AI Humanizer

AI Humanizer — hands-on reviews, top picks, pricing, pros and cons and a practical how-to guide on Aizhi.

Gas (app)

Gas (sometimes stylized in all caps), formerly known as Melt as well as Crush, was an American anonymous social media app. Launched in August 2022, the app is oriented towards high schoolers. The app was developed by Nikita Bier, Isaiah Turner, and former Facebook engineer Dave Schatz. Gas was largely based upon the prior tbh app developed by co-founder Nikita Bier, along with Erik Hazzard, Kyle Zaragoza, and Nicolas Ducdodon in September 2017. tbh was acquired by Facebook inc. (now Meta Platforms) on October 16, 2017, and nearly a year later in July 2018 was dissolved, owing to low usage. Gas follows a similar purpose to tbh in being a social media app oriented towards high schoolers. In the app, users participate in anonymous polls regarding pre-written complimentary statements to their peers, such as "I'd say yes if (blank) asked me out on a date," "I think (blank) is the coolest kid in school," or "would make an ugly face and still look pretty." Winners of said polls receive a "flame." The name of the app is derived from this, with "gassing someone up" being Gen Z slang for complimenting someone. Users can pay a $6.99 subscription that enables "God Mode," which shows hints regarding who voted for them in a poll. Gas overtook TikTok and BeReal as the most downloaded app on the Apple App Store in October 2022 (the app is currently not available for Android). The app has over 5.1 million downloads as of early November 2022, over a million active users and 300 thousand daily downloads as of October 2022. Currently, the app is available in Canada and the majority of the United States. On January 17, 2023, Gas was acquired by Discord, however it would remain a standalone app and its developers became Discord staff members. On October 18, 2023, Discord announced that service for Gas would be permanently ending effective November 7, 2023, due to a steep decline in users. Effective November 7, the app became completely unusable. == Controversy regarding human-trafficking == Beginning in October 2022, rumors spread largely throughout TikTok and Snapchat alleged that the app was linked to human trafficking (in particular sex trafficking). According to Bier, the rumor originated with a single user review from China on October 5, and then was disseminated through TikTok accounts with "few to no US teen followers." Although largely dismissed as a hoax by experts, who cite how the app doesn't log user locations and general anonymity, the hoax became pervasive to the extent that various police departments, school systems, and local news outlets began issuing warnings regarding the app. For instance, on October 31, 2022, the police department of Piedmont, Oklahoma issued a warning to parents, encouraging them to check their children's phones, while on November 3, the Oklahoma Oktaha Public School system stated in a Facebook post that "Children are being kidnapped in other towns and this new app is thought to be the source of predators finding their location." (both statements have since been retracted by Police Chief Scott Singer and Superintendent Jerry Needham respectively). Additionally, local medial outlets such as KOCO in Oklahoma City ran stories making similar statements. The rumor had a negative impact on the app, with downloads plateauing for a two-week period in late October and with 3% of users in a single day reportedly uninstalling the app. Revenue and ratings have also reportedly dropped and the company's social media accounts have been bombarded with comments labeling them as sex-traffickers. Additionally, the four-person development team has reportedly been bombarded with various death threats as a result.
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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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AI Paraphrasing Tools: Free vs Paid (2026)

In search of the best AI paraphrasing tool? An AI paraphrasing tool 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 paraphrasing tool 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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Is an AI Content Generator Worth It in 2026?

Trying to pick the best AI content generator? An AI content generator 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 content generator 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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Data item

A data item describes an atomic state of a particular object concerning a specific property at a certain time point. A collection of data items for the same object at the same time forms an object instance (or table row). Any type of complex information can be broken down to elementary data items (atomic state). Data items are identified by object (o), property (p) and time (t), while the value (v) is a function of o, p and t: v = F(o,p,t). Values typically are represented by symbols like numbers, texts, images, sounds or videos. Values are not necessarily atomic. A value's complexity depends on the complexity of the property and time component. When looking at databases or XML files, the object is usually identified by an object name or other type of object identifier, which is part of the "data". Properties are defined as columns (table row), properties (object instance) or tags (XML). Often, time is not explicitly expressed and is an attribute applying to the complete data set. Other data collections provide time on the instance level (time series), column level, or even attribute/property level.
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Karl Steinbuch

