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  Spike-and-slab regression

  Spike-and-slab regression is a type of Bayesian linear regression in which a particular hierarchical prior distribution for the regression coefficients is chosen such that only a subset of the possible regressors is retained. The technique is particularly useful when the number of possible predictors is larger than the number of observations. The idea of the spike-and-slab model was originally proposed by Mitchell & Beauchamp (1988). The approach was further significantly developed by Madigan & Raftery (1994) and George & McCulloch (1997). A recent and important contribution to this literature is Ishwaran & Rao (2005). == Model description == Suppose we have P possible predictors in some model. Vector γ has a length equal to P and consists of zeros and ones. This vector indicates whether a particular variable is included in the regression or not. If no specific prior information on initial inclusion probabilities of particular variables is available, a Bernoulli prior distribution is a common default choice. Conditional on a predictor being in the regression, we identify a prior distribution for the model coefficient, which corresponds to that variable (β). A common choice on that step is to use a normal prior with a mean equal to zero and a large variance calculated based on ( X T X ) − 1 {\displaystyle (X^{T}X)^{-1}} (where X {\displaystyle X} is a design matrix of explanatory variables of the model). A draw of γ from its prior distribution is a list of the variables included in the regression. Conditional on this set of selected variables, we take a draw from the prior distribution of the regression coefficients (if γi = 1 then βi ≠ 0 and if γi = 0 then βi = 0). βγ denotes the subset of β for which γi = 1. In the next step, we calculate a posterior probability for both inclusion and coefficients by applying a standard statistical procedure. All steps of the described algorithm are repeated thousands of times using the Markov chain Monte Carlo (MCMC) technique. As a result, we obtain a posterior distribution of γ (variable inclusion in the model), β (regression coefficient values) and the corresponding prediction of y. The model got its name (spike-and-slab) due to the shape of the two prior distributions. The "spike" is the probability of a particular coefficient in the model to be zero. The "slab" is the prior distribution for the regression coefficient values. An advantage of Bayesian variable selection techniques is that they are able to make use of prior knowledge about the model. In the absence of such knowledge, some reasonable default values can be used; to quote Scott and Varian (2013): "For the analyst who prefers simplicity at the cost of some reasonable assumptions, useful prior information can be reduced to an expected model size, an expected R2, and a sample size ν determining the weight given to the guess at R2." Some researchers suggest the following default values: R2 = 0.5, ν = 0.01, and π = 0.5 (parameter of a prior Bernoulli distribution).

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  Project Debater

  Project Debater is an IBM artificial intelligence project, designed to participate in a full live debate with expert human debaters. It follows on from the Watson project which played Jeopardy! == Development == Project Debater was developed at IBM's lab in Haifa, Israel. The project was proposed by Noam Slonim in 2011 as the IBM Research next Grand Challenge, following Deep Blue and the victory of Watson in Jeopardy! It was exposed for the first time in a closed media event at June 18, 2018, in San Francisco, under the leadership of Ranit Aharonov and Slonim, both from the IBM Research lab in Haifa, Israel. The AI technology debated two human debaters, Noa Ovadia, who was the 2016 Israeli debate champion and Dan Zafrir. The two debated on the topics "We should subsidize space exploration" and "Should we increase the use of telemedicine." A demonstration of Project Debater also aired on the Discovery Channel in June 2018 debating the question of whether sports gambling should be legalized. == Live Debate == On February 11, 2019, Project Debater was revealed to the world in a live debate in San Francisco. Nonpartisan media group Intelligence Squared U.S. Debates hosted the debate which was moderated by journalist John Donvan. The debate took place between Project Debater and Harish Natarajan, who holds the world record in number of debate competition victories. The motion was “We should subsidize preschools.” == That's Debatable Television Show == Project Debater was featured in a television series called “That’s Debatable” presented by Intelligence Squared U.S. Debates and Bloomberg Media. For each episode of “That’s Debatable,” Project Debater provided insight into three distinct debate topics on the redistribution of wealth, modern monetary theory, and a US-China space race. More than 5,000 arguments were submitted online from around the world across the three topics, which were then analyzed and distilled into key points that were highlighted on the television show and discussed by human debaters. == Artificial Intelligence Capabilities == To develop Project Debater, the IBM Research team had to endow the system with the following AI capabilities: Data-driven speech writing and delivery: Project Debater is the first demonstration of a computer that can digest massive corpora, and given a short description of a controversial topic, write a well-structured speech, and deliver it with clarity and purpose, while even incorporating humor where appropriate. Listening comprehension: the ability to identify the key concepts and claims hidden within long continuous spoken language. Four minutes of persuasive speech: the guarantee of producing four minutes of persuasive speech. Modeling human dilemmas: modeling the world of human controversy and dilemmas in a unique knowledge representation, enabling the system to suggest principled arguments as needed. An article on the project was published in Nature in March 2021.

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  Anytime algorithm

  In computer science, an anytime algorithm is an algorithm that can return a valid solution to a problem even if it is interrupted before it ends. The algorithm is expected to find better and better solutions the longer it keeps running. Most algorithms run to completion: they provide a single answer after performing some fixed amount of computation. In some cases, however, the user may wish to terminate the algorithm prior to completion. The amount of computation required may be substantial, for example, and computational resources might need to be reallocated. Most algorithms either run to completion or they provide no useful solution information. Anytime algorithms, however, are able to return a partial answer, whose quality depends on the amount of computation they were able to perform. The answer generated by anytime algorithms is an approximation of the correct answer. == Names == An anytime algorithm may be also called an "interruptible algorithm". They are different from contract algorithms, which must declare a time in advance; in an anytime algorithm, a process can just announce that it is terminating. == Goals == The goal of anytime algorithms are to give intelligent systems the ability to make results of better quality in return for turn-around time. They are also supposed to be flexible in time and resources. They are important because artificial intelligence or AI algorithms can take a long time to complete results. This algorithm is designed to complete in a shorter amount of time. Also, these are intended to have a better understanding that the system is dependent and restricted to its agents and how they work cooperatively. An example is the Newton–Raphson iteration applied to finding the square root of a number. Another example that uses anytime algorithms is trajectory problems when you're aiming for a target; the object is moving through space while waiting for the algorithm to finish and even an approximate answer can significantly improve its accuracy if given early. What makes anytime algorithms unique is their ability to return many possible outcomes for any given input. An anytime algorithm uses many well defined quality measures to monitor progress in problem solving and distributed computing resources. It keeps searching for the best possible answer with the amount of time that it is given. It may not run until completion and may improve the answer if it is allowed to run longer. This is often used for large decision set problems. This would generally not provide useful information unless it is allowed to finish. While this may sound similar to dynamic programming, the difference is that it is fine-tuned through random adjustments, rather than sequential. Anytime algorithms are designed so that it can be told to stop at any time and would return the best result it has found so far. This is why it is called an interruptible algorithm. Certain anytime algorithms also maintain the last result, so that if they are given more time, they can continue from where they left off to obtain an even better result. == Decision trees == When the decider has to act, there must be some ambiguity. Also, there must be some idea about how to solve this ambiguity. This idea must be translatable to a state to action diagram. == Performance profile == The performance profile estimates the quality of the results based on the input and the amount of time that is allotted to the algorithm. The better the estimate, the sooner the result would be found. Some systems have a larger database that gives the probability that the output is the expected output. One algorithm can have several performance profiles. Most of the time performance profiles are constructed using mathematical statistics using representative cases. For example, in the traveling salesman problem, the performance profile was generated using a user-defined special program to generate the necessary statistics. In this example, the performance profile is the mapping of time to the expected results. This quality can be measured in several ways: certainty: where probability of correctness determines quality accuracy: where error bound determines quality specificity: where the amount of particulars determine quality == Algorithm prerequisites == Initial behavior: While some algorithms start with immediate guesses, others take a more calculated approach and have a start up period before making any guesses. Growth direction: How the quality of the program's "output" or result, varies as a function of the amount of time ("run time") Growth rate: Amount of increase with each step. Does it change constantly, such as in a bubble sort or does it change unpredictably? End condition: The amount of runtime needed

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  AAAI Conference on Artificial Intelligence

  The AAAI Conference on Artificial Intelligence is a leading international academic conference in artificial intelligence held annually. It ranks 4th in terms of H5 Index in Google Scholar's list of top AI publications, after ICLR, NeurIPS, and ICML. It is supported by the Association for the Advancement of Artificial Intelligence (AAAI), after which it is named. Precise dates vary from year to year, but paper submissions are generally due at the end of August to beginning of September, and the conference is generally held during the following February. The first AAAI was held in 1980 at Stanford University, Stanford California. During AAAI-20 conference, AI pioneers and 2018 Turing Award winners (often referred to as the Nobel Prize of Computing) Yann LeCun and Yoshua Bengio, among eight other researchers, were honored as the AAAI 2020 Fellows. Along with other conferences such as NeurIPS and ICML, AAAI uses an artificial-intelligence algorithm to assign papers to reviewers. == Sponsors == Many leading technology companies, including Google, Microsoft, Amazon (company), IBM, Baidu, Bytedance, and Huawei, generously sponsor and participate in AAAI to publish and showcase their latest theoretical and applied research. Sponsoring companies also actively recruit AI talents at the conference. == Locations == AAAI-2026 Singapore Expo, Singapore AAAI-2025 Pennsylvania Convention Center, Philadelphia, Pennsylvania, United States AAAI-2024 Vancouver Convention Centre, Vancouver, British Columbia, Canada AAAI-2023 Washington Convention Center, Washington, D.C., United States AAAI-2022 Virtual Conference AAAI-2021 Virtual Conference AAAI-2020 Hilton New York Midtown, New York, New York, United States AAAI-2019 Hilton Hawaiian Village, Honolulu, Hawaii, United States AAAI-2018 Hilton New Orleans Riverside, New Orleans, Louisiana, United States AAAI-2017 San Francisco, California, United States AAAI-2016 Phoenix, Arizona, United States AAAI-2015 Austin, Texas, United States AAAI-2014 Québec Convention Center, Québec City, Québec, Canada AAAI-2013 Bellevue, Washington, United States AAAI-2012 Toronto, Ontario, Canada AAAI-2011 San Francisco, California, United States AAAI-2010 Westin Peachtree Plaza, Atlanta, Georgia, United States AAAI-2008 Chicago, Illinois, United States AAAI-2007 Toronto, Ontario, Canada AAAI-2006 Boston, Massachusetts, United States AAAI-2005 Pittsburgh, Pennsylvania, United States AAAI-2004 San Jose, California, United States AAAI-2002 Shaw conference center in Edmonton, Alberta, Canada AAAI-2000 Austin, Texas, United States AAAI-1999 Orlando, Florida, United States AAAI-1998 Madison, Wisconsin, United States AAAI-1997 Providence, Rhode Island, United States AAAI-1996 Portland, Oregon, United States AAAI-1994 Seattle, Washington, United States AAAI-1993 Washington Convention Center, Washington, D.C., United States AAAI-1992 San Jose Convention Center, San Jose, California, United States AAAI-1991 Anaheim Convention Center, Anaheim, California, United States AAAI-1990 Boston, Massachusetts, United States AAAI-1988 Saint Paul, Minnesota, United States AAAI-1987 Seattle, Washington, United States AAAI-1986 Philadelphia, Pennsylvania, United States AAAI-1984 University of Texas, Austin, Texas, United States AAAI-1983 Washington, D.C., United States AAAI-1982 Carnegie Mellon University and the University of Pittsburgh, Pittsburgh, Pennsylvania, United States AAAI-1980 Stanford, California, United States

