AI Detector Xero

AI Detector Xero — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  LIVAC Synchronous Corpus

  LIVAC is an uncommon language corpus dynamically maintained since 1995. Different from other existing corpora, LIVAC has adopted a rigorous and regular "Windows" approach in processing and filtering massive media texts from representative Chinese speech communities such as Beijing, Hong Kong, Macau, Taipei, Singapore, Shanghai, as well as Guangzhou, and Shenzhen. The contents are thus deliberately repetitive in most cases, represented by textual samples drawn from editorials, local and international news, cross-Taiwan Strait news, as well as news on finance, sports and entertainment. By 2023, more than 3 billion characters of news media texts have been filtered, of which 700 million characters have been processed and analyzed and have yielded an expanding Pan-Chinese dictionary of 2.5 million words from the Pan-Chinese printed media. Through rigorous analysis based on computational linguistic methodology, LIVAC has at the same time accumulated a large amount of accurate and meaningful statistical data on the Chinese language and on their diverse speech communities in the Pan-Chinese context, and the results show considerable and important long standing as well as evolving variations. The "Windows" approach is the most innovative feature of LIVAC and has enabled Pan-Chinese media texts to be quantitatively analyzed according to various attributes such as locations, time and subject domains. Thus, various types of comparative studies and applications in information technology as well as development of often related innovative applications have been possible. Moreover, LIVAC has allowed longitudinal developments to be taken into account, facilitating Key Word in Context (KWIC) search and comprehensive study of target words and their underlying concepts as well as linguistic structures over the past 25 years, based on the above mentioned variables of location, time and subject. Results from the extensive and accumulative data analysis contained in LIVAC have enabled the cultivation of textual databases of proper names, place names, organization names, new words, and bi-weekly and annual rosters of media figures. Related applications have included the establishment of verb and adjective databases, the formulation of sentiment indices, and related opinion mining, to measure and compare the popularity of global media figures in the Chinese media (LIVAC Annual Pan-Chinese Celebrity Rosters, later renamed as the Pan-Chinese Newsmaker Rosters). Notable among these are the decades long periodic reviews of the 25 years of annual pan-Chinese rosters since 2000 and compilation of new word databases (LIVAC Annual Pan-Chinese New Word Rosters). On this basis, the analysis of the emergence, diffusion and transformation of new words, and the publication of dictionaries of neologisms have been made possible. A recent focus is on the relative balance between disyllabic words and growing trisyllabic words in the Chinese language, and the comparative study of light verbs in three Chinese speech communities. as well as the link between the language use and use of language as a reflection of epochal change in China. A new LIVAC version 3.1 was launched in February 2024. == Corpus data processing == Accessing media texts, manual input, etc. Text unification including conversion from simplified to traditional Chinese characters, stored as Big5 and Unicode versions Automatic word segmentation Automatic alignment of parallel texts Manual verification, part-of-speech tagging Extraction of words and addition to regional sub-corpora Combination of regional sub-corpora to update the LIVAC corpus, and master lexical database == Labeling for data curation == Categories used include general terms and proper names, such as: general names, surnames, semi titles; geographical, organizations and commercial entities, etc.; time, prepositions, locations, etc.; stack-words; loanwords; case-word; numerals, etc. Construction of databases of proper names, place names, and specific terms, etc. Generate rosters: "new word rosters", "celebrity or media personality rosters", "place name rosters", compound words and matched words Other parts of speech tagging for sub-database, such as common nouns, numerals, numeral classifiers, different types of verbs, and of adjectives, pronouns, adverbs, prepositions, conjunctions, particles marking mood, onomatopoeia, interjection, etc. == Applications == Compilation of Pan-Chinese dictionaries or local dictionaries Information technology research, such as predictive Chinese text input for mobile phones, automatic speech to text conversion, opinion mining Comparative studies on linguistic and cultural developments in the Pan-Chinese regions, especially in a critical period of history in modern China. Language teaching and learning research, and speech-to-text conversion Customized service on linguistic research and lexical search for international corporations and government agencies The above applications are provided by the following functions: Word Segmentation Search Phrase Search Example Sentence Selection Multi-word Comparison Word Cloud

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  Artificial Intelligence Cold War

  The Artificial Intelligence Cold War (AI Cold War) is a narrative in which geopolitical tensions between the United States of America (USA) and the People's Republic of China (PRC) could lead to a Second Cold War waged in the area of artificial intelligence technology rather than in the areas of nuclear capabilities or ideology. The context of the AI Cold War narrative is the AI arms race, which involves a build-up of military capabilities using AI technology by the US and China and the usage of increasingly advanced semiconductors which power those capabilities. According to a February 2019 publication by the Center for a New American Security, General Secretary of the Chinese Communist Party Xi Jinping – believes that being at the forefront of AI technology will be critical to the future of China's global military and economic power competition. == Origins of the term == The term AI Cold War first appeared in 2018 in an article in Wired magazine by Nicholas Thompson and Ian Bremmer. The two authors trace the emergence of the AI Cold War narrative to 2017, when China published its AI Development Plan, which included a strategy aimed at becoming the global leader in AI by 2030. While the authors acknowledge the use of AI by China to strengthen its authoritarian (totalitarian) rule, they warn against the perils for the US of engaging in an AI Cold War strategy. Thompson and Bremmer rather advocate for a technological cooperation between the US and China to encourage global standards in privacy and ethical use of AI. Shortly after the publication of the article in Wired magazine, the former U.S. Treasury Secretary Hank Paulson referred to the emergence of an ‘Economic Iron Curtain’ between the US and China, reinforcing the new AI Cold War narrative. == Proponents of the AI Cold War narrative == Politico contributed to reinforcing the AI Cold War narrative. In 2020, the paper argued that because of the increasing AI capabilities of China, the US and other democratic countries have to create an alliance to stay ahead of China. Former Google chief executive Eric Schmidt, together with Graham T. Allison alleged in an article in Project Syndicate that, in the context of the COVID-19 pandemic, the AI capabilities of China are ahead of the US in most critical areas. Scientists who have immigrated to the U.S. play an outsize role in the country's development of AI technology. Many of them were educated in China, prompting debates about national security concerns amid worsening relations between the two countries. Policy and technology experts have pointed to concerns about unethical use of AI which would be primarily associated with China. Ethics would therefore constitute a major ideological divide in the upcoming AI Cold War. Fears around disrupting supply chains and a global semiconductor shortage are linked to Taiwan's critical role in the production of semiconductors. 70% of semiconductors are either produced in Taiwan or transfer through Taiwan, where TSMC, world's largest chipmaker is headquartered. The PRC does not recognize the sovereignty of Taiwan and trade restrictions by the US on companies selling semiconductors to the PRC have disrupted in the past the commercial relationships between TSMC and Huawei. == Reactions to the AI Cold War == === Review of the validity of the AI Cold War narrative === Academics and observers expressed concerns about the validity and soundness of the AI Cold War narrative. Denise Garzia expressed concern in Nature that the AI Cold War narrative will undermine the efforts by the US to establish global rules for AI ethics. Researchers have warned in MIT Technology Review that the breakdown in international collaboration in the area of science because of the threat of the alleged AI Cold War would be detrimental to progress. Additionally, the AI Cold War narrative impacts on many more areas including the planning of supply chains and the proliferation of AI. The dissemination of the AI Cold War narrative could therefore be costly and destructive and exacerbate existing tensions. Joanna Bryson and Helena Malikova have pointed to Big Tech's potential interest in promoting the AI Cold War narrative, as technology companies lobby for less onerous regulation of AI in the US and the EU. A factual assessment of the existing AI capabilities of different countries shows a less binary reality than portrayed by the AI Cold War narrative. The AI Cold War started as a narrative but it could turn into a self-fulfilling prophecy and fuel an arms race, not only because of corporate interests but also because of the existing interests at different national security departments. Regarding cyber power, the International Institute for Strategic Studies published a study in June 2021, which argued that the online capabilities of China have been exaggerated and that Chinese cyber power is at least a decade behind the US, largely due to lingering security issues. === Restrictions to trading with China === US politicians and European industry players have invoked the looming AI Cold War as a reason to ban procurement by public authorities in Europe of Huawei 5G technology due to concerns over the Chinese state-sponsored surveillance industry. In 2019, the Trump administration successfully lobbied the Dutch government into stopping the Netherlands-based company ASML from exporting equipment to China. ASML manufactures a machine called an extreme ultraviolet lithography system used by semiconductor producers, including TSMC and Intel to produce state-of the-art microchips. The Biden administration adopted the same course of action as the Trump administration and requested the Netherlands to restrict sales by ASML to China, invoking national-security concerns. The trade restrictions imposed by the Trump administration affected semiconductors imports from China to the US and raised concerns by the US industry that supply chains will be disrupted in case of an AI Cold War. This prompted US technology companies to develop mitigation strategies including hoarding semiconductors and trying to set up local semiconductor production facilities, with the support of government subsidies. === Industrial policy initiatives === ==== United States ==== In June 2021, the US Senate approved the U.S. Innovation and Competition Act providing around 250 billion US dollars public money support to the US technological and manufacturing industry. The alleged Chinese threat in the area of technology helped secure a strong bipartisan support for the new legislation, amounting to the largest industrial policy move by the US in decades. Chinese authorities reproached to the US that the bill was “full of cold war zero-sum thinking”. The legislative bill is aimed at strengthening capabilities in the area of technology, such as quantum computing and AI specifically to face the competitive threat from China perceived as urgent. Senator Chuck Schumer, the leader of the Senate majority and one of the sponsors of the industrial policy bill invoked the threat of authoritarian regimes that want “grab the mantle of global economic leadership and own the innovations”. In 2022, U.S. Innovation and Competition Act was amended and turned into the Chips and Science Act with planned spending of 280 billion US dollars, 53 billion thereof are allocated directly to subsidies for semiconductors manufacturing. Commentators identified possible positive effects on innovation from the US attempts to compete with China in a perceived rivalry. Among the main beneficiaries of the US CHIPS Act are the semiconductor producers Intel, TSMC and Micron Technology. ==== European Chips Act ==== In February 2022, the European Union introduced its own European Chips Act initiative. The background of the initiative would be the objective of European strategic autonomy. The EU's initiative puts forward subsidies of 30 billion euros to encourage manufacturing of semiconductors in the EU. The US company Intel is one beneficiary of the initiative. The US and European chips acts raise concerns of protectionism and a risk of a subsidies "race to the bottom." === New world order === The AI Cold War heralds a new world order in geopolitics, according to Hemant Taneja and Fareed Zakaria. This new world order is a departure from the unipolar system dominated by the US. It is characterized by existence of two parallel digital ecosystems, ran by China and the US. In order to succeed countries that consider themselves as democracies are to align their technological ecosystems to that of the US, in a process labelled re-globalization.

