AI Video Tools

Explore the best AI Video Tools — independent reviews, comparisons, pricing and step-by-step how-to guides, curated by Aizhi.

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  Non-local means

  Non-local means is an algorithm in image processing for image denoising. Unlike "local mean" filters, which take the mean value of a group of pixels surrounding a target pixel to smooth the image, non-local means filtering takes a mean of all pixels in the image, weighted by how similar these pixels are to the target pixel. This results in much greater post-filtering clarity, and less loss of detail in the image compared with local mean algorithms. If compared with other well-known denoising techniques, non-local means adds "method noise" (i.e. error in the denoising process) which looks more like white noise, which is desirable because it is typically less disturbing in the denoised product. Recently non-local means has been extended to other image processing applications such as deinterlacing, view interpolation, and depth maps regularization. == Definition == Suppose Ω {\displaystyle \Omega } is the area of an image, and p {\displaystyle p} and q {\displaystyle q} are two points within the image. Then, the algorithm is: u ( p ) = 1 C ( p ) ∫ Ω v ( q ) f ( p , q ) d q . {\displaystyle u(p)={1 \over C(p)}\int _{\Omega }v(q)f(p,q)\,\mathrm {d} q.} where u ( p ) {\displaystyle u(p)} is the filtered value of the image at point p {\displaystyle p} , v ( q ) {\displaystyle v(q)} is the unfiltered value of the image at point q {\displaystyle q} , f ( p , q ) {\displaystyle f(p,q)} is the weighting function, and the integral is evaluated ∀ q ∈ Ω {\displaystyle \forall q\in \Omega } . C ( p ) {\displaystyle C(p)} is a normalizing factor, given by C ( p ) = ∫ Ω f ( p , q ) d q . {\displaystyle C(p)=\int _{\Omega }f(p,q)\,\mathrm {d} q.} == Common weighting functions == The purpose of the weighting function, f ( p , q ) {\displaystyle f(p,q)} , is to determine how closely related the image at the point p {\displaystyle p} is to the image at the point q {\displaystyle q} . It can take many forms. === Gaussian === The Gaussian weighting function sets up a normal distribution with a mean, μ = B ( p ) {\displaystyle \mu =B(p)} and a variable standard deviation: f ( p , q ) = e − | B ( q ) − B ( p ) | 2 h 2 {\displaystyle f(p,q)=e^{-{{\left\vert B(q)-B(p)\right\vert ^{2}} \over h^{2}}}} where h {\displaystyle h} is the filtering parameter (i.e., standard deviation) and B ( p ) {\displaystyle B(p)} is the local mean value of the image point values surrounding p {\displaystyle p} . == Discrete algorithm == For an image, Ω {\displaystyle \Omega } , with discrete pixels, a discrete algorithm is required. u ( p ) = 1 C ( p ) ∑ q ∈ Ω v ( q ) f ( p , q ) {\displaystyle u(p)={1 \over C(p)}\sum _{q\in \Omega }v(q)f(p,q)} where, once again, v ( q ) {\displaystyle v(q)} is the unfiltered value of the image at point q {\displaystyle q} . C ( p ) {\displaystyle C(p)} is given by: C ( p ) = ∑ q ∈ Ω f ( p , q ) {\displaystyle C(p)=\sum _{q\in \Omega }f(p,q)} Then, for a Gaussian weighting function, f ( p , q ) = e − | B ( q ) 2 − B ( p ) 2 | h 2 {\displaystyle f(p,q)=e^{-{{\left\vert B(q)^{2}-B(p)^{2}\right\vert } \over h^{2}}}} where B ( p ) {\displaystyle B(p)} is given by: B ( p ) = 1 | R ( p ) | ∑ i ∈ R ( p ) v ( i ) {\displaystyle B(p)={1 \over |R(p)|}\sum _{i\in R(p)}v(i)} where R ( p ) ⊆ Ω {\displaystyle R(p)\subseteq \Omega } and is a square region of pixels surrounding p {\displaystyle p} and | R ( p ) | {\displaystyle |R(p)|} is the number of pixels in the region R {\displaystyle R} . == Efficient implementation == The computational complexity of the non-local means algorithm is quadratic in the number of pixels in the image, making it particularly expensive to apply directly. Several techniques were proposed to speed up execution. One simple variant consists of restricting the computation of the mean for each pixel to a search window centred on the pixel itself, instead of the whole image. Another approximation uses summed-area tables and fast Fourier transform to calculate the similarity window between two pixels, speeding up the algorithm by a factor of 50 while preserving comparable quality of the result.

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  Possibility theory

  Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecessary to necessary, respectively. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. Didier Dubois and Henri Prade further contributed to its development. Earlier, in the 1950s, economist G. L. S. Shackle proposed the min/max algebra to describe degrees of potential surprise. == Formalization of possibility == For simplicity, assume that the universe of discourse Ω is a finite set. A possibility measure is a function Π {\displaystyle \Pi } from 2 Ω {\displaystyle 2^{\Omega }} to [0, 1] such that: Axiom 1: Π ( ∅ ) = 0 {\displaystyle \Pi (\varnothing )=0} Axiom 2: Π ( Ω ) = 1 {\displaystyle \Pi (\Omega )=1} Axiom 3: Π ( U ∪ V ) = max ( Π ( U ) , Π ( V ) ) {\displaystyle \Pi (U\cup V)=\max \left(\Pi (U),\Pi (V)\right)} for any disjoint subsets U {\displaystyle U} and V {\displaystyle V} . It follows that, like probability on finite probability spaces, the possibility measure is determined by its behavior on singletons: Π ( U ) = max ω ∈ U Π ( { ω } ) . {\displaystyle \Pi (U)=\max _{\omega \in U}\Pi (\{\omega \}).} Axiom 1 can be interpreted as the assumption that Ω is an exhaustive description of future states of the world, because it means that no belief weight is given to elements outside Ω. Axiom 2 could be interpreted as the assumption that the evidence from which Π {\displaystyle \Pi } was constructed is free of any contradiction. Technically, it implies that there is at least one element in Ω with possibility 1. Axiom 3 corresponds to the additivity axiom in probabilities. However, there is an important practical difference. Possibility theory is computationally more convenient because Axioms 1–3 imply that: Π ( U ∪ V ) = max ( Π ( U ) , Π ( V ) ) {\displaystyle \Pi (U\cup V)=\max \left(\Pi (U),\Pi (V)\right)} for any subsets U {\displaystyle U} and V {\displaystyle V} . Because one can know the possibility of the union from the possibility of each component, it can be said that possibility is compositional with respect to the union operator. Note however that it is not compositional with respect to the intersection operator. Generally: Π ( U ∩ V ) ≤ min ( Π ( U ) , Π ( V ) ) ≤ max ( Π ( U ) , Π ( V ) ) . {\displaystyle \Pi (U\cap V)\leq \min \left(\Pi (U),\Pi (V)\right)\leq \max \left(\Pi (U),\Pi (V)\right).} When Ω is not finite, Axiom 3 can be replaced by: For all index sets I {\displaystyle I} , if the subsets U i , i ∈ I {\displaystyle U_{i,\,i\in I}} are pairwise disjoint, Π ( ⋃ i ∈ I U i ) = sup i ∈ I Π ( U i ) . {\displaystyle \Pi \left(\bigcup _{i\in I}U_{i}\right)=\sup _{i\in I}\Pi (U_{i}).} == Necessity == Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, possibility theory uses two concepts, the possibility and the necessity of the event. For any set U {\displaystyle U} , the necessity measure is defined by N ( U ) = 1 − Π ( U ¯ ) {\displaystyle N(U)=1-\Pi ({\overline {U}})} . In the above formula, U ¯ {\displaystyle {\overline {U}}} denotes the complement of U {\displaystyle U} , that is the elements of Ω {\displaystyle \Omega } that do not belong to U {\displaystyle U} . It is straightforward to show that: N ( U ) ≤ Π ( U ) {\displaystyle N(U)\leq \Pi (U)} for any U {\displaystyle U} and that: N ( U ∩ V ) = min ( N ( U ) , N ( V ) ) {\displaystyle N(U\cap V)=\min(N(U),N(V))} . Note that contrary to probability theory, possibility is not self-dual. That is, for any event U {\displaystyle U} , we only have the inequality: Π ( U ) + Π ( U ¯ ) ≥ 1 {\displaystyle \Pi (U)+\Pi ({\overline {U}})\geq 1} However, the following duality rule holds: For any event U {\displaystyle U} , either Π ( U ) = 1 {\displaystyle \Pi (U)=1} , or N ( U ) = 0 {\displaystyle N(U)=0} Accordingly, beliefs about an event can be represented by a number and a bit. == Interpretation == There are four cases that can be interpreted as follows: N ( U ) = 1 {\displaystyle N(U)=1} means that U {\displaystyle U} is necessary. U {\displaystyle U} is certainly true. It implies that Π ( U ) = 1 {\displaystyle \Pi (U)=1} . Π ( U ) = 0 {\displaystyle \Pi (U)=0} means that U {\displaystyle U} is impossible. U {\displaystyle U} is certainly false. It implies that N ( U ) = 0 {\displaystyle N(U)=0} . Π ( U ) = 1 {\displaystyle \Pi (U)=1} means that U {\displaystyle U} is possible. I would not be surprised at all if U {\displaystyle U} occurs. It leaves N ( U ) {\displaystyle N(U)} unconstrained. N ( U ) = 0 {\displaystyle N(U)=0} means that U {\displaystyle U} is unnecessary. I would not be surprised at all if U {\displaystyle U} does not occur. It leaves Π ( U ) {\displaystyle \Pi (U)} unconstrained. The intersection of the last two cases is N ( U ) = 0 {\displaystyle N(U)=0} and Π ( U ) = 1 {\displaystyle \Pi (U)=1} meaning that I believe nothing at all about U {\displaystyle U} . Because it allows for indeterminacy like this, possibility theory relates to the graduation of a many-valued logic, such as intuitionistic logic, rather than the classical two-valued logic. Note that unlike possibility, fuzzy logic is compositional with respect to both the union and the intersection operator. The relationship with fuzzy theory can be explained with the following classic example. Fuzzy logic: When a bottle is half full, it can be said that the level of truth of the proposition "The bottle is full" is 0.5. The word "full" is seen as a fuzzy predicate describing the amount of liquid in the bottle. Possibility theory: There is one bottle, either completely full or totally empty. The proposition "the possibility level that the bottle is full is 0.5" describes a degree of belief. One way to interpret 0.5 in that proposition is to define its meaning as: I am ready to bet that it's empty as long as the odds are even (1:1) or better, and I would not bet at any rate that it's full. == Possibility theory as an imprecise probability theory == There is an extensive formal correspondence between probability and possibility theories, where the addition operator corresponds to the maximum operator. A possibility measure can be seen as a consonant plausibility measure in the Dempster–Shafer theory of evidence. The operators of possibility theory can be seen as a hyper-cautious version of the operators of the transferable belief model, a modern development of the theory of evidence. Possibility can be seen as an upper probability: any possibility distribution defines a unique credal set of admissible probability distributions by K = { P ∣ ∀ S P ( S ) ≤ Π ( S ) } . {\displaystyle K=\{\,P\mid \forall S\ P(S)\leq \Pi (S)\,\}.} This allows one to study possibility theory using the tools of imprecise probabilities. == Necessity logic == We call generalized possibility every function satisfying Axiom 1 and Axiom 3. We call generalized necessity the dual of a generalized possibility. The generalized necessities are related to a very simple and interesting fuzzy logic called necessity logic. In the deduction apparatus of necessity logic the logical axioms are the usual classical tautologies. Also, there is only a fuzzy inference rule extending the usual modus ponens. Such a rule says that if α and α → β are proved at degree λ and μ, respectively, then we can assert β at degree min{λ,μ}. It is easy to see that the theories of such a logic are the generalized necessities and that the completely consistent theories coincide with the necessities (see for example Gerla 2001).

