A whitelist or allowlist is a list or register of entities that are being provided a particular privilege, service, mobility, access or recognition. Entities on the list will be accepted, approved and/or recognized. Whitelisting is the reverse of blacklisting, the practice of identifying entities that are denied, unrecognized, or ostracized. == Email whitelists == Spam filters often include the ability to "whitelist" certain sender IP addresses, email addresses or domain names to protect their email from being rejected or sent to a junk mail folder. These can be manually maintained by the user or system administrator - but can also refer to externally maintained whitelist services. === Non-commercial whitelists === Non-commercial whitelists are operated by various non-profit organizations, ISPs, and others interested in blocking spam. Rather than paying fees, the sender must pass a series of tests; for example, their email server must not be an open relay and have a static IP address. The operator of the whitelist may remove a server from the list if complaints are received. === Commercial whitelists === Commercial whitelists are a system by which an Internet service provider allows someone to bypass spam filters when sending email messages to its subscribers, in return for a pre-paid fee, either an annual or a per-message fee. A sender can then be more confident that their messages have reached recipients without being blocked, or having links or images stripped out of them, by spam filters. The purpose of commercial whitelists is to allow companies to reliably reach their customers by email. == Advertising whitelist == Many websites rely on ads as a source of revenue, but the use of ad blockers is increasingly common. Websites that detect an adblocker in use often ask for it to be disabled - or their site to be "added to the whitelist" - a standard feature of most adblockers. == Network whitelists == === LAN whitelists === A use for whitelists is in local area network (LAN) security. Many network admins set up MAC address whitelists, or a MAC address filter, to control who is allowed on their networks. This is used when encryption is not a practical solution or in tandem with encryption. However, it's sometimes ineffective because a MAC address can be faked. === IP whitelist === Firewalls can usually be configured to only allow data-traffic from/to certain (ranges of) IP-addresses. === Application whitelists === One approach in combating viruses and malware is to whitelist software which is considered safe to run, blocking all others. This is particularly attractive in a corporate environment, where there are typically already restrictions on what software is approved. Leading providers of application whitelisting technology include Bit9, Velox, McAfee, Lumension, ThreatLocker, Airlock Digital and SMAC. On Microsoft Windows, recent versions include AppLocker, which allows administrators to control which executable files are denied or allowed to execute. With AppLocker, administrators are able to create rules based on file names, publishers or file location that will allow certain files to execute. Rules can apply to individuals or groups. Policies are used to group users into different enforcement levels. For example, some users can be added to a report-only policy that will allow administrators to understand the impact before moving that user to a higher enforcement level. Linux systems typically have AppArmor and SE Linux features available which can be used to effectively block all applications which are not explicitly whitelisted, and commercial products are also available. On HP-UX introduced a feature called "HP-UX Whitelisting" on 11iv3 version. == Controversy regarding name == In 2018, a journal commentary on a report on predatory publishing was released making claims that "white" and "black" are racially charged terms that need to be avoided in instances such as "whitelist" and "blacklist". The premise of the journal is that "black" and "white" have negative and positive connotations respectively. It states that since "blacklisting" was first referred to during "the time of mass enslavement and forced deportation of Africans to work in European-held colonies in the Americas," the word is therefore related to race. There is no mention of "whitelist" and its origin or relation to race. This issue is most widely disputed in computing industries where "whitelist" and "blacklist" are prevalent (e.g. IP whitelisting). Despite the commentary nature of the journal, some companies and individuals in others have taken to replacing "whitelist" and "blacklist" with new alternatives such as "allow list" and "deny list". Those adopting this change consider using the "whitelist"/"blacklist" names as a code smell. Those that oppose these changes question its attribution to race, citing the same etymology quote that the 2018 journal uses. According to the remark, the term "blacklist" evolved from the term "black book" about a century ago. The term "black book" does not appear to have any etymology or sources that support racial associations, instead originating in the 1400s as a reference to "a list of people who had committed crimes or fallen out of favor with leaders", and popularized by King Henry VIII's literal use of a black book. Others also note the prevalence of positive and negative connotations to "white" and "black" in the Bible, predating attributions to skin tone and slavery. It wasn't until the 1960s Black Power movement that "Black" became a widespread word to refer to one's race as a person of color in America (alternate to African-American) lending itself to the argument that the negative connotation behind "black" and "blacklist" both predate attribution to race.
Spatiotemporal reservoir resampling
Spatiotemporal reservoir resampling, commonly known as ReSTIR (from "Reservoir-based SpatioTemporal Importance Resampling"), is a collection of computer graphics techniques for reusing samples during rendering. It was developed primarily to allow more realistic lighting in real-time rendering, because relatively few rays can be traced per pixel while maintaining an acceptable frame rate. It can also be used to speed up off-line path tracing. The first ReSTIR paper, published in 2020, provided algorithms for direct lighting, allowing scenes containing thousands of lights to be rendered in real time on a high-end GPU. Researchers later proposed versions for rendering indirect lighting (and more recently, motion blur and depth of field) and built up a framework of mathematical concepts and notation conventions that help analyze such algorithms. A major focus of this work is removing or reducing the bias that could be introduced when samples from other pixels or frames are reused—or selectively allowing some bias in order to speed up rendering and reduce variance (visible as "noise" in the image). Versions for path tracing apply transformations called shift mappings to samples, typically reusing parts of paths closer to the light and modifying the portion closer to the camera. ReSTIR-related papers and talks have been presented every year at the SIGGRAPH conference since 2020. One of the first games to incorporate ReSTIR into its rendering was Cyberpunk 2077. == Overview and motivation == According to Chris Wyman, one of the co-authors of the original paper, although