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Nextcloud

Nextcloud is a modular workspace platform designed to provide teams and businesses with a comprehensive environment for digital collaboration. Beyond central data management, it integrates office suites like Collabora Online and EuroOffice office suites. for seamless, cooperative workflows. The platform features built-in tools for chat, videoconferencing, and a privacy-focused AI assistant capable of running entirely on local LLMs. Supported by a rich ecosystem of apps, it can be hosted in the cloud or on premises and can scale up to millions of users. It has been translated into over 100 languages. == Features == Nextcloud files are stored in conventional directory structures, accessible via WebDAV if necessary. A SQLite, MySQL/MariaDB or PostgreSQL database is required to provide additional functionality like permissions, shares, and comments. Nextcloud can synchronize with local clients running Windows (Windows 8.1 and above), macOS (10.14 or later), Linux and FreeBSD. Nextcloud permits user and group administration locally or via different backends like OpenID or LDAP. Content can be shared inside the system by defining granular read/write permissions between users and groups. Nextcloud users can create public URLs when sharing files. Logging of file-related actions, as well as disallowing access based on file access rules is also available. Security options like brute-force protection and multi-factor authentication using TOTP, WebAuthn, Oauth2, and OpenID Connect are available. Nextcloud has planned new features such as monitoring capabilities, full-text search and Kerberos authentication, as well as audio/video conferencing, expanded federation and smaller user interface improvements. == History == In April 2016 Frank Karlitschek and most core contributors left ownCloud Inc. These included some of ownCloud's staff according to sources near to the ownCloud community. Karlitschek and many of these contributors went on to fork ownCloud, creating Nextcloud. The fork was preceded by a blog post of Karlitschek announcing his departure and raising questions about the management of the ownCloud, its community, and priorities between growth, money, and sustainability. There have been no official statements about the reason for the fork. However, Karlitschek mentioned the fork several times in a talk at the 2018 FOSDEM conference and in two appearances on the FLOSS Weekly podcast, emphasizing cultural mismatch between open source developers and business oriented people not used to the open source community. On June 2, within 12 hours of the announcement of the fork, the American entity "ownCloud Inc." announced that it is shutting down with immediate effect, stating that "[...] main lenders in the US have cancelled our credit. Following American law, we are forced to close the doors of ownCloud, Inc. with immediate effect and terminate the contracts of 8 employees." ownCloud Inc. accused Karlitschek of poaching developers, while Nextcloud developers such as Arthur Schiwon stated that he "decided to quit because not everything in the ownCloud Inc. company world evolved as I imagined". ownCloud GmbH continued operations, secured financing from new investors and took over the business of ownCloud Inc. In April 2018 Informationstechnikzentrum Bund (ITZBund) reported Nextcloud won the tender for "Bundescloud" (Germany government cloud) project. In August 2019 it was announced that the governments of France, Sweden and the Netherlands would use Nextcloud for file transfer. In January 2020 Nextcloud 18 "Nextcloud Hub" was released. The major change was direct integration with an Office suite (OnlyOffice) and Nextcloud announced that their goal was to compete with Office 365 and Google Docs. A partnership with Ionos was revealed – its hosting location in Germany and compliance with GDPR should support the goal of data sovereignty. In spring 2020 remote work and web conferencing usage increased due to the COVID-19 pandemic and Nextcloud released version 19 with chat and videoconferencing Talk app integrated into the application core. Communication with an optional "high performance back-end" allows self-hosting of web conferences with more than 10 participants. Collabora Online was introduced as another integrated office suite. In August 2021 Nextcloud was chosen as a collaboration platform for European cloud software GAIA-X. In a September 2021 European Commission report it was mentioned as "the most widely deployed Open Source content collaboration platform" Following the 2025 United States tariffs against the European Union, fear of overreliance on US cloud providers such as Microsoft 365 and Google Workspace increased, with Nextcloud being one of the foremost contenders to replace them. Some governmental organisations including the European Data Protection Supervisor and the German state of Schleswig-Holstein have since switched from Microsoft's Sharepoint to Nextcloud. According to Nextcloud, during the first 5 months of 2025, customer interest in the software had tripled.
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Krzysztof Wołk

Krzysztof Wołk (born 16 August 1986) is a Polish IT researcher who specializes in artificial intelligence, machine learning, mobile applications, linguistic engineering, multimedia, NLP and graphic applications. His research works have been cited in more than 70 international research journals, books and research papers. He is member of scientific committee at the Health and Social Care Information Systems and Technologies (HCist), an international conference which brings in new ideas, new technologies, academic scientists, healthcare IT professionals, managers and solution providers from all over the world. His research in statistical machine learning has been recognized as one of the most cited researches in the world. He is the member of Scientific Committee-Reviewers at Research Conference in Technical Disciplines (RCITD), based in Slovakia, which brings together the academic scientists and researchers from all around the world. == Biography == He obtained the doctorate degree in 2016 from the Polish-Japanese Academy of Information and Technology in Warsaw, Poland. He is currently working as researcher and assistant professor at the Polish-Japanese Computer Science Academy (PJATK) in Warsaw, Poland. == Achievements == He has published three books: Biblia Windows Server 2012, Administrator's Guide, Mac OS X Server 10.8, and MAC OS X Server 10.6 and 10.7 Practical Guide has been cited by many researchers in the scholarly books, research journals and articles. His research work on the Polish-English statistical machine translation has been featured in the book New Research in Multimedia and Internet System. Similarly, his works regarding the machine translation system have been featured in the books New Perspective in Information System and Technologies Volume 1, Multimedia and Network Information System, and Recent Advances in Information Systems and Technologies, Volume 1.
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Top 10 AI Logo Makers Compared (2026)