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  Kuwahara filter

  The Kuwahara filter is a non-linear smoothing filter used in image processing for adaptive noise reduction. Most filters that are used for image smoothing are linear low-pass filters that effectively reduce noise but also blur out the edges. However the Kuwahara filter is able to apply smoothing on the image while preserving the edges. It is named after Michiyoshi Kuwahara, Ph.D., who worked at Kyoto and Osaka Sangyo Universities in Japan, developing early medical imaging of dynamic heart muscle in the 1970s and 80s. == The Kuwahara operator == Suppose that I ( x , y ) {\displaystyle I(x,y)} is a grey scale image and that we take a square window of size 2 a + 1 {\displaystyle 2a+1} centered around a point ( x , y ) {\displaystyle (x,y)} in the image. This square can be divided into four smaller square regions Q i = 1 ⋯ 4 {\displaystyle Q_{i=1\cdots 4}} each of which will be Q i ( x , y ) = { [ x , x + a ] × [ y , y + a ] if i = 1 [ x − a , x ] × [ y , y + a ] if i = 2 [ x − a , x ] × [ y − a , y ] if i = 3 [ x , x + a ] × [ y − a , y ] if i = 4 {\displaystyle Q_{i}(x,y)={\begin{cases}\left[x,x+a\right]\times \left[y,y+a\right]&{\mbox{ if }}i=1\\\left[x-a,x\right]\times \left[y,y+a\right]&{\mbox{ if }}i=2\\\left[x-a,x\right]\times \left[y-a,y\right]&{\mbox{ if }}i=3\\\left[x,x+a\right]\times \left[y-a,y\right]&{\mbox{ if }}i=4\\\end{cases}}} where × {\displaystyle \times } is the cartesian product. Pixels located on the borders between two regions belong to both regions so there is a slight overlap between subregions. The arithmetic mean m i ( x , y ) {\displaystyle m_{i}(x,y)} and standard deviation σ i ( x , y ) {\displaystyle \sigma _{i}(x,y)} of the four regions centered around a pixel (x,y) are calculated and used to determine the value of the central pixel. The output of the Kuwahara filter Φ ( x , y ) {\displaystyle \Phi (x,y)} for any point ( x , y ) {\displaystyle (x,y)} is then given by Φ ( x , y ) = m i ( x , y ) {\textstyle \Phi (x,y)=m_{i}(x,y)} where i = a r g min j ⁡ σ j ( x , y ) {\displaystyle i=\operatorname {arg\min } _{j}\sigma _{j}(x,y)} . This means that the central pixel will take the mean value of the area that is most homogenous. The location of the pixel in relation to an edge plays a great role in determining which region will have the greater standard deviation. If for example the pixel is located on a dark side of an edge it will most probably take the mean value of the dark region. On the other hand, should the pixel be on the lighter side of an edge it will most probably take a light value. On the event that the pixel is located on the edge it will take the value of the more smooth, least textured region. The fact that the filter takes into account the homogeneity of the regions ensures that it will preserve the edges while using the mean creates the blurring effect. Similarly to the median filter, the Kuwahara filter uses a sliding window approach to access every pixel in the image. The size of the window is chosen in advance and may vary depending on the desired level of blur in the final image. Bigger windows typically result in the creation of more abstract images whereas small windows produce images that retain their detail. Typically windows are chosen to be square with sides that have an odd number of pixels for symmetry. However, there are variations of the Kuwahara filter that use rectangular windows. Additionally, the subregions do not need to overlap or have the same size as long as they cover all of the window. == Color images == For color images, the filter should not be performed by applying the filter to each RGB channel separately, and then recombining the three filtered color channels to form the filtered RGB image. The main problem with that is that the quadrants will have different standard deviations for each of the channels. For example, the upper left quadrant may have the lowest standard deviation in the red channel, but the lower right quadrant may have the lowest standard deviation in the green channel. This situation would result in the color of the central pixel to be determined by different regions, which might result in color artifacts or blurrier edges. To overcome this problem, for color images a slightly modified Kuwahara filter must be used. The image is first converted into another color space, the HSV color space. The modified filter then operates on only the "brightness" channel, the Value coordinate in the HSV model. The variance of the "brightness" of each quadrant is calculated to determine the quadrant from which the final filtered color should be taken from. The filter will produce an output for each channel which will correspond to the mean of that channel from the quadrant that had the lowest standard deviation in "brightness". This ensures that only one region will determine the RGB values of the central pixel. ImageMagick uses a similar approach, but using the Rec. 709 Luma as the brightness metric. === Julia Implementation === == Applications == Originally the Kuwahara filter was proposed for use in processing RI-angiocardiographic images of the cardiovascular system. The fact that any edges are preserved when smoothing makes it especially useful for feature extraction and segmentation and explains why it is used in medical imaging. The Kuwahara filter however also finds many applications in artistic imaging and fine-art photography due to its ability to remove textures and sharpen the edges of photographs. The level of abstraction helps create a desirable painting-like effect in artistic photographs especially in the case of the colored image version of the filter. These applications have known great success and have encouraged similar research in the field of image processing for the arts. Although the vast majority of applications have been in the field of image processing there have been cases that use modifications of the Kuwahara filter for machine learning tasks such as clustering. The Kuwahara filter has been implemented in CVIPtools. The Kuwahara filter is present as a shader node in Blender. == Drawbacks and restrictions == The Kuwahara filter despite its capabilities in edge preservation has certain drawbacks. At a first glance it is noticeable that the Kuwahara filter does not take into account the case where two regions have equal standard deviations. This is not often the case in real images since it is rather hard to find two regions with exactly the same standard deviation due to the noise that is always present. In cases where two regions have similar standard deviations the value of the center pixel could be decided at random by the noise in these regions. Again this would not be a problem if the regions had the same mean. However, it is not unusual for regions of very different means to have the same standard deviation. This makes the Kuwahara filter susceptible to noise. Different ways have been proposed for dealing with this issue, one of which is to set the value of the center pixel to ( m 1 + m 2 ) / 2 {\textstyle (m_{1}+m_{2})/2} in cases where the standard deviation of two regions do not differ more than a certain value D {\displaystyle D} . The Kuwahara filter is also known to create block artifacts in the images especially in regions of the image that are highly textured. These blocks disrupt the smoothness of the image and are considered to have a negative effect in the aesthetics of the image. This phenomenon occurs due to the division of the window into square regions. A way to overcome this effect is to take windows that are not rectangular(i.e. circular windows) and separate them into more non-rectangular regions. There have also been approaches where the filter adapts its window depending on the input image. == Extensions of the Kuwahara filter == The success of the Kuwahara filter has spurred an increase the development of edge-enhancing smoothing filters. Several variations have been proposed for similar use most of which attempt to deal with the drawbacks of the original Kuwahara filter. The "Generalized Kuwahara filter" proposed by P. Bakker considers several windows that contain a fixed pixel. Each window is then assigned an estimate and a confidence value. The value of the fixed pixel then takes the value of the estimate of the window with the highest confidence. This filter is not characterized by the same ambiguity in the presence of noise and manages to eliminate the block artifacts. The "Mean of Least Variance"(MLV) filter, proposed by M.A. Schulze also produces edge-enhancing smoothing results in images. Similarly to the Kuwahara filter it assumes a window of size 2 d − 1 × 2 d − 1 {\displaystyle 2d-1\times 2d-1} but instead of searching amongst four subregions of size d × d {\displaystyle d\times d} for the one with minimum variance it searches amongst all possible d × d {\displaystyle d\times d} subregions. This means the central pixel of the window will be assigned the mean of the one subregion out of a poss

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  Willy's Chocolate Experience

