In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs now." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). Markov processes are named in honor of the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes. They provide the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in areas including Bayesian statistics, biology, chemistry, economics, finance, information theory, physics, signal processing, and speech processing. The adjectives Markovian and Markov are used to describe something that is related to a Markov process. == Principles == === Definition === A Markov process is a stochastic process that satisfies the Markov property (sometimes characterized as "memorylessness"). In simpler terms, it is a process for which predictions can be made regarding future outcomes based solely on its present state and—most importantly—such predictions are just as good as the ones that could be made knowing the process's full history. In other words, conditional on the present state of the system, its future and past states are independent. A Markov chain is a type of Markov process that has either a discrete state space or a discrete index set (often representing time), but the precise definition of a Markov chain varies. For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time), but it is also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space). === Types of Markov chains === The system's state space and time parameter index need to be specified. The following table gives an overview of the different instances of Markov processes for different levels of state space generality for both discrete and continuous time: Note that there is no definitive agreement in the literature on the use of some of the terms that signify special cases of Markov processes. Usually the term "Markov chain" is reserved for a process with a discrete set of times, that is, a discrete-time Markov chain (DTMC), but a few authors use the term "Markov process" to refer to a continuous-time Markov chain (CTMC) without explicit mention. In addition, there are other extensions of Markov processes that are referred to as such but do not necessarily fall within any of these four categories (see Markov model). Moreover, the time index need not necessarily be real-valued; like with the state space, there are conceivable processes that move through index sets with other mathematical constructs. Notice that the general state space continuous-time Markov chain is general to such a degree that it has no designated term. While the time parameter is usually discrete, the state space of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space. However, many applications of Markov chains employ finite or countably infinite state spaces, which have a more straightforward statistical analysis. Besides time-index and state-space parameters, there are many other variations, extensions and generalizations (see Variations). For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, unless mentioned otherwise. === Transitions === The changes of state of the system are called transitions. The probabilities associated with various state changes are called transition probabilities. The process is characterized by a state space, a transition matrix describing the probabilities of particular transitions, and an initial state (or initial distribution) across the state space. By convention, we assume all possible states and transitions have been included in the definition of the process, so there is always a next state, and the process does not terminate. A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps. The steps are often thought of as moments in time, but they can equally well refer to physical distance or any other discrete measurement. Formally, the steps are the integers or natural numbers, and the random process is a mapping of these to states. The Markov property states that the conditional probability distribution for the system at the next step (and in fact at all future steps) depends only on the current state of the system, and not additionally on the state of the system at previous steps. Since the system changes randomly, it is generally impossible to predict with certainty the state of a Markov chain at a given point in the future. However, the statistical properties of the system's future can be predicted. In many applications, it is these statistical properties that are important. == History == Andrey Markov studied Markov processes in the early 20th century, publishing his first paper on the topic in 1906. Markov processes in continuous time were discovered long before his work in the early 20th century in the form of the Poisson process. Markov was interested in studying an extension of independent random sequences, motivated by a disagreement with Pavel Nekrasov who claimed independence was necessary for the weak law of large numbers to hold. In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed vector of values, so proving