In search of the best AI logo maker? An AI logo maker 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 logo maker slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.
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Interacting particle system

In probability theory, an interacting particle system (IPS) is a stochastic process ( X ( t ) ) t ∈ R + {\displaystyle (X(t))_{t\in \mathbb {R} ^{+}}} on some configuration space Ω = S G {\displaystyle \Omega =S^{G}} given by a site space, a countably-infinite-order graph G {\displaystyle G} and a local state space, a compact metric space S {\displaystyle S} . More precisely IPS are continuous-time Markov jump processes describing the collective behavior of stochastically interacting components. IPS are the continuous-time analogue of stochastic cellular automata. Among the main examples are the voter model, the contact process, the asymmetric simple exclusion process (ASEP), the Glauber dynamics and in particular the stochastic Ising model. IPS are usually defined via their Markov generator giving rise to a unique Markov process using Markov semigroups and the Hille-Yosida theorem. The generator again is given via so-called transition rates c Λ ( η , ξ ) > 0 {\displaystyle c_{\Lambda }(\eta ,\xi )>0} where Λ ⊂ G {\displaystyle \Lambda \subset G} is a finite set of sites and η , ξ ∈ Ω {\displaystyle \eta ,\xi \in \Omega } with η i = ξ i {\displaystyle \eta _{i}=\xi _{i}} for all i ∉ Λ {\displaystyle i\notin \Lambda } . The rates describe exponential waiting times of the process to jump from configuration η {\displaystyle \eta } into configuration ξ {\displaystyle \xi } . More generally the transition rates are given in form of a finite measure c Λ ( η , d ξ ) {\displaystyle c_{\Lambda }(\eta ,d\xi )} on S Λ {\displaystyle S^{\Lambda }} . The generator L {\displaystyle L} of an IPS has the following form. First, the domain of L {\displaystyle L} is a subset of the space of "observables", that is, the set of real valued continuous functions on the configuration space Ω {\displaystyle \Omega } . Then for any observable f {\displaystyle f} in the domain of L {\displaystyle L} , one has L f ( η ) = ∑ Λ ∫ ξ : ξ Λ c = η Λ c c Λ ( η , d ξ ) [ f ( ξ ) − f ( η ) ] {\displaystyle Lf(\eta )=\sum _{\Lambda }\int _{\xi :\xi _{\Lambda ^{c}}=\eta _{\Lambda ^{c}}}c_{\Lambda }(\eta ,d\xi )[f(\xi )-f(\eta )]} . For example, for the stochastic Ising model we have G = Z d {\displaystyle G=\mathbb {Z} ^{d}} , S = { − 1 , + 1 } {\displaystyle S=\{-1,+1\}} , c Λ = 0 {\displaystyle c_{\Lambda }=0} if Λ ≠ { i } {\displaystyle \Lambda \neq \{i\}} for some i ∈ G {\displaystyle i\in G} and c i ( η , η i ) = exp ⁡ [ − β ∑ j : | j − i | = 1 η i η j ] {\displaystyle c_{i}(\eta ,\eta ^{i})=\exp[-\beta \sum _{j:|j-i|=1}\eta _{i}\eta _{j}]} where η i {\displaystyle \eta ^{i}} is the configuration equal to η {\displaystyle \eta } except it is flipped at site i {\displaystyle i} . β {\displaystyle \beta } is a new parameter modeling the inverse temperature. == The Voter model == The voter model (usually in continuous time, but there are discrete versions as well) is a process similar to the contact process. In this process η ( x ) {\displaystyle \eta (x)} is taken to represent a voter's attitude on a particular topic. Voters reconsider their opinions at times distributed according to independent exponential random variables (this gives a Poisson process locally – note that there are in general infinitely many voters so no global Poisson process can be used). At times of reconsideration, a voter chooses one neighbor uniformly from amongst all neighbors and takes that neighbor's opinion. One can generalize the process by allowing the picking of neighbors to be something other than uniform. === Discrete time process === In the discrete time voter model in one dimension, ξ t ( x ) : Z → { 0 , 1 } {\displaystyle \xi _{t}(x):\mathbb {Z} \to \{0,1\}} represents the state of particle x {\displaystyle x} at time t {\displaystyle t} . Informally each individual is arranged on a line and can "see" other individuals that are within a radius, r {\displaystyle r} . If more than a certain proportion, θ {\displaystyle \theta } of these people disagree then the individual changes her attitude, otherwise she keeps it the same. Durrett and Steif (1993) and Steif (1994) show that for large radii there is a critical value θ c {\displaystyle \theta _{c}} such that if θ > θ c {\displaystyle \theta >\theta _{c}} most individuals never change, and for θ ∈ ( 1 / 2 , θ c ) {\displaystyle \theta \in (1/2,\theta _{c})} in the limit most sites agree. (Both of these results assume the probability of ξ 0 ( x ) = 1 {\displaystyle \xi _{0}(x)=1} is one half.) This process has a natural generalization to more dimensions, some results for this are discussed in Durrett and Steif (1993). === Continuous time process === The continuous time process is similar in that it imagines each individual has a belief at a time and changes it based on the attitudes of its neighbors. The process is described informally by Liggett (1985, 226), "Periodically (i.e., at independent exponential times), an individual reassesses his view in a rather simple way: he chooses a 'friend' at random with certain probabilities and adopts his position." A model was constructed with this interpretation by Holley and Liggett (1975). This process is equivalent to a process first suggested by Clifford and Sudbury (1973) where animals are in conflict over territory and are equally matched. A site is selected to be invaded by a neighbor at a given time.
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Robomart