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  2025 Bilderberg Conference

  The 2025 Bilderberg Conference was held between June 12–June 15, 2025 at the Grand Hôtel in Stockholm, Sweden. The 2025 meeting was the 71st edition of the event. A Bilderberg Group press release listed 121 participants from 23 countries. Established in 1954 by Prince Bernhard of the Netherlands, Bilderberg conferences (or meetings) are an annual private gathering of the European and North American political and business elite. Events are attended by between 120 and 150 people each year invited by the Bilderberg Group's steering committee; including prominent politicians, CEOs, national security experts, academics and journalists. Bilderberg conferences operate under the Chatham House Rule, meaning that participants are sworn to secrecy and cannot disclose the identity or affiliation of any particular speaker. As a result, media are not invited and delegates rarely speak about what was discussed. Permits for two public demonstrations against the meeting were requested, one of which, a march from the Norrmalm Square to the Grand Hôtel, was planned for June 14. == Agenda == The key topics for discussion were announced on the Bilderberg website shortly before the meeting. These topics included: == Participants == A list of 121 participants was published on the Bilderberg website. This list may not be complete, as a source connected to the Bilderberg group told The Daily Telegraph in 2013 that some attendees do not have their names publicized. Prime Minister of Sweden Ulf Kristersson attended the meeting despite his name not appearing on the published participant list.