a weak law of large numbers without the independence assumption, which had been commonly regarded as a requirement for such mathematical laws to hold. Markov later used Markov chains to study the distribution of vowels in Eugene Onegin, written by Alexander Pushkin, and proved a central limit theorem for such chains. In 1912 Henri Poincaré studied Markov chains on finite groups with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced by Paul and Tatyana Ehrenfest in 1907, and a branching process, introduced by Francis Galton and Henry William Watson in 1873, preceding the work of Markov. After the work of Galton and Watson, it was later revealed that their branching process had been independently discovered and studied around three decades earlier by Irénée-Jules Bienaymé. Starting in 1928, Maurice Fréchet became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains. Andrey Kolmogorov developed in a 1931 paper a large part of the early theory of continuous-time Markov processes. Kolmogorov was partly inspired by Louis Bachelier's 1900 work on fluctuations in the stock market as well as Norbert Wiener's work on Einstein's model of Brownian movement. He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes. Independent of Kolmogorov's work, Sydney Chapman derived in a 1928 paper an equation, now called the Chapman–Kolmogorov equation, in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement. The differential equations are now called the Kolmogorov equations or the Kolmogorov–Chapman equations. Other mathematicians who contributed significantly to the foundations of Markov processes include William Feller, starting in 1930s, and then later Eugene Dynkin, starting in the 1950s. == Examples == Mark V. Shaney is a third-order Markov chain program, and a Markov text generator. It ingests the sample text (the Tao Te Ching, or the posts of a Usenet group) and creates a massive list of every sequence of three successive words (triplet) which occurs in the text. It then chooses two words at random, and looks for a word which follows those two in one of the triplets in its massive list. If there is more than one, it picks at random (identical triplets count separately, so a sequence which occurs twice is twice as likely to be picked as one which only occurs once). It then adds that word to the generated text. Then, in the same way, it picks a triplet that starts with the second and third words in the generated text, and that gives a fourth word. It adds the fourth word, then repeats with the third and fourth words, and so on. Random walks based on integers and the gambler's ruin problem are ex
Texture filtering
In computer graphics, texture filtering or texture smoothing is the method used to determine the texture color for a texture mapped pixel, using the colors of nearby texels (ie. pixels of the texture). Filtering describes how a texture is applied at many different shapes, size, angles and scales. Depending on the chosen filter algorithm, the result will show varying degrees of blurriness, detail, spatial aliasing, temporal aliasing and blocking. Depending on the circumstances, filtering can be performed in software (such as a software rendering package) or in hardware, eg. with either real time or GPU accelerated rendering circuits, or in a mixture of both. For most common interactive graphical applications, modern texture filtering is performed by dedicated hardware which optimizes memory access through memory cacheing and pre-fetch, and implements a selection of algorithms available to the user and developer. There are two main categories of texture filtering: magnification filtering and minification filtering. Depending on the situation, texture filtering is either a type of reconstruction filter where sparse data is interpolated to fill gaps (magnification), or a type of anti-aliasing (AA) where texture samples exist at a higher frequency than required for the sample frequency needed for texture fill (minification). There are many methods of texture filtering, which make different trade-offs between computational complexity, memory bandwidth and image quality. == The need for filtering == During the texture mapping process for any arbitrary 3D surface, a texture lookup takes place to find out where on the texture each pixel center falls. For texture-mapped polygonal surfaces composed of triangles typical of most surfaces in 3D games and movies, every pixel (or subordinate pixel sample) of that surface will be associated with some triangle(s) and a set of barycentric coordinates, which are used to provide a position within a texture. Such a position may not lie perfectly on the "pixel grid," necessitating some function to account for these cases. In other words, since the textured surface may be at an arbitrary distance and orientation relative to the viewer, one pixel does not usually correspond directly to one texel. Some form of filtering has to be applied to determine the best color for the