Robomart is an American technology company headquartered in Santa Monica, California that builds autonomous smart shops for cafes, ice cream parlors, and quick-service restaurants. The company’s white label platform gives retailers the option to expand their footprint at a significantly lower cost than traditional brick-and-mortar real-estate. Robomarts are equipped with a proprietary checkout-free system, temperature controlled compartments, sensors for autonomous operation, and external cameras for added security. The company licenses its technology and white label applications to retailers who manage their fleet of stores and deploy them to their consumers’ locations. After consumers have taken goods from the robomart, their order is automatically calculated, their card on file is charged and they are sent a receipt. The company has announced partnerships with Unilever, Mars, and Fatty Mart. == History == Robomart was founded by Ali Ahmed, Tigran Shahverdyan, and Emad Suhail Rahim. The company debuted at CES 2018 where it unveiled its concept of a self-driving store. At GITEX 2018 the company presented its first functional prototype of a fully driverless Robomart. At the 2019 Consumer Electronics Show the company demonstrated the technology behind its autonomous stores and checkout-free shopping experience. In January 2019, Robomart announced its first partnership with U.S. grocery chain Stop & Shop to test its driverless stores. In December 2020, Robomart deployed the Pharmacy Robomart in a trial in West Hollywood. In June 2021, the company launched its commercial service with a fleet of Pharmacy and Snacks Robomarts operating within West Hollywood and Central Hollywood. In August 2023, Robomart announced a $2 million seed round, putting its to-date funding at $3.4 million. == Partnerships == In September 2019, Robomart partnered with Avery Dennison to source the RFID tags used to enable its checkout-free shopping experience. In December 2020, Robomart partnered with Zeeba Vans to provide vehicles for its growing fleet. In June 2021, Robomart partnered with REEF Technology to provide inventory management and restocking services. In addition, REEF's Light Speed grocery division serves as the first merchant selling products through Robomart. == Products == The company currently offers three Robomart types. The frozen Robomart that stocks ice cream, the refrigerated Robomart that stocks perishable foods, and the ambient Robomart that stocks shelf-stable goods.
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AI Content Generators Reviews: What Actually Works in 2026

In search of the best AI content generator? An AI content generator 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 content generator slots into your workflow and pays for itself fast. Below we compare features, pricing, and real output so you can choose with confidence.
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Top 10 AI Art Generators Compared (2026)

Shopping for the best AI art generator? An AI art generator is software that uses machine learning to help you get more done — it keeps getting smarter as the underlying models improve. Pricing, accuracy, and the size of the model behind the tool are the three factors that most affect daily usefulness. Whether you are a beginner or a pro, the right AI art generator slots into your workflow and pays for itself fast. Below we compare features, pricing, and real output so you can choose with confidence.
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AI Sales Assistants: Free vs Paid (2026)

Trying to pick the best AI sales assistant? An AI sales assistant is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI sales assistant slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.
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Inauthentic text

An inauthentic text is a computer-generated expository document meant to appear as genuine, but which is actually meaningless. Frequently they are created in order to be intermixed with genuine documents and thus manipulate the results of search engines, as with Spam blogs. They are also carried along in email in order to fool spam filters by giving the spam the superficial characteristics of legitimate text. Sometimes nonsensical documents are created with computer assistance for humorous effect, as with Dissociated press or Flarf poetry. They have also been used to challenge the veracity of a publication—MIT students submitted papers generated by a computer program called SCIgen to a conference, where they were initially accepted. This led the students to claim that the bar for submissions was too low. With the amount of computer generated text outpacing the ability of people to humans to curate it, there needs some means of distinguishing between the two. Yet automated approaches to determining absolutely whether a text is authentic or not face intrinsic challenges of semantics. Noam Chomsky coined the phrase "Colorless green ideas sleep furiously" giving an example of grammatically correct, but semantically incoherent sentence; some will point out that in certain contexts one could give this sentence (or any phrase) meaning. The first group to use the expression in this regard can be found below from Indiana University. Their work explains in detail an attempt to detect inauthentic texts and identify pernicious problems of inauthentic texts in cyberspace. The site has a means of submitting text that assesses, based on supervised learning, whether a corpus is inauthentic or not. Many users have submitted incorrect types of data and have correspondingly commented on the scores. This application is meant for a specific kind of data; therefore, submitting, say, an email, will not return a meaningful score.
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Bernard Vauquois