  Willy's Chocolate Experience was an unlicensed event based on Charlie and the Chocolate Factory that took place in Glasgow, Scotland, in February 2024. The event was promoted as an immersive and interactive family experience, illustrated on a promotional website with "dreamlike" AI-generated images. Once it was discovered that the event was held in a sparsely decorated warehouse, many customers complained, and the police were called to the venue. The event went viral on the Internet and attracted worldwide media attention. The event drew comparisons to the 2008 Lapland New Forest controversy, the 2014 Tumblr fan convention DashCon, and Billy McFarland's 2017 Fyre Festival. == Background and advertising == The event was stated to take place over the weekend of 24–25 February 2024. Promotional material advertised "stunning and intricately designed settings inspired by Roald Dahl's timeless tale" and "an array of delectable treats scattered throughout the experience". Both the website and promotional material used poor-quality AI-generated images, which included several spelling errors such as "cartchy tuns" and "a pasadise of sweet teats" and nonsensical words such as "catgacating" and "exarserdray". Tickets cost up to £35 per person. While the event was being promoted in early February, a Reddit user who saw Facebook advertisements suspected it to be a scam and was surprised that people were apparently buying tickets based solely on AI-generated images. The event was organised by House of Illuminati, a company registered to Billy Coull which claimed to offer "unparalleled immersive experiences". An investigation by Third Force News conducted after the event described Coull's previous "murky involvement in the charity sector." Coull had previously registered several other companies and claimed to work as a "consultant" for the now-defunct brand Empowerity, formerly known as the charity Gowanbank Community Hub. In 2021, Gowanbank was forced to remove claims of a £95-per-ticket fundraising "gala" at DoubleTree Glasgow which had been falsely advertised to feature TV personalities and performers including Gok Wan and Joe Black. Coull had claimed to be a doctor with a fake degree from a false university that provided "metaphysical degrees", and had attempted to use the charity to win the 2022 Glasgow City Council election in the seat of Greater Pollok, though he never registered for the election. In the summer of 2023, he independently published 17 AI-generated books on various topics, including vaccine conspiracy theories. Rolling Stone concluded that House of Illuminati's websites and event descriptions were likely written by an AI chatbot, such as ChatGPT. Three actors were hired to portray "Willy McDuff", a character based on Willy Wonka. One of them, Paul Connell, said that the cast were given one day to learn the script. Another actor playing Willy McDuff was 18-year-old Michael Archibald; the experience was his first ever acting job, and he was given the script at 6 pm on Friday before the event began on Saturday. Kirsty Paterson, an actress who played one of the Oompa-Loompas (called "Wonkidoodles" in the script), said that the job offer had been posted on Indeed.com and offered £500 for two days of work. The day before the event, the actors attended a dress rehearsal at the sparsely decorated venue. They were told that others would be working through the night on the production. When they returned on the day of the event, the venue was in the same condition. Paterson was given her costume an hour before the event opened, saying that "We were just handed an Amazon box that probably arrived that morning." == Script == The script for the event is titled Wonkidoodles at McDuff's Chocolate Factory: A Script, and describes Willy McDuff leading an audience through the Garden of Enchantment and the Twilight Tunnel. Once there, they are confronted by a character called The Unknown, described as "an evil chocolate maker who lives in the walls" who seeks to steal the magical "Anti-Graffiti Gobstopper" from McDuff's Imagination Lab. The gobstopper is "a sweet so powerful, it can make any room sparkle without lifting a finger". McDuff defeats The Unknown by amplifying the power of the gobstopper and causing his enemy to be "gently swept up by a robotic vacuum, humorously ending the confrontation". The script was unusual in that it included stage directions for the audience, and descriptions of their reactions. Connell described it as "15 pages of AI-generated gibberish of me just monologuing these mad things", and compared the vacuum cleaner plot point to that of the Nintendo video game Luigi's Mansion. Interviewed after the event, Coull claimed to have written the script himself, using AI only to "check spelling, grammar, and continuity" as he said he had dyslexia. == Event == The event was held at the Box Hub Warehouse event space in Whiteinch, an industrial area of Glasgow. Customers described the venue as "little more than an abandoned, empty warehouse", with set dressings including a small bouncy castle, AI-generated backdrop images pinned to some of the walls, and props which were "strewn about on bare concrete floors". The venue's windows were dirty and its air conditioning systems were left exposed. Paterson has stated that by the time she saw the venue, she had already signed her contract and "didn't want to disappoint the kids", and thus chose to proceed with the work. The Unknown was played by a 16-year-old actress named Felicia Dawkins, who wore a silver mask and a black cloak. Young children were frightened by the character, who appeared from behind a large rectangular mirror. Despite the script calling for The Unknown to be defeated with a vacuum cleaner, no such prop was provided, and actors were instead asked to improvise. Connell said that he and other employees were told to give each child "two jelly beans and a quarter of a cup of lemonade", although the limited supply of jelly beans quickly ran out. Paterson and another "Wonkidoodle" actress, Jenny Fogarty, said that after the first three 45-minute performances, the cast were told to abandon the script and instead let guests walk through the venue, a process that Paterson said took "about two minutes". The character of The Unknown, previously introduced as the main antagonist, was now "scaring children for no reason". One of the actors playing McDuff improvised the idea that children should pull a "silly face" at The Unknown to scare them away, but Dawkins said that, in other cases, she "just had to awkwardly walk back to my corner". Connell was told he would be given a 15-minute break every 45 minutes, but on the day of the event, he played Willy McDuff for three and a half hours without a break. After returning from a lunch break, Connell encountered a crowd of customers demanding refunds from Coull, and the other actors were unsure what to do next. After being told that the event was now cancelled halfway through its opening day, the actors left and went to a pub. Upon returning to the venue some time later, Connell said that he felt "the threat of violence had become quite high" and that there were two police vans and two squad cars at the scene. == Customer reviews and response == Willy's Chocolate Experience was widely criticised by those who attended it, many of whom demanded refunds. One customer, who had driven with his children for two hours to reach the event, described it as an "absolute con". Other visitors who arrived after the event was closed and were not informed of its cancellation requested compensation for wasted rail fares. Following the event's cancellation, Coull offered to refund 850 people, a statement repeated by the event's Facebook page. Some Facebook users stated that they had received their money back. Paterson and Fogarty stated that they only received half of their paycheque. Box Hub, the organisation that had rented the warehouse to House of Illuminati, issued an apology on House of Illuminati's behalf, stating that they "either have no regards for the families and young children they have disappointed or are too embarrassed to comment", and offered to provide a venue free of charge for those who attended the event. House of Illuminati later stated that they would not host any future events. Coull deleted his LinkedIn profile, his YouTube channel, and his personal website in response to the controversy. A few days after the event, Connell said he felt that Coull was "probably one of the most disliked people in Glasgow right now". In an interview with The Sunday Times, Coull apologised for how the event turned out, saying he would accept responsibility. == Fundraising == In an interview with Wired magazine, Connell stated that he and the other actors were working with parents to provide a free show for the children who attended. Some items from the event were later auctioned for charity. The venue auctioned the leftover hand-written "even

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  Veo (text-to-video model)

  Veo, or Google Veo, is a text-to-video model developed by Google DeepMind and announced in May 2024. As a generative AI model, it creates videos based on user prompts. Veo 3, released in May 2025, can also generate accompanying audio. == Development == In May 2024, a multimodal video generation model called Veo was announced at Google I/O 2024. Google claimed that it could generate 1080p videos over a minute long. In December 2024, Google released Veo 2, available via VideoFX. It supports 4K resolution video generation and has an improved understanding of physics. In April 2025, Google announced that Veo 2 became available for advanced users on the Gemini app. In May 2025, Google released Veo 3, which not only generates videos but also creates synchronized audio — including dialogue, sound effects, and ambient noise — to match the visuals. Google also announced Flow, a video-creation tool powered by Veo and Imagen. Google DeepMind CEO Demis Hassabis described the release as the moment when AI video generation left the era of the silent film. This was rebranded as Google Flow at the 2026 Google I/O keynote, along with the announcement of Google Flow Music. == Capabilities == Google Veo can be purchased at multiple subscription tiers and through Google "AI credits". The software itself can be run by two different consoles, Google Gemini and Google Flow. Gemini being geared towards shorter, quicker, and faster projects, using the Gemini AI chat model, with Google Flow, which is essentially a movie editor allowing users to create longer projects with continuity, using the same characters and actors. Users can create a maximum of eight seconds per clip. According to Gizmodo Veo 3 users were directing the model to generate low-quality content, such as man on the street interviews or haul videos of people unboxing products. 404 Media reported that the tool tended to repeat the same joke in response to different prompts. Commentators speculated that Google had trained the service on YouTube videos or Reddit posts. Google itself had not stated the source of its training content. In July 2025, Media Matters for America reported that racist and antisemitic videos generated using Veo 3 were being uploaded to TikTok. Ryan Whitwam of Ars Technica commented, "In a perfect world, Veo 3 would refuse to create these videos, but vagueness in the prompt and the AI's inability to understand the subtleties of racist tropes (i.e., the use of monkeys instead of humans in some videos) make it easy to skirt the rules."