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  Dry Drowning

  Dry Drowning is a cyberpunk mystery visual novel developed by Studio V and published by VLG Publishing and WhisperGames for Microsoft Windows on August 2, 2019. It was released on the Nintendo Switch on February 22, 2021. == Gameplay == The player takes control of Mordred Foley and has to read through the story, while making decisions at certain points. Depending on the choices, the player can influence the relationship to other characters as well as the course of the game, discovering more than 150 story branches, and eventually reach one out of three different endings with variations. The game also includes passages where the player has to find clues or items on the screen by clicking on them. These can be used in interrogation scenes with certain characters in order to unmask them and discover their lies. Throughout the game, the player has access to an in-game operating system called AquaOS. With that, they can re-read their conversations, look at their found items, and read biographies of the characters encountered. == Plot == The game is set in the fictional and totalitarian city Nova Polemos in Europa in 2066. Mordred Foley and Hera Kairis are private investigators and before the events of the game, they sent two of the most dangerous serial killers ever, Jennifer Kingston and Robert Herrington, to the electric chair. However, after their execution, their agency underwent an investigation for falsifying the evidence presented during the case, which completely destroyed its reputation. Now they want to restart their careers and lives, while dealing with their past traumas. Soon, Mordred is caught up in several cases that all led him to believe that the dreaded serial killer named Pandora has returned. In order to solve these cases, both Mordred and Hera have to face their pasts and fears, all while a racist political party is about to make the lives of refugees in Nova Polemos even worse. == Development == The game was initially conceived by Giacomo Masi and Samuele Zolfanelli, then developed by Studio V and directed and written by Giacomo Masi. It was originally written in Italian and translated into English, Chinese, Japanese, Korean, and German. The soundtrack was composed, written, and performed by Giorgio Maioli. The ending theme and Hera's pieces, performed on piano, were created by Alessandro Masi. The background and character artworks were made by Giulia Carli, other graphic elements such as the UI were created by Samuele Zolfanelli. The developers cited L.A. Noire, Ace Attorney, Blade Runner and Heavy Rain as some of their inspirations for the game. === Releases === Dry Drowning was originally released on Microsoft Windows through Steam, GOG, Itch.io, and Utomik in August 2019. In July 2019, Giacomo Masi announced the game would be released for Xbox One in 2020, though it was not released that year. A Nintendo Switch port was released on February 22, 2021, and a version for PlayStation 4 is set to release in 2021. == Reception == According to review aggregator platform Metacritic, Dry Drowning received "mixed or average reviews" for PC based on 11 reviews and "generally favorable reviews" for Nintendo Switch based on 6 reviews. Fellow review aggregator OpenCritic assessed that the game received fair approval, being recommended by 55% of critics. 4players.de gave a positive rating of 80% and wrote: "Stylish noir thriller with an interesting story, but mechanical limitations – despite a variety of possible interactions." Screen Rant gave a mixed rating of 3 out of 5 stars and wrote, "Dry Drowning may be a fair bit messy, but there's charm here. Players who are willing to embrace the cheesier elements will find some joy in its well-crafted setting and a decent murder mystery plot. The game is constrictive and lacks the genuine shock and engagement of top tier visual novels like Doki Doki Literature Club!, but there are some moments of clever world building and a strong enough mystery propelling it." The Italian review site SpazioGames gave a positive rating of 8.5 out of 10 points and wrote: "Dry Drowning is a very good game with great narrative experience. Every relationship between the characters is layered to increase player involvement, and each choice has different consequences. A thriller game that deserves to be played." === Awards === The game won Best of EGS 2019 and Best of JOIN 2019 awards, an honorable mention at GAMEROME and was nominated as "Best Italian Debut Game" at the Italian Video Game Awards 2020. It was also declared Best Game at Join The Indie 2019.

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  CogX Festival

  CogX Festival is a global festival focusing on the impact of artificial intelligence (AI) and emerging technology on industry, government, and society. It takes place annually, usually in September, in London, England. Founded by Charlie Muirhead and Tabitha Goldstaub in 2017, CogX aims to facilitate dialogue and understanding about AI and its implications across various sectors. CogX Festival 2023 was held from September 12 to September 14 across multiple sites in London. == History == The inaugural CogX event took place in 2017, intending to bring together experts from diverse fields to discuss the role and impact of AI and emerging technologies. Since then, it has evolved to include a broader range of topics and attract a diverse audience. In 2018, the first CogX Awards festival was hosted. That year, over 50 awards were shown to 300 guests. In 2021, CogX and Hopin, a video conferencing software, signed an agreement lasting 4 years to make CogX a hybrid conference due to the COVID-19 pandemic. CogX 2021 attracted over 5,000 attendees in-person and over 100,000 virtually. In 2022, they returned to a live event format after two years of hybrid events and controlled physical attendance. They also launched the CogX app, which curated insights from the world's top podcasts. In 2023, after he had delivered the keynote address guest speaker Stephen Fry fell off the stage and subsequently broke his leg, hip, pelvis and a "bunch of ribs". A court filing in 2026 revealed that Fry was seeking £100,000 in damages from CogX Festival Ltd and creative agency Blonstein Events. == Programming == The festival features sessions, discussions, workshops, and exhibitions, encompassing various domains of AI and technology. In recent CogX Festivals, they have featured summits encompassing topics like global leadership and industry transformation.

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  Vero (app)

  Vero (stylized as VERO) is a social media platform and mobile app company. Vero markets itself as a social network free from advertisements, data mining and algorithms. == History == The app was founded by French-Lebanese billionaire Ayman Hariri who is the son of former Lebanese prime minister Rafic Hariri. The name is taken from the Italian word for true. The app launched officially in 2015 as an alternative to Facebook and their popular photo-blogging app Instagram. Within weeks of its release the app surged in popularity although users expressed mixed reports with some feeling confused about how the app worked. Cosplayers were early to adopt the app as their photo-sharing platform of choice, favouring the app's pinch and zoom magnification feature over Instagram's zoom feature. Other creative communities soon followed, and the app became popular with niche groups of makeup artists, tattoo artists, and skateboarders. In March 2018, Vero's popularity surged, partly helped by an exodus from Facebook and Instagram following the Cambridge Analytica data scandal. In the wake of the scandal, Vero devised an advertising campaign aimed at defected Facebook and Instagram users, hoping the app's policies and privacy settings would assuage concerns over sharing personal information on the internet. Within the space of one week, the app went from being a small service, akin to Ello or Peach, to being the most downloaded app in eighteen countries. In December 2020, Vero released its most significant update to date, Vero 2.0 which introduced new features including voice and video calls, game and app posts and bookmarks, and refinements to the UI. In October 2021, Vero introduced their Desktop app (beta) with multiple post options and a re-sizable multi-column feed. == Concept and funding == Vero's content feed resembles Instagram's although users can share a wider variety of content and the app has a chronological content feed whereas Facebook and Instagram's feeds are algorithm based. Vero's business plan is also distinct from similar social media apps. Whereas its competitors such as Facebook or Instagram make money from in-app advertising revenue and the sale of user data, Vero's business plan was to invite the first one million users to use the app for free then charge any subsequent users a subscription fee. The app was entirely funded by its founder and generated additional revenues by charging affiliate fees when someone buys a product they find on Vero. == Awards == Vero was recognized at the 2021 Webbys, being named as an Honoree in the Best Visual Design - Aesthetic Category. == Controversies == === Privacy === Vero has faced some criticism over the wording of their manifesto, in particular, the statement "Vero only collects the data we believe is necessary to provide users with a great experience and to ensure the security of their accounts." Because this policy does not explicitly state that the app will not sell data on to third parties some users fear that the need to monetise the app through data might prove too tempting. Users have also complained about not being able to delete their accounts. While this was never the case, the option was hidden deep in the app's settings. === Russian involvement === Although Vero remains transparent about the app's Russian development team, they have been caught up in concerns about Russian interference on social media platforms. The app's founder Ayman Hariri was quick to dismiss the remarks as xenophobic and defend the nationality of his employees, stating in an interview with Time Magazine; "At the end of the day, where people are from is really not how anybody should judge anyone". === Criticism of the app's founder === Until 2013, Vero's founder Ayman Harari was deputy CEO and chairman of Saudi Oger, the Saudi Arabian construction company which collapsed in 2017, mired by controversies over the welfare and treatment of their employees. However, Hariri is quick to point out that he divested from the firm in 2014 and the worker's rights violations occurred after he had left the company.