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  Type-1 OWA operators

  Type-1 OWA operators are a set of aggregation operators that generalise the Yager's OWA (ordered weighted averaging) operators in the interest of aggregating fuzzy sets rather than crisp values in soft decision making and data mining. These operators provide a mathematical technique for directly aggregating uncertain information with uncertain weights via OWA mechanism in soft decision making and data mining, where these uncertain objects are modelled by fuzzy sets. The two definitions for type-1 OWA operators are based on Zadeh's Extension Principle and α {\displaystyle \alpha } -cuts of fuzzy sets. The two definitions lead to equivalent results. == Definitions == === Definition 1 === Let F ( X ) {\displaystyle F(X)} be the set of fuzzy sets with domain of discourse X {\displaystyle X} , a type-1 OWA operator is defined as follows: Given n linguistic weights { W i } i = 1 n {\displaystyle \left\{{W^{i}}\right\}_{i=1}^{n}} in the form of fuzzy sets defined on the domain of discourse U = [ 0 , 1 ] {\displaystyle U=[0,1]} , a type-1 OWA operator is a mapping, Φ {\displaystyle \Phi } , Φ : F ( X ) × ⋯ × F ( X ) ⟶ F ( X ) {\displaystyle \Phi \colon F(X)\times \cdots \times F(X)\longrightarrow F(X)} ( A 1 , ⋯ , A n ) ↦ Y {\displaystyle (A^{1},\cdots ,A^{n})\mapsto Y} such that μ Y ( y ) = sup ∑ k = 1 n w ¯ i a σ ( i ) = y ( μ W 1 ( w 1 ) ∧ ⋯ ∧ μ W n ( w n ) ∧ μ A 1 ( a 1 ) ∧ ⋯ ∧ μ A n ( a n ) ) {\displaystyle \mu _{Y}(y)=\displaystyle \sup _{\displaystyle \sum _{k=1}^{n}{\bar {w}}_{i}a_{\sigma (i)}=y}\left({\begin{array}{{1}l}\mu _{W^{1}}(w_{1})\wedge \cdots \wedge \mu _{W^{n}}(w_{n})\wedge \mu _{A^{1}}(a_{1})\wedge \cdots \wedge \mu _{A^{n}}(a_{n})\end{array}}\right)} where w ¯ i = w i ∑ i = 1 n w i {\displaystyle {\bar {w}}_{i}={\frac {w_{i}}{\sum _{i=1}^{n}{w_{i}}}}} , and σ : { 1 , ⋯ , n } ⟶ { 1 , ⋯ , n } {\displaystyle \sigma \colon \{1,\cdots ,n\}\longrightarrow \{1,\cdots ,n\}} is a permutation function such that a σ ( i ) ≥ a σ ( i + 1 ) , ∀ i = 1 , ⋯ , n − 1 {\displaystyle a_{\sigma (i)}\geq a_{\sigma (i+1)},\ \forall i=1,\cdots ,n-1} , i.e., a σ ( i ) {\displaystyle a_{\sigma (i)}} is the i {\displaystyle i} th highest element in the set { a 1 , ⋯ , a n } {\displaystyle \left\{{a_{1},\cdots ,a_{n}}\right\}} . === Definition 2 === Using the alpha-cuts of fuzzy sets: Given the n linguistic weights { W i } i = 1 n {\displaystyle \left\{{W^{i}}\right\}_{i=1}^{n}} in the form of fuzzy sets defined on the domain of discourse U = [ 0 , 1 ] {\displaystyle U=[0,\;\;1]} , then for each α ∈ [ 0 , 1 ] {\displaystyle \alpha \in [0,\;1]} , an α {\displaystyle \alpha } -level type-1 OWA operator with α {\displaystyle \alpha } -level sets { W α i } i = 1 n {\displaystyle \left\{{W_{\alpha }^{i}}\right\}_{i=1}^{n}} to aggregate the α {\displaystyle \alpha } -cuts of fuzzy sets { A i } i = 1 n {\displaystyle \left\{{A^{i}}\right\}_{i=1}^{n}} is: Φ α ( A α 1 , … , A α n ) = { ∑ i = 1 n w i a σ ( i ) ∑ i = 1 n w i | w i ∈ W α i , a i ∈ A α i , i = 1 , … , n } {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\ldots ,A_{\alpha }^{n}}\right)=\left\{{{\frac {\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}}}{\sum \limits _{i=1}^{n}{w_{i}}}}\left|{w_{i}\in W_{\alpha }^{i},\;a_{i}}\right.\in A_{\alpha }^{i},\;i=1,\ldots ,n}\right\}} where W α i = { w | μ W i ( w ) ≥ α } , A α i = { x | μ A i ( x ) ≥ α } {\displaystyle W_{\alpha }^{i}=\{w|\mu _{W_{i}}(w)\geq \alpha \},A_{\alpha }^{i}=\{x|\mu _{A_{i}}(x)\geq \alpha \}} , and σ : { 1 , ⋯ , n } → { 1 , ⋯ , n } {\displaystyle \sigma :\{\;1,\cdots ,n\;\}\to \{\;1,\cdots ,n\;\}} is a permutation function such that a σ ( i ) ≥ a σ ( i + 1 ) , ∀ i = 1 , ⋯ , n − 1 {\displaystyle a_{\sigma (i)}\geq a_{\sigma (i+1)},\;\forall \;i=1,\cdots ,n-1} , i.e., a σ ( i ) {\displaystyle a_{\sigma (i)}} is the i {\displaystyle i} th largest element in the set { a 1 , ⋯ , a n } {\displaystyle \left\{{a_{1},\cdots ,a_{n}}\right\}} . == Representation theorem of Type-1 OWA operators == Given the n linguistic weights { W i } i = 1 n {\displaystyle \left\{{W^{i}}\right\}_{i=1}^{n}} in the form of fuzzy sets defined on the domain of discourse U = [ 0 , 1 ] {\displaystyle U=[0,\;\;1]} , and the fuzzy sets A 1 , ⋯ , A n {\displaystyle A^{1},\cdots ,A^{n}} , then we have that Y = G {\displaystyle Y=G} where Y {\displaystyle Y} is the aggregation result obtained by Definition 1, and G {\displaystyle G} is the result obtained by in Definition 2. == Programming problems for Type-1 OWA operators == According to the Representation Theorem of Type-1 OWA Operators, a general type-1 OWA operator can be decomposed into a series of α {\displaystyle \alpha } -level type-1 OWA operators. In practice, this series of α {\displaystyle \alpha } -level type-1 OWA operators is used to construct the resulting aggregation fuzzy set. So we only need to compute the left end-points and right end-points of the intervals Φ α ( A α 1 , ⋯ , A α n ) {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)} . Then, the resulting aggregation fuzzy set is constructed with the membership function as follows: μ G ( x ) = ⋁ α : x ∈ Φ α ( A α 1 , ⋯ , A α n ) α ⁡ α {\displaystyle \mu _{G}(x)=\operatorname {\bigvee } \limits _{\alpha :x\in \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{\alpha }}\alpha } For the left end-points, we need to solve the following programming problem: Φ α ( A α 1 , ⋯ , A α n ) − = min W α − i ≤ w i ≤ W α + i A α − i ≤ a i ≤ A α + i ⁡ ∑ i = 1 n w i a σ ( i ) / ∑ i = 1 n w i {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{-}=\operatorname {\min } \limits _{\begin{array}{l}W_{\alpha -}^{i}\leq w_{i}\leq W_{\alpha +}^{i}A_{\alpha -}^{i}\leq a_{i}\leq A_{\alpha +}^{i}\end{array}}\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}/\sum \limits _{i=1}^{n}{w_{i}}}} while for the right end-points, we need to solve the following programming problem: Φ α ( A α 1 , ⋯ , A α n ) + = max W α − i ≤ w i ≤ W α + i A α − i ≤ a i ≤ A α + i ⁡ ∑ i = 1 n w i a σ ( i ) / ∑ i = 1 n w i {\displaystyle \Phi _{\alpha }\left({A_{\alpha }^{1},\cdots ,A_{\alpha }^{n}}\right)_{+}=\operatorname {\max } \limits _{\begin{array}{l}W_{\alpha -}^{i}\leq w_{i}\leq W_{\alpha +}^{i}A_{\alpha -}^{i}\leq a_{i}\leq A_{\alpha +}^{i}\end{array}}\sum \limits _{i=1}^{n}{w_{i}a_{\sigma (i)}/\sum \limits _{i=1}^{n}{w_{i}}}} A fast method has been presented to solve two programming problem so that the type-1 OWA aggregation operation can be performed efficiently, for details, please see the paper. == Alpha-level approach to Type-1 OWA operation == Three-step process: Step 1—To set up the α {\displaystyle \alpha } - level resolution in [0, 1]. Step 2—For each α ∈ [ 0 , 1 ] {\displaystyle \alpha \in [0,1]} , Step 2.1—To calculate ρ α + i 0 ∗ {\displaystyle \rho _{\alpha +}^{i_{0}^{\ast }}} Let i 0 = 1 {\displaystyle i_{0}=1} ; If ρ α + i 0 ≥ A α + σ ( i 0 ) {\displaystyle \rho _{\alpha +}^{i_{0}}\geq A_{\alpha +}^{\sigma (i_{0})}} , stop, ρ α + i 0 {\displaystyle \rho _{\alpha +}^{i_{0}}} is the solution; otherwise go to Step 2.1-3. i 0 ← i 0 + 1 {\displaystyle i_{0}\leftarrow i_{0}+1} , go to Step 2.1-2. Step 2.2 To calculate ρ α − i 0 ∗ {\displaystyle \rho _{\alpha -}^{i_{0}^{\ast }}} Let i 0 = 1 {\displaystyle i_{0}=1} ; If ρ α − i 0 ≥ A α − σ ( i 0 ) {\displaystyle \rho _{\alpha -}^{i_{0}}\geq A_{\alpha -}^{\sigma (i_{0})}} , stop, ρ α − i 0 {\displaystyle \rho _{\alpha -}^{i_{0}}} is the solution; otherwise go to Step 2.2-3. i 0 ← i 0 + 1 {\displaystyle i_{0}\leftarrow i_{0}+1} , go to step Step 2.2-2. Step 3—To construct the aggregation resulting fuzzy set G {\displaystyle G} based on all the available intervals [ ρ α − i 0 ∗ , ρ α + i 0 ∗ ] {\displaystyle \left[{\rho _{\alpha -}^{i_{0}^{\ast }},\;\rho _{\alpha +}^{i_{0}^{\ast }}}\right]} : μ G ( x ) = ⋁ α : x ∈ [ ρ α − i 0 ∗ , ρ α + i 0 ∗ ] ⁡ α {\displaystyle \mu _{G}(x)=\operatorname {\bigvee } \limits _{\alpha :x\in \left[{\rho _{\alpha -}^{i_{0}^{\ast }},\;\rho _{\alpha +}^{i_{0}^{\ast }}}\right]}\alpha } == Some Examples == The type-1 OWA operator with the weights shown in the top figure is used to aggregate the fuzzy sets (solide lines) in the bottom figure, and the dashed line is the aggregation result. == Special cases == Any OWA operators, like maximum, minimum, mean operators; Join operators of (type-1) fuzzy sets, i.e., fuzzy maximum operators; Meet operators of (type-1) fuzzy sets, i.e., fuzzy minimum operators; Join-like operators of (type-1) fuzzy sets; Meet-like operators of (type-1) fuzzy sets. == Generalizations == Type-2 OWA operators have been suggested to aggregate the type-2 fuzzy sets for soft decision making. == Applications == Type-1 OWA operators have been applied to different domains for soft decision making. Improved efficiency of computing approach ; Type reduction of type-2 fuzzy sets ; Group decision making ; Credit risk evaluation ; Information fusion ; Linguistic expressions and symbolic translation ; Sentiment analysis ; Ro

