAE), computer language editing and compiling tools, integrated development environments, and application programmer interfaces. Entertainment Refers to video games, screen savers, programs to display motion pictures or play recorded music, and other forms of entertainment which can be experienced through the use of a computing device. == Taxonomy == This section is a taxonomy of kinds of applications. This organization is but one of many different ways to organize them. A kind is included in only one category even if it logically fits in multiple. === General-purpose === Calculator Spreadsheet Web browser Web mapping E-commerce Social media === Communication === Chat Email Presentation software Phone Messages Networking software Web conferencing === Documentation === Desktop

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  Empirical dynamic modeling

  Empirical dynamic modeling (EDM) is a framework for analysis and prediction of nonlinear dynamical systems. Applications include population dynamics, ecosystem service, medicine, neuroscience, dynamical systems, geophysics, and human-computer interaction. EDM was originally developed by Robert May and George Sugihara. It can be considered a methodology for data modeling, predictive analytics, dynamical system analysis, machine learning and time series analysis. == Description == Mathematical models have tremendous power to describe observations of real-world systems. They are routinely used to test hypothesis, explain mechanisms and predict future outcomes. However, real-world systems are often nonlinear and multidimensional, in some instances rendering explicit equation-based modeling problematic. Empirical models, which infer patterns and associations from the data instead of using hypothesized equations, represent a natural and flexible framework for modeling complex dynamics. Donald DeAngelis and Simeon Yurek illustrated that canonical statistical models are ill-posed when applied to nonlinear dynamical systems. A hallmark of nonlinear dynamics is state-dependence: system states are related to previous states governing transition from one state to another. EDM operates in this space, the multidimensional state-space of system dynamics rather than on one-dimensional observational time series. EDM does not presume relationships among states, for example, a functional dependence, but projects future states from localised, neighboring states. EDM is thus a state-space, nearest-neighbors paradigm where system dynamics are inferred from states derived from observational time series. This provides a model-free representation of the system naturally encompassing nonlinear dynamics. A cornerstone of EDM is recognition that time series observed from a dynamical system can be transformed into higher-dimensional state-spaces by time-delay embedding with Takens's theorem. The state-space models are evaluated based on in-sample fidelity to observations, conventionally with Pearson correlation between predictions and observations. == Methods == Primary EDM algorithms include Simplex projection, Sequential locally weighted global linear maps (S-Map) projection, Multivariate embedding in Simplex or S-Map, Convergent cross mapping (CCM), and Multiview Embeding, described below. Nearest neighbors are found according to: NN ( y , X , k ) = ‖ X N i E − y ‖ ≤ ‖ X N j E − y ‖ if 1 ≤ i ≤ j ≤ k {\displaystyle {\text{NN}}(y,X,k)=\|X_{N_{i}}^{E}-y\|\leq \|X_{N_{j}}^{E}-y\|{\text{ if }}1\leq i\leq j\leq k} === Simplex === Simplex projection is a nearest neighbor projection. It locates the k {\displaystyle k} nearest neighbors to the location in the state-space from which a prediction is desired. To minimize the number of free parameters k {\displaystyle k} is typically set to E + 1 {\displaystyle E+1} defining an E + 1 {\displaystyle E+1} dimensional simplex in the state-space. The prediction is computed as the average of the weighted phase-space simplex projected T p {\displaystyle Tp} points ahead. Each neighbor is weighted proportional to their distance to the projection origin vector in the state-space. Find k {\displaystyle k} nearest neighbor: N k ← NN ( y , X , k ) {\displaystyle N_{k}\gets {\text{NN}}(y,X,k)} Define the distance scale: d ← ‖ X N 1 E − y ‖ {\displaystyle d\gets \|X_{N_{1}}^{E}-y\|} Compute weights: For{ i = 1 , … , k {\displaystyle i=1,\dots ,k} } : w i ← exp ⁡ ( − ‖ X N i E − y ‖ / d ) {\displaystyle w_{i}\gets \exp(-\|X_{N_{i}}^{E}-y\|/d)} Average of state-space simplex: y ^ ← ∑ i = 1 k ( w i X N i + T p ) / ∑ i = 1 k w i {\displaystyle {\hat {y}}\gets \sum _{i=1}^{k}\left(w_{i}X_{N_{i}+T_{p}}\right)/\sum _{i=1}^{k}w_{i}} === S-Map === S-Map extends the state-space prediction in Simplex from an average of the E + 1 {\displaystyle E+1} nearest neighbors to a linear regression fit to all neighbors, but localised with an exponential decay kernel. The exponential localisation function is F ( θ ) = exp ( − θ d / D ) {\displaystyle F(\theta )={\text{exp}}(-\theta d/D)} , where d {\displaystyle d} is the neighbor distance and D {\displaystyle D} the mean distance. In this way, depending on the value of θ {\displaystyle \theta } , neighbors close to the prediction origin point have a higher weight than those further from it, such that a local linear approximation to the nonlinear system is reasonable. This localisation ability allows one to identify an optimal local scale, in-effect quantifying the degree of state dependence, and hence nonlinearity of the system. Another feature of S-Map is that for a properly fit model, the regression coefficients between variables have been shown to approximate the gradient (directional derivative) of variables along the manifold. These Jacobians represent the time-varying interaction strengths between system variables. Find k {\displaystyle k} nearest neighbor: N ← NN ( y , X , k ) {\displaystyle N\gets {\text{NN}}(y,X,k)} Sum of distances: D ← 1 k ∑ i = 1 k ‖ X N i E − y ‖ {\displaystyle D\gets {\frac {1}{k}}\sum _{i=1}^{k}\|X_{N_{i}}^{E}-y\|} Compute weights: For{ i = 1 , … , k {\displaystyle i=1,\dots ,k} } : w i ← exp ⁡ ( − θ ‖ X N i E − y ‖ / D ) {\displaystyle w_{i}\gets \exp(-\theta \|X_{N_{i}}^{E}-y\|/D)} Reweighting matrix: W ← diag ( w i ) {\displaystyle W\gets {\text{diag}}(w_{i})} Design matrix: A ← [ 1 X N 1 X N 1 − 1 … X N 1 − E + 1 1 X N 2 X N 2 − 1 … X N 2 − E + 1 ⋮ ⋮ ⋮ ⋱ ⋮ 1 X N k X N k − 1 … X N k − E + 1 ] {\displaystyle A\gets {\begin{bmatrix}1&X_{N_{1}}&X_{N_{1}-1}&\dots &X_{N_{1}-E+1}\\1&X_{N_{2}}&X_{N_{2}-1}&\dots &X_{N_{2}-E+1}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&X_{N_{k}}&X_{N_{k}-1}&\dots &X_{N_{k}-E+1}\end{bmatrix}}} Weighted design matrix: A ← W A {\displaystyle A\gets WA} Response vector at T p {\displaystyle Tp} : b ← [ X N 1 + T p X N 2 + T p ⋮ X N k + T p ] {\displaystyle b\gets {\begin{bmatrix}X_{N_{1}+T_{p}}\\X_{N_{2}+T_{p}}\\\vdots \\X_{N_{k}+T_{p}}\end{bmatrix}}} Weighted response vector: b ← W b {\displaystyle b\gets Wb} Least squares solution (SVD): c ^ ← argmin c ‖ A c − b ‖ 2 2 {\displaystyle {\hat {c}}\gets {\text{argmin}}_{c}\|Ac-b\|_{2}^{2}} Local linear model c ^ {\displaystyle {\hat {c}}} is prediction: y ^ ← c ^ 0 + ∑ i = 1 E c ^ i y i {\displaystyle {\hat {y}}\gets {\hat {c}}_{0}+\sum _{i=1}^{E}{\hat {c}}_{i}y_{i}} === Multivariate Embedding === Multivariate Embedding recognizes that time-delay embeddings are not the only valid state-space construction. In Simplex and S-Map one can generate a state-space from observational vectors, or time-delay embeddings of a single observational time series, or both. === Convergent Cross Mapping === Convergent cross mapping (CCM) leverages a corollary to the Generalized Takens Theorem that it should be possible to cross predict or cross map between variables observed from the same system. Suppose that in some dynamical system involving variables X {\displaystyle X} and Y {\displaystyle Y} , X {\displaystyle X} causes Y {\displaystyle Y} . Since X {\displaystyle X} and Y {\displaystyle Y} belong to the same dynamical system, their reconstructions (via embeddings) M x {\displaystyle M_{x}} , and M y {\displaystyle M_{y}} , also map to the same system. The causal variable X {\displaystyle X} leaves a signature on the affected variable Y {\displaystyle Y} , and consequently, the reconstructed states based on Y {\displaystyle Y} can be used to cross predict values of X {\displaystyle X} . CCM leverages this property to infer causality by predicting X {\displaystyle X} using the M y {\displaystyle M_{y}} library of points (or vice versa for the other direction of causality), while assessing improvements in cross map predictability as larger and larger random samplings of M y {\displaystyle M_{y}} are used. If the prediction skill of X {\displaystyle X} increases and saturates as the entire M y {\displaystyle M_{y}} is used, this provides evidence that X {\displaystyle X} is casually influencing Y {\displaystyle Y} . === Multiview Embedding === Multiview Embedding is a Dimensionality reduction technique where a large number of state-space time series vectors are combitorially assessed towards maximal model predictability. == Extensions == Extensions to EDM techniques include: Generalized Theorems for Nonlinear State Space Reconstruction Extended Convergent Cross Mapping Dynamic stability S-Map regularization Visual analytics with EDM Convergent Cross Sorting Expert system with EDM hybrid Sliding windows based on the extended convergent cross-mapping Empirical Mode Modeling Accounting for missing data and variable step sizes Accounting for observation noise Hierarchical Bayesian EDM via Gaussian processes Intelligent and Adaptive Control Optimal control via Empirical dynamic programming Multiview distance regularised S-map