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  Sword Art Online

  Sword Art Online (Japanese: ソードアート・オンライン, Hepburn: Sōdo Āto Onrain) is a Japanese light novel series written by Reki Kawahara and illustrated by abec. The series takes place in the 2020s and focuses on protagonists Kazuto "Kirito" Kirigaya and Asuna Yuuki as they play through various virtual reality MMORPG worlds, and later their involvement in the matters of a simulated civilization. Kawahara originally released the series as a web novel on his website from 2002 to 2008. The light novels began publication on ASCII Media Works' Dengeki Bunko imprint from April 10, 2009, with a spin-off series launching in October 2012. The series has spawned twelve manga adaptations published by ASCII Media Works and Kadokawa. The novels and the manga adaptations have been licensed for release in North America by Yen Press. An anime television series produced by A-1 Pictures, known simply as Sword Art Online, aired in Japan between July and December 2012, with a television film Sword Art Online: Extra Edition airing on December 31, 2013, and a second season, titled Sword Art Online II, airing between July and December 2014. An animated film titled Sword Art Online the Movie: Ordinal Scale, featuring an original story by Kawahara, premiered in Japan and Southeast Asia on February 18, 2017, and was released in the United States on March 9, 2017. A spin-off anime series titled Sword Art Online Alternative: Gun Gale Online premiered in April 2018, while a third season titled Sword Art Online: Alicization aired from October 2018 to September 2020. An anime film adaptation of Sword Art Online: Progressive titled Sword Art Online Progressive: Aria of a Starless Night premiered on October 30, 2021. A second film titled Sword Art Online Progressive: Scherzo of Deep Night premiered on October 22, 2022. Many video games based on the series have been released for consoles, PC, and mobile devices. Sword Art Online has achieved widespread commercial success, with the light novels having over 30 million copies sold worldwide. The anime series has received mixed to positive reviews, with praise for its animation, musical score, and exploration of the psychological aspects of virtual reality, but it has also been met with criticisms for its pacing and writing. == Synopsis == === Setting === The light novel series spans several virtual reality worlds, beginning with the game, Sword Art Online (SAO), which is set in a world known as Aincrad. Each world is built on a game engine called Cardinal system, which was initially developed specifically for SAO by Akihiko Kayaba, but was later duplicated for Alfheim Online (ALO), and a consolidated package is later given to Kirito in the form of the World Seed, who had it leaked online with the successful intention of reviving the virtual reality industry. A third world known as Gun Gale Online (GGO) appears in the third arc and is stylized as a first-person shooter game instead of a role-playing game, and is the main setting of Alternative Gun Gale Online. It was created using the World Seed by an American company. A fourth world appears in the fourth arc known as the Underworld (UW). The world itself was created using the World Seed as a base, but it is as realistic as the real world due to using many powerful government resources to keep it running. === Plot === In 2022, a virtual reality massively multiplayer online role-playing game (VRMMORPG) called Sword Art Online (SAO) was released. With the NerveGear, a helmet that stimulates the user's five senses via their brain, players can experience and control their in-game characters with their minds. Both the game and the NerveGear were created by Akihiko Kayaba. On November 6, 10,000 players log into SAO's mainframe cyberspace for the first time, only to discover that they are unable to log out. Kayaba appears and tells the players that they must beat all 100 floors of Aincrad, a steel castle which is the setting of SAO, if they wish to be free. He also states that those who suffer in-game deaths or forcibly remove the NerveGear out-of-game will suffer real-life deaths. A player named Kazuto "Kirito" Kirigaya is one of 1,000 testers in the game's previous closed beta. With the advantage of previous VR gaming experience and a drive to protect other beta testers from discrimination, he isolates himself from the greater groups and plays the game alone, bearing the mantle of "beater", a portmanteau of "beta tester" and "cheater". As the players progress through the game Kirito eventually befriends a young woman named Asuna Yuuki, forming a relationship with and later marrying her in-game. After the duo discover the identity of Kayaba's secret ID, who was playing as "Heathcliff", the leader of the guild Asuna joined in, they confront and destroy him, freeing themselves and the other players from the game. In the real world, Kazuto discovers that 300 SAO players, including Asuna, remain trapped in their NerveGear. As he goes to the hospital to see Asuna, he meets Asuna's father Shouzou Yuuki who is asked by an associate of his, Nobuyuki Sugou, to make a decision, which Sugou later reveals to be his marriage with Asuna, angering Kazuto. Several months later, he is informed by Agil, another SAO survivor, that a figure similar to Asuna was spotted on "The World Tree" in another VRMMORPG cyberspace called Alfheim Online (ALO). Assisted in-game by his cousin and adoptive sister Suguha "Leafa" Kirigaya and Yui, a navigation pixie (originally an AI from SAO), he quickly learns that the trapped players in ALO are part of a plan conceived by Sugou to perform illegal experiments on their minds. The goal is to create the perfect mind-control for financial gain and to subjugate Asuna, whom he intends to marry in the real world, to assume control of her family's corporation. Kirito eventually stops the experiment and rescues the remaining 300 SAO players, foiling Sugou's plans. Before leaving ALO to see Asuna, Kayaba, who has uploaded his mind to the Internet using an experimental, destructively high-powered version of NerveGear at the cost of his life, entrusts Kirito with The Seed – a package program designed to create virtual worlds. Kazuto eventually reunites with Asuna in the real world after thwarting an attack from Sugou and The Seed is released onto the Internet, reviving Aincrad as other VRMMORPGs begin to thrive. One year after the events of SAO, at the prompting of a government official investigating strange occurrences in VR, Kazuto takes on a job to investigate a series of murders involving another VRMMORPG called Gun Gale Online (GGO), the AmuSphere (the successor of the NerveGear), and a player called Death Gun. Aided by a female player named Shino "Sinon" Asada, he participates in a gunfight tournament called the Bullet of Bullets (BoB) and discovers the truth behind the murders, which originated with a player who participated in a player-killing guild in SAO. Through his and Sinon's efforts, two suspects are captured, though the third suspect, Johnny Black, escapes. Kazuto is later recruited to test an experimental FullDive machine, Soul Translator (STL), which has an interface far more realistic and complex than the previous machine he had played, to help RATH, a research and development organization under the Ministry of Defense (MOD), develop an artificial intelligence named A.L.I.C.E. He tests the STL by entering the Underworld (UW), a virtual reality cyberspace created with The Seed package. In the UW, the flow of time proceeds a thousand times faster than in the real world, and Kirito's memories of what happens inside are restricted. However, when Johnny Black ambushes and mortally wounds Kazuto with suxamethonium chloride, RATH recovers Kazuto and places him back into the STL to preserve his mind while attempts are made to save him. During his time in Underworld, Kirito befriends Eugeo, a carver in a small village of Rulid, and helps him on a journey to save Alice Zuberg, his friend who was taken by a group of highly skilled warriors known as the Integrity Knights for accidentally breaking a rule of the Axiom Church, the leaders of the Human Empire. He and Eugeo soon find themselves uncovering the secrets of the Axiom Church, led by a woman only known as "The Administrator", and the true purpose of Underworld itself, while unbeknownst to them, a war against the opposing Dark Territory is brewing on the horizon. They meet Alice, now an Integrity Knight, and though she does not remember them, Kirito helps her remember her true identity: a form of true artificial intelligence known as A.L.I.C.E. In the battle against the Administrator, Kirito manages to slay her, though Eugeo dies in the process, to Kirito's dismay. Meanwhile, in the real world, conflict escalates as American forces raid RATH's facility in the Ocean Turtle in an effort to take A.L.I.C.E. for purposes unknown. Two of the attackers - Gabriel "Vecta" Miller and Vassago "Prince of Hell" Cassals - take contr

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  Manual override

  A manual override (MO) or manual analog override (MAO) is a mechanism where control is taken from an automated system and given to the user. For example, a manual override in photography refers to the ability for the human photographer to turn off the automatic aperture sizing, automatic focusing, or any other automated system on the camera. Some manual overrides can be used to veto an automated system's judgment when the system is in error. An example of this is a printer's ink level detection: in one case, a researcher found that when he overrode the system, up to 38% more pages could be printed at good quality by the printer than the automated system would have allowed. Automated systems are becoming increasingly common and integrated into everyday objects such as automobiles and domestic appliances. This development of ubiquitous computing raises general issues of policy and law about the need for manual overrides for matters of great importance such as life-threatening situations and major economic decisions. The loyalty of such autonomous devices then becomes an issue. If they follow rules installed by the manufacturer or required by law and refuse to cede control in some situations then the owners of the devices may feel disempowered, alienated and lacking true ownership. == Major incidents == China Airlines Flight 140 crashed, causing many deaths, due to a misunderstanding about the manual overrides for the autopilot. The Take-Off/Go Around system had been activated to abort a landing. It was programmed to ignore manual controls in this situation but the human pilots tried to continue the landing. The conflicting control signals from the pilots and autopilot then resulted in the aircraft stalling and crashing. The autopilot for this aircraft type was then reprogrammed so that it would never ignore a manual override.