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  Keyword extraction

  Keyword extraction is tasked with the automatic identification of terms that best describe the subject of a document. Key phrases, key terms, key segments or just keywords are the terminology which is used for defining the terms that represent the most relevant information contained in the document. Although the terminology is different, function is the same: characterization of the topic discussed in a document. The task of keyword extraction is an important problem in text mining, information extraction, information retrieval and natural language processing (NLP). == Keyword assignment vs. extraction == Keyword assignment methods can be roughly divided into: keyword assignment (keywords are chosen from controlled vocabulary or taxonomy) and keyword extraction (keywords are chosen from words that are explicitly mentioned in original text). Methods for automatic keyword extraction can be supervised, semi-supervised, or unsupervised. Unsupervised methods can be further divided into simple statistics, linguistics or graph-based, or ensemble methods that combine some or most of these methods.

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  Ghost in the Shell

  Ghost in the Shell is a Japanese cyberpunk military science fiction media franchise that began with the eponymous manga series, written and illustrated by Masamune Shirow. The manga, first serialized from 1989 to 1991, is set in the mid-21st-century and follows the fictional counter-cyberterrorist organization Public Security Section 9, led by protagonist Major Motoko Kusanagi. Animation studio Production I.G has produced several anime adaptations of the series. These include the 1995 film of the same name and its 2004 sequel, Ghost in the Shell 2: Innocence; the 2002 television series Ghost in the Shell: Stand Alone Complex and its 2020 follow-up, Ghost in the Shell: SAC_2045; and the Ghost in the Shell: Arise original video animation series. In addition, an American-produced live-action film was released in March 2017. == Overview == === Title === The original editor Koichi Yuri says: At first, Ghost in the Shell came from Shirow, but when Yuri asked for "something more flashy", Shirow came up with "攻殻機動隊 Koukaku Kidou Tai (Shell Squad)" for Yuri. But Shirow was attached to including "Ghost in the Shell" as well even if in smaller type. === Setting === Primarily set in the mid-twenty-first century in the fictional Japanese city of Niihama, Niihama Prefecture (新浜県新浜市, Niihama-ken Niihama-shi), otherwise known as New Port City (ニューポートシティ, Nyū Pōto Shiti), the manga and the many anime adaptations follow the members of Public Security Section 9, a task-force consisting of various professionals skilled at solving and preventing crime, mostly with some sort of police background. Political intrigue and counter-terrorism operations are standard fare for Section 9, but the various actions of corrupt officials, companies, and cyber-criminals in each scenario are unique and require the diverse skills of Section 9's staff to prevent a series of incidents from escalating. In this post-cyberpunk iteration of a possible future, computer technology has advanced to the point that many members of the public possess cyberbrains, technology that allows them to interface their biological brain with various networks. The level of cyberization varies from simple minimal interfaces to almost complete replacement of the brain with cybernetic parts, in cases of severe trauma. This can also be combined with various levels of prostheses, with a fully prosthetic body enabling a person to become a cyborg. The main character of Ghost in the Shell, Major Motoko Kusanagi, is such a cyborg, having had a terrible accident befall her as a child that ultimately required her to use a full-body prosthesis to house her cyberbrain. This high level of cyberization, however, opens the brain up to attacks from highly skilled hackers, with the most dangerous being those who will hack a person to bend to their whims. == Media == === Literature === ==== Original manga ==== The original Ghost in the Shell manga ran in Japan from April 1989 to November 1990 in Kodansha's manga anthology Young Magazine, and was released in a tankōbon volume on October 2, 1991. Ghost in the Shell 2: Man-Machine Interface followed in 1997 for nine issues in Young Magazine, and was collected in the Ghost in the Shell: Solid Box on December 1, 2000. Then a standard version with modifications and new pages was published on June 26, 2001. Four stories from Man-Machine Interface that were not released in tankobon format from previous releases were later collected in Ghost in the Shell 1.5: Human-Error Processor, and published by Kodansha on July 17, 2003. Several art books have also been published for the manga. === Films === ==== Animated films ==== Two animated films based on the original manga have been released, both directed by Mamoru Oshii and animated by Production I.G. Ghost in the Shell was released in 1995 and follows the "Puppet Master" storyline from the manga. It was re-released in 2008 as Ghost in the Shell 2.0 with new audio and updated 3D computer graphics in certain scenes. Innocence, otherwise known as Ghost in the Shell 2: Innocence, was released in 2004, with its story based on a chapter from the first manga. ==== Live-action film ==== In 2008, DreamWorks and producer Steven Spielberg acquired the rights to a live-action film adaptation of the original Ghost in the Shell manga. On January 24, 2014, Rupert Sanders was announced as director, with a screenplay by William Wheeler. In April 2016, the full cast was announced, which included Juliette Binoche, Chin Han, Lasarus Ratuere and Kaori Momoi, and Scarlett Johansson in the lead role; the casting of Johansson drew accusations of whitewashing. Principal photography on the film began on location in Wellington, New Zealand, on February 1, 2016. Filming wrapped in June 2016. Ghost in the Shell premiered in Tokyo on March 16, 2017, and was released in the United States on March 31, 2017, in 2D, 3D and IMAX 3D. It received mixed reviews, with praise for its visuals and Johansson's performance but criticism for its script. === Television === ==== Stand Alone Complex TV series, film and ONA ==== In 2002, Ghost in the Shell: Stand Alone Complex premiered on Animax, presenting a new telling of Ghost in the Shell independent from the original manga, focusing on Section 9's investigation of the Laughing Man hacker. It was followed in 2004 by a second season titled Ghost in the Shell: S.A.C. 2nd GIG, which focused on the Individual Eleven terrorist group. The primary storylines of both seasons were compressed into OVAs broadcast as Ghost in the Shell: Stand Alone Complex The Laughing Man in 2005 and Ghost in the Shell: Stand Alone Complex Individual Eleven in 2006. Also in 2006, Ghost in the Shell: Stand Alone Complex - Solid State Society, featuring Section 9's confrontation with a hacker known as the Puppeteer, was broadcast, serving as a finale to the anime series. The extensive score for the series and its films was composed by Yoko Kanno. On April 7, 2017, Kodansha and Production I.G announced that Kenji Kamiyama and Shinji Aramaki would be co-directing a new Kōkaku Kidōtai anime production. On December 7, 2018, it was reported by Netflix that they had acquired the worldwide streaming rights to the original net animation (ONA) anime series, titled Ghost in the Shell: SAC_2045, and that it would premiere on April 23, 2020. The series is in 3DCG and Sola Digital Arts collaborated with Production I.G on the project. Ilya Kuvshinov handled character designs. The series had two seasons of 12 episodes each. In addition to the anime, a series of published books, two separate manga adaptations, and several video games for consoles and mobile phones have been released for Stand Alone Complex. ==== Arise OVA, TV series and film ==== In 2013, a new iteration of the series titled Ghost in the Shell: Arise premiered, taking an original look at the Ghost in the Shell world, set before the original manga. It was released as a series of four original video animation (OVA) episodes (with limited theatrical releases) from 2013 to 2014, then recompiled as a 10-episode television series under the title of Kōkaku Kidōtai: Arise - Alternative Architecture. An additional fifth OVA titled Pyrophoric Cult, originally premiering in the Alternative Architecture broadcast as two original episodes, was released on August 26, 2015. Kazuchika Kise served as the chief director of the series, with Tow Ubukata as head writer. Cornelius was brought onto the project to compose the score for the series, with the Major's new voice actress Maaya Sakamoto also providing vocals for certain tracks. Ghost in the Shell: The New Movie, also known as Ghost in the Shell: Arise − The Movie or New Ghost in the Shell, is a 2015 film directed by Kazuya Nomura that serves as a finale to the Ghost in the Shell: Arise story arc. The film is a continuation to the plot of the Pyrophoric Cult episode of Arise, and ties up loose ends from that arc. A manga adaptation was serialized in Kodansha's Young Magazine, which started on March 13 and ended on August 26, 2013. ==== 2026 anime ==== On May 25, 2024, it was announced that a new anime television series adaptation will be produced by Science Saru for a July 2026 premiere. Saru will be in a production committee with Bandai Namco Filmworks, Kodansha and Production I.G. The series will be directed by Monkochan, with a script by EnJoe Toh. === Video games === Ghost in the Shell was developed by Exact and released for the PlayStation on July 17, 1997, in Japan by Sony Computer Entertainment. It is a third-person shooter featuring an original storyline where the character plays a rookie member of Section 9. The video game's soundtrack Megatech Body features various techno artists, such as Takkyu Ishino, Scan X and Mijk Van Dijk. Several video games were also developed to tie into the Stand Alone Complex television series, in addition to a first-person shooter by Nexon and Neople titled Ghost in the Shell: Stand Alone Complex - First Assault Online,

