How to Choose an AI Pair Programmer

How to Choose an AI Pair Programmer

In search of the best AI pair programmer? An AI pair programmer is software that uses machine learning to help you get more done — it turns a rough idea into a polished result in seconds. When choosing one, weigh output quality, pricing, export formats, and how well it fits the tools you already use. Whether you are a beginner or a pro, the right AI pair programmer slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.

RIPAC (microprocessor)

RIPAC was a VLSI single-chip microprocessor designed for automatic recognition of the connected speech, one of the first of this use. The project of the microprocessor RIPAC started in 1984. RIPAC was aimed to provide efficient real-time speech recognition services to the italian telephone system provided by SIP. The microprocessor was presented in September 1986 at The Hague (Netherlands) at EUSPICO conference. It was composed of 70.000 transistors and structured as Harvard architecture. The name RIPAC is the acronym for "Riconoscimento del PArlato Connesso", that means "Recognition of the connected speech" in Italian. The microprocessor was designed by the Italian companies CSELT and ELSAG and was produced by SGS: a combination of Hidden Markov Model and Dynamic Time Warping algorithms was used for processing speech signals. It was able to do real-time speech recognition of Italian and many languages with a good affordability. The chip, issued by U.S. Patent No. 4,907,278, worked at first run.

Privacy Lost

Privacy Lost is a 2023 short science fiction film directed by Peter Stoel and Robert Berger. It follows a family using augmented reality (AR) and artificial intelligence (AI) devices capable of reading emotional states, raising questions about privacy and manipulation. == Premise == Privacy Lost follows a family using AR glasses that capture and interpret emotions in real time. As the parents argue in a restaurant, their emotional states and even hidden feelings become visible through these glasses. An AI-driven waiter adapts its appearance for each family member, employing emotional data to influence their decisions. == Cast == Brian Kant as Waiter Michael Krass as Husband Estelle Levinson as Waitress Thor van der Linden as Scotty Carlijn van Ramshorst as Wife == Production == Filming took place at HeadQ Productions, a virtual studio located in Amsterdam. The creators sought to depict a near-future scenario in which real-time emotion analysis becomes part of daily interactions. The film was screened at the Augmented World Expo (AWE), where it was noted for its thematic focus on AI-driven manipulation and emotional tracking. The depiction of AR glasses and AI characters integrates modern visual effects to show how devices might analyze emotional responses in real time. It also depicts how AI-driven interactions could influence consumer decisions, pointing to concerns over potential misuse. == Themes == Privacy Lost focuses on the intersection of advanced AI capabilities and AR environments, showing how real-time emotional analysis can be leveraged for targeted persuasion. The film aims to highlight the social and ethical implications of emerging AR and AI technologies, underlining how establishing clear regulatory frameworks for them is necessary to protect individual privacy, govern the storage of emotion-based data, and prevent manipulative practices. Critics describe the film’s theme as dystopian and note that such a reality is unlikely to occur in the near future. However, despite the exaggerated scenario, the film emphasizes the importance of a responsible approach by developers toward emerging technologies.