Karl W. Steinbuch (June 15, 1917 in Stuttgart-Bad Cannstatt – June 4, 2005 in Ettlingen) was a German computer scientist, cyberneticist, and electrical engineer. He was an early and influential researcher in German computer science, and was the developer of the Lernmatrix, an early implementation of artificial neural networks. From the late 1960s onwards the focus of his activity shifted from scientific research to right-wing political activism supporting the Neue Rechte. == Biography == Steinbuch joined the National Socialist German Students' League (NSDStB) and the Nazi Party. Steinbuch studied at the University of Stuttgart and in 1944 he received his PhD in physics. In 1948 he joined Standard Elektrik Lorenz (SEL, part of the ITT group) in Stuttgart, as a computer design engineer and later as a director of research and development, where he filed more than 70 patents. Steinbuch completed the first European fully transistorized computer, the ER 56 marketed by SEL. In 1958 he became professor and director of the Institute of Technology for information processing (ITIV) of the University of Karlsruhe, where he retired in 1980. In 1967 he began publishing books, in which he tried to influence German education policy. Together with books from colleagues like Jean Ziegler from Switzerland, Eric J. Hobsbawm from the UK, and John Naisbitt his books predicted what he regarded as the coming education disaster of the emerging civic lobby society. In 1957, together with Helmut Gröttrup, Steinbuch coined the term Informatik, the German word for computer science, which gave informatics, and the term kybernetische Anthropologie. == Awards and recognition == Wilhelm-Boelsche award - medal in Gold German non-fiction book award Gold medal award of the XXI. International Congresses on Aerospace Medicine Konrad Adenauer award of science Jakob Fugger award medal Medal of merit of the state of Baden-Wuerttemberg member, German Academy of Sciences Leopoldina member, International Academy of Science, Munich. grants from a state government grants program, named "Karl-Steinbuch-Stipendium" Steinbuch Centre for Computing at the Karlsruhe Institute of Technology named after him == Books == Steinbuch wrote several books and articles, including: 1957 Informatik: Automatische Informationsverarbeitung ("Informatics: automatic information processing"). 1963 Learning matrices and their applications (together with U. A. W. Piske) 1965 A critical comparison of two kinds of adaptive classification networks (together with Bernard Widrow) 1966 (1969): Die informierte Gesellschaft. Geschichte und Zukunft der Nachrichtentechnik (The informed society. History and Future of telecommunications) 1989: Die desinformierte Gesellschaft (The disinformed society) 1968: Falsch programmiert. Über das Versagen unserer Gesellschaft in der Gegenwart und vor der Zukunft und was eigentlich geschehen müßte. (as a bestseller listet in: Der Spiegel) (Programmed falsely. About our society's failure in the present and with respect to the future and what should be done.) 1969: Programm 2000. (as a bestseller listet in: Der Spiegel) 1971: Automat und Mensch. Auf dem Weg zu einer kybernetischen Anthropologie (Machine and Man. On the way to a cybernetic anthropology; 4th revised edition) 1971: Mensch Technik Zukunft. Probleme von Morgen (German non-fiction book award) (Man Technology Future. Problems of Tomorrow) 1973: Kurskorrektur (Correcting the Course) 1978: Maßlos informiert. Die Enteignung des Denkens (Excessively informed. The Deprivation of Thinking) 1984: Unsere manipulierte Demokratie. Müssen wir mit der linken Lüge leben? (Our Thought-controlled Democracy. Do we have to live with the leftist lie?)
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Powerset construction