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  Thompson sampling

  Thompson sampling, named after William R. Thompson, is a heuristic for choosing actions that address the exploration–exploitation dilemma in the multi-armed bandit problem. It consists of choosing the action that maximizes the expected reward with respect to a randomly drawn belief. == Description == Consider a set of contexts X {\displaystyle {\mathcal {X}}} , a set of actions A {\displaystyle {\mathcal {A}}} , and rewards in R {\displaystyle \mathbb {R} } . The aim of the player is to play actions under the various contexts, such as to maximize the cumulative rewards. Specifically, in each round, the player obtains a context x ∈ X {\displaystyle x\in {\mathcal {X}}} , plays an action a ∈ A {\displaystyle a\in {\mathcal {A}}} and receives a reward r ∈ R {\displaystyle r\in \mathbb {R} } following a distribution that depends on the context and the issued action. The elements of Thompson sampling are as follows: a likelihood function P ( r | θ , a , x ) {\displaystyle P(r|\theta ,a,x)} ; a set Θ {\displaystyle \Theta } of parameters θ {\displaystyle \theta } of the distribution of r {\displaystyle r} ; a prior distribution P ( θ ) {\displaystyle P(\theta )} on these parameters; past observations triplets D = { ( x ; a ; r ) } {\displaystyle {\mathcal {D}}=\{(x;a;r)\}} ; a posterior distribution P ( θ | D ) ∝ P ( D | θ ) P ( θ ) {\displaystyle P(\theta |{\mathcal {D}})\propto P({\mathcal {D}}|\theta )P(\theta )} , where P ( D | θ ) {\displaystyle P({\mathcal {D}}|\theta )} is the likelihood function. Thompson sampling consists of playing the action a ∗ ∈ A {\displaystyle a^{\ast }\in {\mathcal {A}}} according to the probability that it maximizes the expected reward; action a ∗ {\displaystyle a^{\ast }} is chosen with probability ∫ I [ E ( r | a ∗ , x , θ ) = max a ′ E ( r | a ′ , x , θ ) ] P ( θ | D ) d θ , {\displaystyle \int \mathbb {I} \left[\mathbb {E} (r|a^{\ast },x,\theta )=\max _{a'}\mathbb {E} (r|a',x,\theta )\right]P(\theta |{\mathcal {D}})d\theta ,} where I {\displaystyle \mathbb {I} } is the indicator function. In practice, the rule is implemented by sampling. In each round, parameters θ ∗ {\displaystyle \theta ^{\ast }} are sampled from the posterior P ( θ | D ) {\displaystyle P(\theta |{\mathcal {D}})} , and an action a ∗ {\displaystyle a^{\ast }} chosen that maximizes E [ r | θ ∗ , a ∗ , x ] {\displaystyle \mathbb {E} [r|\theta ^{\ast },a^{\ast },x]} , i.e. the expected reward given the sampled parameters, the action, and the current context. Conceptually, this means that the player instantiates their beliefs randomly in each round according to the posterior distribution, and then acts optimally according to them. In most practical applications, it is computationally onerous to maintain and sample from a posterior distribution over models. As such, Thompson sampling is often used in conjunction with approximate sampling techniques. == History == Thompson sampling was originally described by Thompson in 1933. It was subsequently rediscovered numerous times independently in the context of multi-armed bandit problems. A first proof of convergence for the bandit case has been shown in 1997. The first application to Markov decision processes was in 2000. A related approach (see Bayesian control rule) was published in 2010. In 2010 it was also shown that Thompson sampling is instantaneously self-correcting. Asymptotic convergence results for contextual bandits were published in 2011. Thompson Sampling has been widely used in many online learning problems including A/B testing in website design and online advertising, and accelerated learning in decentralized decision making. A Double Thompson Sampling (D-TS) algorithm has been proposed for dueling bandits, a variant of traditional MAB, where feedback comes in the form of pairwise comparison. == Relationship to other approaches == === Probability matching === Probability matching is a decision strategy in which predictions of class membership are proportional to the class base rates. Thus, if in the training set positive examples are observed 60% of the time, and negative examples are observed 40% of the time, the observer using a probability-matching strategy will predict (for unlabeled examples) a class label of "positive" on 60% of instances, and a class label of "negative" on 40% of instances. === Bayesian control rule === A generalization of Thompson sampling to arbitrary dynamical environments and causal structures, known as Bayesian control rule, has been shown to be the optimal solution to the adaptive coding problem with actions and observations. In this formulation, an agent is conceptualized as a mixture over a set of behaviours. As the agent interacts with its environment, it learns the causal properties and adopts the behaviour that minimizes the relative entropy to the behaviour with the best prediction of the environment's behaviour. If these behaviours have been chosen according to the maximum expected utility principle, then the asymptotic behaviour of the Bayesian control rule matches the asymptotic behaviour of the perfectly rational agent. The setup is as follows. Let a 1 , a 2 , … , a T {\displaystyle a_{1},a_{2},\ldots ,a_{T}} be the actions issued by an agent up to time T {\displaystyle T} , and let o 1 , o 2 , … , o T {\displaystyle o_{1},o_{2},\ldots ,o_{T}} be the observations gathered by the agent up to time T {\displaystyle T} . Then, the agent issues the action a T + 1 {\displaystyle a_{T+1}} with probability: P ( a T + 1 | a ^ 1 : T , o 1 : T ) , {\displaystyle P(a_{T+1}|{\hat {a}}_{1:T},o_{1:T}),} where the "hat"-notation a ^ t {\displaystyle {\hat {a}}_{t}} denotes the fact that a t {\displaystyle a_{t}} is a causal intervention (see Causality), and not an ordinary observation. If the agent holds beliefs θ ∈ Θ {\displaystyle \theta \in \Theta } over its behaviors, then the Bayesian control rule becomes P ( a T + 1 | a ^ 1 : T , o 1 : T ) = ∫ Θ P ( a T + 1 | θ , a ^ 1 : T , o 1 : T ) P ( θ | a ^ 1 : T , o 1 : T ) d θ {\displaystyle P(a_{T+1}|{\hat {a}}_{1:T},o_{1:T})=\int _{\Theta }P(a_{T+1}|\theta ,{\hat {a}}_{1:T},o_{1:T})P(\theta |{\hat {a}}_{1:T},o_{1:T})\,d\theta } , where P ( θ | a ^ 1 : T , o 1 : T ) {\displaystyle P(\theta |{\hat {a}}_{1:T},o_{1:T})} is the posterior distribution over the parameter θ {\displaystyle \theta } given actions a 1 : T {\displaystyle a_{1:T}} and observations o 1 : T {\displaystyle o_{1:T}} . In practice, the Bayesian control amounts to sampling, at each time step, a parameter θ ∗ {\displaystyle \theta ^{\ast }} from the posterior distribution P ( θ | a ^ 1 : T , o 1 : T ) {\displaystyle P(\theta |{\hat {a}}_{1:T},o_{1:T})} , where the posterior distribution is computed using Bayes' rule by only considering the (causal) likelihoods of the observations o 1 , o 2 , … , o T {\displaystyle o_{1},o_{2},\ldots ,o_{T}} and ignoring the (causal) likelihoods of the actions a 1 , a 2 , … , a T {\displaystyle a_{1},a_{2},\ldots ,a_{T}} , and then by sampling the action a T + 1 ∗ {\displaystyle a_{T+1}^{\ast }} from the action distribution P ( a T + 1 | θ ∗ , a ^ 1 : T , o 1 : T ) {\displaystyle P(a_{T+1}|\theta ^{\ast },{\hat {a}}_{1:T},o_{1:T})} . === Upper-confidence-bound (UCB) algorithms === Thompson sampling and upper-confidence bound algorithms share a fundamental property that underlies many of their theoretical guarantees. Roughly speaking, both algorithms allocate exploratory effort to actions that might be optimal and are in this sense "optimistic". Leveraging this property, one can translate regret bounds established for UCB algorithms to Bayesian regret bounds for Thompson sampling or unify regret analysis across both these algorithms and many classes of problems.