developers commonly thought that bias was acceptable for real-time rendering, end users (e.g. gamers) are well-aware of the artifacts caused by bias and many have a negative opinion of common sample-reuse techniques such as temporal anti-aliasing (TAA), which may cause "ghosting" when the camera moves, and denoising, which causes blurring and other artifacts. ReSTIR techniques can reduce or avoid these types of bias by reusing samples of the set of possible paths taken by light to reach the camera, instead of reusing rendered pixel color values (which are typically the average of multiple samples, discarding information such as the direction of the light). While other techniques reuse samples in a generic post-processing step, ReSTIR passes can test for shadowing, and reused samples are converted into pixel color values by rendering code that takes the characteristics of different materials into account (e.g. by implementing BRDFs). However the output of ReSTIR is noisy, and a denoising pass is typically still used. Stochastic ray tracing techniques such as path tracing need to average multiple samples (produced by tracing individual rays) in order to render a visually acceptable image. When using a simple unbiased renderer based on Monte Carlo integration, halving the deviation of the result (apparent as "noise" in the image) requires multiplying the number of samples by four, meaning that a rapidly increasingly number of samples is needed to improve quality, Standard ways to mitigate this problem include importance sampling (which requires finding improved sampling distributions for specific situations), and quasi-Monte Carlo integration (which usually still requires tracing a large number of rays). ReSTIR offers a solution that multiplies the effective number of samples while tracing a fixed number of additional rays per frame. Temporal reuse multiplies the effective sample count by the number of frames rendered. Spatial reuse multiplies the effective count by the number of neighboring pixels examined. These two types of reuse can be combined, allowing spatial reuse to be applied recursively, which appears to offer an exponentially increasing effective sample count, however this is quickly limited by the size of the neighborhood used for spatial reuse. Spatial reuse is also potentially less effective near shadow and object edges, especially for objects with fine geometric detail, and temporal reuse is limited by movement of the camera and scene elements. == Variations == Many variations of ReSTIR have been proposed that generalize or improve the original technique (which builds on an earlier method called RIS), specialize it for particular types of illumination or other visual effects, or allow incorporation into rendering algorithms other than standard path tracing. Some published versions are listed below. == Algorithms == === Basic algorithm === ReSTIR uses a combination of resampled importance sampling (RIS) and weighted reservoir sampling (WRS) which the authors call streaming RIS. RIS processes samples from an initial probability distribution (e.g. a probability distribution for which a cheap sampling method exists) and generates samples in a new probability distribution (e.g. a sampling distribution that is optimal for rendering but is impractical to draw samples from directly). WRS allows this to be done while storing only a small number of samples in memory, which is especially helpful on a GPU. Information about the samples is stored in a data structure called a reservoir. WRS also allows samples from multiple reservoirs to be combined ("merged") into a single reservoir; this is crucial for sample reuse. Each pixel has a reservoir, typically containing only a single sample when ReSTIR is used for real-time rendering (some implementations use a larger number, e.g. four samples). The reservoir is typically initialized to a sample drawn using a simple method and is then updated by RIS steps and by reservoir merging, so that the pixel value produced by shading using the sample(s) currently in the reservoir, times the weight for the sample, is always an unbiased estimate of the correct pixel value. If appropriate resampling steps are used, the variance of this estimate (or some function of it, typically the luminance of the RGB color value) decreases with each step. A possible sequence of steps performed for each frame, suitable for computing unbiased direct illumination (DI) is: Perform reservoir resampling by drawing multiple light samples and using streaming RIS to choose one, using probabilities based on a target function, e.g. the luminance of the sample's contribution to the pixel. A weight is also computed for the sample. Typically, a single visibility check is performed here, after choosing a sample, setting the weight to 0 if the light is shadowed. Resampling (combined with the visibility check) ensures that the expected value of the weight times the sample brightness is the correct (unbiased) value for the pixel. (temporal reuse) For each pixel, merge the sample(s) from the previous frame into the current reservoir. Multiple importance sampling (MIS) weights are used to avoid bias due to the fact that the samples in the previous frame's reservoirs may have a different target probability distribution if the objects, lights, or camera have moved. (spatial reuse) For each pixel, choose one or more neighboring pixels and merge their samples into the current pixel's reservoir. Multiple importance sampling (MIS) weights are used to avoid bias due to the fact that the samples in each pixel's reservoir have a different target probability distribution. Because computing unbiased MIS weights requires tracing additional rays (along with other work such as evaluating BRDFs), real-time rendering often uses only a single neighboring pixel. Use the sample in each pixel's reservoir, along with its weight, to determine the color of the pixel for the current frame. Alternatively, multiple samples examined during the preceding steps may be averaged and used to shade the pixel instead (decoupled shading and sampling). For direct lighting, the initial samples used in step 1 are typically drawn by importance sampling from the set of lights in a scene. The algorithm above (from the original ReSTIR paper) draws many lower-quality light samples (e.g. 32) using a fast method, without considering visibility, and chooses one using streaming RIS. Visibility is then tested for the final chosen sample. Considering visibility for each sample drawn would require tracing 32 rays, which would make it much more expensive. The intent is to reduce the number of rays traced, relying on the sample reuse in steps 2 and 3 to make up for the loss of quality caused by rejecting many of the rays due to shadowing. A large part of the initial efforts to optimize ReSTIR (to make it run in real-time on available hardware) went into reducing the cost of randomly sampling the lights. Glossy surfaces may require a larger number of samples, and combining light sampling with BRDF sampling (using MIS) may increase quality. Step 2 (temporal reuse) is sometimes skipped for off-line rendering, and the output of multiple repetitions of initial sampling and spatial reuse is averaged instead; this helps avoids artifacts due to correlations. Step 3 (spatial reuse) may be repeated multiple times in a single frame.