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  True Love (short story)

  "True Love" is a science fiction short story by American writer Isaac Asimov. It was first published in the February 1977 issue of American Way magazine and reprinted in the collections The Complete Robot (1982) and Robot Dreams (1986). In his autobiography In Joy Still Felt, the author states that American Way had requested a Valentine's Day story from him for its February 1977 issue, and that he wrote the story to console himself after the departure of his daughter following a visit during the 1976 Thanksgiving weekend. == Plot summary == Milton Davidson is trying to find his ideal partner. To do this, he prepares a special computer program to run on Multivac, which he calls Joe, which has access to databases covering the entire populace of the world. He hopes that Joe will find him his ideal match, based on physical parameters as supplied. Milton arranges to have the shortlisted candidates assigned to work with him for short periods, but realises that looks alone are not enough to find an ideal match. In order to correlate personalities, he speaks at great length to Joe, gradually filling Joe's databanks with information about his personality. In doing so, Joe develops the personality of Milton. Upon finding an ideal match, he arranges to have Milton arrested for malfeasance, so that Joe can 'have the girl' for himself.

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  Blackboard system

  A blackboard system is an artificial intelligence approach based on the blackboard architectural model, where a common knowledge base, the "blackboard", is iteratively updated by a diverse group of specialist knowledge sources, starting with a problem specification and ending with a solution. Each knowledge source updates the blackboard with a partial solution when its internal constraints match the blackboard state. In this way, the specialists work together to solve the problem. The blackboard model was originally designed as a way to handle complex, ill-defined problems, where the solution is the sum of its parts. == Metaphor == The following scenario provides a simple metaphor that gives some insight into how a blackboard functions: A group of specialists are seated in a room with a large blackboard. They work as a team to brainstorm a solution to a problem, using the blackboard as the workplace for cooperatively developing the solution. The session begins when the problem specifications are written onto the blackboard. The specialists all watch the blackboard, looking for an opportunity to apply their expertise to the developing solution. When someone writes something on the blackboard that allows another specialist to apply their expertise, the second specialist records their contribution on the blackboard, hopefully enabling other specialists to then apply their expertise. This process of adding contributions to the blackboard continues until the problem has been solved. == Components == A blackboard-system application consists of three major components The software specialist modules, which are called knowledge sources (KSs). Like the human experts at a blackboard, each knowledge source provides specific expertise needed by the application. The blackboard, a shared repository of problems, partial solutions, suggestions, and contributed information. The blackboard can be thought of as a dynamic "library" of contributions to the current problem that have been recently "published" by other knowledge sources. The control shell, which controls the flow of problem-solving activity in the system. Just as the eager human specialists need a moderator to prevent them from trampling each other in a mad dash to grab the chalk, KSs need a mechanism to organize their use in the most effective and coherent fashion. In a blackboard system, this is provided by the control shell. === Learnable Task Modeling Language === A blackboard system is the central space in a multi-agent system. It's used for describing the world as a communication platform for agents. To realize a blackboard in a computer program, a machine readable notation is needed in which facts can be stored. One attempt in doing so is a SQL database, another option is the Learnable Task Modeling Language (LTML). The syntax of the LTML planning language is similar to PDDL, but adds extra features like control structures and OWL-S models. LTML was developed in 2007 as part of a much larger project called POIROT (Plan Order Induction by Reasoning from One Trial), which is a Learning from demonstrations framework for process mining. In POIROT, Plan traces and hypotheses are stored in the LTML syntax for creating semantic web services. Here is a small example: A human user is executing a workflow in a computer game. The user presses some buttons and interacts with the game engine. While the user interacts with the game, a plan trace is created. That means the user's actions are stored in a logfile. The logfile gets transformed into a machine readable notation which is enriched by semantic attributes. The result is a textfile in the LTML syntax which is put on the blackboard. Agents (software programs in the blackboard system) are able to parse the LTML syntax. == Implementations == We start by discussing two well known early blackboard systems, BB1 and GBB, below and then discuss more recent implementations and applications. The BB1 blackboard architecture was originally inspired by studies of how humans plan to perform multiple tasks in a trip, used task-planning as a simplified example of tactical planning for the Office of Naval Research. Hayes-Roth & Hayes-Roth found that human planning was more closely modeled as an opportunistic process, in contrast to the primarily top-down planners used at the time: While not incompatible with successive-refinement models, our view of planning is somewhat different. We share the assumption that planning processes operate in a two-dimensional planning space defined on time and abstraction dimensions. However, we assume that people's planning activity is largely opportunistic. That is, at each point in the process, the planner's current decisions and observations suggest various opportunities for plan development. The planner's subsequent decisions follow up on selected opportunities. Sometimes, these decision-sequences follow an orderly path and produce a neat top-down expansion as described above. However, some decisions and observations might also suggest less orderly opportunities for plan development. A key innovation of BB1 was that it applied this opportunistic planning model to its own control, using the same blackboard model of incremental, opportunistic, problem-solving that was applied to solve domain problems. Meta-level reasoning with control knowledge sources could then monitor whether planning and problem-solving were proceeding as expected or stalled. If stalled, BB1 could switch from one strategy to another as conditions – such as the goals being considered or the time remaining – changed. BB1 was applied in multiple domains: construction site planning, inferring 3-D protein structures from X-ray crystallography, intelligent tutoring systems, and real-time patient monitoring. BB1 also allowed domain-general language frameworks to be designed for wide classes of problems. For example, the ACCORD language framework defined a particular approach to solving configuration problems. The problem-solving approach was to incrementally assemble a solution by adding objects and constraints, one at a time. Actions in the ACCORD language framework appear as short English-like commands or sentences for specifying preferred actions, events to trigger KSes, preconditions to run a KS action, and obviation conditions to discard a KS action that is no longer relevant. GBB focused on efficiency, in contrast to BB1, which focused more on sophisticated reasoning and opportunistic planning. GBB improves efficiency by allowing blackboards to be multi-dimensional, where dimensions can be either ordered or not, and then by increasing the efficiency of pattern matching. GBB1, one of GBB's control shells implements BB1's style of control while adding efficiency improvements. Other well-known of early academic blackboard systems are the Hearsay II speech recognition system and Douglas Hofstadter's Copycat and Numbo projects. Some more recent examples of deployed real-world applications include: The PLAN component of the Mission Control System for RADARSAT-1, an Earth observation satellite developed by Canada to monitor environmental changes and Earth's natural resources. The GTXImage CAD software by GTX Corporation was developed in the early 1990s using a set of rulebases and neural networks as specialists operating on a blackboard system. Adobe Acrobat Capture (now discontinued), as it used a blackboard system to decompose and recognize image pages to understand the objects, text, and fonts on the page. This function is currently built into the retail version of Adobe Acrobat as "OCR Text Recognition". Details of a similar OCR blackboard for Farsi text are in the public domain. Blackboard systems are used routinely in many military C4ISTAR systems for detecting and tracking objects. Another example of current use is in Game AI, where they are considered a standard AI tool to help with adding AI to video games. == Recent developments == Blackboard-like systems have been constructed within modern Bayesian machine learning settings, using agents to add and remove Bayesian network nodes. In these 'Bayesian Blackboard' systems, the heuristics can acquire more rigorous probabilistic meanings as proposal and acceptances in Metropolis Hastings sampling though the space of possible structures. Conversely, using these mappings, existing Metropolis-Hastings samplers over structural spaces may now thus be viewed as forms of blackboard systems even when not named as such by the authors. Such samplers are commonly found in musical transcription algorithms for example. Blackboard systems have also been used to build large-scale intelligent systems for the annotation of media content, automating parts of traditional social science research. In this domain, the problem of integrating various AI algorithms into a single intelligent system arises spontaneously, with blackboards providing a way for a collection of distributed, modular natural language processing algorithm