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  Algorithmic bias

  Algorithmic bias describes systematic and repeatable harmful tendency in a computerized sociotechnical system to create "unfair" outcomes, such as "privileging" one category over another in ways that may or may not be different from the intended function of the algorithm. Bias can emerge from many factors, including intentionally biased design decisions or the unintended or unanticipated use or decisions relating to the way data is coded, collected, selected or used to train the algorithm. For example, algorithmic bias has been observed in search engine results and social media platforms. This bias can have impacts ranging from privacy violations to reinforcing social biases of race, gender, sexuality, and ethnicity. The study of algorithmic bias is most concerned with algorithms that reflect "systematic and unfair" discrimination. This bias has only recently been addressed in legal frameworks, such as the European Union's General Data Protection Regulation (enforced in 2018) and the Artificial Intelligence Act (proposed in 2021 and adopted in 2024). As algorithms expand their ability to organize society, politics, institutions, and behavior, sociologists have become concerned with the ways in which unanticipated output and manipulation of data can impact the physical world. Because algorithms are often considered to be neutral and unbiased, they can inaccurately project greater authority than human expertise (in part due to the psychological phenomenon of automation bias), and in some cases, reliance on algorithms can displace human responsibility for their outcomes, without last mile thinking. Bias can enter into algorithmic systems as a result of pre-existing cultural, social, or institutional expectations; by how features and labels are chosen; because of technical limitations of their design; or by being used in unanticipated contexts or by audiences who are not considered in the software's initial design. Algorithmic bias has been cited in cases ranging from election outcomes to the spread of online hate speech. It has also arisen in criminal justice, healthcare, and hiring, compounding existing racial, socioeconomic, and gender biases. The relative inability of facial recognition technology to accurately identify darker-skinned faces has been linked to multiple wrongful arrests of black men, an issue stemming from imbalanced datasets. Problems in understanding, researching, and discovering algorithmic bias persist due to the proprietary nature of algorithms, which are typically treated as trade secrets. Even when full transparency is provided, the complexity of certain algorithms poses a barrier to understanding their functioning. Furthermore, algorithms may change, or respond to input or output in ways that cannot be anticipated or easily reproduced for analysis. In many cases, even within a single website or application, there is no single "algorithm" to examine, but a network of many interrelated programs and data inputs, even between users of the same service. A 2021 survey identified multiple forms of algorithmic bias, including historical, representation, and measurement biases, each of which can contribute to unfair outcomes. == Definitions == Algorithms are difficult to define, but may be generally understood as lists of instructions that determine how programs read, collect, process, and analyze data to generate a usable output. For a rigorous technical introduction, see Algorithms. Advances in computer hardware and software have led to an increased capability to process, store and transmit data. This has in turn made the design and adoption of technologies such as machine learning and artificial intelligence technically and commercially feasible. By analyzing and processing data, algorithms are the backbone of search engines, social media websites, recommendation engines, online retail, online advertising, and more. Contemporary social scientists are concerned with algorithmic processes embedded into hardware and software applications because of their political and social impact, and question the underlying assumptions of an algorithm's neutrality. The term algorithmic bias describes systematic and repeatable errors that create unfair outcomes, such as privileging one arbitrary group of users over others. For example, a credit score algorithm may deny a loan without being unfair, if it is consistently weighing relevant financial criteria. If the algorithm recommends loans to one group of users, but denies loans to another set of nearly identical users based on unrelated criteria, and if this behavior can be repeated across multiple occurrences, an algorithm can be described as biased. This bias may be intentional or unintentional (for example, it can come from biased data obtained from a worker that previously did the job the algorithm is going to do from now on). == Methods == Bias can be introduced to an algorithm in several ways. During the assemblage of a dataset, data may be collected, digitized, adapted, and entered into a database according to human-designed cataloging criteria. Next, programmers assign priorities, or hierarchies, for how a program assesses and sorts that data. This requires human decisions about how data is categorized, and which data is included or discarded. Some algorithms collect their own data based on human-selected criteria, which can also reflect the bias of human designers. Other algorithms may reinforce stereotypes and preferences as they process and display "relevant" data for human users, for example, by selecting information based on previous choices of a similar user or group of users. Beyond assembling and processing data, bias can emerge as a result of design. For example, algorithms that determine the allocation of resources or scrutiny (such as determining school placements) may inadvertently discriminate against a category when determining risk based on similar users (as in credit scores). Meanwhile, recommendation engines that work by associating users with similar users, or that make use of inferred marketing traits, might rely on inaccurate associations that reflect broad ethnic, gender, socio-economic, or racial stereotypes. Another example comes from determining criteria for what is included and excluded from results. These criteria could present unanticipated outcomes for search results, such as with flight-recommendation software that omits flights that do not follow the sponsoring airline's flight paths. Algorithms may also display an uncertainty bias, offering more confident assessments when larger data sets are available. This can skew algorithmic processes toward results that more closely correspond with larger samples, which may disregard data from underrepresented populations. == History == === Early critiques === The earliest computer programs were designed to mimic human reasoning and deductions, and were deemed to be functioning when they successfully and consistently reproduced that human logic. In his 1976 book Computer Power and Human Reason, artificial intelligence pioneer Joseph Weizenbaum suggested that bias could arise both from the data used in a program, but also from the way a program is coded. Weizenbaum wrote that programs are a sequence of rules created by humans for a computer to follow. By following those rules consistently, such programs "embody law", that is, enforce a specific way to solve problems. The rules a computer follows are based on the assumptions of a computer programmer for how these problems might be solved. That means the code could incorporate the programmer's imagination of how the world works, including their biases and expectations. While a computer program can incorporate bias in this way, Weizenbaum also noted that any data fed to a machine additionally reflects "human decision making processes" as data is being selected. Finally, he noted that machines might also transfer good information with unintended consequences if users are unclear about how to interpret the results. Weizenbaum warned against trusting decisions made by computer programs that a user doesn't understand, comparing such faith to a tourist who can find his way to a hotel room exclusively by turning left or right on a coin toss. Crucially, the tourist has no basis of understanding how or why he arrived at his destination, and a successful arrival does not mean the process is accurate or reliable. An early example of algorithmic bias resulted in as many as 60 women and ethnic minorities denied entry to St. George's Hospital Medical School per year from 1982 to 1986, based on implementation of a new computer-guidance assessment system that denied entry to women and men with "foreign-sounding names" based on historical trends in admissions. While many schools at the time employed similar biases in their selection process, St. George was most notable for automating said bias through the use of an algorithm, thus gaining the attention of people on a much

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  Percept (artificial intelligence)

  A percept is the input that an intelligent agent is perceiving at any given moment. It is essentially the same concept as a percept in psychology, except that it is being perceived not by the brain but by the agent. A percept is detected by a sensor, often a camera, processed accordingly, and acted upon by an actuator. Each percept is added to a "percept sequence", which is a complete history of each percept ever detected. The agent's action at any instant point may depend on the entire percept sequence up to that particular instant point. An intelligent agent chooses how to act not only based on the current percept, but the percept sequence. The next action is chosen by the agent function, which maps every percept to an action. For example, if a camera were to record a gesture, the agent would process the percepts, calculate the corresponding spatial vectors, examine its percept history, and use the agent program (the application of the agent function) to act accordingly. == Examples == Examples of percepts include inputs from touch sensors, cameras, infrared sensors, sonar, microphones, mice, and keyboards. A percept can also be a higher-level feature of the data, such as lines, depth, objects, faces, or gestures.