Constellation model
The constellation model is a probabilistic, generative model for category-level object recognition in computer vision. Like other part-based models, the constellation model attempts to represent an object class by a set of N parts under mutual geometric constraints. Because it considers the geometric relationship between different parts, the constellation model differs significantly from appearance-only, or "bag-of-words" representation models, which explicitly disregard the location of image features. The problem of defining a generative model for object recognition is difficult. The task becomes significantly complicated by factors such as background clutter, occlusion, and variations in viewpoint, illumination, and scale. Ideally, we would like the particular representation we choose to be robust to as many of these factors as possible. In category-level recognition, the problem is even more challenging because of the fundamental problem of intra-class variation. Even if two objects belong to the same visual category, their appearances may be significantly different. However, for structured objects such as cars, bicycles, and people, separate instances of objects from the same category are subject to similar geometric constraints. For this reason, particular parts of an object such as the headlights or tires of a car still have consistent appearances and relative positions. The Constellation Model takes advantage of this fact by explicitly modeling the relative location, relative scale, and appearance of these parts for a particular object category. Model parameters are estimated using an unsupervised learning algorithm, meaning that the visual concept of an object class can be extracted from an unlabeled set of training images, even if that set contains "junk" images or instances of objects from multiple categories. It can also account for the absence of model parts due to appearance variability, occlusion, clutter, or detector error. == History == The idea for a "parts and structure" model was originally introduced by Fischler and Elschlager in 1973. This model has since been built upon and extended in many directions. The Constellation Model, as introduced by Dr. Perona and his colleagues, was a probabilistic adaptation of this approach. In the late '90s, Burl et al. revisited the Fischler and Elschlager model for the purpose of face recognition. In their work, Burl et al. used manual selection of constellation parts in training images to construct a statistical model for a set of detectors and the relative locations at which they should be applied. In 2000, Weber et al. made the significant step of training the model using a more unsupervised learning process, which precluded the necessity for tedious hand-labeling of parts. Their algorithm was particularly remarkable because it performed well even on cluttered and occluded image data. Fergus et al. then improved upon this model by making the learning step fully unsupervised, having both shape and appearance learned simultaneously, and accounting explicitly for the relative scale of parts. == The method of Weber and Welling et al. == In the first step, a standard interest point detection method, such as Harris corner detection, is used to generate interest points. Image features generated from the vicinity of these points are then clustered using k-means or another appropriate algorithm. In this process of vector quantization, one can think of the centroids of these clusters as being representative of the appearance of distinctive object parts. Appropriate feature detectors are then trained using these clusters, which can be used to obtain a set of candidate parts from images. As a result of this process, each image can now be represented as a set of parts. Each part has a type, corresponding to one of the aforementioned appearance clusters, as well as a location in the image space. === Basic generative model === Weber & Welling here introduce the concept of foreground and background. Foreground parts correspond to an instance of a target object class, whereas background parts correspond to background clutter or false detections. Let T be the number of different types of parts. The positions of all parts extracted from an image can then be represented in the following "matrix," X o = ( x 11 , x 12 , ⋯ , x 1 N 1 x 21 , x 22 , ⋯ , x 2 N 2 ⋮ x T 1 , x T 2 , ⋯ , x T N T ) {\displaystyle X^{o}={\begin{pmatrix}x_{11},x_{12},{\cdots },x_{1N_{1}}\\x_{21},x_{22},{\cdots },x_{2N_{2}}\\\vdots \\x_{T1},x_{T2},{\cdots },x_{TN_{T}}\end{pmatrix}}} where N i {\displaystyle N_{i}\,} represents the number of parts of type i ∈ { 1 , … , T } {\displaystyle i\in \{1,\dots ,T\}} observed in the image. The superscript o indicates that these positions are observable, as opposed to missing. The positions of unobserved object parts can be represented by the vector x m {\displaystyle x^{m}\,} . Suppose that the object will be composed of F {\displaystyle F\,} distinct foreground parts. For notational simplicity, we assume here that F = T {\displaystyle F=T\,} , though the model can be generalized to F > T {\displaystyle F>T\,} . A hypothesis h {\displaystyle h\,} is then defined as a set of indices, with h i = j {\displaystyle h_{i}=j\,} , indicating that point x i j {\displaystyle x_{ij}\,} is a foreground point in X o {\displaystyle X^{o}\,} . The generative probabilistic model is defined through the joint probability density p ( X o , x m , h ) {\displaystyle p(X^{o},x^{m},h)\,} . === Model details === The rest of this section summarizes the details of Weber & Welling's model for a single component model. The formulas for multiple component models are extensions of those described here. To parametrize the joint probability density, Weber & Welling introduce the auxiliary variables b {\displaystyle b\,} and n {\displaystyle n\,} , where b {\displaystyle b\,} is a binary vector encoding the presence/absence of parts in detection ( b i = 1 {\displaystyle b_{i}=1\,} if h i > 0 {\displaystyle h_{i}>0\,} , otherwise b i = 0 {\displaystyle b_{i}=0\,} ), and n {\displaystyle n\,} is a vector where n i {\displaystyle n_{i}\,} denotes the number of background candidates included in the i t h {\displaystyle i^{th}} row of X o {\displaystyle X^{o}\,} . Since b {\displaystyle b\,} and n {\displaystyle n\,} are completely determined by h {\displaystyle h\,} and the size of X o {\displaystyle X^{o}\,} , we have p ( X o , x m , h ) = p ( X o , x m , h , n , b ) {\displaystyle p(X^{o},x^{m},h)=p(X^{o},x^{m},h,n,b)\,} . By decomposition, p ( X o , x m , h , n , b ) = p ( X o , x m | h , n , b ) p ( h | n , b ) p ( n ) p ( b ) {\displaystyle p(X^{o},x^{m},h,n,b)=p(X^{o},x^{m}|h,n,b)p(h|n,b)p(n)p(b)\,} The probability density over the number of background detections can be modeled by a Poisson distribution, p ( n ) = ∏ i = 1 T 1 n i ! ( M i ) n i e − M i {\displaystyle p(n)=\prod _{i=1}^{T}{\frac {1}{n_{i}!}}(M_{i})^{n_{i}}e^{-M_{i}}} where M i {\displaystyle M_{i}\,} is the average number of background detections of type i {\displaystyle i\,} per image. Depending on the number of parts F {\displaystyle F\,} , the probability p ( b ) {\displaystyle p(b)\,} can be modeled either as an explicit table of length 2 F {\displaystyle 2^{F}\,} , or, if F {\displaystyle F\,} is large, as F {\displaystyle F\,} independent probabilities, each governing the presence of an individual part. The density p ( h | n , b ) {\displaystyle p(h|n,b)\,} is modeled by p ( h | n , b ) = { 1 ∏ f = 1 F N f b f , if h ∈ H ( b , n ) 0 , for other h {\displaystyle p(h|n,b)={\begin{cases}{\frac {1}{\textstyle \prod _{f=1}^{F}N_{f}^{b_{f}}}},&{\mbox{if }}h\in H(b,n)\\0,&{\mbox{for other }}h\end{cases}}} where H ( b , n ) {\displaystyle H(b,n)\,} denotes the set of all hypotheses consistent with b {\displaystyle b\,} and n {\displaystyle n\,} , and N f {\displaystyle N_{f}\,} denotes the total number of detections of parts of type f {\displaystyle f\,} . This expresses the fact that all consistent hypotheses, of which there are ∏ f = 1 F N f b f {\displaystyle \textstyle \prod _{f=1}^{F}N_{f}^{b_{f}}} , are equally likely in the absence of information on part locations. And finally, p ( X o , x m | h , n ) = p f g ( z ) p b g ( x b g ) {\displaystyle p(X^{o},x^{m}|h,n)=p_{fg}(z)p_{bg}(x_{bg})\,} where z = ( x o x m ) {\displaystyle z=(x^{o}x^{m})\,} are the coordinates of all foreground detections, observed and missing, and x b g {\displaystyle x_{bg}\,} represents the coordinates of the background detections. Note that foreground detections are assumed to be independent of the background. p f g ( z ) {\displaystyle p_{fg}(z)\,} is modeled as a joint Gaussian with mean μ {\displaystyle \mu \,} and covariance Σ {\displaystyle \Sigma \,} . === Classification === The ultimate objective of this model is to classify images into classes "object present" (class C 1 {\displaystyle C_{1}\,} ) and "object absent" (class C 0 {\displaystyle C_{0}\,} ) given t