Grok sexual deepfake scandal

From 2025 onwards, X (formerly Twitter)'s integrated chatbot, Grok, has allowed users to nonconsensually alter images of individuals, including minors, to show them in bikinis or transparent clothing, or in sexually suggestive contexts. The majority of these prompts were targeted at women and girls. Users were able to generate such images by responding to a photo with a request to Grok, such as "put her in a bikini", to which the chatbot would publicly reply with a generated image. The scandal drew significant criticism from lawmakers across the world, and there were calls for bans on X, as well as legal crackdowns on X and xAI for, amongst other reasons, the facilitation of sexual abuse, revenge porn, and child pornography. == Background == Deepfake pornography emerged in the late 2010s with the advent of machine learning. Originally, it was created on a small individual scale using a combination of machine learning algorithms, computer vision techniques, and AI software. However, the production process has significantly evolved since 2018, with the advent of several public apps that have largely automated the process. Since 2023, several AI apps available on Google Play and the Apple App Store are capable of "nudify-ing" user provided photos to generate non-consensual deepfake pornography. Grok would first be proposed by Elon Musk in 2023, when he expressed an intention to create his own AI chatbot to "combat bias". Grok version 2.0, released on August 14, 2024, would introduce image generation capabilities, ones which would be improved over successive updates. == Grok deepfake generation == Cases of Grok being used to remove the clothes from women in pictures, replacing them with bikinis or lingerie, began to surface in May 2025. By late December 2025, a trend of X users requesting such edits to women's photos without permission had taken root, and this received significant media attention in the first few days of January 2026. Some users prompted Grok to edit photos of women into sexualized poses, and others to add blood and bruising, with the chatbot publicly posting these graphic images in response. Grok's X account was restricted on January 9 from posting image generation responses to users who are not paid subscribers, providing a link to "subscribe to unlock these features". All users were still able to generate Grok-altered images using X's "Edit image" feature, and the standalone Grok website and app. However, by March 19, Grok’s Imagine feature was fully restricted to paid subscribers only (SuperGrok tier) for both the standalone Grok website and mobile app. == Analysis == An analysis of 20,000 images generated by Grok between December 25, 2025, and January 1, 2026, showed 2% appeared to be 18 or younger, including 30 of "young or very young" women or girls in bikinis or transparent clothes. A Reuters review of Grok requests over 10 minutes on January 2nd found 102 attempts to put women in bikinis. A separate analysis conducted over 24 hours from January 5 to 6 calculated that users had Grok create 6,700 sexually suggestive or nudified images per hour — 84 times more so than the top 5 deepfake websites combined. Wired reported that far more graphic AI-generated sexual imagery was being created by Grok on its website and app, which are separate to X, including female celebrities removing their clothes and engaging in sexual acts. An analysis of 800 pieces of recovered content by the Paris-based nonprofit AI Forensics found that almost 10% were "instances of photorealistic people, very young, doing sexual activities". AI-generated deepfakes have been described as sexual assault, and as a means to push women out of the public sphere. AI-generated sexually explicit or exploitative image claims are now being treated more like product safety or personal injury harms, not just privacy violations. Because harm may occur the moment an image is generated, some plaintiffs argue liability should focus on the system’s design and safety safeguards. == Reactions == On January 15, the Get Grok Gone campaign delivered letters to Apple and Google, demanding the removal of the app from Apple Store and Google Play Store respectively. The campaign accused both companies of profiting from nonconsensual intimate imagery and child sexual abuse imagery, which were also banned by the companies own policies. The Get Grok Gone campaign argues that the restrictions placed on Grok by xAI are not enough and that Apple and