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  T-norm fuzzy logics

  T-norm fuzzy logics are a family of non-classical logics, informally delimited by having a semantics that takes the real unit interval [0, 1] for the system of truth values and functions called t-norms for permissible interpretations of conjunction. They are mainly used in applied fuzzy logic and fuzzy set theory as a theoretical basis for approximate reasoning. T-norm fuzzy logics belong in broader classes of fuzzy logics and many-valued logics. In order to generate a well-behaved implication, the t-norms are usually required to be left-continuous; logics of left-continuous t-norms further belong in the class of substructural logics, among which they are marked with the validity of the law of prelinearity, (A → B) ∨ (B → A). Both propositional and first-order (or higher-order) t-norm fuzzy logics, as well as their expansions by modal and other operators, are studied. Logics that restrict the t-norm semantics to a subset of the real unit interval (for example, finitely valued Łukasiewicz logics) are usually included in the class as well. Important examples of t-norm fuzzy logics are monoidal t-norm logic (MTL) of all left-continuous t-norms, basic logic (BL) of all continuous t-norms, product fuzzy logic of the product t-norm, or the nilpotent minimum logic of the nilpotent minimum t-norm. Some independently motivated logics belong among t-norm fuzzy logics, too, for example Łukasiewicz logic (which is the logic of the Łukasiewicz t-norm) or Gödel–Dummett logic (which is the logic of the minimum t-norm). == Motivation == As members of the family of fuzzy logics, t-norm fuzzy logics primarily aim at generalizing classical two-valued logic by admitting intermediary truth values between 1 (truth) and 0 (falsity) representing degrees of truth of propositions. The degrees are assumed to be real numbers from the unit interval [0, 1]. In propositional t-norm fuzzy logics, propositional connectives are stipulated to be truth-functional, that is, the truth value of a complex proposition formed by a propositional connective from some constituent propositions is a function (called the truth function of the connective) of the truth values of the constituent propositions. The truth functions operate on the set of truth degrees (in the standard semantics, on the [0, 1] interval); thus the truth function of an n-ary propositional connective c is a function Fc: [0, 1]n → [0, 1]. Truth functions generalize truth tables of propositional connectives known from classical logic to operate on the larger system of truth values. T-norm fuzzy logics impose certain natural constraints on the truth function of conjunction. The truth function ∗ : [ 0 , 1 ] 2 → [ 0 , 1 ] {\displaystyle \colon [0,1]^{2}\to [0,1]} of conjunction is assumed to satisfy the following conditions: Commutativity, that is, x ∗ y = y ∗ x {\displaystyle xy=yx} for all x and y in [0, 1]. This expresses the assumption that the order of fuzzy propositions is immaterial in conjunction, even if intermediary truth degrees are admitted. Associativity, that is, ( x ∗ y ) ∗ z = x ∗ ( y ∗ z ) {\displaystyle (xy)z=x(yz)} for all x, y, and z in [0, 1]. This expresses the assumption that the order of performing conjunction is immaterial, even if intermediary truth degrees are admitted. Monotony, that is, if x ≤ y {\displaystyle x\leq y} then x ∗ z ≤ y ∗ z {\displaystyle xz\leq yz} for all x, y, and z in [0, 1]. This expresses the assumption that increasing the truth degree of a conjunct should not decrease the truth degree of the conjunction. Neutrality of 1, that is, 1 ∗ x = x {\displaystyle 1x=x} for all x in [0, 1]. This assumption corresponds to regarding the truth degree 1 as full truth, conjunction with which does not decrease the truth value of the other conjunct. Together with the previous conditions this condition ensures that also 0 ∗ x = 0 {\displaystyle 0x=0} for all x in [0, 1], which corresponds to regarding the truth degree 0 as full falsity, conjunction with which is always fully false. Continuity of the function ∗ {\displaystyle } (the previous conditions reduce this requirement to the continuity in either argument). Informally this expresses the assumption that microscopic changes of the truth degrees of conjuncts should not result in a macroscopic change of the truth degree of their conjunction. This condition, among other things, ensures a good behavior of (residual) implication derived from conjunction; to ensure the good behavior, however, left-continuity (in either argument) of the function ∗ {\displaystyle } is sufficient. In general t-norm fuzzy logics, therefore, only left-continuity of ∗ {\displaystyle } is required, which expresses the assumption that a microscopic decrease of the truth degree of a conjunct should not macroscopically decrease the truth degree of conjunction. These assumptions make the truth function of conjunction a left-continuous t-norm, which explains the name of the family of fuzzy logics (t-norm based). Particular logics of the family can make further assumptions about the behavior of conjunction (for example, Gödel–Dummett logic requires its idempotence) or other connectives (for example, the logic IMTL (involutive monoidal t-norm logic) requires the involutiveness of negation). All left-continuous t-norms ∗ {\displaystyle } have a unique residuum, that is, a binary function ⇒ {\displaystyle \Rightarrow } such that for all x, y, and z in [0, 1], x ∗ y ≤ z {\displaystyle xy\leq z} if and only if x ≤ y ⇒ z . {\displaystyle x\leq y\Rightarrow z.} The residuum of a left-continuous t-norm can explicitly be defined as ( x ⇒ y ) = sup { z ∣ z ∗ x ≤ y } . {\displaystyle (x\Rightarrow y)=\sup\{z\mid zx\leq y\}.} This ensures that the residuum is the pointwise largest function such that for all x and y, x ∗ ( x ⇒ y ) ≤ y . {\displaystyle x(x\Rightarrow y)\leq y.} The latter can be interpreted as a fuzzy version of the modus ponens rule of inference. The residuum of a left-continuous t-norm thus can be characterized as the weakest function that makes the fuzzy modus ponens valid, which makes it a suitable truth function for implication in fuzzy logic. Left-continuity of the t-norm is the necessary and sufficient condition for this relationship between a t-norm conjunction and its residual implication to hold. Truth functions of further propositional connectives can be defined by means of the t-norm and its residuum, for instance the residual negation ¬ x = ( x ⇒ 0 ) {\displaystyle \neg x=(x\Rightarrow 0)} or bi-residual equivalence x ⇔ y = ( x ⇒ y ) ∗ ( y ⇒ x ) . {\displaystyle x\Leftrightarrow y=(x\Rightarrow y)(y\Rightarrow x).} Truth functions of propositional connectives may also be introduced by additional definitions: the most usual ones are the minimum (which plays a role of another conjunctive connective), the maximum (which plays a role of a disjunctive connective), or the Baaz Delta operator, defined in [0, 1] as Δ x = 1 {\displaystyle \Delta x=1} if x = 1 {\displaystyle x=1} and Δ x = 0 {\displaystyle \Delta x=0} otherwise. In this way, a left-continuous t-norm, its residuum, and the truth functions of additional propositional connectives determine the truth values of complex propositional formulae in [0, 1]. Formulae that always evaluate to 1 are called tautologies with respect to the given left-continuous t-norm ∗ , {\displaystyle ,} or ∗ - {\displaystyle {\mbox{-}}} tautologies. The set of all ∗ - {\displaystyle {\mbox{-}}} tautologies is called the logic of the t-norm ∗ , {\displaystyle ,} as these formulae represent the laws of fuzzy logic (determined by the t-norm) that hold (to degree 1) regardless of the truth degrees of atomic formulae. Some formulae are tautologies with respect to a larger class of left-continuous t-norms; the set of such formulae is called the logic of the class. Important t-norm logics are the logics of particular t-norms or classes of t-norms, for example: Łukasiewicz logic is the logic of the Łukasiewicz t-norm x ∗ y = max ( x + y − 1 , 0 ) {\displaystyle xy=\max(x+y-1,0)} Gödel–Dummett logic is the logic of the minimum t-norm x ∗ y = min ( x , y ) {\displaystyle xy=\min(x,y)} Product fuzzy logic is the logic of the product t-norm x ∗ y = x ⋅ y {\displaystyle xy=x\cdot y} Monoidal t-norm logic MTL is the logic of (the class of) all left-continuous t-norms Basic fuzzy logic BL is the logic of (the class of) all continuous t-norms It turns out that many logics of particular t-norms and classes of t-norms are axiomatizable. The completeness theorem of the axiomatic system with respect to the corresponding t-norm semantics on [0, 1] is then called the standard completeness of the logic. Besides the standard real-valued semantics on [0, 1], the logics are sound and complete with respect to general algebraic semantics, formed by suitable classes of prelinear commutative bounded integral residuated lattices. == History == Some particular t-norm fuzzy logics have been introduced and investigated long before the family was re