Mutation (evolutionary algorithm)
Mutation is a genetic operator used to maintain genetic diversity of the chromosomes of a population of an evolutionary algorithm (EA), including genetic algorithms in particular. It is analogous to biological mutation. The classic example of a mutation operator of a binary coded genetic algorithm (GA) involves a probability that an arbitrary bit in a genetic sequence will be flipped from its original state. A common method of implementing the mutation operator involves generating a random variable for each bit in a sequence. This random variable tells whether or not a particular bit will be flipped. This mutation procedure, based on the biological point mutation, is called single point mutation. Other types of mutation operators are commonly used for representations other than binary, such as floating-point encodings or representations for combinatorial problems. The purpose of mutation in EAs is to introduce diversity into the sampled population. Mutation operators are used in an attempt to avoid local minima by preventing the population of chromosomes from becoming too similar to each other, thus slowing or even stopping convergence to the global optimum. This reasoning also leads most EAs to avoid only taking the fittest of the population in generating the next generation, but rather selecting a random (or semi-random) set with a weighting toward those that are fitter. The following requirements apply to all mutation operators used in an EA: every point in the search space must be reachable by one or more mutations. there must be no preference for parts or directions in the search space (no drift). small mutations should be more probable than large ones. For different genome types, different mutation types are suitable. Some mutations are Gaussian, Uniform, Zigzag, Scramble, Insertion, Inversion, Swap, and so on. An overview and more operators than those presented below can be found in the introductory book by Eiben and Smith or in. == Bit string mutation == The mutation of bit strings ensue through bit flips at random positions. Example: The probability of a mutation of a bit is 1 l {\displaystyle {\frac {1}{l}}} , where l {\displaystyle l} is the length of the binary vector. Thus, a mutation rate of 1 {\displaystyle 1} per mutation and individual selected for mutation is reached. == Mutation of real numbers == Many EAs, such as the evolution strategy or the real-coded genetic algorithms, work with real numbers instead of bit strings. This is due to the good experiences that have been made with this type of coding. The value of a real-valued gene can either be changed or redetermined. A mutation that implements the latter should only ever be used in conjunction with the value-changing mutations and then only with comparatively low probability, as it can lead to large changes. In practical applications, the respective value range of the decision variables to be changed of the optimisation problem to be solved is usually limited. Accordingly, the values of the associated genes are each restricted to an interval [ x min , x max ] {\displaystyle [x_{\min },x_{\max }]} . Mutations may or may not take these restrictions into account. In the latter case, suitable post-treatment is then required as described below. === Mutation without consideration of restrictions === A real number x {\displaystyle x} can be mutated using normal distribution N ( 0 , σ ) {\displaystyle {\mathcal {N}}(0,\sigma )} by adding the generated random value to the old value of the gene, resulting in the mutated value x ′ {\displaystyle x'} : x ′ = x + N ( 0 , σ ) {\displaystyle x'=x+{\mathcal {N}}(0,\sigma )} In the case of genes with a restricted range of values, it is a good idea to choose the step size of the mutation σ {\displaystyle \sigma } so that it reasonably fits the range [ x min , x max ] {\displaystyle [x_{\min },x_{\max }]} of the gene to be changed, e.g.: σ = x max − x min 6 {\displaystyle \sigma ={\frac {x_{\text{max}}-x_{\text{min}}}{6}}} The step size can also be adjusted to the smaller permissible change range depending on the current value. In any case, however, it is likely that the new value x ′ {\displaystyle x'} of the gene will be outside the permissible range of values. Such a case must be considered a lethal mutation, since the obvious repair by using the respective violated limit as the new value of the gene would lead to a drift. This is because the limit value would then be selected with the entire probability of the values beyond the limit of the value range. The evolution strategy works with real numbers and mutation based on normal distribution. The step sizes are part of the chromosome and are subject to evolution together with the actual decision variables. === Mutation with consideration of restrictions === One possible form of changing the value of a gene while taking its value range [ x min , x max ] {\displaystyle [x_{\min },x_{\max }]} into account is the mutation relative parameter change of the evolutionary algorithm GLEAM (General Learning Evolutionary Algorithm and Method), in which, as with the mutation presented earlier, small changes are more likely than large ones. First, an equally distributed decision is made as to whether the current value x {\displaystyle x} should be increased or decreased and then the corresponding total change interval is determined. Without loss of generality, an increase is assumed for the explanation and the total change interval is then [ x , x max ] {\displaystyle [x,x_{\max }]} . It is divided into k {\displaystyle k} sub-areas of equal size with the width δ {\displaystyle \delta } , from which k {\displaystyle k} sub-change intervals of different size are formed: i {\displaystyle i} -th sub-change interval: [ x , x + δ ⋅ i ] {\displaystyle [x,x+\delta \cdot i]} with δ = ( x max − x ) k {\displaystyle \delta ={\frac {(x_{\text{max}}-x)}{k}}} and i = 1 , … , k {\displaystyle i=1,\dots ,k} Subsequently, one of the k {\displaystyle k} sub-change intervals is selected in equal distribution and a random number, also equally distributed, is drawn from it as the new value x ′ {\displaystyle x'} of the gene. The resulting summed probabilities of the sub-change intervals result in the probability distribution of the k {\displaystyle k} sub-areas shown in the adjacent figure for the exemplary case of k = 10 {\displaystyle k=10} . This is not a normal distribution as before, but this distribution also clearly favours small changes over larger ones. This mutation for larger values of k {\displaystyle k} , such as 10, is less well suited for tasks where the optimum lies on one of the value range boundaries. This can be remedied by significantly reducing k {\displaystyle k} when a gene value approaches its limits very closely. === Common properties === For both mutation operators for real-valued numbers, the probability of an increase and decrease is independent of the current value and is 50% in each case. In addition, small changes are considerably more likely than large ones. For mixed-integer optimization problems, rounding is usually used. == Mutation of permutations == Mutations of permutations are specially designed for genomes that are themselves permutations of a set. These are often used to solve combinatorial tasks. In the two mutations presented, parts of the genome are rotated or inverted. === Rotation to the right === The presentation of the procedure is illustrated by an example on the right: === Inversion === The presentation of the procedure is illustrated by an example on the right: === Variants with preference for smaller changes === The requirement raised at the beginning for mutations, according to which small changes should be more probable than large ones, is only inadequately fulfilled by the two permutation mutations presented, since the lengths of the partial lists and the number of shift positions are determined in an equally distributed manner. However, the longer the partial list and the shift, the greater the change in gene order. This can be remedied by the following modifications. The end index j {\displaystyle j} of the partial lists is determined as the distance d {\displaystyle d} to the start index i {\displaystyle i} : j = ( i + d ) mod | P 0 | {\displaystyle j=(i+d){\bmod {\left|P_{0}\right|}}} where d {\displaystyle d} is determined randomly according to one of the two procedures for the mutation of real numbers from the interval [ 0 , | P 0 | − 1 ] {\displaystyle \left[0,\left|P_{0}\right|-1\right]} and rounded. For the rotation, k {\displaystyle k} is determined similarly to the distance d {\displaystyle d} , but the value 0 {\displaystyle 0} is forbidden. For the inversion, note that i ≠ j {\displaystyle i\neq j} must hold, so for d {\displaystyle d} the value 0 {\displaystyle 0} must be excluded.