Bernard Vauquois ((1929-06-14)June 14, 1929 — (1985-09-30)September 30, 1985) was a French mathematician and computer scientist. He was a pioneer of computer science and machine translation (MT) in France. An astronomer-turned-computer scientist, he is known for his work on the programming language ALGOL 60, and later for extensive work on the theoretical and practical problems of MT, of which the eponymous Vauquois triangle is one of the most widely-known contributions. He was a professor at what would become the Grenoble Alpes University. == Biography == Bernard Vauquois was initially a researcher at French National Centre for Scientific Research (CNRS) from 1952 to 1958 at the Astrophysics Institute of the Meudon Observatory, after completing studies in mathematics, physics, and astronomy. Since 1957, his research program has also focused on methods applied to physics from the perspective of electronic computers, and he has taught programming to physicists. This double interest in astrophysics and electronic computers is reflected in the subject of his thesis and that of the complementary thesis in physical sciences that he defended in 1958. In 1960, at 31 years old, he was appointed professor of computer science at Grenoble University, where, alongside professors Jean Kuntzmann and Noël Gastinel, he began work in the field. At that time, he was also contributing to the definition of the language ALGOL 60. Also in 1960, he founded the Centre d'Étude pour la Traduction Automatique (CETA), later renamed as Groupe d'Étude pour la Traduction Automatique (GETA) and currently known as GETALP, a team at the Laboratoire d'informatique de Grenoble, and soon showed his gift for rapid understanding, synthesis, and innovation, and his taste for personal communication across linguistic borders and barriers. After visiting a number of centers, mainly in the United States, where machine translation research was conducted, he analyzed the shortcomings of the "first-generation" approach and evaluated the potential of a new generation based on grammar and formal language theory, and proposed a new approach based on a representational "pivot" and the use of (declarative) rule systems that transform a sequential sentence from one level of representation to another. He led the GETA in constructing the first large second-generation system, applied to Russian–French, from 1962 to 1971. At the end of this period, the accumulated experience led him to correct some defects of the "pure" declarative and interlingual approach, and to use heuristic programming methods, implemented with procedural grammars written in LSPLs ("specialized languages for linguistic programming", langages spécialisés pour la programmation linguistique) that were developed under his direction, and integrated into the ARIANE-78 machine translation system. In 1974, when he cofounded the Leibniz laboratory, he proposed "multilevel structure descriptors" (descripteurs de structures multiniveaux) for units larger than sentence translation. This idea, premonitory of later theoretical work (Ray Jackendoff, Gerald Gazdar) is still the cornerstone of all machine translation software built by GETA and the French national TA project. Bernard Vauquois' last contribution was "static grammar" (grammaire statique) in 1982–83, during the ESOPE project, the preparatory phase of the French national MT project. He was a key figure in the field of computational linguistics in France. At CNRS, he was a member of section 22 of the National Committee in 1963: "General Linguistics, Modern Languages and Comparative Literature", and then, in 1969, of section 28: "General Linguistics, Foreign Languages and Literature". Since 1965, he has been vice-president of the Association for Natural Language Processing (ATALA). He was its president from 1966 to 1971. He was also one of the founders, in 1965, of the ICCL (International Committee on Computational Linguistics), which organizes COLING conferences. He was its president from 1969 to 1984. From France, he often collaborated with other countries (notably Canada, the United States, the USSR, Czechoslovakia, Japan, China, Brazil, Malaysia, and Thailand), working on the specification and implementation of grammars and dictionaries. He began cooperating with Malaysia, for example, in 1979, which led to the creation of the Automatic Terjemaan Project, with a first prototype of an English-Malay MT system demonstrated in 1980. == Vauquois triangle == The Vauquois triangle is a conceptual model and diagram illustrating possible approaches to the design of machine translation systems, first proposed in 1968. == Legacy == Bernard Vauquois is regarded as a pioneer of machine translation in France. He played a key role in developing the first large-scale second-generation machine translation system, and his work influenced the field of machine translation for many years. He supervised some twenty doctoral theses, most of them concerning formal aspects of natural and artificial languages, with an emphasis on machine translation. The Center for Studies on Automatic Translation, which Vauquois founded in 1960, later became the Group for the Study of Machine Translation and Automated Processing of Languages and Speech (GETALP). It is still a research institution in natural language processing. Vauquois was a prolific writer and speaker, disseminating knowledge about machine translation and related topics. His papers and presentations were instrumental in establishing the field of machine translation in France and beyond. == Publications == Vauquois, Bernard (1973). Traduction automatique (in French). Paris: Gauthier-Villars. Vauquois, Bernard (1967). Introduction à la traduction automatique (in French). Paris: Gauthier-Villars.
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Additive smoothing