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  Kleene star

  In formal language theory, the Kleene star (or Kleene operator or Kleene closure) refers to two related unary operations, that can be applied either to an alphabet of symbols or to a formal language, a set of strings (finite sequences of symbols). The Kleene star operator on an alphabet V generates the set V of all finite-length strings over V, that is, finite sequences whose elements belong to V; in mathematics, it is more commonly known as the free monoid construction. The Kleene star operator on a language L generates another language L, the set of all strings that can be obtained as a concatenation of zero or more members of L. In both cases, repetitions are allowed. The Kleene star operators are named after American mathematician Stephen Cole Kleene, who first introduced and widely used it to characterize automata for regular expressions. == Of an alphabet == Given an alphabet V {\displaystyle V} , define V 0 = { ε } {\displaystyle V^{0}=\{\varepsilon \}} (the set consists only of the empty string), V 1 = V , {\displaystyle V^{1}=V,} and define recursively the set V i + 1 = { w v : w ∈ V i and v ∈ V } {\displaystyle V^{i+1}=\{wv:w\in V^{i}{\text{ and }}v\in V\}} for each i > 0 , {\displaystyle i>0,} where w v {\displaystyle wv} denotes the string obtained by appending the single character v {\displaystyle v} to the end of w {\displaystyle w} . Here, V i {\displaystyle V^{i}} can be understood to be the set of all strings of length exactly i {\displaystyle i} , with characters from V {\displaystyle V} . The definition of Kleene star on V {\displaystyle V} is V ∗ = ⋃ i ≥ 0 V i = V 0 ∪ V 1 ∪ V 2 ∪ V 3 ∪ V 4 ∪ ⋯ . {\displaystyle V^{}=\bigcup _{i\geq 0}V^{i}=V^{0}\cup V^{1}\cup V^{2}\cup V^{3}\cup V^{4}\cup \cdots .} == Of a language == Given a language L {\displaystyle L} (any finite or infinite set of strings), define L 0 = { ε } {\displaystyle L^{0}=\{\varepsilon \}} (the language consisting only of the empty string), L 1 = L , {\displaystyle L^{1}=L,} and define recursively the set L i + 1 = { w v : w ∈ L i and v ∈ L } {\displaystyle L^{i+1}=\{wv:w\in L^{i}{\text{ and }}v\in L\}} for each i > 0 , {\displaystyle i>0,} where w v {\displaystyle wv} denotes the string obtained by concatenating w {\displaystyle w} and v {\displaystyle v} . Here, L i {\displaystyle L^{i}} can be understood to be the set of all strings that can be obtained by concatenating exactly i {\displaystyle i} strings from L {\displaystyle L} , allowing repetitions. The definition of Kleene star on L {\displaystyle L} is L ∗ = ⋃ i ≥ 0 L i = L 0 ∪ L 1 ∪ L 2 ∪ L 3 ∪ L 4 ∪ ⋯ . {\displaystyle L^{}=\bigcup _{i\geq 0}L^{i}=L^{0}\cup L^{1}\cup L^{2}\cup L^{3}\cup L^{4}\cup \cdots .} == Kleene plus == In some formal language studies, (e.g. AFL theory) a variation on the Kleene star operation called the Kleene plus is used. The Kleene plus omits the V 0 {\displaystyle V^{0}} or L 0 {\displaystyle L^{0}} term in the above unions. In other words, the Kleene plus on V {\displaystyle V} is V + = ⋃ i ≥ 1 V i = V 1 ∪ V 2 ∪ V 3 ∪ ⋯ , {\displaystyle V^{+}=\bigcup _{i\geq 1}V^{i}=V^{1}\cup V^{2}\cup V^{3}\cup \cdots ,} or V + = V ∗ V . {\displaystyle V^{+}=V^{}V.} == Examples == Example of Kleene star applied to a set of strings: {"ab","c"} = { ε, "ab", "c", "abab", "abc", "cab", "cc", "ababab", "ababc", "abcab", "abcc", "cabab", "cabc", "ccab", "ccc", ...}. Example of Kleene star applied to a set of strings without the prefix property: {"a","ab","b"} = { ε, "a", "ab", "b", "aa", "aab", "aba", "abab", "abb", "ba", "bab", "bb", ...};In this example, the string "aab" can be obtained in two different ways. The Sardinas-Patterson algorithm can be used to check for a given V whether any member of V can be obtained in more than one way. Example of Kleene and Kleene plus applied to a set of characters (following the C programming language convention where a character is denoted by single quotes and a string is denoted by double quotes): {'a', 'b', 'c'} = { ε, "a", "b", "c", "aa", "ab", "ac", "ba", "bb", "bc", "ca", "cb", "cc", "aaa", "aab", ...}. {'a', 'b', 'c'}+ = { "a", "b", "c", "aa", "ab", "ac", "ba", "bb", "bc", "ca", "cb", "cc", "aaa", "aab", ...}. == Properties == If V {\displaystyle V} is any finite or countably infinite set of characters, then V ∗ {\displaystyle V^{}} is a countably infinite set. As a result, each formal language over a finite or countably infinite alphabet Σ {\displaystyle \Sigma } is countable, since it is a subset of the countably infinite set Σ ∗ {\displaystyle \Sigma ^{}} . ( L ∗ ) ∗ = L ∗ {\displaystyle (L^{})^{}=L^{}} , which means that the Kleene star operator is an idempotent unary operator, as ( L ∗ ) i = L ∗ {\displaystyle (L^{})^{i}=L^{}} for every i ≥ 1 {\displaystyle i\geq 1} . V ∗ = { ε } {\displaystyle V^{}=\{\varepsilon \}} , if V {\displaystyle V} is the empty set ∅. For the version of the Kleene star operator on languages, L ∗ = { ε } {\displaystyle L^{}=\{\varepsilon \}} when L {\displaystyle L} is either the empty set ∅ or the singleton set { ε } {\displaystyle \{\varepsilon \}} . == Generalization == Strings form a monoid with concatenation as the binary operation and ε the identity element. In addition to strings, the Kleene star is defined for any monoid. More precisely, let (M, ⋅) be a monoid, and S ⊆ M. Then S is the smallest submonoid of M containing S; that is, S contains the neutral element of M, the set S, and is such that if x,y ∈ S, then x⋅y ∈ S. Furthermore, the Kleene star is generalized by including the -operation (and the union) in the algebraic structure itself by the notion of complete star semiring.

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  Instance selection

  Instance selection (or dataset reduction, or dataset condensation) is an important data pre-processing step that can be applied in many machine learning (or data mining) tasks. Approaches for instance selection can be applied for reducing the original dataset to a manageable volume, leading to a reduction of the computational resources that are necessary for performing the learning process. Algorithms of instance selection can also be applied for removing noisy instances, before applying learning algorithms. This step can improve the accuracy in classification problems. Algorithm for instance selection should identify a subset of the total available data to achieve the original purpose of the data mining (or machine learning) application as if the whole data had been used. Considering this, the optimal outcome of IS would be the minimum data subset that can accomplish the same task with no performance loss, in comparison with the performance achieved when the task is performed using the whole available data. Therefore, every instance selection strategy should deal with a trade-off between the reduction rate of the dataset and the classification quality. == Instance selection algorithms == The literature provides several different algorithms for instance selection. They can be distinguished from each other according to several different criteria. Considering this, instance selection algorithms can be grouped in two main classes, according to what instances they select: algorithms that preserve the instances at the boundaries of classes and algorithms that preserve the internal instances of the classes. Within the category of algorithms that select instances at the boundaries it is possible to cite DROP3, ICF and LSBo. On the other hand, within the category of algorithms that select internal instances, it is possible to mention ENN and LSSm. In general, algorithm such as ENN and LSSm are used for removing harmful (noisy) instances from the dataset. They do not reduce the data as the algorithms that select border instances, but they remove instances at the boundaries that have a negative impact on the data mining task. They can be used by other instance selection algorithms, as a filtering step. For example, the ENN algorithm is used by DROP3 as the first step, and the LSSm algorithm is used by LSBo. There is also another group of algorithms that adopt different selection criteria. For example, the algorithms LDIS, CDIS and XLDIS select the densest instances in a given arbitrary neighborhood. The selected instances can include both, border and internal instances. The LDIS and CDIS algorithms are very simple and select subsets that are very representative of the original dataset. Besides that, since they search by the representative instances in each class separately, they are faster (in terms of time complexity and effective running time) than other algorithms, such as DROP3 and ICF. Besides that, there is a third category of algorithms that, instead of selecting actual instances of the dataset, select prototypes (that can be synthetic instances). In this category it is possible to include PSSA, PSDSP and PSSP. The three algorithms adopt the notion of spatial partition (a hyperrectangle) for identifying similar instances and extract prototypes for each set of similar instances. In general, these approaches can also be modified for selecting actual instances of the datasets. The algorithm ISDSP adopts a similar approach for selecting actual instances (instead of prototypes).