Determining the number of clusters in a data set
Determining the number of clusters in a data set, a quantity often labelled k as in the k-means algorithm, is a frequent problem in data clustering, and is a distinct issue from the process of actually solving the clustering problem. For a certain class of clustering algorithms (in particular k-means, k-medoids and expectation–maximization algorithm), there is a parameter commonly referred to as k that specifies the number of clusters to detect. Other algorithms such as DBSCAN and OPTICS algorithm do not require the specification of this parameter; hierarchical clustering avoids the problem altogether. The correct choice of k is often ambiguous, with interpretations depending on the shape and scale of the distribution of points in a data set and the desired clustering resolution of the user. In addition, increasing k without penalty will always reduce the amount of error in the resulting clustering, to the extreme case of zero error if each data point is considered its own cluster (i.e., when k equals the number of data points, n). Intuitively then, the optimal choice of k will strike a balance between maximum compression of the data using a single cluster, and maximum accuracy by assigning each data point to its own cluster. If an appropriate value of k is not apparent from prior knowledge of the properties of the data set, it must be chosen somehow. There are several categories of methods for making this decision. == Elbow method == The elbow method looks at the percentage of explained variance as a function of the number of clusters: One should choose a number of clusters so that adding another cluster does not give much better modeling of the data. More precisely, if one plots the percentage of variance explained by the clusters against the number of clusters, the first clusters will add much information (explain a lot of variance), but at some point the marginal gain will drop, giving an angle in the graph. The number of clusters is chosen at this point, hence the "elbow criterion". In most datasets, this "elbow" is ambiguous, making this method subjective and unreliable. Because the scale of the axes is arbitrary, the concept of an angle is not well-defined, and even on uniform random data, the curve produces an "elbow", making the method rather unreliable. Percentage of variance explained is the ratio of the between-group variance to the total variance, also known as an F-test. A slight variation of this method plots the curvature of the within group variance. The method can be traced to speculation by Robert L. Thorndike in 1953. While the idea of the elbow method sounds simple and straightforward, other methods (as detailed below) give better results. == X-means clustering == In statistics and data mining, X-means clustering is a variation of k-means clustering that refines cluster assignments by repeatedly attempting subdivision, and keeping the best resulting splits, until a criterion such as the Akaike information criterion (AIC) or Bayesian information criterion (BIC) is reached. == Information criterion approach == Another set of methods for determining the number of clusters are information criteria, such as the Akaike information criterion (AIC), Bayesian information criterion (BIC), or the deviance information criterion (DIC) — if it is possible to make a likelihood function for the clustering model. For example: The k-means model is "almost" a Gaussian mixture model and one can construct a likelihood for the Gaussian mixture model and thus also determine information criterion values. == Information–theoretic approach == Rate distortion theory has been applied to choosing k called the "jump" method, which determines the number of clusters that maximizes efficiency while minimizing error by information-theoretic standards. The strategy of the algorithm is to generate a distortion curve for the input data by running a standard clustering algorithm such as k-means for all values of k between 1 and n, and computing the distortion (described below) of the resulting clustering. The distortion curve is then transformed by a negative power chosen based on the dimensionality of the data. Jumps in the resulting values then signify reasonable choices for k, with the largest jump representing the best choice. The distortion of a clustering of some input data is formally defined as follows: Let the data set be modeled as a p-dimensional random variable, X, consisting of a mixture distribution of G components with common covariance, Γ. If we let c 1 … c K {\displaystyle c_{1}\ldots c_{K}} be a set of K cluster centers, with c X {\displaystyle c_{X}} the closest center to a given sample of X, then the minimum average distortion per dimension when fitting the K centers to the data is: d K = 1 p min c 1 … c K E [ ( X − c X ) T Γ − 1 ( X − c X ) ] {\displaystyle d_{K}={\frac {1}{p}}\min _{c_{1}\ldots c_{K}}{E[(X-c_{X})^{T}\Gamma ^{-1}(X-c_{X})]}} This is also the average Mahalanobis distance per dimension between X and the closest cluster center c X {\displaystyle c_{X}} . Because the minimization over all possible sets of cluster centers is prohibitively complex, the distortion is computed in practice by generating a set of cluster centers using a standard clustering algorithm and computing the distortion using the result. The pseudo-code for the jump method with an input set of p-dimensional data points X is: JumpMethod(X): Let Y = (p/2) Init a list D, of size n+1 Let D[0] = 0 For k = 1 ... n: Cluster X with k clusters (e.g., with k-means) Let d = Distortion of the resulting clustering D[k] = d^(-Y) Define J(i) = D[i] - D[i-1] Return the k between 1 and n that maximizes J(k) The choice of the transform power Y = ( p / 2 ) {\displaystyle Y=(p/2)} is motivated by asymptotic reasoning using results from rate distortion theory. Let the data X have a single, arbitrarily p-dimensional Gaussian distribution, and let fixed