pixel. Insufficient or incorrect filtering will show up in the image as artifacts (errors in the image), such as 'blockiness', jaggies, or shimmering. There can be different types of correspondence between a pixel and the texel/texels it represents on the screen. These depend on the position of the textured surface relative to the viewer, and different forms of filtering are needed in each case. Given a square texture mapped on to a square surface in the world, at some viewing distance the size of one screen pixel is exactly the same as one texel. Closer than that, the texels are larger than screen pixels, and need to be scaled up appropriately — a process known as texture magnification. Farther away, each texel is smaller than a pixel, and so one pixel covers multiple texels. In this case an appropriate color has to be picked based on the covered texels, via texture minification. Graphics APIs such as OpenGL allow the programmer to set different choices for minification and magnification filters. Note that even in the case where the pixels and texels are exactly the same size, one pixel will not necessarily match up exactly to one texel. It may be misaligned or rotated, and cover parts of up to four neighboring texels. Hence some form of filtering is still required. == Mipmapping == Mipmapping is a standard technique used to save some of the filtering work needed during texture minification. It is also highly beneficial for cache coherency - without it the memory access pattern during sampling from distant textures will exhibit extremely poor locality, adversely affecting performance even if no filtering is performed. During texture magnification, the number of texels that need to be looked up for any pixel is always four or fewer; during minification, however, as the textured polygon moves farther away potentially the entire texture might fall into a single pixel. This would necessitate reading all of its texels and combining their values to correctly determine the pixel color, a prohibitively expensive operation. Mipmapping avoids this by prefiltering the texture and storing it in smaller sizes down to a single pixel. As the textured surface moves farther away, the texture being applied switches to the prefiltered smaller size. Different sizes of the mipmap are referred to as 'levels', with Level 0 being the largest size (used closest to the viewer), and increasing levels used at increasing distances. == Filtering methods == This section lists the most common texture filtering methods, in increasing order of computational cost and image quality. === Nearest-neighbor interpolation === Nearest-neighbor interpolation is the simplest and crudest filtering method — it simply uses the color of the texel closest to the pixel center for the pixel color. While simple, this results in a large number of artifacts - texture 'blockiness' during magnification, and aliasing and shimmering during minification. This method is fast during magnification but during minification the stride through memory becomes arbitrarily large and it can often be less efficient than MIP-mapping due to the lack of spatially coherent texture access and cache-line reuse. === Nearest-neighbor with mipmapping === This method still uses nearest neighbor interpolation, but adds mipmapping — first the nearest mipmap level is chosen according to distance, then the nearest texel center is sampled to get the pixel color. This reduces the aliasing and shimmering significantly during minification but does not eliminate it entirely. In doing so it improves texture memory access and cache-line reuse through avoiding arbitrarily large access strides through texture memory during rasterization. This does not help with blockiness during magnification as each magnified texel will still appear as a large rectangle. === Linear mipmap filtering === Less commonly used, OpenGL and other APIs support nearest-neighbor sampling from individual mipmaps whilst linearly interpolating the two nearest mipmaps relevant to the sample. === Bilinear filtering === In Bilinear filtering, the four nearest texels to the pixel center are sampled (at the closest mipmap level), and their colors are combined by weighted average according to distance. This removes the 'blockiness' seen during magnification, as there is now a smooth gradient of color change from one texel to the next, instead of an abrupt jump as the pixel center crosses the texel boundary. Bilinear filtering for magnification filtering is common. When used for minification it is often used with mipmapping; though it can be used without, it would suffer the same aliasing and shimmering problems as nearest-neighbor filtering when minified too much. For modest minification ratios, however, it can be used as an inexpensive hardware accelerated weighted texture supersample. The Nintendo 64 used an unusual version of bilinear filtering where only three pixels are used known as 3-point texture filtering, instead of four due to