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  Rademacher complexity

  In computational learning theory (machine learning and theory of computation), Rademacher complexity, named after Hans Rademacher, measures richness of a class of sets with respect to a probability distribution. The concept can also be extended to real valued functions. == Definitions == === Rademacher complexity of a set === Given a set A ⊆ R m {\displaystyle A\subseteq \mathbb {R} ^{m}} , the Rademacher complexity of A is defined as follows: Rad ⁡ ( A ) := 1 m E σ [ sup a ∈ A ∑ i = 1 m σ i a i ] {\displaystyle \operatorname {Rad} (A):={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{a\in A}\sum _{i=1}^{m}\sigma _{i}a_{i}\right]} where σ 1 , σ 2 , … , σ m {\displaystyle \sigma _{1},\sigma _{2},\dots ,\sigma _{m}} are independent random variables drawn from the Rademacher distribution i.e. Pr ( σ i = + 1 ) = Pr ( σ i = − 1 ) = 1 / 2 {\displaystyle \Pr(\sigma _{i}=+1)=\Pr(\sigma _{i}=-1)=1/2} for i ∈ { 1 , 2 , … , m } {\displaystyle i\in \{1,2,\dots ,m\}} , and a = ( a 1 , … , a m ) ∈ A {\displaystyle a=(a_{1},\ldots ,a_{m})\in A} . Some authors take the absolute value of the sum before taking the supremum, but if A {\displaystyle A} is symmetric this makes no difference. === Rademacher complexity of a function class === Let S = { z 1 , z 2 , … , z m } ⊆ Z {\displaystyle S=\{z_{1},z_{2},\dots ,z_{m}\}\subseteq Z} be a sample of points and consider a function class F {\displaystyle {\mathcal {F}}} of real-valued functions over Z {\displaystyle Z} . Then, the empirical Rademacher complexity of F {\displaystyle {\mathcal {F}}} given S {\displaystyle S} is defined as: Rad S ⁡ ( F ) = 1 m E σ [ sup f ∈ F | ∑ i = 1 m σ i f ( z i ) | ] {\displaystyle \operatorname {Rad} _{S}({\mathcal {F}})={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{f\in {\mathcal {F}}}\left|\sum _{i=1}^{m}\sigma _{i}f(z_{i})\right|\right]} This can also be written using the previous definition: Rad S ⁡ ( F ) = Rad ⁡ ( F ∘ S ) {\displaystyle \operatorname {Rad} _{S}({\mathcal {F}})=\operatorname {Rad} ({\mathcal {F}}\circ S)} where F ∘ S {\displaystyle {\mathcal {F}}\circ S} denotes function composition, i.e.: F ∘ S := { ( f ( z 1 ) , … , f ( z m ) ) ∣ f ∈ F } {\displaystyle {\mathcal {F}}\circ S:=\{(f(z_{1}),\ldots ,f(z_{m}))\mid f\in {\mathcal {F}}\}} The worst case empirical Rademacher complexity is Rad ¯ m ( F ) = sup S = { z 1 , … , z m } Rad S ⁡ ( F ) {\displaystyle {\overline {\operatorname {Rad} }}_{m}({\mathcal {F}})=\sup _{S=\{z_{1},\dots ,z_{m}\}}\operatorname {Rad} _{S}({\mathcal {F}})} Let P {\displaystyle P} be a probability distribution over Z {\displaystyle Z} . The Rademacher complexity of the function class F {\displaystyle {\mathcal {F}}} with respect to P {\displaystyle P} for sample size m {\displaystyle m} is: Rad P , m ⁡ ( F ) := E S ∼ P m [ Rad S ⁡ ( F ) ] {\displaystyle \operatorname {Rad} _{P,m}({\mathcal {F}}):=\mathbb {E} _{S\sim P^{m}}\left[\operatorname {Rad} _{S}({\mathcal {F}})\right]} where the above expectation is taken over an identically independently distributed (i.i.d.) sample S = ( z 1 , z 2 , … , z m ) {\displaystyle S=(z_{1},z_{2},\dots ,z_{m})} generated according to P {\displaystyle P} . == Intuition == The Rademacher complexity is typically applied on a function class of models that are used for classification, with the goal of measuring their ability to classify points drawn from a probability space under arbitrary labellings. When the function class is rich enough, it contains functions that can appropriately adapt for each arrangement of labels, simulated by the random draw of σ i {\displaystyle \sigma _{i}} under the expectation, so that this quantity in the sum is maximized. The Rademacher complexity of a set A {\displaystyle A} can be rewritten as Rad ⁡ ( A ) := 1 m E σ [ sup a ∈ A ∑ i = 1 m σ i a i ] = 1 m 2 m ∑ σ ∈ { − 1 / m , + 1 / m } m [ sup a ∈ A ⟨ σ , a ⟩ ] . {\displaystyle \operatorname {Rad} (A):={\frac {1}{m}}\mathbb {E} _{\sigma }\left[\sup _{a\in A}\sum _{i=1}^{m}\sigma _{i}a_{i}\right]={\frac {1}{{\sqrt {m}}2^{m}}}\sum _{\sigma \in \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}}\left[\sup _{a\in A}\langle \sigma ,a\rangle \right].} Each term in the summation is the farthest distance of the set A {\displaystyle A} from the origin, along a unit-length direction σ {\displaystyle \sigma } . The directions are along the vertices of a hypercube. Thus, we can also write it as Rad ⁡ ( A ) = 1 2 m 1 2 m − 1 ∑ σ ∈ { − 1 / m , + 1 / m } m / { − 1 , + 1 } [ sup a ∈ A ⟨ σ , a ⟩ − inf a ∈ A ⟨ σ , a ⟩ ] {\displaystyle \operatorname {Rad} (A)={\frac {1}{2{\sqrt {m}}}}{\frac {1}{2^{m-1}}}\sum _{\sigma \in \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}/\{-1,+1\}}\left[\sup _{a\in A}\langle \sigma ,a\rangle -\inf _{a\in A}\langle \sigma ,a\rangle \right]} Here, the set { − 1 / m , + 1 / m } m / { − 1 , + 1 } {\displaystyle \{-1/{\sqrt {m}},+1/{\sqrt {m}}\}^{m}/\{-1,+1\}} denotes half of the vertices of a hypercube, selected so that each diagonal has exactly one vertex selected. In words, this states that 2 m Rad ⁡ ( A ) {\displaystyle 2{\sqrt {m}}\operatorname {Rad} (A)} is precisely the average width of the set A {\displaystyle A} along all diagonal directions of a hypercube. == Examples == A singleton set has 0 width in any direction, so it has Rademacher complexity 0. The set A = { ( 1 , 1 ) , ( 1 , 2 ) } ⊆ R 2 {\displaystyle A=\{(1,1),(1,2)\}\subseteq \mathbb {R} ^{2}} has average width 1 / 2 {\displaystyle 1/{\sqrt {2}}} along the two diagonal directions of the square, so it has Rademacher complexity 1 / 4 {\displaystyle 1/4} . The unit cube [ 0 , 1 ] m {\displaystyle [0,1]^{m}} has constant width m {\displaystyle {\sqrt {m}}} along the diagonal directions, so it has Rademacher complexity 1 / 2 {\displaystyle 1/2} . Similarly, the unit cross-polytope { x ∈ R m : ‖ x ‖ 1 ≤ 1 } {\displaystyle \{x\in \mathbb {R} ^{m}:\|x\|_{1}\leq 1\}} has constant width 2 / m {\displaystyle 2/{\sqrt {m}}} along the diagonal directions, so it has Rademacher complexity 1 / m {\displaystyle 1/m} . == Using the Rademacher complexity == The Rademacher complexity can be used to derive data-dependent upper-bounds on the learnability of function classes. Intuitively, a function-class with smaller Rademacher complexity is easier to learn. === Bounding the representativeness === In machine learning, it is desired to have a training set that represents the true distribution of some sample data S {\displaystyle S} . This can be quantified using the notion of representativeness. Denote by P {\displaystyle P} the probability distribution from which the samples are drawn. Denote by H {\displaystyle H} the set of hypotheses (potential classifiers) and denote by F {\displaystyle {\mathcal {F}}} the corresponding set of error functions, i.e., for every hypothesis h ∈ H {\displaystyle h\in H} , there is a function f h ∈ F {\displaystyle f_{h}\in F} , that maps each training sample (features,label) to the error of the classifier h {\displaystyle h} (note in this case hypothesis and classifier are used interchangeably). For example, in the case that h {\displaystyle h} represents a binary classifier, the error function is a 0–1 loss function, i.e. the error function f h {\displaystyle f_{h}} returns 0 if h {\displaystyle h} correctly classifies a sample and 1 else. We omit the index and write f {\displaystyle f} instead of f h {\displaystyle f_{h}} when the underlying hypothesis is irrelevant. Define: L P ( f ) := E z ∼ P [ f ( z ) ] {\displaystyle L_{P}(f):=\mathbb {E} _{z\sim P}[f(z)]} – the expected error of some error function f ∈ F {\displaystyle f\in {\mathcal {F}}} on the real distribution P {\displaystyle P} ; L S ( f ) := 1 m ∑ i = 1 m f ( z i ) {\displaystyle L_{S}(f):={1 \over m}\sum _{i=1}^{m}f(z_{i})} – the estimated error of some error function f ∈ F {\displaystyle f\in {\mathcal {F}}} on the sample S {\displaystyle S} . The representativeness of the sample S {\displaystyle S} , with respect to P {\displaystyle P} and F {\displaystyle {\mathcal {F}}} , is defined as: Rep P ⁡ ( F , S ) := sup f ∈ F ( L P ( f ) − L S ( f ) ) {\displaystyle \operatorname {Rep} _{P}({\mathcal {F}},S):=\sup _{f\in F}(L_{P}(f)-L_{S}(f))} Smaller representativeness is better, since it provides a way to avoid overfitting: it means that the true error of a classifier is not much higher than its estimated error, and so selecting a classifier that has low estimated error will ensure that the true error is also low. Note however that the concept of representativeness is relative and hence can not be compared across distinct samples. The expected representativeness of a sample can be bounded above by the Rademacher complexity of the function class: If F {\displaystyle {\mathcal {F}}} is a set of functions with range within [ 0 , 1 ] {\displaystyle [0,1]} , then Rad P , m ⁡ ( F ) − ln ⁡ 2 2 m ≤ E S ∼ P m [ Rep P ⁡ ( F , S ) ] ≤ 2 Rad P , m ⁡ ( F ) {\displaystyle \operatorname {Rad} _{P,m}({\mathcal {F}})-{\sqrt {\frac {\ln 2}{2m}}}\leq \mathbb {E} _{S\sim P^{m}}[\operatorname {Rep} _{P}({\