Google are enabling the distribution of harmful material by hosting the apps. === Elon Musk and xAI === xAI responded to requests for comment from media organizations with the automated reply, "Legacy Media Lies." On January 2, Elon Musk reacted "Not sure why, but I couldn’t stop laughing about this one 🤣🤣" to an image of a toaster dressed in a bikini by Grok. Later, on January 14, Elon Musk said that he was "not aware of any naked underage images generated by Grok. Literally zero." Later that same day, xAI announced that X users will no longer be able to use Grok to alter images of real people to portray them in revealing clothing. However, verified X users, as well as users of the standalone Grok app and website, were still able to generate such images. ==== Elon Musk's family ==== Ashley St. Clair, mother of one of Elon Musk's children, reported that Grok users were creating fake sexualized images from her photos, including a photo of her as a child. She considers the photos to be a form of revenge porn, and considered suing under the Take It Down Act. A spokesperson for X stated, "We take action against illegal content on X, including child sexual abuse material (CSAM), by removing it, permanently suspending accounts, and working with local governments and law enforcement as necessary. Anyone using or prompting Grok to make illegal content will suffer the same consequences as if they upload illegal content." However, Grok continued to post non-consensual sexual images. On January 15, St. Clair filed a lawsuit against xAI in the New York Supreme Court. === Canada === In response to the Grok deepfake scandal, individuals have asked that the government of Canada boycott X. On January 10, 2026, Canadian MP and Minister of AI Evan Solomon declared that Canada "is not considering a ban on X". In April 2026, Bill C-16, An Act to amend certain Acts in relation to criminal and correctional matters (child protection, gender-based violence, delays and other measures), was amended following a proposal by Conservative MP Andrew Lawton to ensure that AI-generated images and "nearly nude" intimate images are criminalized. A further proposal by NDP MP Leah Gazan to encompass "sexualized or humiliating contexts, such transparent bathing suits or being covered in blood or bruises" was voted down. === France === On January 2, 2026, French ministers reported the AI tool to prosecutors, calling the content "manifestly illegal", and also asked regulators to check compliance with the Digital Services Act. On February 3, Paris prosecutors office, a cybercrime team employed by them and Europol searched the Paris offices of X. The investigation started as one into allegations of abuse of algorithms and fraudulent data extraction, but has expanded into spreading Holocaust denial and sexual deepfakes. Elon Musk and former CEO Linda Yaccarino have been summoned to a hearing on April 20, with other X staff as witnesses. On April 20, Musk did not turn up for the hearing. The Paris prosecutors office told the BBC on April 20 that it had "taken note of the absence of the people summoned", adding "the presence or absence (of the people summoned) is not an obstacle to continuing the investigation". === India === Indian Member of Parliament Priyanka Chaturvedi filed a complaint to India's IT ministry, demanding a review of Grok's safety mechanisms. === Indonesia === On January 10, Indonesia announced that Grok will be temporarily blocked, becoming the first country to do so. Meutya Hafid, the Minister of Communication and Digital Affairs, stated that "the government views the practice of non-consensual sexual deepfakes as a serious violation of human rights, dignity, and the security of citizens in the digital space." Access to Grok in the country was later restored on February 1. === Ireland === On January 6, Coimisiún na Meán, the Irish media commission, said they were consulting with the European Commission about concerns that Grok was generating sexualized images of women and children. The same day, Ofcom of the United Kingdom contacted X concerning complaints about these images. On January 13, Micheál Martin, Taoiseach of Ireland, announced he would talk with Rossa Fanning, the country's Attorney General, about the Grok chatbot being used to produce sexually explicit images of women and minors. On January 14, the Garda Síochána announced there are 200 investigations into child sex abuse images generated by Grok. The Garda National Cyber Crime Bureau has al