hardware optimization concerns. This introduces a noticeable "triangulation bias" in some textures. === Trilinear filtering === Trilinear filtering is a remedy to a common artifact seen in mipmapped bilinearly filtered images: an abrupt and very noticeable change in quality at boundaries where the renderer switches from one mipmap level to the next. Trilinear filtering solves this by doing a texture lookup and bilinear filtering on the two closest mipmap levels (one higher and one lower quality), and then linearly interpolating the results. This results in a smooth degradation of texture quality as distance from the viewer increases, rather than a series of sudden drops. Of course, closer than Level 0 there is only one mipmap level available, and the algorithm reverts to bilinear filtering. === Anisotropic filtering === Anisotropic filtering is the highest quality filtering available in current consumer 3D graphics cards. Simpler, "isotropic" techniques use only square mipmaps which are then interpolated using bi– or trilinear filtering. (Isotropic means same in all directions, and hence is used to describe a system in which all the maps are squares rather than rectangles or other quadrilaterals.) When a surface is at a high angle relative to the camera, the fill area for a texture will not be approximately square. Consider the common case of a floor in a game: the fill area is far wider than it is tall. In this case, none of the square maps are a good fit. The result is blurriness and/or shimmering, depending on how the fit is chosen. Anisotropic filtering corrects this by sampling the texture as a non-square shape. The goal is
Netflix Prize
The Netflix Prize was an open competition for the best collaborative filtering algorithm to predict user ratings for films, based on previous ratings without any other information about the users or films, i.e. without the users being identified except by numbers assigned for the contest. The competition was held by Netflix, a video streaming service, and was open to anyone who was neither connected with Netflix (current and former employees, agents, close relatives of Netflix employees, etc.) nor a resident of certain blocked countries (such as Cuba or North Korea). On September 21, 2009, the grand prize of US$1,000,000 was given to the BellKor's Pragmatic Chaos team which bested Netflix's own algorithm for predicting ratings by 10.06%. == Problem and data sets == Netflix provided a training data set of 100,480,507 ratings that 480,189 users gave to 17,770 movies. Each training rating is a quadruplet of the form
T-norm
In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersection in a lattice and conjunction in logic. The name triangular norm refers to the fact that in the framework of probabilistic metric spaces t-norms are used to generalize the triangle inequality of ordinary metric spaces. == Definition == A t-norm is a function T: [0, 1] × [0, 1] → [0, 1] that satisfies the following properties: Commutativity: T(a, b) = T(b, a) Monotonicity: T(a, b) ≤ T(c, d) if a ≤ c and b ≤ d Associativity: T(a, T(b, c)) = T(T(a, b), c) The number 1 acts as identity element: T(a, 1) = a Since a t-norm is a binary algebraic operation on the interval [0, 1], infix algebraic notation is also common, with the t-norm usually denoted by ∗ {\displaystyle } . The defining conditions of the t-norm are exactly those of a partially ordered abelian monoid on the real unit interval [0, 1]. (Cf. ordered group.) The monoidal operation of any partially ordered abelian monoid L is therefore by some authors called a triangular norm on L. === Classification of t-norms === A t-norm is called continuous if it is continuous as a function, in the usual interval topology on [0, 1]2. (Similarly for left- and right-continuity.) A t-norm is called strict if it is continuous and strictly monotone. A t-norm is called nilpotent if it is continuous and each x in the open interval (0, 1) is nilpotent, that is, there is a natural number n such that x ∗ {\displaystyle } ... ∗ {\displaystyle } x (n times) equals 0. A t-norm ∗ {\displaystyle } is called Archimedean if it has the Archimedean property, that is, if for each x, y in the open interval (0, 1) there is a natural number n such that x ∗ {\displaystyle } ... ∗ {\displaystyle } x (n times) is less than or equal to y. The usual partial ordering of t-norms is pointwise, that is, T1 ≤ T2 if T1(a, b) ≤ T2(a, b) for all a, b in [0, 1]. As functions, pointwise larger t-norms are sometimes called stronger than those pointwise smaller. In the semantics of t-norm fuzzy logics, however, the larger a t-norm, the weaker (in terms of logical strength) conjunction it represents. == Prominent examples == Minimum t-norm ⊤ m i n ( a , b ) = min { a , b } , {\displaystyle \top _{\mathrm {min} }(a,b)=\min\{a,b\},} also called the Gödel t-norm, as it is the standard semantics for conjunction in Gödel fuzzy logic. Besides that, it occurs in most t-norm based fuzzy logics as the standard semantics for weak conjunction. It is the pointwise largest t-norm (see the properties of t-norms below). Product t-norm ⊤ p r o d ( a , b ) = a ⋅ b {\displaystyle \top _{\mathrm {prod} }(a,b)=a\cdot b} (the ordinary product of real numbers). Besides other uses, the product t-norm is the standard semantics for strong conjunction in product fuzzy logic. It is a strict Archimedean t-norm. Łukasiewicz t-norm ⊤ L u k ( a , b ) = max { 0 , a + b − 1 } . {\displaystyle \top _{\mathrm {Luk} }(a,b)=\max\{0,a+b-1\}.