In statistics, additive smoothing, also called Laplace smoothing or Lidstone smoothing, is a technique used to smooth count data, eliminating issues caused by certain values having 0 occurrences. Given a set of observation counts x = ⟨ x 1 , x 2 , … , x d ⟩ {\displaystyle \mathbf {x} =\langle x_{1},x_{2},\ldots ,x_{d}\rangle } from a d {\displaystyle d} -dimensional multinomial distribution with N {\displaystyle N} trials, a "smoothed" version of the counts gives the estimator θ ^ i = x i + α N + α d ( i = 1 , … , d ) , {\displaystyle {\hat {\theta }}_{i}={\frac {x_{i}+\alpha }{N+\alpha d}}\qquad (i=1,\ldots ,d),} where the smoothed count x ^ i = N θ ^ i {\displaystyle {\hat {x}}_{i}=N{\hat {\theta }}_{i}} , and the "pseudocount" α > 0 is a smoothing parameter, with α = 0 corresponding to no smoothing (this parameter is explained in § Pseudocount below). Additive smoothing is a type of shrinkage estimator, as the resulting estimate will be between the empirical probability (relative frequency) x i / N {\displaystyle x_{i}/N} and the uniform probability 1 / d . {\displaystyle 1/d.} Common choices for α are 0 (no smoothing), +1⁄2 (the Jeffreys prior), or 1 (Laplace's rule of succession), but the parameter may also be set empirically based on the observed data. From a Bayesian point of view, this corresponds to the expected value of the posterior distribution, using a symmetric Dirichlet distribution with parameter α as a prior distribution. In the special case where the number of categories is 2, this is equivalent to using a beta distribution as the conjugate prior for the parameters of the binomial distribution. == History == Laplace came up with this smoothing technique when he tried to estimate the chance that the sun will rise tomorrow. His rationale was that even given a large sample of days with the rising sun, we still can not be completely sure that the sun will still rise tomorrow (known as the sunrise problem). == Pseudocount == A pseudocount is an amount (not generally an integer, despite its name) added to the number of observed cases in order to change the expected probability in a model of those data, when not known to be zero. It is so named because, roughly speaking, a pseudo-count of value α {\displaystyle \alpha } weighs into the posterior distribution similarly to each category having an additional count of α {\displaystyle \alpha } . If the number of occurrences of each item i {\displaystyle i} is x i {\displaystyle x_{i}} out of N {\displaystyle N} samples, the empirical probability of event i {\displaystyle i} is p i , empirical = x i N , {\displaystyle p_{i,{\text{empirical}}}={\frac {x_{i}}{N}},} but the posterior probability when additively smoothed is p i , α -smoothed = x i + α N + α d , {\displaystyle p_{i,\alpha {\text{-smoothed}}}={\frac {x_{i}+\alpha }{N+\alpha d}},} as if to increase each count x i {\displaystyle x_{i}} by α {\displaystyle \alpha } a priori. Depending on the prior knowledge, which is sometimes a subjective value, a pseudocount may have any non-negative finite value. It may only be zero (or the possibility ignored) if impossible by definition, such as the possibility of a decimal digit of π being a letter, or a physical possibility that would be rejected and so not counted, such as a computer printing a letter when a valid program for π is run, or excluded and not counted because of no interest, such as if only interested in the zeros and ones. Generally, there is also a possibility that no value may be computable or observable in a finite time (see the halting problem). But at least one possibility must have a non-zero pseudocount, otherwise no prediction could be computed before the first observation. The relative values of pseudocounts represent the relative prior expected probabilities of their possibilities. The sum of the pseudocounts, which may be very large, represents the estimated weight of the prior knowledge compared with all the actual observations (one for each) when determining the expected probability. In any observed data set or sample there is the possibility, especially with low-probability events and with small data sets, of a possible event not occurring. Its observed frequency is therefore zero, apparently implying a probability of zero. This oversimplification is inaccurate and often unhelpful, particularly in probability-based machine learning techniques such as artificial neural networks and hidden Markov models. By artificially adjusting the probability of rare (but not impossible) events so those probabilities are not exactly zero, zero-frequency problems are avoided. Also see Cromwell's rule. === Choice of pseudocount === ==== Weakly informative prior ==== One common approach is to add 1 to each observed number of events, including the zero-count possibilities. This is sometimes called Laplace's rule of succession. This approach is equivalent to assuming a uniform prior distribution over the probabilities for each possible event (spanning the simplex where each probability is between 0 and 1, and they all sum to 1). Using the Jeffreys prior approach, a pseudocount of one half should be added to each possible outcome. Pseudocounts should be set to one or one-half only when there is no prior knowledge at all – see the principle of indifference. However, given appropriate prior knowledge, the sum should be adjusted in proportion to the expectation that the prior probabilities should be considered correct, despite evidence to the contrary – see further analysis. Higher values are appropriate inasmuch as there is prior knowledge of the true values (for a mint-condition coin, say); lower values inasmuch as there is prior knowledge that there is probable bias, but of unknown degree (for a bent coin, say). ==== Frequentist interval ==== One way to motivate pseudocounts, particularly for binomial data, is via a formula for the midpoint of an interval estimate, particularly a binomial proportion confidence interval. The best-known is due to Edwin Bidwell Wilson, in Wilson (1927): the midpoint of the Wilson score interval corresponding to ⁠ z {\displaystyle z} ⁠ standard deviations on either side is n S + z n + 2 z {\displaystyle {\frac {n_{S}+z}{n+2z}}} Taking z = 2 {\displaystyle z=2} standard deviations to approximate a 95% confidence interval (⁠ z ≈ 1.96 {\displaystyle z\approx 1.96} ⁠) yields pseudocount of 2 for each outcome, so 4 in total, colloquially known as the "plus four rule": n S + 2 n + 4 {\displaystyle {\frac {n_{S}+2}{n+4}}} This is also the midpoint of the Agresti–Coull interval (Agresti & Coull 1998). ==== Known incidence rates ==== Often the bias of an unknown trial population is tested against a control population with known parameters (incidence rates) μ = ⟨ μ 1 , μ 2 , … , μ d ⟩ . {\displaystyle {\boldsymbol {\mu }}=\langle \mu _{1},\mu _{2},\ldots ,\mu _{d}\rangle .} In this case the uniform probability 1 / d {\displaystyle 1/d} should be replaced by the known incidence rate of the control population μ i {\displaystyle \mu _{i}} to calculate the smoothed estimator: θ ^ i = x i + μ i α d N + α d ( i = 1 , … , d ) . {\displaystyle {\hat {\theta }}_{i}={\frac {x_{i}+\mu _{i}\alpha d}{N+\alpha d}}\qquad (i=1,\ldots ,d).} As a consistency check, if the empirical estimator happens to equal the incidence rate, i.e. μ i = x i / N , {\displaystyle \mu _{i}=x_{i}/N,} the smoothed estimator is independent of α {\displaystyle \alpha } and also equals the incidence rate. == Applications == === Classification === Additive smoothing is commonly a component of naive Bayes classifiers. === Statistical language modelling === In a bag of words model of natural language processing and information retrieval, the data consists of the number of occurrences of each word in a document. Additive smoothing allows the assignment of non-zero probabilities to words which do not occur in the sample. Studies have shown that additive smoothing is more effective than other probability smoothing methods in several retrieval tasks such as language-model-based pseudo-relevance feedback and recommender systems.
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Radford M. Neal