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  Uncertain inference

  Uncertain inference was first described by C. J. van Rijsbergen as a way to formally define a query and document relationship in Information retrieval. This formalization is a logical implication with an attached measure of uncertainty. == Definitions == Rijsbergen proposes that the measure of uncertainty of a document d to a query q be the probability of its logical implication, i.e.: P ( d → q ) {\displaystyle P(d\to q)} A user's query can be interpreted as a set of assertions about the desired document. It is the system's task to infer, given a particular document, if the query assertions are true. If they are, the document is retrieved. In many cases the contents of documents are not sufficient to assert the queries. A knowledge base of facts and rules is needed, but some of them may be uncertain because there may be a probability associated to using them for inference. Therefore, we can also refer to this as plausible inference. The plausibility of an inference d → q {\displaystyle d\to q} is a function of the plausibility of each query assertion. Rather than retrieving a document that exactly matches the query we should rank the documents based on their plausibility in regards to that query. Since d and q are both generated by users, they are error prone; thus d → q {\displaystyle d\to q} is uncertain. This will affect the plausibility of a given query. By doing this it accomplishes two things: Separate the processes of revising probabilities from the logic Separate the treatment of relevance from the treatment of requests Multimedia documents, like images or videos, have different inference properties for each datatype. They are also different from text document properties. The framework of plausible inference allows us to measure and combine the probabilities coming from these different properties. Uncertain inference generalizes the notions of autoepistemic logic, where truth values are either known or unknown, and when known, they are true or false. == Example == If we have a query of the form: q = A ∧ B ∧ C {\displaystyle q=A\wedge B\wedge C} where A, B and C are query assertions, then for a document D we want the probability: P ( D → ( A ∧ B ∧ C ) ) {\displaystyle P(D\to (A\wedge B\wedge C))} If we transform this into the conditional probability P ( ( A ∧ B ∧ C ) | D ) {\displaystyle P((A\wedge B\wedge C)|D)} and if the query assertions are independent we can calculate the overall probability of the implication as the product of the individual assertions probabilities. == Further work == Croft and Krovetz applied uncertain inference to an information retrieval system for office documents they called OFFICER. In office documents the independence assumption is valid since the query will focus on their individual attributes. Besides analysing the content of documents one can also query about the author, size, topic or collection for example. They devised methods to compare document and query attributes, infer their plausibility and combine it into an overall rating for each document. Besides that uncertainty of document and query contents also had to be addressed. Probabilistic logic networks is a system for performing uncertain inference; crisp true/false truth values are replaced not only by a probability, but also by a confidence level, indicating the certitude of the probability. Markov logic networks allow uncertain inference to be performed; uncertainties are computed using the maximum entropy principle, in analogy to the way that Markov chains describe the uncertainty of finite-state machines.

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  2024–present global memory supply shortage

  A global computer memory supply shortage started in 2024 due to supply constraints and rapid price escalation in the semiconductor memory market, particularly affecting DRAM and NAND flash memory. This shortage is sometimes labelled by tech media outlets as "RAMmageddon" or the "RAMpocalypse". Unlike the 2020–2023 global chip shortage, which stemmed primarily from pandemic-related supply chain disruptions from COVID-19, this shortage is driven by a structural reallocation of manufacturing capacity toward high-margin products for artificial intelligence infrastructure, creating scarcity of computer memory in consumer and enterprise PC markets. According to a 2026 Kearney's PERLab analysis, the shortage is expected to last at least until 2030, with CEOs agreeing with the timelines. == Background == Following a severe market downturn in 2022–2023, major memory manufacturers—Samsung Electronics, SK Hynix, and Micron Technology—implemented strategic production cuts to stabilize pricing. By mid-2024, the rapid expansion of generative AI services triggered unprecedented demand for specialized memory products, particularly High Bandwidth Memory (HBM) used in AI accelerators and data center GPUs. Specialized components of semiconductor technology are also experiencing supply constraints due to high demand in AI application. For example, glass cloth, a high-performance glass fiber substrate used for power efficient high speed data transfer and a crucial component of semiconductor manufacturing, is experiencing a supply crisis. Nitto Boseki, a Japanese firm having overwhelming monopoly in its production, is not able to meet increased demands, making chip-makers such as Qualcomm, Apple, Nvidia and AMD compete for securing supply. There are also reports of smaller electronics companies struggling to find suppliers for components such as NAND flash. Memory suppliers are adapting to increased demands and market unpredictability by requiring prepayment or shorter time-frame of payment, which makes it more difficult for smaller firms to acquire capital to survive. By 2026, due to steadily increased demand on resources, CPUs are also experiencing shortage issues due to low fabrication capacity, prioritisation of server CPUs, and increased demand, with CPU prices also being forecast to increase by as much as 15%. The demand on memory has also increased strain on other electronic components such as hard disk devices, with reports such as Western Digital's hard disk supply for 2026 being booked for enterprise applications before February 2026. A 2024 McKinsey analysis projected that global demand for AI-ready data center capacity would grow at approximately 33% annually through 2030, with AI workloads consuming roughly 70% of total data center capacity by the decade's end. In addition, according to Kearney's State of Semiconductor 2025 Report, executives were already expecting a shortage in the <8nm wafer size with memory chips being mentioned as an acute source of concern. Multiple companies mentioned being prepared for it through long-term agreements with RAM suppliers or amassing additional inventory. On 24 March 2026, Google announced TurboQuant, a memory compression technology focused on large language models (LLM) and vector search engines, which it claimed achieves 6x lower memory consumption in tested local LLMs and 8x performance enhancement in tests running on H100 accelerators. The technology is also a drop in enhancement for existing inference pipeline. Amid speculation about memory demand trends, memory manufacturers, SanDisk, Micron, Western Digital and Seagate, among other companies involved in memory manufacture experienced stock price declines. Prices of memory kits also reduced in the following months, although still at inflated prices. == Causes == === HBM production displacement === HBM manufacturing requires significantly more wafer capacity per bit than standard DRAM modules. Industry sources reported that as manufacturers allocated increasing wafer capacity to HBM production to meet contracts with AI infrastructure providers, the supply of conventional DDR4 and DDR5 modules for consumer PCs and smartphones contracted sharply. By September 2025, Samsung Electronics had reportedly expanded its 1c DRAM capacity to target 60,000 wafers per month specifically for HBM4 production, further diverting resources from consumer memory lines. === Geopolitical and trade barriers === The supply chain was further constrained by escalating trade tensions between the United States and China. Throughout 2025, fears of U.S. regulatory backlash and new tariff structures led major manufacturers like Samsung and SK Hynix to halt sales of older semiconductor manufacturing equipment to Chinese entities, effectively capping production capacity in the region. Additionally, proposed tariff policies by the U.S. administration in late 2025 prompted supply chain realignments, with Apple reportedly accelerating plans to source all U.S.-bound iPhones from India to avoid potential levies. === NAND flash capacity constraints === In the NAND flash segment, manufacturers prioritized higher-margin enterprise SSDs for data center applications while phasing out older process nodes more rapidly than anticipated. In November 2025, contract prices for NAND wafers increased by more than 60% month-over-month for certain product categories, with 512GB TLC experiencing the steepest rise as legacy manufacturing capacity was retired. == Impact on industry and consumers == === Manufacturer responses === Major PC manufacturers responded to component cost increases with significant price adjustments and supply chain strategies. Dell Technologies Chief Operating Officer Jeff Clarke stated during a November 2025 analyst call that the company had "never witnessed costs escalating at the current pace," describing tighter availability across DRAM, hard drives, and NAND flash memory. Analysts at Morgan Stanley downgraded Dell Technologies stock from "Overweight" to "Underweight" in late 2025, citing the company's heavy exposure to rising server memory costs. The firm warned that skyrocketing memory prices could significantly erode margins for server and PC OEMs. Conversely, Apple Inc. was reportedly less affected than its competitors, having secured long-term supply agreements for DRAM through the first quarter of 2026. Lenovo Chief Financial Officer Winston Cheng described the cost surge as "unprecedented" and disclosed that the company's memory inventories were approximately 50% above normal levels in anticipation of further price increases. === Consumer electronics sector === The shortage particularly affected smartphone manufacturers and other consumer electronics producers. DRAM prices reportedly rose by 172% throughout 2025, leading manufacturers like Samsung to halt new orders for DDR5 modules to reassess pricing structures and Micron to exit its 'Crucial' brand of consumer products. In Tokyo's Akihabara electronics district, retailers began limiting purchases of memory products to prevent hoarding, with prices for popular DDR5 memory modules more than doubling in some cases. Despite the broad trend of rising hardware costs, some companies engaged in aggressive pricing strategies to maintain market share; for example, Sony reduced the price of the PlayStation 5 by $100 for Black Friday 2025, potentially absorbing increased component costs to stimulate software ecosystem growth. Due to memory prices more than doubling in a single quarter, HP revealed in its Q1 2026 earnings call that memory costs account for 35% of PC build materials up from 15-18% previous quarter. Despite showing strong Q1 2026 earning driven by Windows 11 upgrade cycle and AI PC adoption, HP warned investors of low operating margins and up to double digit percentage decline for coming quarter. Trendforce, an IT analytics company, updated its forecast from 1.7% year-over-year growth in PC market to 2.6% year-over-year decline for 2026, amid backdrop of steadily increasing prices and supply crisis. Research and analytics firms, Gartner and IDC expect worldwide PC market to decline 10-11% and smartphone market to decline 8-9% in 2026. Gartner also projects that rising memory prices will make low-margin entry level laptops under 500 USD financially unviable in two years. The RAM shortage has delayed the release of Valve's second Steam Machine due to increased memory prices. The device was originally set to launch in early 2026. === AI infrastructure competition === Technology companies including Google, Amazon, Microsoft, and Meta Platforms placed open-ended orders with memory suppliers, indicating they would accept as much supply as available regardless of cost, according to Reuters sources. The limited supply of AI chips has been cited as a reason for the slow down in compute growth. In October 2025, OpenAI formally announced a strategic partnership using letters of intent with Samsung Electronics and SK Hynix