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  Construction of t-norms

  In mathematics, t-norms are a special kind of binary operations on the real unit interval [0, 1]. Various constructions of t-norms, either by explicit definition or by transformation from previously known functions, provide a plenitude of examples and classes of t-norms. This is important, e.g., for finding counter-examples or supplying t-norms with particular properties for use in engineering applications of fuzzy logic. The main ways of construction of t-norms include using generators, defining parametric classes of t-norms, rotations, or ordinal sums of t-norms. Relevant background can be found in the article on t-norms. == Generators of t-norms == The method of constructing t-norms by generators consists in using a unary function (generator) to transform some known binary function (most often, addition or multiplication) into a t-norm. In order to allow using non-bijective generators, which do not have the inverse function, the following notion of pseudo-inverse function is employed: Let f: [a, b] → [c, d] be a monotone function between two closed subintervals of extended real line. The pseudo-inverse function to f is the function f (−1): [c, d] → [a, b] defined as f ( − 1 ) ( y ) = { sup { x ∈ [ a , b ] ∣ f ( x ) < y } for f non-decreasing sup { x ∈ [ a , b ] ∣ f ( x ) > y } for f non-increasing. {\displaystyle f^{(-1)}(y)={\begin{cases}\sup\{x\in [a,b]\mid f(x)y\}&{\text{for }}f{\text{ non-increasing.}}\end{cases}}} === Additive generators === The construction of t-norms by additive generators is based on the following theorem: Let f: [0, 1] → [0, +∞] be a strictly decreasing function such that f(1) = 0 and f(x) + f(y) is in the range of f or in [f(0+), +∞] for all x, y in [0, 1]. Then the function T: [0, 1]2 → [0, 1] defined as T(x, y) = f (-1)(f(x) + f(y)) is a t-norm. Alternatively, one may avoid using the notion of pseudo-inverse function by having T ( x , y ) = f − 1 ( min ( f ( 0 + ) , f ( x ) + f ( y ) ) ) {\displaystyle T(x,y)=f^{-1}\left(\min \left(f(0^{+}),f(x)+f(y)\right)\right)} . The corresponding residuum can then be expressed as ( x ⇒ y ) = f − 1 ( max ( 0 , f ( y ) − f ( x ) ) ) {\displaystyle (x\Rightarrow y)=f^{-1}\left(\max \left(0,f(y)-f(x)\right)\right)} . And the biresiduum as ( x ⇔ y ) = f − 1 ( | f ( x ) − f ( y ) | ) {\displaystyle (x\Leftrightarrow y)=f^{-1}\left(\left|f(x)-f(y)\right|\right)} . If a t-norm T results from the latter construction by a function f which is right-continuous in 0, then f is called an additive generator of T. Examples: The function f(x) = 1 – x for x in [0, 1] is an additive generator of the Łukasiewicz t-norm. The function f defined as f(x) = –log(x) if 0 < x ≤ 1 and f(0) = +∞ is an additive generator of the product t-norm. The function f defined as f(x) = 2 – x if 0 ≤ x < 1 and f(1) = 0 is an additive generator of the drastic t-norm. Basic properties of additive generators are summarized by the following theorem: Let f: [0, 1] → [0, +∞] be an additive generator of a t-norm T. Then: T is an Archimedean t-norm. T is continuous if and only if f is continuous. T is strictly monotone if and only if f(0) = +∞. Each element of (0, 1) is a nilpotent element of T if and only if f(0) < +∞. The multiple of f by a positive constant is also an additive generator of T. T has no non-trivial idempotents. (Consequently, e.g., the minimum t-norm has no additive generator.) === Multiplicative generators === The isomorphism between addition on [0, +∞] and multiplication on [0, 1] by the logarithm and the exponential function allow two-way transformations between additive and multiplicative generators of a t-norm. If f is an additive generator of a t-norm T, then the function h: [0, 1] → [0, 1] defined as h(x) = e−f (x) is a multiplicative generator of T, that is, a function h such that h is strictly increasing h(1) = 1 h(x) · h(y) is in the range of h or equal to 0 or h(0+) for all x, y in [0, 1] h is right-continuous in 0 T(x, y) = h (−1)(h(x) · h(y)). Vice versa, if h is a multiplicative generator of T, then f: [0, 1] → [0, +∞] defined by f(x) = −log(h(x)) is an additive generator of T. == Parametric classes of t-norms == Many families of related t-norms can be defined by an explicit formula depending on a parameter p. This section lists the best known parameterized families of t-norms. The following definitions will be used in the list: A family of t-norms Tp parameterized by p is increasing if Tp(x, y) ≤ Tq(x, y) for all x, y in [0, 1] whenever p ≤ q (similarly for decreasing and strictly increasing or decreasing). A family of t-norms Tp is continuous with respect to the parameter p if lim p → p 0 T p = T p 0 {\displaystyle \lim _{p\to p_{0}}T_{p}=T_{p_{0}}} for all values p0 of the parameter. === Schweizer–Sklar t-norms === The family of Schweizer–Sklar t-norms, introduced by Berthold Schweizer and Abe Sklar in the early 1960s, is given by the parametric definition T p S S ( x , y ) = { T min ( x , y ) if p = − ∞ ( x p + y p − 1 ) 1 / p if − ∞ < p < 0 T p r o d ( x , y ) if p = 0 ( max ( 0 , x p + y p − 1 ) ) 1 / p if 0 < p < + ∞ T D ( x , y ) if p = + ∞ . {\displaystyle T_{p}^{\mathrm {SS} }(x,y)={\begin{cases}T_{\min }(x,y)&{\text{if }}p=-\infty \\(x^{p}+y^{p}-1)^{1/p}&{\text{if }}-\infty −∞ Continuous if and only if p < +∞ Strict if and only if −∞ < p ≤ 0 (for p = −1 it is the Hamacher product) Nilpotent if and only if 0 < p < +∞ (for p = 1 it is the Łukasiewicz t-norm). The family is strictly decreasing for p ≥ 0 and continuous with respect to p in [−∞, +∞]. An additive generator for T p S S {\displaystyle T_{p}^{\mathrm {SS} }} for −∞ < p < +∞ is f p S S ( x ) = { − log ⁡ x if p = 0 1 − x p p otherwise. {\displaystyle f_{p}^{\mathrm {SS} }(x)={\begin{cases}-\log x&{\text{if }}p=0\\{\frac {1-x^{p}}{p}}&{\text{otherwise.}}\end{cases}}} === Hamacher t-norms === The family of Hamacher t-norms, introduced by Horst Hamacher in the late 1970s, is given by the following parametric definition for 0 ≤ p ≤ +∞: T p H ( x , y ) = { T D ( x , y ) if p = + ∞ 0 if p = x = y = 0 x y p + ( 1 − p ) ( x + y − x y ) otherwise. {\displaystyle T_{p}^{\mathrm {H} }(x,y)={\begin{cases}T_{\mathrm {D} }(x,y)&{\text{if }}p=+\infty \\0&{\text{if }}p=x=y=0\\{\frac {xy}{p+(1-p)(x+y-xy)}}&{\text{otherwise.}}\end{cases}}} The t-norm T 0 H {\displaystyle T_{0}^{\mathrm {H} }} is called the Hamacher product. Hamacher t-norms are the only t-norms which are rational functions. The Hamacher t-norm T p H {\displaystyle T_{p}^{\mathrm {H} }} is strict if and only if p < +∞ (for p = 1 it is the product t-norm). The family is strictly decreasing and continuous with respect to p. An additive generator of T p H {\displaystyle T_{p}^{\mathrm {H} }} for p < +∞ is f p H ( x ) = { 1 − x x if p = 0 log ⁡ p + ( 1 − p ) x x otherwise. {\displaystyle f_{p}^{\mathrm {H} }(x)={\begin{cases}{\frac {1-x}{x}}&{\text{if }}p=0\\\log {\frac {p+(1-p)x}{x}}&{\text{otherwise.}}\end{cases}}} === Frank t-norms === The family of Frank t-norms, introduced by M.J. Frank in the late 1970s, is given by the parametric definition for 0 ≤ p ≤ +∞ as follows: T p F ( x , y ) = { T m i n ( x , y ) if p = 0 T p r o d ( x , y ) if p = 1 T L u k ( x , y ) if p = + ∞ log p ⁡ ( 1 + ( p x − 1 ) ( p y − 1 ) p − 1 ) otherwise. {\displaystyle T_{p}^{\mathrm {F} }(x,y)={\begin{cases}T_{\mathrm {min} }(x,y)&{\text{if }}p=0\\T_{\mathrm {prod} }(x,y)&{\text{if }}p=1\\T_{\mathrm {Luk} }(x,y)&{\text{if }}p=+\infty \\\log _{p}\left(1+{\frac {(p^{x}-1)(p^{y}-1)}{p-1}}\right)&{\text{otherwise.}}\end{cases}}} The Frank t-norm T p F {\displaystyle T_{p}^{\mathrm {F} }} is strict if p < +∞. The family is strictly decreasing and continuous with respect to p. An additive generator for T p F {\displaystyle T_{p}^{\mathrm {F} }} is f p F ( x ) = { − log ⁡ x if p = 1 1 − x if p = + ∞ log ⁡ p − 1 p x − 1 otherwise. {\displaystyle f_{p}^{\mathrm {F} }(x)={\begin{cases}-\log x&{\text{if }}p=1\\1-x&{\text{if }}p=+\infty \\\log {\frac {p-1}{p^{x}-1}}&{\text{otherwise.}}\end{cases}}} === Yager t-norms === The family of Yager t-norms, introduced in the early 1980s by Ronald R. Yager, is given for 0 ≤ p ≤ +∞ by T p Y ( x , y ) = { T D ( x , y ) if p = 0 max ( 0 , 1 − ( ( 1 − x ) p + ( 1 − y ) p ) 1 / p ) if 0 < p < + ∞ T m i n ( x , y ) if p = + ∞ {\displaystyle T_{p}^{\mathrm {Y} }(x,y)={\begin{cases}T_{\mathrm {D} }(x,y)&{\text{if }}p=0\\\max \left(0,1-((1-x)^{p}+(1-y)^{p})^{1/p}\right)&{\text{if }}0 Read more →

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  Resistance Database Initiative

  HIV Resistance Response Database Initiative (RDI) was formed in 2002 to use artificial intelligence (AI) to predict how patients will respond to HIV drugs using data from more 250,000 patients from around 50 countries around the world. The RDI used its models to power its HIV Treatment Response Prediction System (HIV-TRePS). Launched in 2010, this free online tool enabled healthcare professionals to upload their patient’s data and obtain highly accurate predictions of how they would respond to different combinations of the 30 or more drugs available. The tool enabled physicians to individualize their patients’ treatment, using these predictions based on more than a million patient-years of treatment experience. HIV-TRePS was possibly the first ever AI-based system for medical decision-making to be developed, successfully tested, and used in clinical practice. It has since been used by thousands of healthcare professionals to optimise the treatment of tens of thousands of patients. Since the RDI’s inception the treatment of HIV infection has progressed enormously, with more effective and better tolerated drugs available in ever more convenient combination formulations. In most countries HIV is now considered a chronic, manageable condition. Moreover, the success of the drugs in reducing the amount of virus is substantially reducing the onward transmission of the virus and cases of new infections are falling in many settings. This improvement in HIV treatment means the need for sophisticated AI to support HIV treatment decisions has significantly reduced. In response, the RDI ceased development of further models and, in March 2024, withdrew its HIV-TRePS system. == Background == Human immunodeficiency virus (HIV) is the virus that causes acquired immunodeficiency syndrome (AIDS), a condition in which the immune system begins to fail, leading to life-threatening opportunistic infections. There are approximately 30 HIV antiretroviral drugs that have been approved for the treatment of HIV infection, from six different classes, based on the point in the HIV life-cycle at which they act. They are used in combination; typically 3 or more drugs from 2 or more different classes, a form of therapy known as highly active antiretroviral therapy (HAART). The aim of therapy is to suppress the virus to very low, ideally undetectable, levels in the blood. This prevents the virus from depleting the immune cells that it preferentially attacks CD4 cells and prevents or delays illness and death. Despite the expanding availability of these drugs and the impact of their use, treatments continue to fail, often involving to the development of resistance. During drug therapy, low-level virus replication may still occur, particularly when a patient misses a dose. HIV makes errors in copying its genetic material and, if a mutation makes the virus resistant to one or more of the drugs in the patient's treatment, it may begin to replicate more successfully in the presence of that drug and undermine the effect of the treatment. If this happens, the treatment needs to be changed to re-establish control over the virus. == RDI's Approach == The RDI’s approach was to use artificial intelligence (including neural network and random forest models), trained with data from hundreds of thousands of patients, treated with different drugs in a variety of clinical settings all over the world, to predict how an individual patient will respond to any new combination of HIV drugs. The models were tested with independent data sets and consistently achieved accuracy of approximately 80%.