Radford M. Neal (born September 12, 1956) is a professor emeritus at the Department of Statistics and Department of Computer Science at the University of Toronto, where he held a Canada research chair in statistics and machine learning. == Education and career == Neal studied computer science at the University of Calgary, where he received his B.Sc. in 1977 and M.Sc. in 1980, with thesis work supervised by David Hill. He worked for several years as a sessional instructor at the University of Calgary and as a statistical consultant in the industry before coming back to the academia. Neal continued his study at the University of Toronto, where he received his Ph.D. in 1995 under the supervision of Geoffrey Hinton. Neal became an assistant professor at the University of Toronto in 1995, an associated professor in 1999 and a full professor since 2001. He was the Canada Research Chair in Statistics and Machine Learning from 2003 to 2016 and retired in 2017. Neal has made great contributions in the area of machine learning and statistics, where he is particularly well known for his work on Markov chain Monte Carlo, error correcting codes and Bayesian learning for neural networks. He is also known for his blog and as the developer of pqR: a new version of the R interpreter.
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Gödel machine

A Gödel machine is a hypothetical self-improving computer program that solves problems in an optimal way. It uses a recursive self-improvement protocol in which it rewrites its own code when it can prove the new code provides a better strategy. The machine was invented by Jürgen Schmidhuber (first proposed in 2003), but is named after Kurt Gödel who inspired the mathematical theories. The Gödel machine is often discussed when dealing with issues of meta-learning, also known as "learning to learn." Applications include automating human design decisions and transfer of knowledge between multiple related tasks, and may lead to design of more robust and general learning architectures. Though theoretically possible, no full implementation has been created. The Gödel machine is often compared with Marcus Hutter's AIXI, another formal specification for an artificial general intelligence. Schmidhuber points out that the Gödel machine could start out by implementing AIXItl as its initial sub-program, and self-modify after it finds proof that another algorithm for its search code will be better. == Limitations == Traditional problems solved by a computer only require one input and provide some output. Computers of this sort had their initial algorithm hardwired. This does not take into account the dynamic natural environment, and thus was a goal for the Gödel machine to overcome. The Gödel machine has limitations of its own, however. According to Gödel's First Incompleteness Theorem, any formal system that encompasses arithmetic is either flawed or allows for statements that cannot be proved in the system. Hence even a Gödel machine with unlimited computational resources must ignore those self-improvements whose effectiveness it cannot prove. == Variables of interest == There are three variables that are particularly useful in the run time of the Gödel machine. At some time t {\displaystyle t} , the variable time {\displaystyle {\text{time}}} will have the binary equivalent of t {\displaystyle t} . This is incremented steadily throughout the run time of the machine. Any input meant for the Gödel machine from the natural environment is stored in variable x {\displaystyle x} . It is likely the case that x {\displaystyle x} will hold different values for different values of variable time {\displaystyle {\text{time}}} . The outputs of the Gödel machine are stored in variable y {\displaystyle y} , where y ( t ) {\displaystyle y(t)} would be the output bit-string at some time t {\displaystyle t} . At any given time t {\displaystyle t} , where ( 1 ≤ t ≤ T ) {\displaystyle (1\leq t\leq T)} , the goal is to maximize future success or utility. A typical utility function follows the pattern u ( s , E n v ) : S × E → R {\displaystyle u(s,\mathrm {Env} ):S\times E\rightarrow \mathbb {R} } : u ( s , E n v ) = E μ [ ∑ τ = time T r ( τ ) ∣ s , E n v ] {\displaystyle u(s,\mathrm {Env} )=E_{\mu }{\Bigg [}\sum _{\tau ={\text{time}}}^{T}r(\tau )\mid s,\mathrm {Env} {\Bigg ]}} where r ( t ) {\displaystyle r(t)} is a real-valued reward input (encoded within s ( t ) {\displaystyle s(t)} ) at time t {\displaystyle t} , E μ [ ⋅ ∣ ⋅ ] {\displaystyle