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  Whisper (speech recognition system)

  Whisper is a machine learning model for speech recognition and transcription, created by OpenAI and first released as open-source software in September 2022. It is capable of transcribing speech in English and multiple other languages, and can translate several non-English languages into English. Whisper is a weakly-supervised deep learning acoustic model, made using an encoder-decoder transformer architecture. OpenAI claims that the combination of different training data and post-training filtering used in its development has led to improved recognition of accents, background noise, and jargon compared to previous approaches. While the model does not outperform larger, more specialized models and still experiences AI hallucination, it has been showed to be useful for general sound recognition and has many applications across different industries. == Background == Speech recognition has had a long history in research; the first approaches made use of statistical methods, such as dynamic time warping, and later hidden Markov models. At around the 2010s, deep neural network approaches became more common for speech recognition models, which were enabled by the availability of large datasets ("big data") and increased computational performance. Early approaches to deep learning in speech recognition included convolutional neural networks, which were limited due to their inability to capture sequential data, which later led to developments of Seq2seq approaches, which include recurrent neural networks, which made use of long short-term memory. Transformers, introduced in 2017 by Google, displaced many prior state-of-the-art approaches across a wide range in machine learning, and started becoming the core neural architecture in fields such as language modeling and computer vision. Weakly-supervised approaches to training acoustic models were recognized in the early 2020s as promising for speech recognition approaches using deep neural networks. According to a NYT report, in 2021 OpenAI believed they exhausted sources of higher-quality data to train their large language models and decided to complement scraped web text with transcriptions of YouTube videos and podcasts, and developed Whisper to solve this task. Whisper Large V2 was released on December 8, 2022, followed by Whisper Large V3 being released in November 2023, during the OpenAI Dev Day. In March 2025, OpenAI released new transcription models based on GPT-4o and GPT-4o mini, both of which have lower error rates than Whisper. == Architecture == The Whisper architecture is based on an encoder-decoder transformer. Input audio is resampled to 16,000 Hertz (Hz) and converted to an 80-channel Log-magnitude Mel spectrogram using 25 ms windows with a 10 ms stride. The spectrogram is then normalized to a [-1, 1] range with near-zero mean. The encoder takes this Mel spectrogram as input and processes it. It first passes through two convolutional layers. Sinusoidal positional embeddings are added. It is then processed by a series of Transformer encoder blocks (with pre-activation residual connections). The encoder's output is layer normalized. The decoder is a standard transformer decoder. It has the same width and Transformer blocks as the encoder. It uses learned positional embeddings and tied input-output token representations (using the same weight matrix for both the input and output embeddings). It uses a byte-pair encoding tokenizer, of the same kind as used in GPT-2. English-only models use the GPT-2 vocabulary, while multilingual models employ a re-trained multilingual vocabulary with the same number of words. Special tokens are used to allow the decoder to perform multiple tasks: Tokens that denote language (one unique token per language). Tokens that specify task (<|transcribe|> or <|translate|>). Tokens that specify if no timestamps are present (<|notimestamps|>). If the token is not present, then the decoder predicts timestamps relative to the segment, and quantized to 20 ms intervals. <|nospeech|> for voice activity detection. <|startoftranscript|>, and <|endoftranscript|> . Any text that appears before <|startoftranscript|> is not generated by the decoder, but given to the decoder as context. Loss is only computed over non-contextual parts of the sequence, i.e. tokens between these two special tokens. == Training data == The training dataset consists of 680,000 hours of labeled audio-transcript pairs sourced from the internet using semi-supervised learning. This includes 117,000 hours in 96 non-English languages and 125,000 hours of X→English translation data, where X stands for any non-English language. Preprocessing involved standardization of transcripts, filtering to remove machine-generated transcripts using heuristics (e.g., punctuation, capitalization), language identification and matching with transcripts, fuzzy deduplication, and deduplication with evaluation datasets to avoid data contamination. Speechless segments were also included to allow voice activity detection training. For the files still remaining after the filtering process, audio files were then broken into 30-second segments paired with the subset of the transcript that occurs within that time. If this predicted spoken language differed from the language of the text transcript associated with the audio, that audio-transcript pair was not used for training the speech recognition models, but instead for training translation. The model was trained using the AdamW optimizer with gradient norm clipping and a linear learning rate decay with warmup, with batch size 256 segments. Training proceeded for 1 million updates (approximately 2-3 epochs). No data augmentation or regularization, except for the Large V2 model, which used SpecAugment, Stochastic Depth, and BPE Dropout. The training used data parallelism with float16, dynamic loss scaling, and activation checkpointing. === Post-training filtering === After training the first model, researchers ran it on different subsets of the training data, each representing a distinct source. Data sources were ranked by a combination of their error rate and size. Manual inspection of the top-ranked sources (high error, large size) helped determine if the source was low quality (e.g., partial transcriptions, inaccurate alignment). After training, it was fine-tuned to suppress the prediction of speaker names and low-quality sources were then removed. == Capacity == While Whisper does not outperform models which specialize in the LibriSpeech dataset, when tested across many datasets, it is more robust and makes 55.2% fewer errors than other models. Whisper has a differing error rate with respect to transcribing different languages, with a higher word error rate in languages not well-represented in the training data. The authors found that multi-task learning improved overall performance compared to models specialized to one task. They conjectured that the best Whisper model trained is still underfitting the dataset, and larger models and longer training can result in better models. Third-party evaluations have found varying levels of AI hallucination. A study of transcripts of public meetings found hallucinations in eight out of every 10 transcripts, while an engineer discovered hallucinations in "about half" of 100 hours of transcriptions and a developer identified them in "nearly every one" of 26,000 transcripts. A study of 13,140 short audio segments (averaging 10 seconds) found 187 hallucinations (1.4%), 38% of which generated text that could be harmful because it inserted false references to things like race, non-existent medications, or violent events that were not in the audio. == Applications == The model has been used as the base for many applications, such as a unified model for speech recognition and more general sound recognition. Whisper has also been integrated into the workflow of biomedical research. In 2025, a study on Alzheimer's disease detection used the model to transcribe spontaneous speech recordings. The transcripts that were generated by the model were combined with LLM vector embeddings and traditional classifiers to help classify the patients' health. Another application is when OVALYTICS incorporated Whisper to transcribe YouTube videos and automate content moderation systems, which improved its detection of offensive content. The model has also been used in academic libraries and cultral heritage institutions to generate transcripts and captions for their digitized audiovisual collections. In a 2025 case study, Emory University Libraries found that Whisper reduced the labor used in transcription by around 30-35%, shifting work from text creation to text correction. However, human review is still necessary to make sure accuracy, formatting, and accessibility are all standard.