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  Argumentation framework

  In artificial intelligence and related fields, an argumentation framework is a way to deal with contentious information and draw conclusions from it using formalized arguments. In an abstract argumentation framework, entry-level information is a set of abstract arguments that, for instance, represent data or a proposition. Conflicts between arguments are represented by a binary relation on the set of arguments. In concrete terms, an argumentation framework is represented with a directed graph such that the nodes are the arguments, and the arrows represent the attack relation. There exist some extensions of the Dung's framework, like the logic-based argumentation frameworks or the value-based argumentation frameworks. == Abstract argumentation frameworks == === Formal framework === Abstract argumentation frameworks, also called argumentation frameworks à la Dung, are defined formally as a pair: A set of abstract elements called arguments, denoted A {\displaystyle A} A binary relation on A {\displaystyle A} , called attack relation, denoted R {\displaystyle R} For instance, the argumentation system S = ⟨ A , R ⟩ {\displaystyle S=\langle A,R\rangle } with A = { a , b , c , d } {\displaystyle A=\{a,b,c,d\}} and R = { ( a , b ) , ( b , c ) , ( d , c ) } {\displaystyle R=\{(a,b),(b,c),(d,c)\}} contains four arguments ( a , b , c {\displaystyle a,b,c} and d {\displaystyle d} ) and three attacks ( a {\displaystyle a} attacks b {\displaystyle b} , b {\displaystyle b} attacks c {\displaystyle c} and d {\displaystyle d} attacks c {\displaystyle c} ). Dung defines some notions : an argument a ∈ A {\displaystyle a\in A} is acceptable with respect to E ⊆ A {\displaystyle E\subseteq A} if and only if E {\displaystyle E} defends a {\displaystyle a} , that is ∀ b ∈ A {\displaystyle \forall b\in A} such that ( b , a ) ∈ R , ∃ c ∈ E {\displaystyle (b,a)\in R,\exists c\in E} such that ( c , b ) ∈ R {\displaystyle (c,b)\in R} , a set of arguments E {\displaystyle E} is conflict-free if there is no attack between its arguments, formally : ∀ a , b ∈ E , ( a , b ) ∉ R {\displaystyle \forall a,b\in E,(a,b)\not \in R} , a set of arguments E {\displaystyle E} is admissible if and only if it is conflict-free and all its arguments are acceptable with respect to E {\displaystyle E} . === Different semantics of acceptance === ==== Extensions ==== To decide if an argument can be accepted or not, or if several arguments can be accepted together, Dung defines several semantics of acceptance that allows, given an argumentation system, sets of arguments (called extensions) to be computed. For instance, given S = ⟨ A , R ⟩ {\displaystyle S=\langle A,R\rangle } , E {\displaystyle E} is a complete extension of S {\displaystyle S} only if it is an admissible set and every acceptable argument with respect to E {\displaystyle E} belongs to E {\displaystyle E} , E {\displaystyle E} is a preferred extension of S {\displaystyle S} only if it is a maximal element (with respect to the set-theoretical inclusion) among the admissible sets with respect to S {\displaystyle S} , E {\displaystyle E} is a stable extension of S {\displaystyle S} only if it is a conflict-free set that attacks every argument that does not belong in E {\displaystyle E} (formally, ∀ a ∈ A ∖ E , ∃ b ∈ E {\displaystyle \forall a\in A\backslash E,\exists b\in E} such that ( b , a ) ∈ R {\displaystyle (b,a)\in R} , E {\displaystyle E} is the (unique) grounded extension of S {\displaystyle S} only if it is the smallest element (with respect to set inclusion) among the complete extensions of S {\displaystyle S} . There exists some inclusions between the sets of extensions built with these semantics : Every stable extension is preferred, Every preferred extension is complete, The grounded extension is complete, If the system is well-founded (there exists no infinite sequence a 0 , a 1 , … , a n , … {\displaystyle a_{0},a_{1},\dots ,a_{n},\dots } such that ∀ i > 0 , ( a i + 1 , a i ) ∈ R {\displaystyle \forall i>0,(a_{i+1},a_{i})\in R} ), all these semantics coincide—only one extension is grounded, stable, preferred, and complete. Some other semantics have been defined. One introduce the notation E x t σ ( S ) {\displaystyle Ext_{\sigma }(S)} to note the set of σ {\displaystyle \sigma } -extensions of the system S {\displaystyle S} . In the case of the system S {\displaystyle S} in the figure above, E x t σ ( S ) = { { a , d } } {\displaystyle Ext_{\sigma }(S)=\{\{a,d\}\}} for every Dung's semantic—the system is well-founded. That explains why the semantics coincide, and the accepted arguments are: a {\displaystyle a} and d {\displaystyle d} . ==== Labellings ==== Labellings are a more expressive way than extensions to express the acceptance of the arguments. Concretely, a labelling is a mapping that associates every argument with a label in (the argument is accepted), out (the argument is rejected), or undec (the argument is undefined—not accepted or refused). One can also note a labelling as a set of pairs ( a r g u m e n t , l a b e l ) {\displaystyle ({\mathit {argument}},{\mathit {label}})} . Such a mapping does not make sense without additional constraint. The notion of reinstatement labelling guarantees the sense of the mapping. L {\displaystyle L} is a reinstatement labelling on the system S = ⟨ A , R ⟩ {\displaystyle S=\langle A,R\rangle } if and only if : ∀ a ∈ A , L ( a ) = i n {\displaystyle \forall a\in A,L(a)={\mathit {in}}} if and only if ∀ b ∈ A {\displaystyle \forall b\in A} such that ( b , a ) ∈ R , L ( b ) = o u t {\displaystyle (b,a)\in R,L(b)={\mathit {out}}} ∀ a ∈ A , L ( a ) = o u t {\displaystyle \forall a\in A,L(a)={\mathit {out}}} if and only if ∃ b ∈ A {\displaystyle \exists b\in A} such that ( b , a ) ∈ R {\displaystyle (b,a)\in R} and L ( b ) = i n {\displaystyle L(b)={\mathit {in}}} ∀ a ∈ A , L ( a ) = u n d e c {\displaystyle \forall a\in A,L(a)={\mathit {undec}}} if and only if L ( a ) ≠ i n {\displaystyle L(a)\neq {\mathit {in}}} and L ( a ) ≠ o u t {\displaystyle L(a)\neq {\mathit {out}}} One can convert every extension into a reinstatement labelling: the arguments of the extension are in, those attacked by an argument of the extension are out, and the others are undec. Conversely, one can build an extension from a reinstatement labelling just by keeping the arguments in. Indeed, Caminada proved that the reinstatement labellings and the complete extensions can be mapped in a bijective way. Moreover, the other Datung's semantics can be associated to some particular sets of reinstatement labellings. Reinstatement labellings distinguish arguments not accepted because they are attacked by accepted arguments from undefined arguments—that is, those that are not defended cannot defend themselves. An argument is undec if it is attacked by at least another undec. If it is attacked only by arguments out, it must be in, and if it is attacked some argument in, then it is out. The unique reinstatement labelling that corresponds to the system S {\displaystyle S} above is L = { ( a , i n ) , ( b , o u t ) , ( c , o u t ) , ( d , i n ) } {\displaystyle L=\{(a,{\mathit {in}}),(b,{\mathit {out}}),(c,{\mathit {out}}),(d,{\mathit {in}})\}} . === Inference from an argumentation system === In the general case when several extensions are computed for a given semantic σ {\displaystyle \sigma } , the agent that reasons from the system can use several mechanisms to infer information: Credulous inference: the agent accepts an argument if it belongs to at least one of the σ {\displaystyle \sigma } -extensions—in which case, the agent risks accepting some arguments that are not acceptable together ( a {\displaystyle a} attacks b {\displaystyle b} , and a {\displaystyle a} and b {\displaystyle b} each belongs to an extension) Skeptical inference: the agent accepts an argument only if it belongs to every σ {\displaystyle \sigma } -extension. In this case, the agent risks deducing too little information (if the intersection of the extensions is empty or has a very small cardinal). For these two methods to infer information, one can identify the set of accepted arguments, respectively C r σ ( S ) {\displaystyle Cr_{\sigma }(S)} the set of the arguments credulously accepted under the semantic σ {\displaystyle \sigma } , and S c σ ( S ) {\displaystyle Sc_{\sigma }(S)} the set of arguments accepted skeptically under the semantic σ {\displaystyle \sigma } (the σ {\displaystyle \sigma } can be missed if there is no possible ambiguity about the semantic). Of course, when there is only one extension (for instance, when the system is well-founded), this problem is very simple: the agent accepts arguments of the unique extension and rejects others. The same reasoning can be done with labellings that correspond to the chosen semantic : an argument can be accepted if it is in for each labelling and refused if it is out for each labelling, the others being in an undecided state (the status of the arguments can remind the