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  The Eye of Mexico

  The Eye of Mexico (Spanish: El Ojo de México) is an outdoor sculpture in Mexico City. It is located in Ampliación Granada, Miguel Hidalgo, at the mixed-use development Neuchâtel Polanco, developed by the Canadian real estate company Ivanhoé Cambridge. The artwork was created by the Turkish artist Ferdi Alıcı and it was selected from among 350 proposals from artists from 35 countries. The project for The Eye of Mexico was developed by MIRA, a real estate investment and development company, and MASSIVart, a creative consulting agency. According to MIRA, upon its inauguration it became the first artwork in Latin America to use artificial intelligence (AI). The sculpture can read environmental and urban data using AI algorithms and transform the results into videos related to arts, science and technology. The ring was inaugurated on 20 May 2022 and it is 10 meters (33 ft) high and 3 meters (9.8 ft) wide.

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  PCVC Speech Dataset

  The PCVC (Persian Consonant Vowel Combination) Speech Dataset is a Modern Persian speech corpus for speech recognition and also speaker recognition. The dataset contains sound samples of Modern Persian combination of vowel and consonant phonemes from different speakers. Every sound sample contains just one consonant and one vowel So it is somehow labeled in phoneme level. This dataset consists of 23 Persian consonants and 6 vowels. The sound samples are all possible combinations of vowels and consonants (138 samples for each speaker). The sample rate of all speech samples is 48000 which means there are 48000 sound samples in every 1 second. Every sound sample starts with consonant then continues with vowel. In each sample, in average, 0.5 second of each sample is speech and the rest is silence. Each sound sample ends with silence. All of sound samples are denoised with "Adaptive noise reduction" algorithm. Compared to Farsdat speech dataset and Persian speech corpus it is more easy to use because it is prepared in .mat data files. Also it is more based on phoneme based separation and all samples are denoised. == Contents == The corpus is downloadable from its Kaggle web page, and contains the following: .mat data files of sound samples in a 23630000 matrix, in which 23 is number of consonants, 6 is the number of vowels and 30000 is the length of sound sample.

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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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  Orange (software)

  Orange is an open-source data visualization, machine learning and data mining toolkit. It features a visual programming front-end for exploratory qualitative data analysis and interactive data visualization. == Description == Orange is a component-based visual programming software package for data visualization, machine learning, data mining, and data analysis. Orange components are called widgets. They range from simple data visualization, subset selection, and preprocessing to empirical evaluation of learning algorithms and predictive modeling. Visual programming is implemented through an interface in which workflows are created by linking predefined or user-designed widgets, while advanced users can use Orange as a Python library for data manipulation and widget alteration. == Software == Orange is an open-source software package released under GPL and hosted on GitHub. Versions up to 3.0 include core components in C++ with wrappers in Python. From version 3.0 onwards, Orange uses common Python open-source libraries for scientific computing, such as numpy, scipy and scikit-learn, while its graphical user interface operates within the cross-platform Qt framework. The default installation includes a number of machine learning, preprocessing and data visualization algorithms in 6 widget sets (data, transform, visualize, model, evaluate and unsupervised). Additional functionalities are available as add-ons (text-mining, image analytics, bioinformatics, etc.). Orange is supported on macOS, Windows and Linux and can also be installed from the Python Package Index repository (pip install Orange3). == Features == Orange consists of a canvas interface onto which the user places widgets and creates a data analysis workflow. Widgets offer basic functionalities such as reading the data, showing a data table, selecting features, training predictors, comparing learning algorithms, visualizing data elements, etc. The user can interactively explore visualizations or feed the selected subset into other widgets. Canvas: graphical front-end for data analysis Widgets: Data: widgets for data input, data filtering, sampling, imputation, feature manipulation and feature selection Visualize: widgets for common visualization (box plot, histograms, scatter plot) and multivariate visualization (mosaic display, sieve diagram). Classify: a set of supervised machine learning algorithms for classification Regression: a set of supervised machine learning algorithms for regression Evaluate: cross-validation, sampling-based procedures, reliability estimation and scoring of prediction methods Unsupervised: unsupervised learning algorithms for clustering (k-means, hierarchical clustering) and data projection techniques (multidimensional scaling, principal component analysis, correspondence analysis). == Add-ons == Orange users can extend their core set of components with components in the add-ons. Supported add-ons include: Associate: components for mining frequent itemsets and association rule learning. Bioinformatics: components for gene expression analysis, enrichment, and access to expression databases (e.g., Gene Expression Omnibus) and pathway libraries. Data fusion: components for fusing different data sets, collective matrix factorization, and exploration of latent factors. Educational: components for teaching machine learning concepts, such as k-means clustering, polynomial regression, stochastic gradient descent, ... Explain: provides an extension with components for the model explanation, including Shapley value analysis Geo: components for working with geospatial data. Image analytics: components for working with images and ImageNet embeddings Network: components for graph and network analysis. Text mining: components for natural language processing and text mining. Time series: widget components for time series analysis and modeling. Single-cell: support for single-cell gene expression analysis, including components for loading single-cell data, filtering and batch effect removal, marker genes discovery, scoring of cells and genes, and cell type prediction. Spectroscopy: components for analyzing and visualization of (hyper)spectral datasets. Survival analysis: add-on for data analysis dealing with survival data. It includes widgets for standard survival analysis techniques, such as the Kaplan-Meier plot, the Cox regression model, and several derivative widgets. World Happiness: support for downloading socioeconomic data from a database, including OECD and World Development Indicators. Provides access to thousands of country indicators from various economic databases. Fairness: add-on for evaluation and creation of fair machine learning models without discrimination. Widgets range from computing fairness metrics like statistical parity to post-, pre-, in-processing methods to build fair models. == Objectives == The program provides a platform for experiment selection, recommendation systems, and predictive modelling and is used in biomedicine, bioinformatics, genomic research, and teaching. In science, it is used as a platform for testing new machine learning algorithms and for implementing new techniques in genetics and bioinformatics. In education, it was used for teaching machine learning and data mining methods to students of biology, biomedicine, and informatics. == Extensions == Various projects build on Orange either by extending the core components with add-ons or using only the Orange Canvas to exploit the implemented visual programming features and GUI. OASYS — ORange SYnchrotron Suite scOrange — single cell biostatistics Quasar — data analysis in natural sciences == History == In 1996, the University of Ljubljana and Jožef Stefan Institute started development of ML, a machine learning framework in C++, and Python bindings were developed for this framework in 1997, which, together with emerging Python modules, formed a joint framework called Orange. Over the following years, most contemporary major algorithms for data mining and machine learning were implemented in C++ (Orange's core) or Python modules. In 2002, first prototypes to create a flexible graphical user interface were designed using Pmw Python megawidgets. In 2003, the graphical user interface was redesigned and re-developed for Qt framework using PyQt Python bindings. The visual programming framework was defined, and the development of widgets (graphical components of the data analysis pipeline) began. In 2005, extensions for data analysis in bioinformatics was created. In 2008, Mac OS X DMG and Fink-based installation packages were developed. In 2009, over 100 widgets were created and maintained. In 2009, Orange 2.0 beta was released, offering installation packages on the website based on the daily compiling cycle. In 2012, a new object hierarchy was imposed, replacing the old module-based structure. In 2013, a significant redesign of the graphical user interface included a new toolbox and depiction of workflows. In 2015, Orange 3.0 was released. Orange stores the data in NumPy arrays; machine learning algorithms mostly use scikit-learn. In 2015, a text analysis add-on for Orange3 was released. In 2016, Orange released version 3.3. Development scheduled a monthly cycle for stable releases. In 2016, Orange began development and release of an Image Analytics add-on, with server-side deep neural networks for image embedding In 2017, a Spectroscopy add-on for the analysis of spectral data was introduced. In 2017, Geo, an add-on for dealing with geo-location data and visualisation of geo maps was introduced In 2018, Orange began development and release of an add-on for single-cell data analysis. In 2019, Orange separated its graphical interface for development as a separate project, orange-canvas-core In 2020, Orange introduced the Explain add-on with widgets for explaining classification models and regression models, highlighting the strength and contributions specific features make towards predicting a specific class. In 2022, World Happiness, an add-on for the Orange3 data mining suite, was introduced, providing widgets for accessing socioeconomic data from various databases such as World Happiness Report, World Development Indicators, OECD. In 2022, Orange extended the Explain add-on with an Individual Conditional Expectation plot and the Permutation Feature Importance technique. In 2023, Orange introduced the Fairness add-on, including widgets to calculate bias metrics, as well as widgets for pre-, post-, and in-processing methods, allowing the creation of models less susceptible to systematic error due to the vagaries of the data set.