} The name comes from the fact that the t-norm is the standard semantics for strong conjunction in Łukasiewicz fuzzy logic. It is a nilpotent Archimedean t-norm, pointwise smaller than the product t-norm. Drastic t-norm ⊤ D ( a , b ) = { b if a = 1 a if b = 1 0 otherwise. {\displaystyle \top _{\mathrm {D} }(a,b)={\begin{cases}b&{\mbox{if }}a=1\\a&{\mbox{if }}b=1\\0&{\mbox{otherwise.}}\end{cases}}} The name reflects the fact that the drastic t-norm is the pointwise smallest t-norm (see the properties of t-norms below). It is a right-continuous Archimedean t-norm. Nilpotent minimum ⊤ n M ( a , b ) = { min ( a , b ) if a + b > 1 0 otherwise {\displaystyle \top _{\mathrm {nM} }(a,b)={\begin{cases}\min(a,b)&{\mbox{if }}a+b>1\\0&{\mbox{otherwise}}\end{cases}}} is a standard example of a t-norm that is left-continuous, but not continuous. Despite its name, the nilpotent minimum is not a nilpotent t-norm. Hamacher product ⊤ H 0 ( a , b ) = { 0 if a = b = 0 a b a + b − a b otherwise {\displaystyle \top _{\mathrm {H} _{0}}(a,b)={\begin{cases}0&{\mbox{if }}a=b=0\\{\frac {ab}{a+b-ab}}&{\mbox{otherwise}}\end{cases}}} is a strict Archimedean t-norm, and an important representative of the parametric classes of Hamacher t-norms and Schweizer–Sklar t-norms. == Properties of t-norms == The drastic t-norm is the pointwise smallest t-norm and the minimum is the pointwise largest t-norm: ⊤ D ( a , b ) ≤ ⊤ ( a , b ) ≤ ⊤ m i n ( a , b ) , {\displaystyle \top _{\mathrm {D} }(a,b)\leq \top (a,b)\leq \mathrm {\top _{min}} (a,b),} for any t-norm ⊤ {\displaystyle \top } and all a, b in [0, 1]. In particular, we have that: ⊤ D ( a , b ) ≤ ⊤ L u k ( a , b ) ≤ ⊤ p r o d ( a , b ) ≤ ⊤ m i n ( a , b ) , {\displaystyle \top _{\mathrm {D} }(a,b)\leq \top _{\mathrm {Luk} }(a,b)\leq \top _{\mathrm {prod} }(a,b)\leq \mathrm {\top _{min}} (a,b),} for all a, b in [0, 1]. For every t-norm T, the number 0 acts as null element: T(a, 0) = 0 for all a in [0, 1]. A t-norm T has zero divisors if and only if it has nilpotent elements; each nilpotent element of T is also a zero divisor of T. The set of all nilpotent elements is an interval [0, a] or [0, a), for some a in [0, 1]. === Properties of continuous t-norms === Although real functions of two variables can be continuous in each variable without being continuous on [0, 1]2, this is not the case with t-norms: a t-norm T is continuous if and only if it is continuous in one variable, i.e., if and only if the functions fy(x) = T(x, y) are continuous for each y in [0, 1]. Analogous theorems hold for left- and right-continuity of a t-norm. A continuous t-norm is Archimedean if and only if 0 and 1 are its only idempotents. A continuous Archimedean t-norm is strict if 0 is its only nilpotent element; otherwise it is nilpotent. By definition, moreover, a continuous Archimedean t-norm T is nilpotent if and only if each x < 1 is a nilpotent element of T. Thus with a continuous Archimedean t-norm T, either all or none of the elements of (0, 1) are nilpotent. If it is the case that all elements in (0, 1) are nilpotent, then the t-norm is isomorphic to the Łukasiewicz t-norm; i.e., there is a strictly increasing function f such that ⊤ ( x , y ) = f − 1 ( ⊤ L u k ( f ( x ) , f ( y ) ) ) . {\displaystyle \top (x,y)=f^{-1}(\top _{\mathrm {Luk} }(f(x),f(y))).} If on the other hand it is the case that there are no nilpotent elements of T, the t-norm is isomorphic to the product t-norm. In other words, all nilpotent t-norms are isomorphic, the Łukasiewicz t-norm being their prototypical representative; and all strict t-norms are isomorphic, with the product t-norm as their prototypical example. The Łukasiewicz t-norm is itself isomorphic to the product t-norm undercut at 0.25, i.e., to the function p(x, y) = max(0.25, x ⋅ y) on [0.25, 1]2. For each continuous t-norm, the set of its idempotents is a closed subset of [0, 1]. Its complement—the set of all elements that are not idempotent—is therefore a union of countably many non-overlapping open intervals. The restriction of the t-norm to any of these intervals (including its endpoints) is Archimedean, and thus isomorphic either to the Łukasiewicz t-norm or the product t-norm. For such x, y that do not fall into the same open interval of non-idempotents, the t-norm evaluates to the minimum of x and y. These conditions actually give a characterization of continuous t-norms, called the Mostert–Shields theorem, since every continuous t-norm can in this way be decomposed, and