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  Kai's Power Tools

  Kai's Power Tools (KPT) are a set of API plugins created by the German computer scientist Kai Krause in 1992 that were designed for use with Adobe Photoshop and Corel Photo-Paint. Kai's Power Tools were sold to Corel in 2000 when MetaCreations was closed. There are various versions of Kai's Power Tools. KPT 3, 5, 6, and X sets are compilations of different filters. The program interface features a reward-based function in which a bonus function is revealed as the user moves towards more complex aspects of the tool. == Filters == The KPT Convolver is a mathematics based filter; the level of precision and varying effects can be achieved by using numerical values of colour, tint, hue, saturation, contrast, brightness, luminosity, and posterize. The KPT Projector takes the current image or selection and offers a number of interactive perspective warp effects. To a large extent, with its draggable distortion handles and its moving, scaling and rotating options, this simply duplicates Adobe Photoshop's Free Transform capabilities. What is completely different is the ability to rotate the bitmap image in 3D space and to tile the results if desired. It can also animate the distortions by dragging keyframes from the preview window into an animation palette. KPT 6 will then preview the animation and output it to various sizes in avi or mov format. This animation capability is even more useful with the KPT Turbulence filter. This is another distortion filter, but one that treats the image as if it was completely liquid. The preview panel shows the animation in real time. The KPT Goo filter is used to produce a single frame freeform liquid distortion. This filter is available both with KPT 6 and the standalone version. It works by effectively turning a bitmap image into a liquid that can be interactively smeared, smudged, twirled, and pinched with the range of tools on offer. The obvious use is to distort photographic portraits into caricatures. KPT Materializer can create advanced surface textures based on bump maps that define troughs and peaks. It can use any external image for the basis of the bump map or alternatively the user can pick out the hue, saturation, luminance or red, green, or blue channel of the current image. It can then offset, scale and rotate the texture map, control its lighting, and even blend in a reflection map. The filter can be used for anything from providing an oil-painting feel to an entire image, to giving the illusion of depth to a selection. Also producing the impression of depth is the KPT Gel filter which uses various paint tools to synthesize photo-realistic 3D materials such as metals, liquids, or plastics. Gel painting is very different from traditional 2D painting as the brush strokes pool together when they touch and refract the underlying image. It can also manipulate 3D paint—once it has been added—by twirling, pinching, and carving it. The opposite is true of the Equalizer filter, which is used for applying variations on sharpening effects. The filter has three modes. The first mode, Equalizer, looks and works rather like the graphic equalizer on a stereo system, enabling adjustment of the level of pixel contrast within nine bands of different visual frequencies. The second mode, Contrast Sharpen, allows for increasing the contrast between light and dark areas in an image. The third mode, Bounded Sharpen, can sharpen an image without causing oversharpening, which can lead to halo effects. This feature is particularly useful when pulling out the detail in an image softened by resizing. KPT SceneBuilder is used for producing photorealistic 3D scenes by importing and rendering 3DS files. The main image window offers three tabs for editing in 2D and 3D mode and for setting up the object's final texture. Many users regard this filter as being the most impressive because it acts as a standalone 3D rendering tool and provides control over everything from transparency, reflection, refraction, bump mapping through to multiple light sources, and so on but without the ability to create or edit objects. The final filter, KPT SkyEffects, also has its roots in Metacreations' experience with 3D programs such as Bryce and RayDream. This filter is designed to simulate the interaction between the light from the sun or moon with no less than six atmospheric layers of haze, fog and cloud. The filter is typical of the KPT 6 collection as a whole: at times the interface is inspired and offers the ability to create beautiful reddening sunsets simply by interactively dragging the sun toward the horizon, producing realistic sunsets and moonscapes. == Other effects == Kai's Power Tools 6 features a lens flare effect for precisely managing the type of glow, halo, streaks, and reflection. The addition of a library of preset effects helps to overcome this by allowing the user to choose a standard effect and then interactively position the flare in the image preview. KPT 6 provides a new engine in the form of the KPT Reaction, which takes a reaction seed and turns it into a seamlessly tiling pattern based on a reaction diffusion process. It offers random noise, regular dots or reticulated voronoi patterns or a bitmap image itself as the seed. Corel has no plans for any updates.

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  Inductive programming

  Inductive programming (IP) is a special area of automatic programming, covering research from artificial intelligence and programming, which addresses learning of typically declarative (logic or functional) and often recursive programs from incomplete specifications, such as input/output examples or constraints. Depending on the programming language used, there are several kinds of inductive programming. Inductive functional programming, which uses functional programming languages such as Lisp or Haskell, and most especially inductive logic programming, which uses logic programming languages such as Prolog and other logical representations such as description logics, have been more prominent, but other (programming) language paradigms have also been used, such as constraint programming or probabilistic programming. == Definition == Inductive programming incorporates all approaches which are concerned with learning programs or algorithms from incomplete (formal) specifications. Possible inputs in an IP system are a set of training inputs and corresponding outputs or an output evaluation function, describing the desired behavior of the intended program, traces or action sequences which describe the process of calculating specific outputs, constraints for the program to be induced concerning its time efficiency or its complexity, various kinds of background knowledge such as standard data types, predefined functions to be used, program schemes or templates describing the data flow of the intended program, heuristics for guiding the search for a solution or other biases. Output of an IP system is a program in some arbitrary programming language containing conditionals and loop or recursive control structures, or any other kind of Turing-complete representation language. In many applications the output program must be correct with respect to the examples and partial specification, and this leads to the consideration of inductive programming as a special area inside automatic programming or program synthesis, usually opposed to 'deductive' program synthesis, where the specification is usually complete. In other cases, inductive programming is seen as a more general area where any declarative programming or representation language can be used and we may even have some degree of error in the examples, as in general machine learning, the more specific area of structure mining or the area of symbolic artificial intelligence. A distinctive feature is the number of examples or partial specification needed. Typically, inductive programming techniques can learn from just a few examples. The diversity of inductive programming usually comes from the applications and the languages that are used: apart from logic programming and functional programming, other programming paradigms and representation languages have been used or suggested in inductive programming, such as functional logic programming, constraint programming, probabilistic programming, abductive logic programming, modal logic, action languages, agent languages and many types of imperative languages. == History == The early works of Plotkin, and his "relative least general generalization (rlgg)", had an enormous impact in inductive logic programming. There were some encouraging results on learning recursive Prolog programs such as quicksort from examples together with suitable background knowledge, for example with GOLEM. However, after initial success, the community got disappointed by limited progress about the induction of recursive programs with ILP less and less focusing on recursive programs and leaning more and more towards a machine learning setting with applications in relational data mining and knowledge discovery. In parallel to work in ILP, Koza proposed genetic programming in the early 1990s as a generate-and-test based approach to learning programs. The idea of genetic programming was further developed into the inductive programming system ADATE and the systematic-search-based system MagicHaskeller. Here again, functional programs are learned from sets of positive examples together with an output evaluation (fitness) function which specifies the desired input/output behavior of the program to be learned. The early work in grammar induction (also known as grammatical inference) is related to inductive programming, as rewriting systems or logic programs can be used to represent production rules. In fact, early works in inductive inference considered grammar induction and Lisp program inference as basically the same problem. The results in terms of learnability were related to classical concepts, such as identification-in-the-limit, as introduced in the seminal work of Gold. More recently, the language learning problem was addressed by the inductive programming community. In the recent years, the classical approaches have been resumed and advanced with great success. Therefore, the synthesis problem has been reformulated on the background of constructor-based term rewriting systems taking into account modern techniques of functional programming, as well as moderate use of search-based strategies and usage of background knowledge as well as automatic invention of subprograms. Many new and successful applications have recently appeared beyond program synthesis, most especially in the area of data manipulation, programming by example and cognitive modelling (see below). Other ideas have also been explored with the common characteristic of using declarative languages for the representation of hypotheses. For instance, the use of higher-order features, schemes or structured distances have been advocated for a better handling of recursive data types and structures; abstraction has also been explored as a more powerful approach to cumulative learning and function invention. One powerful paradigm that has been recently used for the representation of hypotheses in inductive programming (generally in the form of generative models) is probabilistic programming (and related paradigms, such as stochastic logic programs and Bayesian logic programming). == Application areas == The first workshop on Approaches and Applications of Inductive Programming (AAIP) Archived 2016-03-03 at the Wayback Machine held in conjunction with ICML 2005 identified all applications where "learning of programs or recursive rules are called for, [...] first in the domain of software engineering where structural learning, software assistants and software agents can help to relieve programmers from routine tasks, give programming support for end users, or support of novice programmers and programming tutor systems. Further areas of application are language learning, learning recursive control rules for AI-planning, learning recursive concepts in web-mining or for data-format transformations". Since then, these and many other areas have shown to be successful application niches for inductive programming, such as end-user programming, the related areas of programming by example and programming by demonstration, and intelligent tutoring systems. Other areas where inductive inference has been recently applied are knowledge acquisition, artificial general intelligence, reinforcement learning and theory evaluation, and cognitive science in general. There may also be prospective applications in intelligent agents, games, robotics, personalisation, ambient intelligence and human interfaces.

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  Instance (computer science)

  In computer science, an instance or token (from metalogic and metamathematics) is a specific occurrence of a software element that is based on a type definition. When created, an occurrence is said to have been instantiated, and both the creation process and the result of creation are called instantiation. == Examples == Chat AI instance In chat-based AI systems, an assistant can be invoked across many independent conversation sessions (often called a thread), each with its own message history. A specific execution of the assistant over that session may be represented as a run (an execution on a thread). Class instance In object-oriented programming, an object created from a class type. Each instance of a class shares the class-defined structure and behavior but has its own identity and state. Procedural instance In some contexts (including Simula), each procedure call can be viewed as an instance of that procedure—an activation with its own parameters and local variables. Computer instance In cloud computing and virtualization, an instance commonly refers to a provisioned virtual machine or virtual server with an allocated combination of compute, memory, network, and storage resources. Polygonal model In computer graphics, a model may be instanced so it can be drawn multiple times with different transforms and parameters, improving performance by reusing shared geometry data. Program instance In a POSIX-oriented operating system, a running process is an instance of a program. It can be instantiated via system calls such as fork() and exec(). Each executing process is an instance of a program it has been instantiated from.