K = ⌊ α p ⌋ {\displaystyle K=\lfloor \alpha ^{p}\rfloor } , for some α greater than zero. Then the distortion of a clustering of K clusters in the limit as p goes to infinity is α − 2 {\displaystyle \alpha ^{-2}} . It can be seen that asymptotically, the distortion of a clustering to the power ( − p / 2 ) {\displaystyle (-p/2)} is proportional to α p {\displaystyle \alpha ^{p}} , which by definition is approximately the number of clusters K. In other words, for a single Gaussian distribution, increasing K beyond the true number of clusters, which should be one, causes a linear growth in distortion. This behavior is important in the general case of a mixture of multiple distribution components. Let X be a mixture of G p-dimensional Gaussian distributions with common covariance. Then for any fixed K less than G, the distortion of a clustering as p goes to infinity is infinite. Intuitively, this means that a clustering of less than the correct number of clusters is unable to describe asymptotically high-dimensional data, causing the distortion to increase without limit. If, as described above, K is made an increasing function of p, namely, K = ⌊ α p ⌋ {\displaystyle K=\lfloor \alpha ^{p}\rfloor } , the same result as above is achieved, with the value of the distortion in the limit as p goes to infinity being equal to α − 2 {\displaystyle \alpha ^{-2}} . Correspondingly, there is the same proportional relationship between the transformed distortion and the number of clusters, K. Putting the results above together, it can be seen that for sufficiently high values of p, the transformed distortion d K − p / 2 {\displaystyle d_{K}^{-p/2}} is approximately zero for K < G, then jumps suddenly and begins increasing linearly for K ≥ G. The jump algorithm for choosing K makes use of these behaviors to identify the most likely value for the true number of clusters. Although the mathematical support for the method is given in terms of asymptotic results, the algorithm has been empirically verified to work well in a variety of data sets with reasonable dimensionality. In addition to the localized jump method described above, there exists a second algorithm for choosing K using the same transformed distortion values known as the broken line method. The broken line method identifies the jump point in the graph of the transformed distortion by doing a simple least squares error line fit of two line segments, which in theory will fall along the x-axis for K < G, and along the linearly increasing phase of the transformed distortion plot for K ≥ G. The broken line method is more robust than the jump method in that its decision is global rather than local, but it also relies on the assumption of Gaussian mixture components, whereas the jump method is fully non-parametric and has been shown to be viable for general mixture distributions. == Silhouette method == The average silhouette of the data is another useful criterion for assessing the natural number of clusters. The silhouette of a data instance is a measure of how closely it is match
Empirical risk minimization
In statistical learning theory, the principle of empirical risk minimization defines a family of learning algorithms based on evaluating performance over a known and fixed dataset. The core idea is based on an application of the law of large numbers; more specifically, we cannot know exactly how well a predictive algorithm will work in practice (i.e. the "true risk") because we do not know the true distribution of the data, but we can instead estimate and optimize the performance of the algorithm on a known set of training data. The performance over the known set of training data is referred to as the "empirical risk". == Background == The following situation is a general setting of many supervised learning problems. There are two spaces of objects X {\displaystyle X} and Y {\displaystyle Y} and we would like to learn a function h : X → Y {\displaystyle \ h:X\to Y} (often called hypothesis) which outputs an object y ∈ Y {\displaystyle y\in Y} , given x ∈ X {\displaystyle x\in X} . To do so, there is a training set of n {\displaystyle n} examples ( x 1 , y 1 ) , … , ( x n , y n ) {\displaystyle \ (x_{1},y_{1}),\ldots ,(x_{n},y_{n})} where x i ∈ X {\displaystyle x_{i}\in X} is an input and y i ∈ Y {\displaystyle y_{i}\in Y} is the corresponding response that is desired from h ( x i ) {\displaystyle h(x_{i})} . To put it more formally, assuming that there is a joint probability distribution P ( x , y ) {\displaystyle P(x,y)} over X {\displaystyle X} and Y {\displaystyle Y} , and that the training set consists of n {\displaystyle n} instances ( x 1 , y 1 ) , … , ( x n , y n ) {\displaystyle \ (x_{1},y_{1}),\ldots ,(x_{n},y_{n})} drawn i.i.d. from P ( x , y ) {\displaystyle P(x,y)} . The assumption of a joint probability distribution allows for the modelling of uncertainty in predictions (e.g. from noise in data) because y {\displaystyle y} is not a deterministic function of x {\displaystyle x} , but rather a random variable with conditional distribution P ( y | x ) {\displaystyle P(y|x)} for a fixed x {\displaystyle x} . It is also assumed that there is a non-negative real-valued loss function L ( y ^ , y ) {\displaystyle L({\hat {y}},y)} which measures how different the prediction y ^ {\displaystyle {\hat {y}}} of a hypothesis is from the true outcome y {\displaystyle y} . For classification tasks, these loss functions can be scoring rules. The risk associated with hypothesis h ( x ) {\displaystyle h(x)} is then defined as the expectation of the loss function: R ( h ) = E [ L ( h ( x ) , y ) ] = ∫ L ( h ( x ) , y ) d P ( x , y ) . {\displaystyle R(h)=\mathbf {E} [L(h(x),y)]=\int L(h(x),y)\,dP(x,y).