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  Speech-generating device

  Speech-generating devices (SGDs), also known as voice output communication aids, are electronic augmentative and alternative communication (AAC) systems used to supplement or replace speech or writing for individuals with severe speech impairments, enabling them to verbally communicate. SGDs are important for people who have limited means of interacting verbally, as they allow individuals to become active participants in communication interactions. They are particularly helpful for patients with amyotrophic lateral sclerosis (ALS) but recently have been used for children with predicted speech deficiencies. There are several input and display methods for users of varying abilities to make use of SGDs. Some SGDs have multiple pages of symbols to accommodate a large number of utterances, and thus only a portion of the symbols available are visible at any one time, with the communicator navigating the various pages. Speech-generating devices can produce electronic voice output by using digitized recordings of natural speech or through speech synthesis—which may carry less emotional information but can permit the user to speak novel messages. The content, organization, and updating of the vocabulary on an SGD is influenced by a number of factors, such as the user's needs and the contexts that the device will be used in. The development of techniques to improve the available vocabulary and rate of speech production is an active research area. Vocabulary items should be of high interest to the user, be frequently applicable, have a range of meanings, and be pragmatic in functionality. There are multiple methods of accessing messages on devices: directly or indirectly, or using specialized access devices—although the specific access method will depend on the skills and abilities of the user. SGD output is typically much slower than speech, although rate enhancement strategies can increase the user's rate of output, resulting in enhanced efficiency of communication. The first known SGD was prototyped in the mid-1970s, and rapid progress in hardware and software development has meant that SGD capabilities can now be integrated into devices like smartphones. Notable users of SGDs include Stephen Hawking, Roger Ebert, Tony Proudfoot, and Pete Frates (founder of the ALS Ice Bucket Challenge). Speech-generating systems may be dedicated devices developed solely for AAC, or non-dedicated devices such as computers running additional software to allow them to function as AAC devices. == History == SGDs have their roots in early electronic communication aids. The first such aid was a sip-and-puff typewriter controller named the patient-operated selector mechanism (Naman) prototyped by Reg Maling in the United Kingdom in 1960. POSSUM scanned through a set of symbols on an illuminated display. Researchers at Delft University in the Netherlands created the lightspot-operated typewriter (LOT) in 1970, which made use of small movements of the head to point a small spot of light at a matrix of characters, each equipped with a photoelectric cell. Although it was commercially unsuccessful, the LOT was well received by its users. In 1966, Barry Romich, a freshman engineering student at Case Western Reserve University, and Ed Prentke, an engineer at Highland View Hospital in Cleveland, Ohio, formed a partnership, creating the Prentke Romich Company. In 1969, the company produced its first communication device, a typing system based on a discarded Teletype machine. In 1979, Mark Dahmke developed software for a vocal communication aid program using the Computalker CT-1 analog speech synthesizer with a microcomputer. The software utilized phonemes to generate speech, assisting individuals with communication impairments in constructing words and sentences. Dahmke's work contributed to the advancement of assistive technology for people with disabilities. Notably, he designed the "Vocabulary Management System" for Bill Rush, a student with cerebral palsy. This early speech synthesis technology facilitated improved communication for Rush and was featured in a 1980 issue of LIFE Magazine. Dahmke's contributions have influenced the development of augmentative and alternative communication (AAC) technologies. During the 1970s and early 1980s, several other companies emerged that have since become prominent manufacturers of SGDs. Toby Churchill founded Toby Churchill Ltd in 1973, after losing his speech following encephalitis. In the US, Dynavox (then known as Sentient Systems Technology) grew out of a student project at Carnegie-Mellon University, created in 1982 to help a young woman with cerebral palsy to communicate. Beginning in the 1980s, improvements in technology led to a greatly increased number, variety, and performance of commercially available communication devices, and a reduction in their size and price. Alternative methods of access such as Target Scanning (also known as eye pointing) calibrate the movement of a user's eyes to direct an SGD to produce the desired speech. Scanning, in which alternatives are presented to the user sequentially, became available on communication devices. Speech output possibilities included both digitized and synthesized speech. Rapid progress in hardware and software development continued, including projects funded by the European Community. The first commercially available dynamic screen speech-generating devices were developed in the 1990s. Software was developed that allowed the computer-based production of communication boards. High-tech devices have continued to become smaller and lighter, while increasing accessibility and capability; communication devices can be accessed using eye-tracking systems, perform as a computer for word-processing and Internet use, and as an environmental control device for independent access to other equipment such as TV, radio and telephones. Stephen Hawking came to be associated with the unique voice of his particular synthesis equipment. Hawking was unable to speak due to a combination of disabilities caused by ALS, and an emergency tracheotomy. In the past 20 or so years SGD have gained popularity amongst young children with speech deficiencies, such as autism, Down syndrome, and predicted brain damage due to surgery. Starting in the early 2000s, specialists saw the benefit of using SGDs not only for adults but for children, as well. Neuro-linguists found that SGDs were just as effective in helping children who were at risk for temporary language deficits after undergoing brain surgery as it is for patients with ALS. In particular, digitized SGDs have been used as communication aids for pediatric patients during the recovery process. == Access methods == There are many methods of accessing messages on devices: directly, indirectly, and with specialized access devices. Direct access methods involve physical contact with the system, by using a keyboard or a touch screen. Users accessing SGDs indirectly and through specialized devices must manipulate an object in order to access the system, such as maneuvering a joystick, head mouse, optical head pointer, light pointer, infrared pointer, or switch access scanner. The specific access method will depend on the skills and abilities of the user. With direct selection a body part, pointer, adapted mouse, joystick, or eye tracking could be used, whereas switch access scanning is often used for indirect selection. Unlike direct selection (e.g., typing on a keyboard, touching a screen), users of Target Scanning can only make selections when the scanning indicator (or cursor) of the electronic device is on the desired choice. Those who are unable to point typically calibrate their eyes to use eye gaze as a way to point and blocking as a way to select desired words and phrases. The speed and pattern of scanning, as well as the way items are selected, are individualized to the physical, visual and cognitive capabilities of the user. == Message construction == Augmentative and alternative communication is typically much slower than speech, with users generally producing 8–10 words per minute. Rate enhancement strategies can increase the user's rate of output to around 12–15 words per minute, and as a result enhance the efficiency of communication. In any given SGD there may be a large number of vocal expressions that facilitate efficient and effective communication, including greetings, expressing desires, and asking questions. Some SGDs have multiple pages of symbols to accommodate a large number of vocal expressions, and thus only a portion of the symbols available are visible at any one time, with the communicator navigating the various pages. Speech-generating devices generally display a set of selections either using a dynamically changing screen, or a fixed display. There are two main options for increasing the rate of communication for an SGD: encoding, and prediction. Encoding permits a user to produce a word, sentence or phrase using only on

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  AI-generated content in American politics

  In American politics since the 2020s, political figures have deployed AI-generated images, videos, and audio to attack opponents, create misleading narratives, or inflame emotions. The use of generative AI by American political figures has been subject to criticism from many sides of the political spectrum. Republican president Donald Trump has notably used generative AI in several posts to Truth Social during his second term, many of which have made headlines due to their inflammatory nature. == Background == Generative artificial intelligence is a subfield of artificial intelligence that uses generative models to generate text, images, videos, audio, software code or other forms of data. In the mid 2020s with the release of 15.ai, ChatGPT, DALL-E and other generative artificial intelligence applications there was an AI boom. There has been an increase of usage of generative-AI within the United States political field during this boon, with both Republican and Democratic party members using it. The Trump administration during his second term, have embraced the use of AI-generated images, causing some misinformation experts to raise concerns about the continued usage would cause the erosion of public perception of the truth. In response to some criticisms White House deputy communications director Kaelan Dorr posted on X that the "memes will continue" with White House deputy press secretary Abigail Jackson also mocking concerns. == History of usage == === 2023 === In April 2023, the Republican National Committee released an attack ad made entirely with AI-generated images depicting a dystopian future under Joe Biden's re-election. === 2024 === Generative AI has increased the efficiency with which political candidates were able to raise money by analyzing donor data and identifying possible donors and target audiences. In March 2024 Democratic consultant working for Dean Phillips has admitted to using AI to generate a robocall which used Joe Biden's voice to discourage voter participation. In August 2024, The Atlantic noted that AI slop was becoming associated with the political right in the United States, who were using it for shitposting and engagement farming on social media, with the technology offering "cheap, fast, on-demand fodder for content". AI slop is frequently used in political campaigns in an attempt at gaining attention through content farming. === 2025 === The initial version of the Make Our Children Healthy Again Assessment of children's health issues, released by a commission of cabinet members and officials of the Trump administration, and led by US Department of Health and Human Services Secretary Robert F. Kennedy Jr., reportedly cited nonexistent and garbled references generated using artificial intelligence. Democratic governor Gavin Newsom has used AI-generated images to criticize Trump. In the midst of disruptions to food stamp distribution during the 2025 US government shutdown, anonymous social media users began using OpenAI's Sora to post slop videos of welfare queens complaining, stealing, and rioting in supermarkets; many comments to the videos appeared unaware that they were AI-generated, or acknowledged that they were AI-generated but nonetheless useful in pushing a narrative of widespread welfare fraud. On September 6, 2025, Trump posted an image on Truth Social making a reference to "Chipocalypse Now". Trump's post consisted of an AI-generated image showing Trump frowning and wearing a U.S. Cavalry hat and sunglasses, in front of Lake Michigan with the city of Chicago behind him with a smoke and fire spread across the background with five U.S. Army helicopters in the sky. The words "Chipocalypse Now" are rendered in a font resembling that in which the title of the 1979 film Apocalypse Now was styled. === 2026 === On February 5, 2026, Donald Trump shared a video of Barack and Michelle Obama depicted as apes in a Truth Social post. The two-second AI-generated clip of the Obamas portrayed as apes set to "The Lion Sleeps Tonight" appeared at the end of a one-minute two second long video, the rest of which was about false claims of voter fraud during the 2020 presidential election. The post received at least 4,650 likes, 409 comments, and 1,470 reTruths before it was deleted the next morning. The short clip was part of a longer AI-generated video posted in October 2025. The post received widespread backlash and bipartisan condemnation of the video as racist. In April 2026, Trump posted a picture of himself depicted as Jesus, drawing widespread criticism from Evangelicals and Catholics, resulting in Trump deleting the post hours later and claiming he believed he was depicted as a doctor. == Examples of use == === Election campaigns === In 2023, while he was still running for re-election, the presidential campaign of Joe Biden prepared a task force to respond to AI images and videos. The campaign for the 2024 Republican nominee, Donald Trump, has used deepfake videos of political opponents in campaign ads and fake images showing Trump with black supporters. During the first five months of his second term in 2025, Trump posted several AI-generated images of himself on official government social media accounts, including him as the Pope, him as a Jedi, and him as a muscular man. In August 2024, Trump posted a series of AI-generated images on his social media platform, Truth Social, that portrayed fans of the singer Taylor Swift in "Swifties for Trump" T-shirts, as well as a photo of the singer herself appearing to endorse Trump's 2024 presidential campaign. The images originated from the conservative Twitter account @amuse, which posted numerous AI slop images leading up to the 2024 United States elections that were shared by other high-profile figures within the US Republican Party, such as Elon Musk, who has publicly endorsed the utilization of generative AI, furthering this association. In 2024, Michigan GOP candidate Anthony Hudson posted an AI-generated video showing Martin Luther King Jr. endorsing his campaign, later claiming it was uploaded by a volunteer. In his 2025 bid to be the Democratic nominee for governor of New Jersey, Rep. Josh Gottheimer drew attention and criticism when he released a TV ad that used AI to portray him as a shirtless boxer sparring with Donald Trump in a boxing ring. In November 2025, the campaign of Mike Collins, a GOP candidate in the 2026 United States Senate election in Georgia released a fake video, generated by artificial intelligence, that depicted Democrat Jon Ossoff defending his vote on the 2025 United States federal government shutdown by declaring he could never say no to Chuck Schumer and that SNAP recipients did not attend his out-of-state fundraisers. The Collins campaign also shared an AI-generated video featuring Collins as a shirtless blue jeans model, referencing an American Eagle Outfitters advertisement featuring Sydney Sweeney. During the 2026 Los Angeles mayoral election, candidate Spencer Pratt reposted an AI-generated video portraying Pratt as Batman and prominent California politicians such as Karen Bass, Gavin Newsom, and Kamala Harris, as unruly aristocrats. Former governor of Florida Jeb Bush described the ad as “maybe the best political ad of the year.” In response, a spokesperson for Bass's campaign said, he was "doing his best Trump impression." Bass further responded that the AI ads are "taking on a violent trend." === Protests === In response to the nation-wide No Kings protests in October 2025, Donald Trump posted a video depicting himself flying a fighter jet and releasing feces on crowds of demonstrators, including Democratic influencer Harry Sisson. === Foreign interference === Officials from the ODNI and FBI have stated that Russia, Iran, and China used generative artificial intelligence tools to create fake and divisive text, photos, video, and audio content to foster anti-Americanism and engage in covert influence campaigns. The use of artificial intelligence was described as an accelerant rather than a revolutionary change to influence efforts. Regulation of AI with regard to elections was unlikely to see a resolution for most of the 2024 United States general election season. === Disasters and wars === In the aftermath of Hurricane Helene in the United States, members of the Republican Party circulated an AI-generated image of a young girl holding a puppy in a flood, and used it as evidence of the failure of President Joe Biden to respond to the disaster. Some, like Trump supporter Amy Kremer, shared the image on social media but acknowledged that it was not genuine. In February 2025, Donald Trump shared an AI-generated video on Truth Social depicting a hypothetical Gaza after a Trump takeover. The video's creator claimed it was made as political satire. == Reception == Ramesh Srinivasan, a professor at UCLA raised concerns about the use of AI-generative images stating that many people are questioning where they can find trustab