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  Network Abstraction Layer

  The Network Abstraction Layer (NAL) is a part of the H.264/AVC and HEVC video coding standards. The main goal of the NAL is the provision of a "network-friendly" video representation addressing "conversational" (video telephony) and "non conversational" (storage, broadcast, or streaming) applications. NAL has achieved a significant improvement in application flexibility relative to prior video coding standards. == Introduction == An increasing number of services and growing popularity of high definition TV are creating greater needs for higher coding efficiency. Moreover, other transmission media such as cable modem, xDSL, or UMTS offer much lower data rates than broadcast channels, and enhanced coding efficiency can enable the transmission of more video channels or higher quality video representations within existing digital transmission capacities. Video coding for telecommunication applications has diversified from ISDN and T1/E1 service to embrace PSTN, mobile wireless networks, and LAN/Internet network delivery. Throughout this evolution, continued efforts have been made to maximize coding efficiency while dealing with the diversification of network types and their characteristic formatting and loss/error robustness requirements. The H.264/AVC and HEVC standards are designed for technical solutions including areas like broadcasting (over cable, satellite, cable modem, DSL, terrestrial, etc.) interactive or serial storage on optical and magnetic devices, conversational services, video-on-demand or multimedia streaming, multimedia messaging services, etc. Moreover, new applications may be deployed over existing and future networks. This raises the question about how to handle this variety of applications and networks. To address this need for flexibility and customizability, the design covers a NAL that formats the Video Coding Layer (VCL) representation of the video and provides header information in a manner appropriate for conveyance by a variety of transport layers or storage media. The NAL is designed in order to provide "network friendliness" to enable simple and effective customization of the use of VCL for a broad variety of systems. The NAL facilitates the ability to map VCL data to transport layers such as: RTP/IP for any kind of real-time wire-line and wireless Internet services. File formats, e.g., ISO MP4 for storage and MMS. H.32X for wireline and wireless conversational services. MPEG-2 systems for broadcasting services, etc. The full degree of customization of the video content to fit the needs of each particular application is outside the scope of the video coding standardization effort, but the design of the NAL anticipates a variety of such mappings. Some key concepts of the NAL are NAL units, byte stream, and packet formats uses of NAL units, parameter sets, and access units. A short description of these concepts is given below. == NAL units == The coded video data is organized into NAL units, each of which is effectively a packet that contains an integer number of bytes. The first byte of each H.264/AVC NAL unit is a header byte that contains an indication of the type of data in the NAL unit. For HEVC the header was extended to two bytes. All the remaining bytes contain payload data of the type indicated by the header. The NAL unit structure definition specifies a generic format for use in both packet-oriented and bitstream-oriented transport systems, and a series of NAL units generated by an encoder is referred to as a NAL unit stream. == NAL Units in Byte-Stream Format Use == Some systems require delivery of the entire or partial NAL unit stream as an ordered stream of bytes or bits within which the locations of NAL unit boundaries need to be identifiable from patterns within the coded data itself. For use in such systems, the H.264/AVC and HEVC specifications define a byte stream format. In the byte stream format, each NAL unit is prefixed by a specific pattern of three bytes called a start code prefix. The boundaries of the NAL unit can then be identified by searching the coded data for the unique start code prefix pattern. The use of emulation prevention bytes guarantees that start code prefixes are unique identifiers of the start of a new NAL unit. A small amount of additional data (one byte per video picture) is also added to allow decoders that operate in systems that provide streams of bits without alignment to byte boundaries to recover the necessary alignment from the data in the stream. Additional data can also be inserted in the byte stream format that allows expansion of the amount of data to be sent and can aid in achieving more rapid byte alignment recovery, if desired. == NAL Units in Packet-Transport System Use == In other systems (e.g., IP/RTP systems), the coded data is carried in packets that are framed by the system transport protocol, and identification of the boundaries of NAL units within the packets can be established without use of start code prefix patterns. In such systems, the inclusion of start code prefixes in the data would be a waste of data carrying capacity, so instead the NAL units can be carried in data packets without start code prefixes. == VCL and Non-VCL NAL Units == NAL units are classified into VCL and non-VCL NAL units. VCL NAL units contain the data that represents the values of the samples in the video pictures. Non-VCL NAL units contain any associated additional information such as parameter sets (important header data that can apply to a large number of VCL NAL units) and supplemental enhancement information (timing information and other supplemental data that may enhance usability of the decoded video signal but are not necessary for decoding the values of the samples in the video pictures). == Parameter Sets == A parameter set contains shared configuration data that is carried in non-VCL NAL units. Parameter sets are typically reused when decoding many coded pictures within a video sequence. Each VCL NAL unit references a picture parameter set (PPS), which in turn references a sequence parameter set (SPS). There are two types of parameter sets: Sequence parameter set (SPS), which specifies mostly constant configuration such as resolution, bit depth, or chroma format. (For a concrete implementation, see FFmpeg's SPS struct.) Picture parameter set (PPS), which applies on top of an SPS, and specifies configuration such as QP offsets. (For a concrete implementation, see FFmpeg's PPS struct.) The sequence and picture parameter-set mechanism decouples the transmission of infrequently changing information from the transmission of coded representations of the values of the samples in the video pictures. Each VCL NAL unit contains an identifier that refers to the content of the relevant picture parameter set and each picture parameter set contains an identifier that refers to the content of the relevant sequence parameter set. In this manner, a small amount of data (the identifier) can be used to refer to a larger amount of information (the parameter set) without repeating that information within each VCL NAL unit. Sequence and picture parameter sets can be sent well ahead of the VCL NAL units that they apply to, and can be repeated to provide robustness against data loss. In some applications, parameter sets may be sent within the channel that carries the VCL NAL units (termed "in-band" transmission). In other applications, it can be advantageous to convey the parameter sets "out-of-band" using a more reliable transport mechanism than the video channel itself. == Access Units == A set of NAL units in a specified form is referred to as an access unit. The decoding of each access unit results in one decoded picture. Each access unit contains a set of VCL NAL units that together compose a primary coded picture. It may also be prefixed with an access unit delimiter to aid in locating the start of the access unit. Some supplemental enhancement information containing data such as picture timing information may also precede the primary coded picture. The primary coded picture consists of a set of VCL NAL units consisting of slices or slice data partitions that represent the samples of the video picture. Following the primary coded picture may be some additional VCL NAL units that contain redundant representations of areas of the same video picture. These are referred to as redundant coded pictures, and are available for use by a decoder in recovering from loss or corruption of the data in the primary coded pictures. Decoders are not required to decode redundant coded pictures if they are present. Finally, if the coded picture is the last picture of a coded video sequence (a sequence of pictures that is independently decodable and uses only one sequence parameter set), an end of sequence NAL unit may be present to indicate the end of the sequence; and if the coded picture is the last coded picture in the entire NAL unit stream, an end of stream NAL unit may be present to

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  Squirrel AI

  Squirrel Ai Learning is an international educational technology company that specializes in intelligent adaptive learning and was one of the first companies in the world to offer large scale AI-powered adaptive education solutions. == Methodology == Squirrel Ai Learning uses artificial intelligence to tailor lesson plans to each individual student. The company's AI researchers have access to the world's largest student databases, which are used to train the AI algorithms. Squirrel Ai Learning works with teachers to identify the most fine-grained possible concepts ("knowledge points") for a course in order to precisely target learning gaps. For example, middle school mathematics is broken into over 10,000 points such as rational numbers, the properties of a triangle, and the Pythagorean theorem. Each point is linked to related items, forming a "knowledge graph". Each knowledge point is addressed by videos, examples and practice problems. A textbook might address 3,000 points; ALEKS, another adaptive learning platform, uses 1,000. Each student begins with a diagnostic test to identify where to begin their learning. The system continues to refine its graph as more students proceed. Learning is not student-directed. The system decides the order of topics. == History and milestones == Squirrel Ai Learning was founded by Derek Haoyang Li in 2014. In March, 2017, The Squirrel Ai Intelligent Adaptive Learning System (IALS) was launched. IALS utilizes artificial intelligence to customize lessons, practice and evaluations for each individual student. In 2018, Squirrel Ai Learning established a joint research lab of AI adaptive learning with the institute of Automation of the Chinese Academy of Sciences. By 2019, Squirrel Ai Learning had opened 2,000 learning centers in 200 cities and registered over a million students in Asia. In 2019, Squirrel Ai Learning opened a research lab in partnership with Carnegie Mellon University. As of 2019, Squirrel Ai Learning had raised over $180 million in funding and in 2018 it surpassed $1 billion in valuation. In 2020, Squirrel Ai Learning launched the $1 million AAAI Squirrel AI Award for Artificial Intelligence for the Benefit of Humanity in partnership with AAAI. The inaugural award was given to Regina Barzilay for her work developing machine learning models to address drug synthesis and early-stage breast cancer diagnosis. In 2020, Squirrel Ai Learning established strategic partnership with DingTalk, Alibaba Group. As of 2021, Squirrel Ai Learning had served over 60,000 public schools, in over 1200 cities in Asia. Squirrel Ai plans to start offering its services in the United States in 2026. The American arm is separate from the Chinese company to avoid regulatory hurdles. As of January 2026, it had set up an "independent technology platform" in the US. == Recognition == Squirrel Ai Learning has gained recognition both in Asia and internationally including: Squirrel Ai Learning was named one of the World's Top 30 AI application case in the 2018 Synced Machine Intelligence Awards. In June 2019, Squirrel Ai Learning was named as one of the 50 smartest companies in China by MIT technology review. Squirrel Ai Learning won the GITEX 2019 Best Education Technology Award. In 2020, Squirrel Ai Learning won the UNESCO AI Innovation Award. Squirrel Ai Learning was listed in the 2020 CB Insight's AI 100, CB Insights' annual ranking of the 100 most promising AI startups in the world. Squirrel Ai Learning won Edtech Review's Best AI in Education Company of the Year award 2020.