} A loss function commonly used in theory is the 0-1 loss function: L ( y ^ , y ) = { 1 if y ^ ≠ y 0 if y ^ = y {\displaystyle L({\hat {y}},y)={\begin{cases}1&{\mbox{ if }}\quad {\hat {y}}\neq y\\0&{\mbox{ if }}\quad {\hat {y}}=y\end{cases}}} . The ultimate goal of a learning algorithm is to find a hypothesis h ∗ {\displaystyle h^{}} among a fixed class of functions H {\displaystyle {\mathcal {H}}} for which the risk R ( h ) {\displaystyle R(h)} is minimal: h ∗ = a r g m i n h ∈ H R ( h ) . {\displaystyle h^{}={\underset {h\in {\mathcal {H}}}{\operatorname {arg\,min} }}\,{R(h)}.} For classification problems, the Bayes classifier is defined to be the classifier minimizing the risk defined with the 0–1 loss function. == Formal definition == In general, the risk R ( h ) {\displaystyle R(h)} cannot be computed because the distribution P ( x , y ) {\displaystyle P(x,y)} is unknown to the learning algorithm. However, given a sample of iid training data points, we can compute an estimate, called the empirical risk, by computing the average of the loss function over the training set; more formally, computing the expectation with respect to the empirical measure: R emp ( h ) = 1 n ∑ i = 1 n L ( h ( x i ) , y i ) . {\displaystyle \!R_{\text{emp}}(h)={\frac {1}{n}}\sum _{i=1}^{n}L(h(x_{i}),y_{i}).} The empirical risk minimization principle states that the learning algorithm should choose a hypothesis h ^ {\displaystyle {\hat {h}}} which minimizes the empirical risk over the hypothesis class H {\displaystyle {\mathcal {H}}} : h ^ = a r g m i n h ∈ H R emp ( h ) . {\displaystyle {\hat {h}}={\underset {h\in {\mathcal {H}}}{\operatorname {arg\,min} }}\,R_{\text{emp}}(h).} Thus, the learning algorithm defined by the empirical risk minimization principle consists in solving the above optimization problem. == Properties == Guarantees for the performance of empirical risk minimization depend strongly on the function class selected as well as the distributional assumptions made. In general, distribution-free methods are too coarse, and do not lead to practical bounds. However, they are still useful in deriving asymptotic properties of learning algorithms, such as consistency. In particular, distribution-free bounds on the performance of empirical risk minimization given a fixed function class can be derived using bounds on the VC complexity of the function class. For simplicity, considering the case of binary classification tasks, it is possible to bound the probability of the selected classifier, ϕ n {\displaystyle \phi _{n}} being much worse than the best possible classifier ϕ ∗ {\displaystyle \phi ^{}} . Consider the risk L {\displaystyle L} defined over the hypothesis class C {\displaystyle {\mathcal {C}}} with growth function S ( C , n ) {\displaystyle {\mathcal {S}}({\mathcal {C}},n)} given a dataset of size n {\displaystyle n} . Then, for every ϵ > 0 {\displaystyle \epsilon >0} : P ( L ( ϕ n ) − L ( ϕ ∗ ) > ϵ ) ≤ 8 S ( C , n ) exp { − n ϵ 2 / 32 } {\displaystyle \mathbb {P} \left(L(\phi _{n})-L(\phi ^{})>\epsilon \right)\leq {\mathcal {8}}S({\mathcal {C}},n)\exp\{-n\epsilon ^{2}/32\}} Similar results hold for regression tasks. These results are often based on uniform laws of large numbers, which control the deviation of the empirical risk from the true risk, uniformly over the hypothesis class. === Impossibility results === It is also possible to show lower bounds on algorithm performance if no distributional assumptions are made. This is sometimes referred to as the No free lunch theorem. Even though a specific learning algorithm may provide the asymptotically optimal performance for any distribution, the finite sample performance is always poor for at least one data distribution. This means that no classifier can improve on the error for a given sample size for all distributions. Specifically, let ϵ > 0 {\displaystyle \epsilon >0} and consider a sample size n {\displaystyle n} and classification rule ϕ n {\displaystyle \phi _{n}} , there exists a distribution of ( X , Y ) {\displaystyle (X,Y)} with risk L ∗ = 0 {\displaystyle L^{}=0} (meaning that perfect prediction is possible) such that: E L n ≥ 1 / 2 − ϵ . {\displaystyle \mathbb {E} L_{n}\geq 1/2-\epsilon .} It is further possible to show that the convergence rate of a learning algorithm is poor for some distributions. Specifically, given a sequence of decreasing positive numbers a i {\displaystyle a_{i}} converging to zero, it is possible to find a distribution such that: E L n ≥ a i {\displaystyle \mathbb {E} L_{n}\geq a_{i}} for all n {\displaystyle n} . This result shows that universally good classification rules do not exist, in the sense that the rule must be low quality for at least one distribution. === Computational complexity === Empirical risk minimization for a classification problem with a 0-1 loss function is known to be an NP-hard problem even for a relatively simple class of functions such as linear classifiers. Nevertheless, it can be solved efficiently when the minimal empirical risk is zero, i.e., data is linearly separable. In practice, machine learning algorithms cope with this issue either by employing a convex approximation to the 0–1 loss function (like hinge loss for SVM), which is easier to optimize, or by imposing assumptions on the distribution P ( x , y ) {\displaystyle P(x,y)} (and thus stop being agnostic learning algorithms to which the above result applies). In the case of convexification, Zhang's lemma majors the excess risk of the original problem using the excess risk of the convexified problem. Minimizing the latter using convex optimization also allow to control the former. == Tilted empirical risk minimization == Tilted empirical risk minimization is a machine learning technique used to modify standard loss functions like squared error, by introducing a tilt parameter. This parameter dynamically adjusts the weight of data points during training, allowing the algorithm to focus on specific regions or characteristics of the data distribution. Tilted empirical risk minimization is particularly useful in scenarios with imbalanced data or when there is a need to emphasize errors in certain parts of the prediction space.