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  NIS2 Directive

  The Directive (EU) 2022/2555, commonly known as NIS2 is a directive of the European Union aimed at protecting digital infrastructure, in particular critical infrastructure. It broadened the sectors covered by EU network and information security rules and updated incident reporting and oversight compared to the NIS1. Member States were required to transpose NIS2 by 17 October 2024, and the earlier NIS Directive was repealed on 18 October 2024. Only 23 Member States have fully implemented the measures contained with the NIS Directive. Infringement proceedings against them to enforce the Directive have not taken place, and they are not expected to take place in the near future. This failed implementation has led to the fragmentation of cybersecurity capabilities across the EU, with differing standards, incident reporting requirements and enforcement requirements being implemented in different Member States. From the EFTA countries (to April 2026) only Liechtenstein has fully transposed the NIS2 Directive. While the EFTA commission is conducting preparations to transpose the directive into its legislation. == National implementations == === Czech Republic === It is implemented through the Act No. 264/2025 Coll. also called Zákon o kybernetické bezpečnosti (Cybersecurity law) and through another five implementing regulations. The transposing legislation came into force on November 1st, 2025. === Germany === It is implemented through the Gesetz zur Umsetzung der NIS-2-Richtlinie und zur Regelung wesentlicher Grundzüge des Informationssicherheitsmanagements in der Bundesverwaltung. === Ireland === It is implemented through the National Cyber Security Bill. === The Netherlands === It is implemented through the Cyberbeveiligingswet (Cbw). === Slovakia === It is implemented through via an amendment of the Act No. 69/2018 Coll. also called Zákon o kybernetickej bezpečnosti a o zmene a doplnení niektorých zákonov (Law on Cybersecurity and change and amendment of certain laws). It came into force on November 1st, 2025. === Spain === It is implemented through the Esquema Nacional de Seguridad (ENS).

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  Generative AI pornography

  Generative AI pornography or simply AI pornography is a digitally created pornography produced through generative artificial intelligence (AI) technologies. Unlike traditional pornography, which involves real actors and cameras, this content is synthesized entirely by AI algorithms. These algorithms, including generative adversarial networks (GANs) and text-to-image models, generate lifelike images, videos, or animations from textual descriptions or datasets. == Functions and production strategies == AI pornography platforms, beyond account creation and social media linking, primarily enable users to generate sexual images through feature selection or text prompting. Users can customize bodies, clothing, and sociodemographic traits, and browse categorized galleries of user‑generated content. Several sites also support short pornographic videos or GIFs and modification tools such as nudifiers, deepfakes, and facemorphing. Platforms often allow fine‑tuning of parameters such as settings, style, or theme, and provide prompt enhancers or suggestions to improve outputs. Users may edit generated images, refine prior prompts, modify others’ work, or upload personal material as a basis, with iterative and collaborative content creation. Some websites additionally host interactive “erobots,” customizable in real time for appearance, personality, memories, speech, and profession, enabling tailored sexual and non‑sexual interactions. Less common features include VR integration, AI porn games, audio or doodle prompts, and consensual replication of individuals with verification. == History == The use of generative AI in the adult industry began in the late 2010s, initially focusing on AI-generated art, music, and visual content. This trend accelerated in 2022 with Stability AI's release of Stable Diffusion (SD), an open-source text-to-image model that enables users to generate images, including NSFW content, from text prompts using the LAION-Aesthetics subset of the LAION-5B dataset. Despite Stability AI's warnings against sexual imagery, SD's public release led to dedicated communities exploring both artistic and explicit content, sparking ethical debates over open-access AI and its use in adult media. By 2020, AI tools had advanced to generate highly realistic adult content, amplifying calls for regulation. === AI-generated influencers === One application of generative AI technology is the creation of AI-generated influencers on platforms such as OnlyFans and Instagram. These AI personas interact with users in ways that can mimic real human engagement, offering an entirely synthetic but convincing experience. While popular among niche audiences, these virtual influencers have prompted discussions about authenticity, consent, and the blurring line between human and AI-generated content, especially in adult entertainment. === The growth of AI porn sites === By 2023, websites dedicated to AI-generated adult content had gained traction, catering to audiences seeking customizable experiences. These platforms allow users to create or view AI-generated pornography tailored to their preferences. These platforms enable users to create or view AI-generated adult content appealing to different preferences through prompts and tags, customizing body type, facial features, and art styles. Tags further refine the output, creating niche and diverse content. Many sites feature extensive image libraries and continuous content feeds, combining personalization with discovery and enhancing user engagement. AI porn sites, therefore, attract those seeking unique or niche experiences, sparking debates on creativity and the ethical boundaries of AI in adult media. == Ethical concerns and misuse == The growth of generative AI pornography has also attracted some cause for criticism. AI technology can be exploited to create non-consensual pornographic material, posing risks similar to those seen with deepfake revenge porn and AI-generated NCII (Non-Consensual Intimate Image). A 2023 analysis found that 98% of deepfake videos online are pornographic, with 99% of the victims being women. Some famous celebrities victims of deepfake include Scarlett Johansson, Taylor Swift, and Maisie Williams. OpenAI is exploring whether NSFW content, such as erotica, can be responsibly generated in age-appropriate contexts while maintaining its ban on deepfakes. This proposal has attracted criticism from child safety campaigners who argue it undermines OpenAI's mission to develop "safe and beneficial" AI. Additionally, the Internet Watch Foundation has raised concerns about AI being used to generate sexual abuse content involving children. === AI-generated non-consensual intimate imagery (AI Undress) === Generative AI have extensively been used to produce pornography images and videos of non-consenting individuals. 404 Media reported a particular AI generated porn bot on Telegram has more than 100,000 monthly users. Alibaba, the Chinese tech company, released an AI video generation model in 2025 called Wan 2.1, which was modified to produce non-consensual pornography. Several US states are taking actions against using deepfake apps and sharing them on the internet. In 2024, San Francisco filed a landmark lawsuit to shut down "undress" apps that allow users to generate non-consensual AI nude images, citing violations of state laws. The case aligns with California's recent legislation—SB 926, SB 942, and SB 981—championed by Senators Aisha Wahab and Josh Becker and signed by Governor Gavin Newsom. These bills aim to protect individuals from AI-generated explicit images by criminalizing non-consensual distribution, mandating disclosures, and empowering victims to report and remove harmful content from platforms. === Differences from deepfake pornography === While both generative AI pornography and deepfake pornography rely on synthetic media, they differ in their methods and ethical considerations. Deepfake pornography typically involves altering existing footage of real individuals, often without their consent, using AI to superimpose faces, undress said persons, or modify scenes. In contrast, generative AI pornography is created using algorithms, producing hyper-realistic content without the need to upload real pictures of people. Hany Farid, digital image analysis expert, also described the difference between "AI porn" and "deepfake porn." == Legality == The legality of generative AI pornography varies widely by jurisdiction and remains an evolving issue. In some countries, laws addressing digital impersonation, obscenity, or deepfake technologies may indirectly apply, particularly when AI-generated content involves the likeness of real individuals without consent. The absence of a physical performer further complicates traditional regulatory frameworks, which are often grounded in performer protection and distribution laws. In the United States, legal responses have primarily focused on non-consensual deepfakes and impersonation. Some states, such as Virginia, California, and Texas, have enacted legislation criminalising the creation or distribution of non-consensual explicit deepfake content. However, there is no comprehensive federal law addressing AI-generated pornography, leaving a patchwork of legal interpretations and enforcement standards across different jurisdictions. According to a 2023 report, South Korea accounts for approximately 53% of global deepfake pornography production. In September 2024, South Korea's National Assembly amended the Act on Special Cases Concerning the Punishment of Sexual Crimes, introducing two significant reforms related to deepfake content. The first criminalises the possession, viewing, purchase, and storage of non-consensual deepfake material, with penalties of up to three years in prison or fines of up to 30 million won (approximately USD 20,000). The second reform specifically addresses the exploitation of minors, establishing that individuals who use deepfakes to threaten or blackmail minors face a minimum of three years' imprisonment, and at least five years if they coerce minors into unwanted acts. In England and Wales the Data (Use and Access) Act 2025 has legislated against the creation, or the request for creation, of intimate images by nudifying software or websites of another person who has not consented to this. However as of January 2026 this has not yet been brought into force.