the described construction always yields a continuous t-norm. The theorem can also be formulated as follows: A t-norm is continuous if and only if it is isomorphic to an ordinal sum of the minimum, Łukasiewicz, and product t-norm. A similar characterization theorem for non-continuous t-norms is not known (not even for left-continuous ones), only some non-exhaustive methods for the construction of t-norms have been found. == Residuum == For any left-continuous t-norm ⊤ {\displaystyle \top } , there is a unique binary operation ⇒ {\displaystyle \Rightarrow } on [0, 1] such that ⊤ ( z , x ) ≤ y {\displaystyle \top (z,x)\leq y} if and only if z ≤ ( x ⇒ y ) {\displaystyle z\leq (x\Rightarrow y)} for all x, y, z in [0, 1]. This operation is called the residuum of the t-norm. In prefix notation, the residuum of a t-norm ⊤ {\displaystyle \top } is often denoted by ⊤ → {\displaystyle {\vec {\top }}} or by the letter R. The interval [0, 1] equipped with a t-norm and its residuum forms a residuated lattice. The relation between a t-norm T and its residuum R is an instance of adjunction (specifically, a Galois connection): the residuum forms a right adjoint R(x, –) to the functor T(–, x) for each x in the lattice [0, 1] taken as a poset category. In the standard semantics of t-norm based fuzzy logics, where conjunction is interpreted by a t-norm, the residuum plays the role of implication (often
Oasis (Minecraft clone)
Oasis is a 2024 video game that attempts to replicate the 2011 sandbox game Minecraft, run entirely using generative artificial intelligence. The project, which began development in 2022 between the AI company Decart and the computer hardware startup Etched, was released by Decart to the public on October 31, 2024. The AI-driven simulation uses "next-frame prediction" to anticipate player actions based on keyboard and mouse inputs, trained on millions of hours of gameplay footage. Without memory or code, the game often outputs unpredictable changes in scenery and inventory, limiting its functionality as a traditional video game. Critics noted its lack of sound, low frame rate, and "dream-like" appearance, though some praised its unpredictability as entertaining. The project is seen as a potential proof of concept for AI-driven video games. == Creation and gameplay == The demo "proof of concept" version of the game was developed by Israeli San Francisco–based AI company Decart and Silicon Valley hardware startup Etched. The idea originated in 2022 when Robert Wachen, a Harvard graduate and co-founder of Etched, met Dean Leitersdorf, an Israel Institute of Technology graduate and co-founder of Decart. Sharing an interest in OpenAI's GPT-3, they collaborated to create the game, naming it after the setting of the novel and film Ready Player One. It was funded by a $21 million grant from Israeli-American billionaire Oren Zeev and New York–based Sequoia Capital. Decart released the game to the public for free on October 31, 2024. The AI replicates Minecraft's gameplay without code using "next-frame prediction", in which the AI tries to predict what the player will see after each keyboard and mouse input, which it was trained to do on millions of hours of Minecraft footage. The game used Nvidia graphics processing units or GPUs for its demo but plans to transition to more energy-efficient Sohu GPUs, under development by Etched, capable of supporting up to 4K graphics. Etched has also suggested the possibility of making the game open source in the future. Alongside Oasis, the company is co-developing AI-generated video and educational content. == Reception == Upon its launch, many players posted videos of their experience with the game online, which often showed Oasis could not maintain coherent logic in its actions or setting. The game also presented low-quality graphics, running between 360p and 720p consistently at 20 FPS, no in-game sound, and could only be played for five minutes at a time before restarting. These issues led some news outlets to refer to the game as a "nightmarish hallucination", and drawing comparisons to dementia and dreams. Despite the negative reviews, Leitersdorf, as well as a number of commentators, have commented that while the game may have fallen short of replicating Minecraft in its demo launch, it was the first step towards something more advanced, which could one day resemble Minecraft or any other game. Online publication The Backdash commented the game could be a "glimpse at the future of game development", while others like Tom's Hardware expressed doubts a game without code could ever look as good as one with, arguing they fail to capture "the point of what makes games fun—or even coherent". In terms of legality, Decart and Etched did not receive permission from Microsoft to create a copy of their game using generative artificial intelligence. No legal actions have been taken by the latter, however, as artificial intelligence and copyright remains largely vague legally.