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  Sugeno integral

  In mathematics, the Sugeno integral, introduced by Michio Sugeno as a fuzzy integral in work on fuzzy measures at the Tokyo Institute of Technology, is a type of integral with respect to a fuzzy measure. Let ( X , Ω ) {\displaystyle (X,\Omega )} be a measurable space and let h : X → [ 0 , 1 ] {\displaystyle h:X\to [0,1]} be an Ω {\displaystyle \Omega } -measurable function. The Sugeno integral over the crisp set A ⊆ X {\displaystyle A\subseteq X} of the function h {\displaystyle h} with respect to the fuzzy measure g {\displaystyle g} is defined by: ∫ A h ( x ) ∘ g = sup E ⊆ X [ min ( min x ∈ E h ( x ) , g ( A ∩ E ) ) ] = sup α ∈ [ 0 , 1 ] [ min ( α , g ( A ∩ F α ) ) ] {\displaystyle \int _{A}h(x)\circ g={\sup _{E\subseteq X}}\left[\min \left(\min _{x\in E}h(x),g(A\cap E)\right)\right]={\sup _{\alpha \in [0,1]}}\left[\min \left(\alpha ,g(A\cap F_{\alpha })\right)\right]} where F α = { x | h ( x ) ≥ α } {\displaystyle F_{\alpha }=\left\{x|h(x)\geq \alpha \right\}} . The Sugeno integral over the fuzzy set A ~ {\displaystyle {\tilde {A}}} of the function h {\displaystyle h} with respect to the fuzzy measure g {\displaystyle g} is defined by: ∫ A h ( x ) ∘ g = ∫ X [ h A ( x ) ∧ h ( x ) ] ∘ g {\displaystyle \int _{A}h(x)\circ g=\int _{X}\left[h_{A}(x)\wedge h(x)\right]\circ g} where h A ( x ) {\displaystyle h_{A}(x)} is the membership function of the fuzzy set A ~ {\displaystyle {\tilde {A}}} . == Usage and Relationships == Sugeno integral is related to h-index.

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  Nagarik App

  Nagarik App (translation: Citizen App) is a mobile application launched by the Government of Nepal to provide government-related services in a single online platform. The app was developed to facilitate an easier, systematic, and simplified delivery of government services to Nepali citizens digitally. The app was launched to play a pivotal role in revolutionizing the way citizens interact with the government. It offers government services through a single unified platform, minimizing the need for citizens to navigate multiple channels or physical offices for their diverse needs of government services. The services are added gradually according to the needs and services required. The government aims to reduce the physical queues and the need to be physically present to get services from the different government offices. One can get services online round-the-clock even during holidays. As of now, 25 services are included in the app, ranging from Police Clearance Report to Voters Card. The app contains and provides a vast range of government services. The app was launched on the occasion of the fourth National Information and Communication Technology Day, 2021 (2078 BS). The event marked a significant milestone in Nepal’s digital transformation journey. It aims to reduce all the bureaucratic hurdles that the citizens have been facing and make government services more efficient and convenient. In Oct 20, 2024, a E-Chalan was introduced for managing traffic violations in initially piloting in Kathmandu Valley. The Kathmandu Valley Traffic Police Office announced that physical licenses would no longer be confiscated for traffic rule violations. Instead, a "Digital Chit (E-Chalan)" system was implemented, allowing drivers to pay fines electronically. Integrated with the NagarikApp, the system enables police to access drivers' licenses, record violations, and update details directly in the app. == Features and Services == Inland Revenue Department (Nepal) PAN Registration Election Commission (Nepal) Voter Card Pre-Registration and Details Nepal Police Online Clearance Report Traffic Violations and Fine Payment Nepal Passport, Driving License, National Identity Card (NID), Citizenship, and Voter ID link details My Municipality (Includes contact info of the representatives, services such as ambulance, nearby police, and budget programs and plans) The Government Press ID card PF/PAN/SST/CIT statements can be viewed Nagarik Pahichan Dwar (Online bank accounts can be opened and KYC can be verified for selected banks using the QR) == Awards and honors == Each year, World Summit Award honors outstanding digital applications and solutions across various categories. The winners of the World Summit Award represent the pinnacle of innovation in their respective categories. Nagarik App was selected among 180 participants and won the World Summit Award of 2022 in Government and Citizen Engagement category. == Latest Statistics & Usage Trends (2082 BS / 2025 AD) == As of August 2025, over 1.5 million Nepali citizens have registered and actively use the Nagarik App, according to the National Information Technology Center (NITC). The majority of daily logins come from: Kathmandu Valley – 37% of total users Province 1 (Koshi) – 19% of total users Bagmati Province – 15% of total users On average, 45,000+ transactions (service requests, document verifications, and payments) are processed through the app each day. The most-used services include: PAN Card Registration – 28% of total requests Police Clearance Report – 22% Driving License Linking & E-Chalan Payment – 18% Vehicle Tax Payment – 14% Source: Internal report from NITC, July 2025 == Step-by-Step: How to Link Your Driving License with Nagarik App == Update the App – Install the latest version from Play Store or App Store. Login or Register – Ensure your SIM is registered in your own name. Go to “Transport Services” in the menu. Select “Driving License” – Enter your license number and date of birth. Verify via OTP – Sent to your registered mobile number. Confirmation – Your digital license will appear inside the app. This guide is continuously updated to reflect the latest rules from the Kathmandu Valley Traffic Police Office and changes in NITC’s backend system. For in-depth details, step-by-step tutorials, and the most recent Nagarik App updates, visit the full article on The Bipin Blog.