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  Construction of t-norms

  In mathematics, t-norms are a special kind of binary operations on the real unit interval [0, 1]. Various constructions of t-norms, either by explicit definition or by transformation from previously known functions, provide a plenitude of examples and classes of t-norms. This is important, e.g., for finding counter-examples or supplying t-norms with particular properties for use in engineering applications of fuzzy logic. The main ways of construction of t-norms include using generators, defining parametric classes of t-norms, rotations, or ordinal sums of t-norms. Relevant background can be found in the article on t-norms. == Generators of t-norms == The method of constructing t-norms by generators consists in using a unary function (generator) to transform some known binary function (most often, addition or multiplication) into a t-norm. In order to allow using non-bijective generators, which do not have the inverse function, the following notion of pseudo-inverse function is employed: Let f: [a, b] → [c, d] be a monotone function between two closed subintervals of extended real line. The pseudo-inverse function to f is the function f (−1): [c, d] → [a, b] defined as f ( − 1 ) ( y ) = { sup { x ∈ [ a , b ] ∣ f ( x ) < y } for f non-decreasing sup { x ∈ [ a , b ] ∣ f ( x ) > y } for f non-increasing. {\displaystyle f^{(-1)}(y)={\begin{cases}\sup\{x\in [a,b]\mid f(x)y\}&{\text{for }}f{\text{ non-increasing.}}\end{cases}}} === Additive generators === The construction of t-norms by additive generators is based on the following theorem: Let f: [0, 1] → [0, +∞] be a strictly decreasing function such that f(1) = 0 and f(x) + f(y) is in the range of f or in [f(0+), +∞] for all x, y in [0, 1]. Then the function T: [0, 1]2 → [0, 1] defined as T(x, y) = f (-1)(f(x) + f(y)) is a t-norm. Alternatively, one may avoid using the notion of pseudo-inverse function by having T ( x , y ) = f − 1 ( min ( f ( 0 + ) , f ( x ) + f ( y ) ) ) {\displaystyle T(x,y)=f^{-1}\left(\min \left(f(0^{+}),f(x)+f(y)\right)\right)} . The corresponding residuum can then be expressed as ( x ⇒ y ) = f − 1 ( max ( 0 , f ( y ) − f ( x ) ) ) {\displaystyle (x\Rightarrow y)=f^{-1}\left(\max \left(0,f(y)-f(x)\right)\right)} . And the biresiduum as ( x ⇔ y ) = f − 1 ( | f ( x ) − f ( y ) | ) {\displaystyle (x\Leftrightarrow y)=f^{-1}\left(\left|f(x)-f(y)\right|\right)} . If a t-norm T results from the latter construction by a function f which is right-continuous in 0, then f is called an additive generator of T. Examples: The function f(x) = 1 – x for x in [0, 1] is an additive generator of the Łukasiewicz t-norm. The function f defined as f(x) = –log(x) if 0 < x ≤ 1 and f(0) = +∞ is an additive generator of the product t-norm. The function f defined as f(x) = 2 – x if 0 ≤ x < 1 and f(1) = 0 is an additive generator of the drastic t-norm. Basic properties of additive generators are summarized by the following theorem: Let f: [0, 1] → [0, +∞] be an additive generator of a t-norm T. Then: T is an Archimedean t-norm. T is continuous if and only if f is continuous. T is strictly monotone if and only if f(0) = +∞. Each element of (0, 1) is a nilpotent element of T if and only if f(0) < +∞. The multiple of f by a positive constant is also an additive generator of T. T has no non-trivial idempotents. (Consequently, e.g., the minimum t-norm has no additive generator.) === Multiplicative generators === The isomorphism between addition on [0, +∞] and multiplication on [0, 1] by the logarithm and the exponential function allow two-way transformations between additive and multiplicative generators of a t-norm. If f is an additive generator of a t-norm T, then the function h: [0, 1] → [0, 1] defined as h(x) = e−f (x) is a multiplicative generator of T, that is, a function h such that h is strictly increasing h(1) = 1 h(x) · h(y) is in the range of h or equal to 0 or h(0+) for all x, y in [0, 1] h is right-continuous in 0 T(x, y) = h (−1)(h(x) · h(y)). Vice versa, if h is a multiplicative generator of T, then f: [0, 1] → [0, +∞] defined by f(x) = −log(h(x)) is an additive generator of T. == Parametric classes of t-norms == Many families of related t-norms can be defined by an explicit formula depending on a parameter p. This section lists the best known parameterized families of t-norms. The following definitions will be used in the list: A family of t-norms Tp parameterized by p is increasing if Tp(x, y) ≤ Tq(x, y) for all x, y in [0, 1] whenever p ≤ q (similarly for decreasing and strictly increasing or decreasing). A family of t-norms Tp is continuous with respect to the parameter p if lim p → p 0 T p = T p 0 {\displaystyle \lim _{p\to p_{0}}T_{p}=T_{p_{0}}} for all values p0 of the parameter. === Schweizer–Sklar t-norms === The family of Schweizer–Sklar t-norms, introduced by Berthold Schweizer and Abe Sklar in the early 1960s, is given by the parametric definition T p S S ( x , y ) = { T min ( x , y ) if p = − ∞ ( x p + y p − 1 ) 1 / p if − ∞ < p < 0 T p r o d ( x , y ) if p = 0 ( max ( 0 , x p + y p − 1 ) ) 1 / p if 0 < p < + ∞ T D ( x , y ) if p = + ∞ . {\displaystyle T_{p}^{\mathrm {SS} }(x,y)={\begin{cases}T_{\min }(x,y)&{\text{if }}p=-\infty \\(x^{p}+y^{p}-1)^{1/p}&{\text{if }}-\infty −∞ Continuous if and only if p < +∞ Strict if and only if −∞ < p ≤ 0 (for p = −1 it is the Hamacher product) Nilpotent if and only if 0 < p < +∞ (for p = 1 it is the Łukasiewicz t-norm). The family is strictly decreasing for p ≥ 0 and continuous with respect to p in [−∞, +∞]. An additive generator for T p S S {\displaystyle T_{p}^{\mathrm {SS} }} for −∞ < p < +∞ is f p S S ( x ) = { − log ⁡ x if p = 0 1 − x p p otherwise. {\displaystyle f_{p}^{\mathrm {SS} }(x)={\begin{cases}-\log x&{\text{if }}p=0\\{\frac {1-x^{p}}{p}}&{\text{otherwise.}}\end{cases}}} === Hamacher t-norms === The family of Hamacher t-norms, introduced by Horst Hamacher in the late 1970s, is given by the following parametric definition for 0 ≤ p ≤ +∞: T p H ( x , y ) = { T D ( x , y ) if p = + ∞ 0 if p = x = y = 0 x y p + ( 1 − p ) ( x + y − x y ) otherwise. {\displaystyle T_{p}^{\mathrm {H} }(x,y)={\begin{cases}T_{\mathrm {D} }(x,y)&{\text{if }}p=+\infty \\0&{\text{if }}p=x=y=0\\{\frac {xy}{p+(1-p)(x+y-xy)}}&{\text{otherwise.}}\end{cases}}} The t-norm T 0 H {\displaystyle T_{0}^{\mathrm {H} }} is called the Hamacher product. Hamacher t-norms are the only t-norms which are rational functions. The Hamacher t-norm T p H {\displaystyle T_{p}^{\mathrm {H} }} is strict if and only if p < +∞ (for p = 1 it is the product t-norm). The family is strictly decreasing and continuous with respect to p. An additive generator of T p H {\displaystyle T_{p}^{\mathrm {H} }} for p < +∞ is f p H ( x ) = { 1 − x x if p = 0 log ⁡ p + ( 1 − p ) x x otherwise. {\displaystyle f_{p}^{\mathrm {H} }(x)={\begin{cases}{\frac {1-x}{x}}&{\text{if }}p=0\\\log {\frac {p+(1-p)x}{x}}&{\text{otherwise.}}\end{cases}}} === Frank t-norms === The family of Frank t-norms, introduced by M.J. Frank in the late 1970s, is given by the parametric definition for 0 ≤ p ≤ +∞ as follows: T p F ( x , y ) = { T m i n ( x , y ) if p = 0 T p r o d ( x , y ) if p = 1 T L u k ( x , y ) if p = + ∞ log p ⁡ ( 1 + ( p x − 1 ) ( p y − 1 ) p − 1 ) otherwise. {\displaystyle T_{p}^{\mathrm {F} }(x,y)={\begin{cases}T_{\mathrm {min} }(x,y)&{\text{if }}p=0\\T_{\mathrm {prod} }(x,y)&{\text{if }}p=1\\T_{\mathrm {Luk} }(x,y)&{\text{if }}p=+\infty \\\log _{p}\left(1+{\frac {(p^{x}-1)(p^{y}-1)}{p-1}}\right)&{\text{otherwise.}}\end{cases}}} The Frank t-norm T p F {\displaystyle T_{p}^{\mathrm {F} }} is strict if p < +∞. The family is strictly decreasing and continuous with respect to p. An additive generator for T p F {\displaystyle T_{p}^{\mathrm {F} }} is f p F ( x ) = { − log ⁡ x if p = 1 1 − x if p = + ∞ log ⁡ p − 1 p x − 1 otherwise. {\displaystyle f_{p}^{\mathrm {F} }(x)={\begin{cases}-\log x&{\text{if }}p=1\\1-x&{\text{if }}p=+\infty \\\log {\frac {p-1}{p^{x}-1}}&{\text{otherwise.}}\end{cases}}} === Yager t-norms === The family of Yager t-norms, introduced in the early 1980s by Ronald R. Yager, is given for 0 ≤ p ≤ +∞ by T p Y ( x , y ) = { T D ( x , y ) if p = 0 max ( 0 , 1 − ( ( 1 − x ) p + ( 1 − y ) p ) 1 / p ) if 0 < p < + ∞ T m i n ( x , y ) if p = + ∞ {\displaystyle T_{p}^{\mathrm {Y} }(x,y)={\begin{cases}T_{\mathrm {D} }(x,y)&{\text{if }}p=0\\\max \left(0,1-((1-x)^{p}+(1-y)^{p})^{1/p}\right)&{\text{if }}0 Read more →