NNDB
The Notable Names Database (NNDB) is an online database of biographical details of over 40,000 people. Soylent Communications, a sole proprietorship that also hosted the later defunct Rotten.com, describes NNDB as an "intelligence aggregator" of noteworthy persons, highlighting their interpersonal connections. The Rotten.com domain was registered in 1996 by former Apple and Netscape software engineer Thomas E. Dell, who was also known by his internet alias, "Soylent". == Entries == Each entry has an executive summary followed by a brief narrative about their life. It also lists date and cause of death if deceased. Businesspeople and government officials are listed with chronologies of their posts, positions, and board memberships. As of 2022, the site is no longer updated. == NNDB Mapper == The NNDB Mapper, a visual tool for exploring connections between people, was made available in May 2008. It required Adobe Flash 7.
Fuzzy classification
Fuzzy classification is the process of grouping elements into fuzzy sets whose membership functions are defined by the truth value of a fuzzy propositional function. A fuzzy propositional function is analogous to an expression containing one or more variables, such that when values are assigned to these variables, the expression becomes a fuzzy proposition. Accordingly, fuzzy classification is the process of grouping individuals having the same characteristics into a fuzzy set. A fuzzy classification corresponds to a membership function μ C ~ : P F ~ × U → T ~ {\textstyle \mu _{\tilde {C}}:{\tilde {PF}}\times U\to {\tilde {T}}} that indicates the degree to which an individual i ∈ U {\textstyle i\in U} is a member of the fuzzy class C ~ {\textstyle {\tilde {C}}} , given its fuzzy classification predicate Π ~ C ~ ∈ P F ~ {\textstyle {\tilde {\Pi }}_{\tilde {C}}\in {\tilde {PF}}} . Here, T ~ {\textstyle {\tilde {T}}} is the set of fuzzy truth values, i.e., the unit interval [ 0 , 1 ] {\textstyle [0,1]} . The fuzzy classification predicate Π ~ C ~ ( i ) {\textstyle {\tilde {\Pi }}_{\tilde {C}}(i)} corresponds to the fuzzy restriction " i {\textstyle i} is a member of C ~ {\textstyle {\tilde {C}}} ". == Classification == Intuitively, a class is a set that is defined by a certain property, and all objects having that property are elements of that class. The process of classification evaluates for a given set of objects whether they fulfill the classification property, and consequentially are a member of the corresponding class. However, this intuitive concept has some logical subtleties that need clarification. A class logic is a logical system which supports set construction using logical predicates with the class operator { ⋅ | ⋅ } {\textstyle \{\cdot |\cdot \}} . A class C = { i | Π ( i ) } {\displaystyle C=\{i|\Pi (i)\}} is defined as a set C of individuals i satisfying a classification predicate Π which is a propositional function. The domain of the class operator { .| .} is the set of variables V and the set of propositional functions PF, and the range is the powerset of this universe P(U) that is, the set of possible subsets: { ⋅ | ⋅ } : V × P F → P ( U ) {\displaystyle \{\cdot |\cdot \}:V\times PF\rightarrow P(U)} Here is an explanation of the logical elements that constitute this definition: An individual is a real object of reference. A universe of discourse is the set of all possible individuals considered. A variable V :→ R {\textstyle V:\rightarrow R} is a function which maps into a predefined range R without any given function arguments: a zero-place function. A propositional function is "an expression containing one or more undetermined constituents, such that, when values are assigned to these constituents, the expression becomes a proposition". In contrast, classification is the process of grouping individuals having the same characteristics into a set. A classification corresponds to a membership function μ that indicates whether an individual is a member of a class, given its classification predicate Π. μ : P F × U → T {\displaystyle \mu :PF\times U\rightarrow T} The membership function maps from the set of propositional functions PF and the universe of discourse U into the set of truth values T. The membership μ of individual i in Class C is defined by the truth value τ of the classification predicate Π. μ C ( i ) := τ ( Π ( i ) ) {\displaystyle \mu C(i):=\tau (\Pi (i))} In classical logic the truth values are certain. Therefore a classification is crisp, since the truth values are either exactly true or exactly false.