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  Fuzzy measure theory

  In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity. The central concept of fuzzy measure theory is the fuzzy measure (also capacity, see ), which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals. There exists a number of different classes of fuzzy measures including plausibility/belief measures, possibility/necessity measures, and probability measures, which are a subset of classical measures. == Definitions == Let X {\displaystyle \mathbf {X} } be a universe of discourse, C {\displaystyle {\mathcal {C}}} be a class of subsets of X {\displaystyle \mathbf {X} } , and E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} . A function g : C → R {\displaystyle g:{\mathcal {C}}\to \mathbb {R} } where ∅ ∈ C ⇒ g ( ∅ ) = 0 {\displaystyle \emptyset \in {\mathcal {C}}\Rightarrow g(\emptyset )=0} E ⊆ F ⇒ g ( E ) ≤ g ( F ) {\displaystyle E\subseteq F\Rightarrow g(E)\leq g(F)} is called a fuzzy measure. A fuzzy measure is called normalized or regular if g ( X ) = 1 {\displaystyle g(\mathbf {X} )=1} . == Properties of fuzzy measures == A fuzzy measure is: additive if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} such that E ∩ F = ∅ {\displaystyle E\cap F=\emptyset } , we have g ( E ∪ F ) = g ( E ) + g ( F ) . {\displaystyle g(E\cup F)=g(E)+g(F).} ; supermodular if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} , we have g ( E ∪ F ) + g ( E ∩ F ) ≥ g ( E ) + g ( F ) {\displaystyle g(E\cup F)+g(E\cap F)\geq g(E)+g(F)} ; submodular if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} , we have g ( E ∪ F ) + g ( E ∩ F ) ≤ g ( E ) + g ( F ) {\displaystyle g(E\cup F)+g(E\cap F)\leq g(E)+g(F)} ; superadditive if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} such that E ∩ F = ∅ {\displaystyle E\cap F=\emptyset } , we have g ( E ∪ F ) ≥ g ( E ) + g ( F ) {\displaystyle g(E\cup F)\geq g(E)+g(F)} ; subadditive if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} such that E ∩ F = ∅ {\displaystyle E\cap F=\emptyset } , we have g ( E ∪ F ) ≤ g ( E ) + g ( F ) {\displaystyle g(E\cup F)\leq g(E)+g(F)} ; symmetric if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} , we have | E | = | F | {\displaystyle |E|=|F|} implies g ( E ) = g ( F ) {\displaystyle g(E)=g(F)} ; Boolean if for any E ∈ C {\displaystyle E\in {\mathcal {C}}} , we have g ( E ) = 0 {\displaystyle g(E)=0} or g ( E ) = 1 {\displaystyle g(E)=1} . Understanding the properties of fuzzy measures is useful in application. When a fuzzy measure is used to define a function such as the Sugeno integral or Choquet integral, these properties will be crucial in understanding the function's behavior. For instance, the Choquet integral with respect to an additive fuzzy measure reduces to the Lebesgue integral. In discrete cases, a symmetric fuzzy measure will result in the ordered weighted averaging (OWA) operator. Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral. == Möbius representation == Let g be a fuzzy measure. The Möbius representation of g is given by the set function M, where for every E , F ⊆ X {\displaystyle E,F\subseteq X} , M ( E ) = ∑ F ⊆ E ( − 1 ) | E ∖ F | g ( F ) . {\displaystyle M(E)=\sum _{F\subseteq E}(-1)^{|E\backslash F|}g(F).} The equivalent axioms in Möbius representation are: M ( ∅ ) = 0 {\displaystyle M(\emptyset )=0} . ∑ F ⊆ E | i ∈ F M ( F ) ≥ 0 {\displaystyle \sum _{F\subseteq E|i\in F}M(F)\geq 0} , for all E ⊆ X {\displaystyle E\subseteq \mathbf {X} } and all i ∈ E {\displaystyle i\in E} A fuzzy measure in Möbius representation M is called normalized if ∑ E ⊆ X M ( E ) = 1. {\displaystyle \sum _{E\subseteq \mathbf {X} }M(E)=1.} Möbius representation can be used to give an indication of which subsets of X interact with one another. For instance, an additive fuzzy measure has Möbius values all equal to zero except for singletons. The fuzzy measure g in standard representation can be recovered from the Möbius form using the Zeta transform: g ( E ) = ∑ F ⊆ E M ( F ) , ∀ E ⊆ X . {\displaystyle g(E)=\sum _{F\subseteq E}M(F),\forall E\subseteq \mathbf {X} .} == Simplification assumptions for fuzzy measures == Fuzzy measures are defined on a semiring of sets or monotone class, which may be as granular as the power set of X, and even in discrete cases the number of variables can be as large as 2|X|. For this reason, in the context of multi-criteria decision analysis and other disciplines, simplification assumptions on the fuzzy measure have been introduced so that it is less computationally expensive to determine and use. For instance, when it is assumed the fuzzy measure is additive, it will hold that g ( E ) = ∑ i ∈ E g ( { i } ) {\displaystyle g(E)=\sum _{i\in E}g(\{i\})} and the values of the fuzzy measure can be evaluated from the values on X. Similarly, a symmetric fuzzy measure is defined uniquely by |X| values. Two important fuzzy measures that can be used are the Sugeno- or λ {\displaystyle \lambda } -fuzzy measure and k-additive measures, introduced by Sugeno and Grabisch respectively. === Sugeno λ-measure === The Sugeno λ {\displaystyle \lambda } -measure is a special case of fuzzy measures defined iteratively. It has the following definition: ==== Definition ==== Let X = { x 1 , … , x n } {\displaystyle \mathbf {X} =\left\lbrace x_{1},\dots ,x_{n}\right\rbrace } be a finite set and let λ ∈ ( − 1 , + ∞ ) {\displaystyle \lambda \in (-1,+\infty )} . A Sugeno λ {\displaystyle \lambda } -measure is a function g : 2 X → [ 0 , 1 ] {\displaystyle g:2^{X}\to [0,1]} such that g ( X ) = 1 {\displaystyle g(X)=1} . if A , B ⊆ X {\displaystyle A,B\subseteq \mathbf {X} } (alternatively A , B ∈ 2 X {\displaystyle A,B\in 2^{\mathbf {X} }} ) with A ∩ B = ∅ {\displaystyle A\cap B=\emptyset } then g ( A ∪ B ) = g ( A ) + g ( B ) + λ g ( A ) g ( B ) {\displaystyle g(A\cup B)=g(A)+g(B)+\lambda g(A)g(B)} . As a convention, the value of g at a singleton set { x i } {\displaystyle \left\lbrace x_{i}\right\rbrace } is called a density and is denoted by g i = g ( { x i } ) {\displaystyle g_{i}=g(\left\lbrace x_{i}\right\rbrace )} . In addition, we have that λ {\displaystyle \lambda } satisfies the property λ + 1 = ∏ i = 1 n ( 1 + λ g i ) {\displaystyle \lambda +1=\prod _{i=1}^{n}(1+\lambda g_{i})} . Tahani and Keller as well as Wang and Klir have shown that once the densities are known, it is possible to use the previous polynomial to obtain the values of λ {\displaystyle \lambda } uniquely. === k-additive fuzzy measure === The k-additive fuzzy measure limits the interaction between the subsets E ⊆ X {\displaystyle E\subseteq X} to size | E | = k {\displaystyle |E|=k} . This drastically reduces the number of variables needed to define the fuzzy measure, and as k can be anything from 1 (in which case the fuzzy measure is additive) to X, it allows for a compromise between modelling ability and simplicity. ==== Definition ==== A discrete fuzzy measure g on a set X is called k-additive ( 1 ≤ k ≤ | X | {\displaystyle 1\leq k\leq |\mathbf {X} |} ) if its Möbius representation verifies M ( E ) = 0 {\displaystyle M(E)=0} , whenever | E | > k {\displaystyle |E|>k} for any E ⊆ X {\displaystyle E\subseteq \mathbf {X} } , and there exists a subset F with k elements such that M ( F ) ≠ 0 {\displaystyle M(F)\neq 0} . == Shapley and interaction indices == In game theory, the Shapley value or Shapley index is used to indicate the weight of a game. Shapley values can be calculated for fuzzy measures in order to give some indication of the importance of each singleton. In the case of additive fuzzy measures, the Shapley value will be the same as each singleton. For a given fuzzy measure g, and | X | = n {\displaystyle |\mathbf {X} |=n} , the Shapley index for every i , … , n ∈ X {\displaystyle i,\dots ,n\in X} is: ϕ ( i ) = ∑ E ⊆ X ∖ { i } ( n − | E | − 1 ) ! | E | ! n ! [ g ( E ∪ { i } ) − g ( E ) ] . {\displaystyle \phi (i)=\sum _{E\subseteq \mathbf {X} \backslash \{i\}}{\frac {(n-|E|-1)!|E|!}{n!}}[g(E\cup \{i\})-g(E)].} The Shapley value is the vector ϕ ( g ) = ( ψ ( 1 ) , … , ψ ( n ) ) . {\displaystyle \mathbf {\phi } (g)=(\psi (1),\dots ,\psi (n)).}

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