Inductive logic programming
Inductive logic programming (ILP) is a subfield of symbolic artificial intelligence which uses logic programming as a uniform representation for examples, background knowledge and hypotheses. The term "inductive" here refers to philosophical (i.e. suggesting a theory to explain observed facts) rather than mathematical (i.e. proving a property for all members of a well-ordered set) induction. Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples. Schema: positive examples + negative examples + background knowledge ⇒ hypothesis. Bioinformatics and drug design have been highlighted as a principal application area of inductive logic programming techniques. == History == Building on earlier work on Inductive inference, Gordon Plotkin was the first to formalise induction in a clausal setting around 1970, adopting an approach of generalising from examples. In 1981, Ehud Shapiro introduced several ideas that would shape the field in his new approach of model inference, an algorithm employing refinement and backtracing to search for a complete axiomatisation of given examples. His first implementation was the Model Inference System in 1981: a Prolog program that inductively inferred Horn clause logic programs from positive and negative examples. The term Inductive Logic Programming was first introduced in a paper by Stephen Muggleton in 1990, defined as the intersection of machine learning and logic programming. Muggleton and Wray Buntine introduced predicate invention and inverse resolution in 1988. Several inductive logic programming systems that proved influential appeared in the early 1990s. FOIL, introduced by Ross Quinlan in 1990 was based on upgrading propositional learning algorithms AQ and ID3. Golem, introduced by Muggleton and Feng in 1990, went back to a restricted form of Plotkin's least generalisation algorithm. The Progol system, introduced by Muggleton in 1995, first implemented inverse entailment, and inspired many later systems. Aleph, a descendant of Progol introduced by Ashwin Srinivasan in 2001, is still one of the most widely used systems as of 2022. At around the same time, the first practical applications emerged, particularly in bioinformatics, where by 2000 inductive logic programming had been successfully applied to drug design, carcinogenicity and mutagenicity prediction, and elucidation of the structure and function of proteins. Unlike the focus on automatic programming inherent in the early work, these fields used inductive logic programming techniques from a viewpoint of relational data mining. The success of those initial applications and the lack of progress in recovering larger traditional logic programs shaped the focus of the field. Recently, classical tasks from automated programming have moved back into focus, as the introduction of meta-interpretative learning makes predicate invention and learning recursive programs more feasible. This technique was pioneered with the Metagol system introduced by Muggleton, Dianhuan Lin, Niels Pahlavi and Alireza Tamaddoni-Nezhad in 2014. This allows ILP systems to work with fewer examples, and brought successes in learning string transformation programs, answer set grammars and general algorithms. == Setting == Inductive logic programming has adopted several different learning settings, the most common of which are learning from entailment and learning from interpretations. In both cases, the input is provided in the form of background knowledge B, a logical theory (commonly in the form of clauses used in logic programming), as well as positive and negative examples, denoted E + {\textstyle E^{+}} and E − {\textstyle E^{-}} respectively. The output is given as a hypothesis H, itself a logical theory that typically consists of one or more clauses. The two settings differ in the format of examples presented. === Learning from entailment === As of 2022, learning from entailment is by far the most popular setting for inductive logic programming. In this setting, the positive and negative examples are given as finite sets E + {\textstyle E^{+}} and E − {\textstyle E^{-}} of positive and negated ground literals, respectively. A correct hypothesis H is a set of clauses satisfying the following requirements, where the turnstile symbol ⊨ {\displaystyle \models } stands for logical entailment: Completeness: B ∪ H ⊨ E + Consistency: B ∪ H ∪ E − ⊭ false {\displaystyle {\begin{array}{llll}{\text{Completeness:}}&B\cup H&\models &E^{+}\\{\text{Consistency: }}&B\cup H\cup E^{-}&\not \models &{\textit {false}}\end{array}}} Completeness requires any generated hypothesis H to explain all positive examples E + {\textstyle E^{+}} , and consistency forbids generation of any hypothesis H that is inconsistent with the negative examples E − {\textstyle E^{-}} , both given the background knowledge B. In Muggleton's setting of concept learning, "completeness" is referred to as "sufficiency", and "consistency" as "strong consistency". Two further conditions are added: "Necessity", which postulates that B does not entail E + {\textstyle E^{+}} , does not impose a restriction on H, but forbids any generation of a hypothesis as long as the positive facts are explainable without it. "Weak consistency", which states that no contradiction can be derived from B ∧ H {\textstyle B\land H} , forbids generation of any hypothesis H that contradicts the background knowledge B. Weak consistency is implied by strong consistency; if no negative examples are given, both requirements coincide. Weak consistency is particularly important in the case of noisy data, where completeness and strong consistency cannot be guaranteed. === Learning from interpretations === In learning from interpretations, the positive and negative examples are given as a set of complete or partial Herbrand structures, each of which are themselves a finite set of ground literals. Such a structure e is said to be a model of the set of clauses B ∪ H {\textstyle B\cup H} if for any substitution θ {\textstyle \theta } and any clause h e a d ← b o d y {\textstyle \mathrm {head} \leftarrow \mathrm {body} } in B ∪ H {\textstyle B\cup H} such that b o d y θ ⊆ e {\textstyle \mathrm {body} \theta \subseteq e} , h e a d θ ⊆ e {\displaystyle \mathrm {head} \theta \subseteq e} also holds. The goal is then to output a hypothesis that is complete, meaning every positive example is a model of B ∪ H {\textstyle B\cup H} , and consistent, meaning that no negative example is a model of B ∪ H {\textstyle B\cup H} . == Approaches to ILP == An inductive logic programming system is a program that takes as an input logic theories B , E + , E − {\displaystyle B,E^{+},E^{-}} and outputs a correct hypothesis H with respect to theories B , E + , E − {\displaystyle B,E^{+},E^{-}} . A system is complete if and only if for any input logic theories B , E + , E − {\displaystyle B,E^{+},E^{-}} any correct hypothesis H with respect to these input theories can be found with its hypothesis search procedure. Inductive logic programming systems can be roughly divided into two classes, search-based and meta-interpretative systems. Search-based systems exploit that the space of possible clauses forms a complete lattice under the subsumption relation, where one clause C 1 {\textstyle C_{1}} subsumes another clause C 2 {\textstyle C_{2}} if there is a substitution θ {\textstyle \theta } such that C 1 θ {\textstyle C_{1}\theta } , the result of applying θ {\textstyle \theta } to C 1 {\textstyle C_{1}} , is a subset of C 2 {\textstyle C_{2}} . This lattice can be traversed either bottom-up or top-down. === Bottom-up search === Bottom-up methods to search the subsumption lattice have been investigated since Plotkin's first work on formalising induction in clausal logic in 1970. Techniques used include least general generalisation, based on anti-unification, and inverse resolution, based on inverting the resolution inference rule. ==== Least general generalisation ==== A least general generalisation algorithm takes as input two clauses C 1 {\textstyle C_{1}} and C 2 {\textstyle C_{2}} and outputs the least general generalisation of C 1 {\textstyle C_{1}} and C 2 {\textstyle C_{2}} , that is, a clause C {\textstyle C} that subsumes C 1 {\textstyle C_{1}} and C 2 {\textstyle C_{2}} , and that is subsumed by every other clause that subsumes C 1 {\textstyle C_{1}} and C 2 {\textstyle C_{2}} . The least general generalisation can be computed by first computing all selections from C 1 {\textstyle C_{1}} and C 2 {\textstyle C_{2}} , which are pairs of literals ( L , M ) ∈ ( C 1 × C 2 ) {\displaystyle (L,M)\in (C_{1}\times C_{2})} sharing the same predicate symbol and negated/unnegated status. Then, the least general generalisation is obtained as the disjunction of the least general generalisations of the indi