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  Surrogate model

  A surrogate model is an engineering method used when an outcome of interest cannot be easily measured or computed, so an approximate mathematical model of the outcome is used instead. Most engineering design problems require experiments and/or simulations to evaluate design objective and constraint functions as a function of design variables. For example, in order to find the optimal airfoil shape for an aircraft wing, an engineer simulates the airflow around the wing for different shape variables (e.g., length, curvature, material, etc.). For many real-world problems, however, a single simulation can take many minutes, hours, or even days to complete. As a result, routine tasks such as design optimization, design space exploration, sensitivity analysis and "what-if" analysis become impossible since they require thousands or even millions of simulation evaluations. One way of alleviating this burden is by constructing approximation models, known as surrogate models, metamodels or emulators, that mimic the behavior of the simulation model as closely as possible while being computationally cheaper to evaluate. Surrogate models are constructed using a data-driven, bottom-up approach. The exact, inner working of the simulation code is not assumed to be known (or even understood), relying solely on the input-output behavior. A model is constructed based on modeling the response of the simulator to a limited number of intelligently chosen data points. This approach is also known as behavioral modeling or black-box modeling, though the terminology is not always consistent. When only a single design variable is involved, the process is known as curve fitting. Though using surrogate models in lieu of experiments and simulations in engineering design is more common, surrogate modeling may be used in many other areas of science where there are expensive experiments and/or function evaluations. == Goals == The scientific challenge of surrogate modeling is the generation of a surrogate that is as accurate as possible, using as few simulation evaluations as possible. The process comprises three major steps which may be interleaved iteratively: Sample selection (also known as sequential design, optimal experimental design (OED) or active learning) Construction of the surrogate model and optimizing the model parameters (i.e., bias-variance tradeoff) Appraisal of the accuracy of the surrogate. The accuracy of the surrogate depends on the number and location of samples (expensive experiments or simulations) in the design space. A systematic data representation during training can improve model scalability, thereby reducing the need for expensive simulations. Various design of experiments (DOE) techniques cater to different sources of errors, in particular, errors due to noise in the data or errors due to an improper surrogate model. == Types of surrogate models == Popular surrogate modeling approaches are: polynomial response surfaces; kriging; more generalized Bayesian approaches; gradient-enhanced kriging (GEK); radial basis function; support vector machines; space mapping; artificial neural networks and Bayesian networks. Other methods recently explored include Fourier surrogate modeling , random forests, convolutional neural networks, and generative adversarial networks. For some problems, the nature of the true function is not known a priori, and therefore it is not clear which surrogate model will be the most accurate one. In addition, there is no consensus on how to obtain the most reliable estimates of the accuracy of a given surrogate. Many other problems have known physics properties. In these cases, physics-based surrogates such as space-mapping based models are commonly used. == Invariance properties == Recently proposed comparison-based surrogate models (e.g., ranking support vector machines) for evolutionary algorithms, such as CMA-ES, allow preservation of some invariance properties of surrogate-assisted optimizers: Invariance with respect to monotonic transformations of the function (scaling) Invariance with respect to orthogonal transformations of the search space (rotation) == Applications == An important distinction can be made between two different applications of surrogate models: design optimization and design space approximation (also known as emulation). In surrogate model-based optimization, an initial surrogate is constructed using some of the available budgets of expensive experiments and/or simulations. The remaining experiments/simulations are run for designs which the surrogate model predicts may have promising performance. The process usually takes the form of the following search/update procedure. Initial sample selection (the experiments and/or simulations to be run) Construct surrogate model Search surrogate model (the model can be searched extensively, e.g., using a genetic algorithm, as it is cheap to evaluate) Run and update experiment/simulation at new location(s) found by search and add to sample Iterate steps 2 to 4 until out of time or design is "good enough" Depending on the type of surrogate used and the complexity of the problem, the process may converge on a local or global optimum, or perhaps none at all. In design space approximation, one is not interested in finding the optimal parameter vector, but rather in the global behavior of the system. Here the surrogate is tuned to mimic the underlying model as closely as needed over the complete design space. Such surrogates are a useful, cheap way to gain insight into the global behavior of the system. Optimization can still occur as a post-processing step, although with no update procedure (see above), the optimum found cannot be validated. == Surrogate modeling software == Surrogate Modeling Toolbox (SMT: https://github.com/SMTorg/smt) is a Python package that contains a collection of surrogate modeling methods, sampling techniques, and benchmarking functions. This package provides a library of surrogate models that is simple to use and facilitates the implementation of additional methods. SMT is different from existing surrogate modeling libraries because of its emphasis on derivatives, including training derivatives used for gradient-enhanced modeling, prediction derivatives, and derivatives with respect to the training data. It also includes new surrogate models that are not available elsewhere: kriging by partial-least squares reduction and energy-minimizing spline interpolation. Python library SAMBO Optimization supports sequential optimization with arbitrary models, with tree-based models and Gaussian process models built in. Surrogates.jl is a Julia packages which offers tools like random forests, radial basis methods and kriging. == Surrogate-Assisted Evolutionary Algorithms (SAEAs) == SAEAs are an advanced class of optimization techniques that integrate evolutionary algorithms (EAs) with surrogate models. In traditional EAs, evaluating the fitness of candidate solutions often requires computationally expensive simulations or experiments. SAEAs address this challenge by building a surrogate model, which is a computationally inexpensive approximation of the objective function or constraint functions. The surrogate model serves as a substitute for the actual evaluation process during the evolutionary search. It allows the algorithm to quickly estimate the fitness of new candidate solutions, thereby reducing the number of expensive evaluations needed. This significantly speeds up the optimization process, especially in cases where the objective function evaluations are time-consuming or resource-intensive. SAEAs typically involve three main steps: (1) building the surrogate model using a set of initial sampled data points, (2) performing the evolutionary search using the surrogate model to guide the selection, crossover, and mutation operations, and (3) periodically updating the surrogate model with new data points generated during the evolutionary process to improve its accuracy. By balancing exploration (searching new areas in the solution space) and exploitation (refining known promising areas), SAEAs can efficiently find high-quality solutions to complex optimization problems. They have been successfully applied in various fields, including engineering design, machine learning, and computational finance, where traditional optimization methods may struggle due to the high computational cost of fitness evaluations.

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  Backend as a service

  Backend as a service (BaaS), sometimes also referred to as mobile backend as a service (MBaaS), is a service for providing web app and mobile app developers with a way to easily build a backend to their frontend applications. Features available include user management, push notifications, and integration with social networking services. These services are provided via the use of custom software development kits (SDKs) and application programming interfaces (APIs). BaaS is a relatively recent development in cloud computing, with most BaaS startups dating from 2011 or later. Some of the most popular service providers are AWS Amplify and Firebase. == Purpose == Web and mobile apps require a similar set of features on the backend, including notification service, integration with social networks, and cloud storage. Each of these services has its own API that must be individually incorporated into an app, a process that can be time-consuming and complicated for app developers. BaaS providers form a bridge between the frontend of an application and various cloud-based backends via a unified API and SDK. Providing a consistent way to manage backend data means that developers do not need to redevelop their own backend for each of the services that their apps need to access, potentially saving both time and money. Although similar to other cloud-computing business models, such as serverless computing, software as a service (SaaS), infrastructure as a service (IaaS), and platform as a service (PaaS), BaaS is distinct from these other services in that it specifically addresses the cloud-computing needs of web and mobile app developers by providing a unified means of connecting their apps to cloud services. == Features == BaaS providers offer different set of features and backend tools. Some of the most common features include: Database management. Most BaaS solutions provide SQL and/or NoSQL database management services for applications. Developers can store their app data without deploying and managing databases themselves. BaaS usually provides client SDKs, REST and GraphQL APIs for the frontend to interact with databases. File storage. BaaS providers often offer storage solutions for media files, user uploads, and other binary data. Applications can upload, download, and delete files through provided SDKs and APIs. Authentication and authorization. Some BaaS offer authentication and authorization services that allow developers to easily manage app users. This includes user sign-up, login, password reset, social media login integration through OAuth, user group and permission management etc. Notification service. Some BaaS providers such as Firebase and AWS Amplify have notification services that can send custom emails to users and push native notifications on mobile platforms. This is especially useful for applications that need to send messages, alerts, and reminders. Cloud functions. Some BaaS allow developers to deploy and run serverless functions. The functions are usually stateless and can be triggered by various ways including HTTP requests, SDK invocation, background server events, and cloud scheduled executions. Different providers offer runtime support for different languages, some of the popular languages are JavaScript/TypeScript (Node.js, Deno), Python, Java/Kotlin. Cloud functions extend the potential and flexibility of BaaS by allowing developers to write custom functionalities for their apps, working in a way similar to a traditional REST API backend framework. Usage analytics. Analytics data about application usage is often included in BaaS. This allows developers to monitor user behaviors and make decisions correspondingly in marketing strategies and performance optimizations. UI design. Some BaaS providers, such as AWS Amplify and Backendless, offer user interface designing tools that help developers design the frontend UI of web and mobile apps. While this may be useful for small teams and individual developers, UI design assistance may not be conventional in BaaS as it goes beyond the scope of backend infrastructure. Real-Time. Real-time features in a BaaS platform ensure that data updates and synchronizations occur instantly across all clients, making changes immediately visible to users. This is crucial for applications like live chat and collaborative tools, using technologies like WebSockets to maintain continuous server-client connections. == Service providers == BaaS providers have a broad focus, providing SDKs and APIs that work for app development on multiple platforms with different technology stacks, such as JavaScript (for Web apps), Flutter, Java/Kotlin (for Android apps), Swift/Objective-C (for iOS/MacOS/WatchOS/TvOS apps), .NET (for Windows) and others. BaaS providers also come in different types, suiting developers of different needs. === Cloud-based BaaS === Most BaaS providers host backend platforms on their cloud servers. They also manage the infrastructure, security, and scalability of the platforms. Developers can access the backend services via a web interface or the provided APIs. Some examples of cloud-based BaaS include Firebase (hosted on Google Cloud Platform), AWS Amplify (hosted on Amazon Web Services), and Microsoft Azure Mobile Apps (hosted on Microsoft Azure). === Self-hosted BaaS === Self-hosted BaaS allow developers to host backend on their own servers, providing more flexibility and potential to customization compared to cloud-based BaaS, which often is more difficult to migrate from. However, developers are also in charge of managing the infrastructure, security, and scalability of their servers. === Mobile BaaS === Mobile backend as a service (MBaaS) is a type of BaaS specifically for applications deployed in mobile systems. While some references use MBaaS interchangeably for BaaS, BaaS can have a wider variety of support such as for web apps and desktop apps. == Business model == BaaS providers generate revenue from their services in various ways, often using a freemium model. Under this model, a client receives a certain number of free active users or API calls per month, and pays a fee for each user or call over this limit. Alternatively, clients can pay a set fee for a package which allows for a greater number of calls or active users per month. There are also flat fee plans that make the pricing more predictable. Some of the providers offer the unlimited API calls inside their free plan offerings. Another business model that has been used by a lot of BaaS providers is PAYG (pay as you go), which has a flexible cost based on developers' usage of database, storage, bandwidth, function calls, user numbers etc.