E_{\mu }[\cdot \mid \cdot ]} denotes the conditional expectation operator with respect to some possibly unknown distribution μ {\displaystyle \mu } from a set M {\displaystyle M} of possible distributions ( M {\displaystyle M} reflects whatever is known about the possibly probabilistic reactions of the environment), and the above-mentioned time = time ⁡ ( s ) {\displaystyle {\text{time}}=\operatorname {time} (s)} is a function of state s {\displaystyle s} which uniquely identifies the current cycle. Note that we take into account the possibility of extending the expected lifespan through appropriate actions. == Instructions used by proof techniques == The nature of the six proof-modifying instructions below makes it impossible to insert an incorrect theorem into proof, thus trivializing proof verification. === get-axiom(n) === Appends the n-th axiom as a theorem to the current theorem sequence. Below is the initial axiom scheme: Hardware Axioms formally specify how components of the machine could change from one cycle to the next. Reward Axioms define the computational cost of hardware instruction and the physical cost of output actions. Related Axioms also define the lifetime of the Gödel machine as scalar quantities representing all rewards/costs. Environment Axioms restrict the way new inputs x are produced from the environment, based on previous sequences of inputs y. Uncertainty Axioms/String Manipulation Axioms are standard axioms for arithmetic, calculus, probability theory, and string manipulation that allow for the construction of proofs related to future variable values within the Gödel machine. Initial State Axioms contain information about how to reconstruct parts or all of the initial state. Utility Axioms describe the overall goal in the form of utility function u. === apply-rule(k, m, n) === Takes in the index k of an inference rule (such as Modus tollens, Modus ponens), and attempts to apply it to the two previously proved theorems m and n. The resulting theorem is then added to the proof. === delete-theorem(m) === Deletes the theorem stored at index m in the current proof. This helps to mitigate storage constraints caused by redundant and unnecessary theorems. Deleted theorems can no longer be referenced by the above apply-rule function. === set-switchprog(m, n) === Replaces switchprog S pm:n, provided it is a non-empty substring of S p. === check() === Verifies whether the goal of the proof search has been reached. A target theorem states that given the current axiomatized utility function u (Item 1f), the utility of a switch from p to the current switchprog would be higher than the utility of continuing the execution of p (which would keep searching for alternative switchprogs). === state2theorem(m, n) === Takes in two arguments, m and n, and attempts to convert the contents of Sm:n into a theorem. == Example applications == === Time-limited NP-hard optimization === The initial input to the Gödel machine is the representation of a connected graph with a large number of nodes linked by edges of various lengths. Within given time T it should find a cyclic path connecting all nodes. The only real-valued reward will occur at time T. It equals 1 divided by the length of the best path found so far (0 if none was found). There are no other inputs. The by-product of maximizing expected reward is to find the shortest path findable within the limited time, given the initial bias. === Fast theorem proving === Prove or disprove as quickly as possible that all even integers > 2 are the sum of two primes (Goldbach’s conjecture). The reward is 1/t, where t is the time required to produce and verify the first such proof. === Maximizing expected reward with bounded resources === A cognitive robot that needs at least 1 liter of gasoline per hour interacts with a partially unknown environment, trying to find hidden, limited gasoline depots to occasionally refuel its tank. It is rewarded in proportion to its lifetime, and dies after at most 100 years or as soon as its tank is empty or it falls off a cliff, and so on. The probabilistic environmental reactions are initially unknown but assumed to be sampled from the axiomatized Speed Prior, according to which hard-to-compute environmental reactions are unlikely. This permits a computable strategy for making near-optimal predictions. One by-product of maximizing expected reward is to maximize expected lifetime.
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Thompson's construction