Knights of Sidonia

Knights of Sidonia (Japanese: シドニアの騎士, Hepburn: Shidonia no Kishi) is a Japanese manga series written and illustrated by Tsutomu Nihei. It was serialized by Kodansha's seinen manga magazine Monthly Afternoon between April 2009 and September 2015, with its chapters collected in 15 tankōbon volumes. It tells the story of Nagate Tanikaze, an "under-dweller" destined to become a Garde pilot, whose mission is to defend the generation ship Sidonia from a hostile alien species called Gauna. The manga was licensed for English release in North America by Vertical. An anime television series adaptation was produced by Polygon Pictures. The first season aired from April to June 2014; the second between April and June 2015. An anime film sequel titled Knights of Sidonia: Love Woven in the Stars premiered in June 2021. In 2015, Knights of Sidonia received the 39th Kodansha Manga Award in the general category, as well as the 47th Seiun Award in the Best Comic category in 2016. == Plot == === Setting === The story is set in the year 3394, a thousand years after mankind flees from Earth after it was destroyed by a race of shapeshifting aliens called the Gauna (奇居子(ガウナ)), aboard hundreds of colossal spacecraft created from the remains of the planet. One such ship is the Sidonia, which has developed its own human culture closely based on that of Japan where human cloning, asexual reproduction, and human genetic engineering, such as granting humans photosynthesis, are commonplace. It is also revealed that the top echelons of this society have secretly been granted immortality. With a population of over 500,000 people, Sidonia is possibly the last human settlement remaining, as the fates of the other ships are unknown. Little is known about the true nature of the Gauna or their motivation for attacking humanity. At any given time, a Gauna consists of a nearly impenetrable core protected by a dense layer of malleable flesh known as "placenta" (胞衣, ena). Once the ena is shed away and the core is destroyed, the Gauna's body disintegrates. While Sidonia itself is heavily armed with an arsenal of high-output beam cannons and mass cannons including slow but powerful planet-destroying warheads, it is primarily defended by large mechanized weapons called Gardes (衛人, Morito) whose weaponry and mobility is powered by "Higgs particles" (ヘイグス粒子, Heigusu Ryūshi), armed with a high-output beam cannon for long range assaults and a special spear known as "Kabizashi" for close combat. The tip of the kabizashi is made of a rare and little-understood material which has the unique property of being able to destroy a Gauna's core. Later the Gardes are also equipped with firearms whose ammunition have the same material of the Kabizashi after a means to artificially mass-produce it is discovered. Most people in the surviving human population are screened and drafted as Garde pilots at a young age, if they are shown to be capable of piloting them. === Story === The story follows the adventures of Garde pilot Nagate Tanikaze, who lived in the underground layer of Sidonia since birth and was raised by his grandfather. Never having met anyone else, he trains himself in an old Guardian pilot simulator every day, eventually mastering it. After his grandfather's death, he emerges to the surface and is selected as a Garde pilot, just as Sidonia is once again threatened by the Gauna. == Media == === Manga === Written and illustrated by Tsutomu Nihei, Knights of Sidonia was serialized in Kodansha's seinen manga magazine Monthly Afternoon from April 25, 2009, to September 25, 2015. It was compiled in 15 tankōbon volumes. The manga has been licensed in North America by Vertical, who released all 15 volumes in English between February 5, 2013, and April 26, 2016. === Anime === An anime television series adaptation, produced by Polygon Pictures, aired its first season from April 10 to June 26, 2014, on MBS and later on TBS, CBC and BS-TBS. The series was directed by Kōbun Shizuno, assisted by Hiroyuki Seshita, with scripts by Sadayuki Murai and character designs by Yuki Moriyama. The opening theme song is "Sidonia" (シドニア, Shidonia), performed by Angela, while the ending theme song is "Show" (掌 -show-, Shō), performed by Eri Kitamura. A second season aired from April 11 to June 26, 2015. For the second season, the opening theme song is "Kishi Kōshinkyoku" (騎士行進曲, Knight March), performed by Angela, while the ending theme song is "Requiem" (鎮魂歌 -レクイエム-, Rekuiemu), performed by CustomiZ. The series was localized and streamed by Netflix in all of its territories since July 4, 2014, becoming the service's first original anime, as well as the first anime series on Netflix available in Dolby Vision/HDR. The first season has been licensed for home video release by Sentai Filmworks. The second season was released on Netflix on July 3, 2015, and has been licensed by Sentai Filmworks for home video distribution. In July 2021, Funimation announced they acquired the streaming rights from Netflix to both seasons. === Films === A compilation film of the first season with additional scenes and re-edited sound effects was released on March 6, 2015. A new anime film, titled Knights of Sidonia: Love Woven in the Stars, was announced on July 3, 2020. Hiroyuki Seshita served as chief director, while Tadahiro Yoshihira served as director for the new film, with Polygon Pictures returning for production. Sadayuki Murai and Tetsuya Yamada returned to write scripts, while Shūji Katayama composed the music. The rest of the staff and cast returned to reprise their roles. The first four minutes of the film were shown on YouTube on April 28, 2021. The film was set to premiere on May 14, 2021, but was delayed to June 4, 2021, due to the COVID-19 pandemic. Funimation screened the film in international theaters starting on September 13, 2021. == Reception == === Manga === Knights of Sidonia won the 39th Kodansha Manga Award in the general category in 2015. The manga won the 47th Seiun Award in the Best Comic category in 2016. It also won the Best Seinen category at the 26th Salón del Manga de Barcelona in 2020. It was one of the Jury Recommended works in the Manga Division at the 17th Japan Media Arts Festival in 2013. The Young Adult Library Services Association listed Knights of Sidonia in its 2014 list of Top 10 Graphic Novels for Teens. Carlo Santos from Anime News Network gave the first manga volume a B, stating, "It is got a young man piloting a giant robot against alien enemies, but Knight of Sidonia is no Neon Genesis Evangelion. Yet it is not as bleak or incomprehensible as Tsutomu Nihei works like Blame! or Biomega, either—rather, it is the best of both worlds, bringing Nihei's hard sci-fi mentality into a more conventional space-adventure environment". === Anime === The anime series received positive reviews, even from famous members of the Japanese anime/game industry, like Hideo Kojima, creator of the Metal Gear series, who claims that "It's a kind of anime that we haven't seen for a while that has that sci-fi spirit. Using digital technology cultivated through games, it creates animation that encapsulates Japan's cultural assets like manga, cel animation, kanji, giant robots, etc. What's born is a unique made-in-Japan work that could never be cooked up in Hollywood. Japanese culture has lost its 'cool', and Knights of Sidonia will be the white knight that saves it". Other industry pros left acknowledgements as well, including Akiko Higashimura, Digitarou and Yoshinao Dao.