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  Evolvability (computer science)

  The term evolvability is a framework of computational learning introduced by Leslie Valiant in his paper of the same name. The aim of this theory is to model biological evolution and categorize which types of mechanisms are evolvable. Evolution is an extension of PAC learning and learning from statistical queries. == General framework == Let F n {\displaystyle F_{n}\,} and R n {\displaystyle R_{n}\,} be collections of functions on n {\displaystyle n\,} variables. Given an ideal function f ∈ F n {\displaystyle f\in F_{n}} , the goal is to find by local search a representation r ∈ R n {\displaystyle r\in R_{n}} that closely approximates f {\displaystyle f\,} . This closeness is measured by the performance Perf ⁡ ( f , r ) {\displaystyle \operatorname {Perf} (f,r)} of r {\displaystyle r\,} with respect to f {\displaystyle f\,} . As is the case in the biological world, there is a difference between genotype and phenotype. In general, there can be multiple representations (genotypes) that correspond to the same function (phenotype). That is, for some r , r ′ ∈ R n {\displaystyle r,r'\in R_{n}} , with r ≠ r ′ {\displaystyle r\neq r'\,} , still r ( x ) = r ′ ( x ) {\displaystyle r(x)=r'(x)\,} for all x ∈ X n {\displaystyle x\in X_{n}} . However, this need not be the case. The goal then, is to find a representation that closely matches the phenotype of the ideal function, and the spirit of the local search is to allow only small changes in the genotype. Let the neighborhood N ( r ) {\displaystyle N(r)\,} of a representation r {\displaystyle r\,} be the set of possible mutations of r {\displaystyle r\,} . For simplicity, consider Boolean functions on X n = { − 1 , 1 } n {\displaystyle X_{n}=\{-1,1\}^{n}\,} , and let D n {\displaystyle D_{n}\,} be a probability distribution on X n {\displaystyle X_{n}\,} . Define the performance in terms of this. Specifically, Perf ⁡ ( f , r ) = ∑ x ∈ X n f ( x ) r ( x ) D n ( x ) . {\displaystyle \operatorname {Perf} (f,r)=\sum _{x\in X_{n}}f(x)r(x)D_{n}(x).} Note that Perf ⁡ ( f , r ) = Prob ⁡ ( f ( x ) = r ( x ) ) − Prob ⁡ ( f ( x ) ≠ r ( x ) ) . {\displaystyle \operatorname {Perf} (f,r)=\operatorname {Prob} (f(x)=r(x))-\operatorname {Prob} (f(x)\neq r(x)).} In general, for non-Boolean functions, the performance will not correspond directly to the probability that the functions agree, although it will have some relationship. Throughout an organism's life, it will only experience a limited number of environments, so its performance cannot be determined exactly. The empirical performance is defined by Perf s ⁡ ( f , r ) = 1 s ∑ x ∈ S f ( x ) r ( x ) , {\displaystyle \operatorname {Perf} _{s}(f,r)={\frac {1}{s}}\sum _{x\in S}f(x)r(x),} where S {\displaystyle S\,} is a multiset of s {\displaystyle s\,} independent selections from X n {\displaystyle X_{n}\,} according to D n {\displaystyle D_{n}\,} . If s {\displaystyle s\,} is large enough, evidently Perf s ⁡ ( f , r ) {\displaystyle \operatorname {Perf} _{s}(f,r)} will be close to the actual performance Perf ⁡ ( f , r ) {\displaystyle \operatorname {Perf} (f,r)} . Given an ideal function f ∈ F n {\displaystyle f\in F_{n}} , initial representation r ∈ R n {\displaystyle r\in R_{n}} , sample size s {\displaystyle s\,} , and tolerance t {\displaystyle t\,} , the mutator Mut ⁡ ( f , r , s , t ) {\displaystyle \operatorname {Mut} (f,r,s,t)} is a random variable defined as follows. Each r ′ ∈ N ( r ) {\displaystyle r'\in N(r)} is classified as beneficial, neutral, or deleterious, depending on its empirical performance. Specifically, r ′ {\displaystyle r'\,} is a beneficial mutation if Perf s ⁡ ( f , r ′ ) − Perf s ⁡ ( f , r ) ≥ t {\displaystyle \operatorname {Perf} _{s}(f,r')-\operatorname {Perf} _{s}(f,r)\geq t} ; r ′ {\displaystyle r'\,} is a neutral mutation if − t < Perf s ⁡ ( f , r ′ ) − Perf s ⁡ ( f , r ) < t {\displaystyle -t<\operatorname {Perf} _{s}(f,r')-\operatorname {Perf} _{s}(f,r) 0 {\displaystyle \epsilon >0\,} , for all ideal functions f ∈ F n {\displaystyle f\in F_{n}} and representations r 0 ∈ R n {\displaystyle r_{0}\in R_{n}} , with probability at least 1 − ϵ {\displaystyle 1-\epsilon \,} , Perf ⁡ ( f , r g ( n , 1 / ϵ ) ) ≥ 1 − ϵ , {\displaystyle \operatorname {Perf} (f,r_{g(n,1/\epsilon )})\geq 1-\epsilon ,} where the sizes of neighborhoods N ( r ) {\displaystyle N(r)\,} for r ∈ R n {\displaystyle r\in R_{n}\,} are at most p ( n , 1 / ϵ ) {\displaystyle p(n,1/\epsilon )\,} , the sample size is s ( n , 1 / ϵ ) {\displaystyle s(n,1/\epsilon )\,} , the tolerance is t ( 1 / n , ϵ ) {\displaystyle t(1/n,\epsilon )\,} , and the generation size is g ( n , 1 / ϵ ) {\displaystyle g(n,1/\epsilon )\,} . F {\displaystyle F\,} is evolvable over D {\displaystyle D\,} if it is evolvable by some R {\displaystyle R\,} over D {\displaystyle D\,} . F {\displaystyle F\,} is evolvable if it is evolvable over all distributions D {\displaystyle D\,} . == Results == The class of conjunctions and the class of disjunctions are evolvable over the uniform distribution for short conjunctions and disjunctions, respectively. The class of parity functions (which evaluate to the parity of the number of true literals in a given subset of literals) are not evolvable, even for the uniform distribution. Evolvability implies PAC learnability.

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  Self-management (computer science)

  Self-management is the process by which computer systems manage their own operation without human intervention. Self-management technologies are expected to pervade the next generation of network management systems. The growing complexity of modern networked computer systems is a limiting factor in their expansion. The increasing heterogeneity of corporate computer systems, the inclusion of mobile computing devices, and the combination of different networking technologies like WLAN, cellular phone networks, and mobile ad hoc networks make the conventional, manual management difficult, time-consuming, and error-prone. More recently, self-management has been suggested as a solution to increasing complexity in cloud computing. An industrial initiative towards realizing self-management is the Autonomic Computing Initiative (ACI) started by IBM in 2001. The ACI defines the following four functional areas: Self-configuration Auto-configuration of components Self-healing Automatic discovery, and correction of faults; automatically applying all necessary actions to bring system back to normal operation Self-optimization Automatic monitoring and control of resources to ensure the optimal functioning with respect to the defined requirements Self-protection Proactive identification and protection from arbitrary attacks

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