In computer science, Thompson's construction algorithm, also called the McNaughton–Yamada–Thompson algorithm, is a method of transforming a regular expression into an equivalent nondeterministic finite automaton (NFA). This NFA can be used to match strings against the regular expression. This algorithm is credited to Ken Thompson. Regular expressions and nondeterministic finite automata are two representations of formal languages. For instance, text processing utilities use regular expressions to describe advanced search patterns, but NFAs are better suited for execution on a computer. Hence, this algorithm is of practical interest, since it can compile regular expressions into NFAs. From a theoretical point of view, this algorithm is a part of the proof that they both accept exactly the same languages, that is, the regular languages. An NFA can be made deterministic by the powerset construction and then be minimized to get an optimal automaton corresponding to the given regular expression. However, an NFA may also be interpreted directly. To decide whether two given regular expressions describe the same language, each can be converted into an equivalent minimal deterministic finite automaton via Thompson's construction, powerset construction, and DFA minimization. If, and only if, the resulting automata agree up to renaming of states, the regular expressions' languages agree. == The algorithm == The algorithm works recursively by splitting an expression into its constituent subexpressions, from which the NFA will be constructed using a set of rules. More precisely, from a regular expression E, the obtained automaton A with the transition function Δ respects the following properties: A has exactly one initial state q0, which is not accessible from any other state. That is, for any state q and any letter a, Δ ( q , a ) {\displaystyle \Delta (q,a)} does not contain q0. A has exactly one final state qf, which is not co-accessible from any other state. That is, for any letter a, Δ ( q f , a ) = ∅ {\displaystyle \Delta (q_{f},a)=\emptyset } . Let c be the number of concatenation of the regular expression E and let s be the number of symbols apart from parentheses — that is, |, , a and ε. Then, the number of states of A is 2s − c (linear in the size of E). The number of transitions leaving any state is at most two. Since an NFA of m states and at most e transitions from each state can match a string of length n in time O(emn), a Thompson NFA can do pattern matching in linear time, assuming a fixed-size alphabet. === Rules === The following rules are depicted according to Aho et al. (2007), p. 122. In what follows, N(s) and N(t) are the NFA of the subexpressions s and t, respectively. The empty-expression ε is converted to A symbol a of the input alphabet is converted to The union expression s|t is converted to State q goes via ε either to the initial state of N(s) or N(t). Their final states become intermediate states of the whole NFA and merge via two ε-transitions into the final state of the NFA. The concatenation expression st is converted to The initial state of N(s) is the initial state of the whole NFA. The final state of N(s) becomes the initial state of N(t). The final state of N(t) is the final state of the whole NFA. The Kleene star expression s is converted to An ε-transition connects initial and final state of the NFA with the sub-NFA N(s) in between. Another ε-transition from the inner final to the inner initial state of N(s) allows for repetition of expression s according to the star operator. The parenthesized expression (s) is converted to N(s) itself. With these rules, using the empty expression and symbol rules as base cases, it is possible to prove with structural induction that any regular expression may be converted into an equivalent NFA. == Example == Two examples are now given, a small informal one with the result, and a bigger with a step by step application of the algorithm. === Small Example === The picture below shows the result of Thompson's construction on (ε|ab). The purple oval corresponds to a, the teal oval corresponds to a, the green oval corresponds to b, the orange oval corresponds to ab, and the blue oval corresponds to ε. === Application of the algorithm === As an example, the picture shows the result of Thompson's construction algorithm on the regular expression (0|(1(01(00)0)1)) that denotes the set of binary numbers that are multiples of 3: { ε, "0", "00", "11", "000", "011", "110", "0000", "0011", "0110", "1001", "1100", "1111", "00000", ... }. The upper right part shows the logical structure (syntax tree) of the expression, with "." denoting concatenation (assumed to have variable arity); subexpressions are named a-q for reference purposes. The left part shows the nondeterministic finite automaton resulting from Thompson's algorithm, with the entry and exit state of each subexpression colored in magenta and cyan, respectively. An ε as transition label is omitted for clarity — unlabelled transitions are in fact ε transitions. The entry and exit state corresponding to the root expression q is the start and accept state of the automaton, respectively. The algorithm's steps are as follows: An equivalent minimal deterministic automaton is shown below. == Relation to other algorithms == Thompson's is one of several algorithms for constructing NFAs from regular expressions; an earlier algorithm was given by McNaughton and Yamada. Converse to Thompson's construction, Kleene's algorithm transforms a finite automaton into a regular expression. Glushkov's construction algorithm is similar to Thompson's construction, once the ε-transitions are removed. == Use in string pattern matching == Regular expressions are often used to specify patterns that software is then asked to match. Generating an NFA by Thompson's construction, and using an appropriate algorithm to simulate it, it is possible to create pattern-matching software with performance that is ⁠ O ( m n ) {\displaystyle O(mn)} ⁠, where m is the length of the regular expression and n is the length of the string being matched. This is much better than is achieved by many popular programming-language implementations; however, it is restricted to purely regular expressions and does not support patterns for non-regular languages like backreferences.
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Best AI Avatar Generators in 2026

Looking for the best AI avatar generator? An AI avatar generator is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI avatar generator slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.
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