Open Syllabus Project

The Open Syllabus Project (OSP) is an online open-source platform that catalogs and analyzes millions of college syllabi. Founded by researchers from the American Assembly at Columbia University, the OSP has amassed the most extensive collection of searchable syllabi. Since its beta launch in 2016, the OSP has collected over 7 million course syllabi from over 80 countries, primarily by scraping publicly accessible university websites. The project is directed by Joe Karaganis. == History == The OSP was formed by a group of data scientists, sociologists, and digital-humanities researchers at the American Assembly, a public-policy institute based at Columbia University. The OSP was partly funded by the Sloan Foundation and the Arcadia Fund. Joe Karaganis, former vice-president of the American Assembly, serves as the project director of the OSP. The project builds on prior attempts to archive syllabi, such as H-Net, MIT OpenCourseWare, and historian Dan Cohen's defunct Syllabus Finder website (Cohen now sits on the OSP's advisory board). The OSP became a non-profit and independent of the American Assembly in November 2019. In January 2016, the OSP launched a beta version of their "Syllabus Explorer," which they had collected data for since 2013. The Syllabus Explorer allows users to browse and search texts from over one million college course syllabi. The OSP launched a more comprehensive version 2.0 of the Syllabus Explorer in July 2019. The newer version includes an interactive visualization that displays texts as dots on a knowledge map. As of 2022, the OSP has collected over 7 million course syllabi. The Syllabus Explorer represents the "largest collection of searchable syllabi ever amassed." == Methodology == The OSP has collected syllabi data from over 80 countries dating to 2000. The syllabi stem from over 4,000 worldwide institutions. Most of the OSP's data originates from the United States. Canada, Australia, and the U.K also have large datasets. The OSP primarily collects syllabi by scraping publicly accessible university websites. The OSP also allows syllabi submissions from faculty, students, and administrators. The OSP developers use machine learning and natural language processing to extract metadata from such syllabi. Since only metadata is collected, no individual syllabus or personal identifying information is found in the OSP database. The OSP classifies the syllabi into 62 subject fields – corresponding to the U.S. Department of Education's Classification of Instructional Programs (CIP). Additionally, the OSP assigns each text a "teaching score" from 0–100. This score represents the text's percentile rank among citations in the total citation count and is a numerical indicator of the relative frequency of which a particular work is taught. The OSP also has data on which texts are most likely to be assigned together. The developers behind the OSP admit that the database is incomplete and likely contains "a fair number of errors." Karaganis estimates that 80–100 million syllabi exist in the United States alone. The OSP is unable to access syllabi behind private course-management software like Blackboard. == Notable findings == === Anthropology === Using data from the OSP, anthropologist Laurence Ralph uncovered that black anthropologists are "woefully under-represented in (if not erased from) most anthropology syllabi." Black authors wrote less than 1 percent of the top 1,000 assigned works. === Economics === The database indicates Greg Mankiw is the most frequently cited author for college economics courses. === English literature === The OSP found that Mary Shelley's Frankenstein was the most widely taught novel in college courses. Additionally, the majority of novels published after 1945 taught in English classes were historical fiction. === Female writers === The most read female writer on college campuses is Kate L. Turabian for her A Manual for Writers of Research Papers, Theses, and Dissertations . Turabian is followed by Diana Hacker, Toni Morrison, Jane Austen, and Virginia Woolf. === Film === The most assigned film according to the OSP is the 1929 Soviet documentary film, Man with a Movie Camera. English filmmaker Alfred Hitchcock is the most assigned director in college courses. === History === Historians George Brown Tindall and David Emory Shi's America: A Narrative History is the number one assigned textbook for history, followed by Anne Moody's memoir, Coming of Age in Mississippi. === Philosophy === The most assigned texts in the field of philosophy include Aristotle's Nicomachean Ethics, John Stuart Mill's Utilitarianism, and Plato's Republic. Plato's Republic was also the second most assigned text in universities in the English-speaking world (only behind Strunk and White's Elements of Style). === Physics === David Halliday's et al. Fundamentals of Physics is the number one ranked physics textbook in the OSP's database. === Political science === Data from the OSP indicates that the dominant political science texts are written almost exclusively by white men and scholars based in the West. In the top 200 most-frequently assigned works, 15 are authored by at least one woman. === Public administration === American president Woodrow Wilson's article "The Study of Administration" was the most frequently assigned text in public affairs and administration syllabi. == Reception == According to William Germano et al., the OSP is a "fascinating resource but is also prone to misrepresenting or at least distracting us from the most important business of a syllabus: communicating with students." Historian William Caferro remarks that the OSP is a "tacit experience of sharing, but a useful one." English professor Bart Beaty writes that, "Despite the many reservations about the completeness of its data, the OSP provides a rare opportunity for scholars to move beyond the anecdotal in discussions of canon-formation in teaching." Media theorist Elizabeth Losh opines that "big data approaches", like the OSP, may "raise troubling questions for instructors about informed consent, pedagogical privacy, and quantified metrics."

Construction of t-norms

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