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  Type-2 fuzzy sets and systems

  Type-2 fuzzy sets and systems generalize standard type-1 fuzzy sets and systems so that more uncertainty can be handled. From the beginning of fuzzy sets, criticism was made about the fact that the membership function of a type-1 fuzzy set has no uncertainty associated with it, something that seems to contradict the word fuzzy, since that word has the connotation of much uncertainty. So, what does one do when there is uncertainty about the value of the membership function? The answer to this question was provided in 1975 by the inventor of fuzzy sets, Lotfi A. Zadeh, when he proposed more sophisticated kinds of fuzzy sets, the first of which he called a "type-2 fuzzy set". A type-2 fuzzy set lets us incorporate uncertainty about the membership function into fuzzy set theory, and is a way to address the above criticism of type-1 fuzzy sets head-on. And, if there is no uncertainty, then a type-2 fuzzy set reduces to a type-1 fuzzy set, which is analogous to probability reducing to determinism when unpredictability vanishes. Type1 fuzzy systems are working with a fixed membership function, while in type-2 fuzzy systems the membership function is fluctuating. A fuzzy set determines how input values are converted into fuzzy variables. == Overview == In order to symbolically distinguish between a type-1 fuzzy set and a type-2 fuzzy set, a tilde symbol is put over the symbol for the fuzzy set; so, A denotes a type-1 fuzzy set, whereas à denotes the comparable type-2 fuzzy set. When the latter is done, the resulting type-2 fuzzy set is called a "general type-2 fuzzy set" (to distinguish it from the special interval type-2 fuzzy set). Zadeh didn't stop with type-2 fuzzy sets, because in that 1976 paper he also generalized all of this to type-n fuzzy sets. The present article focuses only on type-2 fuzzy sets because they are the next step in the logical progression from type-1 to type-n fuzzy sets, where n = 1, 2, ... . Although some researchers are beginning to explore higher than type-2 fuzzy sets, as of early 2009, this work is in its infancy. The membership function of a general type-2 fuzzy set, Ã, is three-dimensional (Fig. 1), where the third dimension is the value of the membership function at each point on its two-dimensional domain that is called its "footprint of uncertainty"(FOU). For an interval type-2 fuzzy set that third-dimension value is the same (e.g., 1) everywhere, which means that no new information is contained in the third dimension of an interval type-2 fuzzy set. So, for such a set, the third dimension is ignored, and only the FOU is used to describe it. It is for this reason that an interval type-2 fuzzy set is sometimes called a first-order uncertainty fuzzy set model, whereas a general type-2 fuzzy set (with its useful third-dimension) is sometimes referred to as a second-order uncertainty fuzzy set model. The FOU represents the blurring of a type-1 membership function, and is completely described by its two bounding functions (Fig. 2), a lower membership function (LMF) and an upper membership function (UMF), both of which are type-1 fuzzy sets! Consequently, it is possible to use type-1 fuzzy set mathematics to characterize and work with interval type-2 fuzzy sets. This means that engineers and scientists who already know type-1 fuzzy sets will not have to invest a lot of time learning about general type-2 fuzzy set mathematics in order to understand and use interval type-2 fuzzy sets. Work on type-2 fuzzy sets languished during the 1980s and early-to-mid 1990s, although a small number of articles were published about them. People were still trying to figure out what to do with type-1 fuzzy sets, so even though Zadeh proposed type-2 fuzzy sets in 1976, the time was not right for researchers to drop what they were doing with type-1 fuzzy sets to focus on type-2 fuzzy sets. This changed in the latter part of the 1990s as a result of Jerry Mendel and his student's works on type-2 fuzzy sets and systems. Since then, more researchers around the world are writing articles about type-2 fuzzy sets and systems. == Interval type-2 fuzzy sets == Interval type-2 fuzzy sets have received the most attention because the mathematics that is needed for such sets—primarily Interval arithmetic—is much simpler than the mathematics that is needed for general type-2 fuzzy sets. The literature about interval type-2 fuzzy sets is large, whereas the literature about general type-2 fuzzy sets is much smaller. Both kinds of fuzzy sets are being actively researched by an ever-growing number of researchers around the world and have resulted in successful employment in a variety of domains such as robot control. Formally, the following have already been worked out for interval type-2 fuzzy sets: Fuzzy set operations: union, intersection and complement Centroid (a very widely used operation by practitioners of such sets, and also an important uncertainty measure for them) Other uncertainty measures [fuzziness, cardinality, variance and skewness and uncertainty bounds Similarity Subsethood Embedded fuzzy sets Fuzzy set ranking Fuzzy rule ranking and selection Type-reduction methods Firing intervals for an interval type-2 fuzzy logic system Fuzzy weighted average Linguistic weighted average Synthesizing an FOU from data that are collected from a group of subject == Interval type-2 fuzzy logic systems == Type-2 fuzzy sets are finding very wide applicability in rule-based fuzzy logic systems (FLSs) because they let uncertainties be modeled by them whereas such uncertainties cannot be modeled by type-1 fuzzy sets. A block diagram of a type-2 FLS is depicted in Fig. 3. This kind of FLS is used in fuzzy logic control, fuzzy logic signal processing, rule-based classification, etc., and is sometimes referred to as a function approximation application of fuzzy sets, because the FLS is designed to minimize an error function. The following discussions, about the four components in Fig. 3 rule-based FLS, are given for an interval type-2 FLS, because to-date they are the most popular kind of type-2 FLS; however, most of the discussions are also applicable for a general type-2 FLS. Rules, that are either provided by subject experts or are extracted from numerical data, are expressed as a collection of IF-THEN statements, e.g., IF temperature is moderate and pressure is high, then rotate the valve a bit to the right. Fuzzy sets are associated with the terms that appear in the antecedents (IF-part) or consequents (THEN-part) of rules, and with the inputs to and the outputs of the FLS. Membership functions are used to describe these fuzzy sets, and in a type-1 FLS they are all type-1 fuzzy sets, whereas in an interval type-2 FLS at least one membership function is an interval type-2 fuzzy set. An interval type-2 FLS lets any one or all of the following kinds of uncertainties be quantified: Words that are used in antecedents and consequents of rules—because words can mean different things to different people. Uncertain consequents—because when rules are obtained from a group of experts, consequents will often be different for the same rule, i.e. the experts will not necessarily be in agreement. Membership function parameters—because when those parameters are optimized using uncertain (noisy) training data, the parameters become uncertain. Noisy measurements—because very often it is such measurements that activate the FLS. In Fig. 3, measured (crisp) inputs are first transformed into fuzzy sets in the Fuzzifier block because it is fuzzy sets and not numbers that activate the rules which are described in terms of fuzzy sets and not numbers. Three kinds of fuzzifiers are possible in an interval type-2 FLS. When measurements are: Perfect, they are modeled as a crisp set; Noisy, but the noise is stationary, they are modeled as a type-1 fuzzy set; and, Noisy, but the noise is non-stationary, they are modeled as an interval type-2 fuzzy set (this latter kind of fuzzification cannot be done in a type-1 FLS). In Fig. 3, after measurements are fuzzified, the resulting input fuzzy sets are mapped into fuzzy output sets by the Inference block. This is accomplished by first quantifying each rule using fuzzy set theory, and by then using the mathematics of fuzzy sets to establish the output of each rule, with the help of an inference mechanism. If there are M rules then the fuzzy input sets to the Inference block will activate only a subset of those rules, where the subset contains at least one rule and usually way fewer than M rules. The inference is done one rule at a time. So, at the output of the Inference block, there will be one or more fired-rule fuzzy output sets. In most engineering applications of an FLS, a number (and not a fuzzy set) is needed as its final output, e.g., the consequent of the rule given above is "Rotate the valve a bit to the right." No automatic valve will know what this means because "a bit to the right" is a linguistic expression, and a valv

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  Construction of t-norms

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