In the theory of computation and automata theory, the powerset construction or subset construction is a standard method for converting a nondeterministic finite automaton (NFA) into a deterministic finite automaton (DFA) that recognizes the same formal language. It is important in theory because it establishes that NFAs, despite their additional flexibility, are unable to recognize any language that cannot be recognized by some DFA. It is also important in practice for converting easier-to-construct NFAs into more efficiently executable DFAs. However, if the NFA has n states, the resulting DFA may have up to 2n states, an exponentially larger number, which sometimes makes the construction impractical for large NFAs. The construction, sometimes called the Rabin–Scott powerset construction (or subset construction) to distinguish it from similar constructions for other types of automata, was first published by Michael O. Rabin and Dana Scott in 1959. == Intuition == To simulate the operation of a DFA on a given input string, one needs to keep track of a single state at any time: the state that the automaton will reach after seeing a prefix of the input. In contrast, to simulate an NFA, one needs to keep track of a set of states: all of the states that the automaton could reach after seeing the same prefix of the input, according to the nondeterministic choices made by the automaton. If, after a certain prefix of the input, a set S of states can be reached, then after the next input symbol x the set of reachable states is a deterministic function of S and x. Therefore, the sets of reachable NFA states play the same role in the NFA simulation as single DFA states play in the DFA simulation, and in fact the sets of NFA states appearing in this simulation may be re-interpreted as being states of a DFA. == Construction == The powerset construction applies most directly to an NFA that does not allow state transformations without consuming input symbols (aka: "ε-moves"). Such an automaton may be defined as a 5-tuple (Q, Σ, T, q0, F), in which Q is the set of states, Σ is the set of input symbols, T is the transition function (mapping a state and an input symbol to a set of states), q0 is the initial state, and F is the set of accepting states. The corresponding DFA has states corresponding to subsets of Q. The initial state of the DFA is {q0}, the (one-element) set of initial states. The transition function of the DFA maps a state S (representing a subset of Q) and an input symbol x to the set T(S,x) = ∪{T(q,x) | q ∈ S}, the set of all states that can be reached by an x-transition from a state in S. A state S of the DFA is an accepting state if and only if at least one member of S is an accepting state of the NFA. In the simplest version of the powerset construction, the set of all states of the DFA is the powerset of Q, the set of all possible subsets of Q. However, many states of the resulting DFA may be useless as they may be unreachable from the initial state. An alternative version of the construction creates only the states that are actually reachable. === NFA with ε-moves === For an NFA with ε-moves (also called an ε-NFA), the construction must be modified to deal with these by computing the ε-closure of states: the set of all states reachable from some given state using only ε-moves. Van Noord recognizes three possible ways of incorporating this closure computation in the powerset construction: Compute the ε-closure of the entire automaton as a preprocessing step, producing an equivalent NFA without ε-moves, then apply the regular powerset construction. This version, also discussed by Hopcroft and Ullman, is straightforward to implement, but impractical for automata with large numbers of ε-moves, as commonly arise in natural language processing application. During the powerset computation, compute the ε-closure { q ′ | q → ε ∗ q ′ } {\displaystyle \{q'~|~q\to _{\varepsilon }^{}q'\}} of each state q that is considered by the algorithm (and cache the result). During the powerset computation, compute the ε-closure { q ′ | ∃ q ∈ Q ′ , q → ε ∗ q ′ } {\displaystyle \{q'~|~\exists q\in Q',q\to _{\varepsilon }^{}q'\}} of each subset of states Q' that is considered by the algorithm, and add its elements to Q'. === Multiple initial states === If NFAs are defined to allow for multiple initial states, the initial state of the corresponding DFA is the set of all initial states of the NFA, or (if the NFA also has ε-moves) the set of all states reachable from initial states by ε-moves. == Example == The NFA below has four states; state 1 is initial, and states 3 and 4 are accepting. Its alphabet consists of the two symbols 0 and 1, and it has ε-moves. The initial state of the DFA constructed from this NFA is the set of all NFA states that are reachable from state 1 by ε-moves; that is, it is the set {1,2,3}. A transition from {1,2,3} by input symbol 0 must follow either the arrow from state 1 to state 2, or the arrow from state 3 to state 4. Additionally, neither state 2 nor state 4 have outgoing ε-moves. Therefore, T({1,2,3},0) = {2,4}, and by the same reasoning the full DFA constructed from the NFA is as shown below. As can be seen in this example, there are five states reachable from the start state of the DFA; the remaining 11 sets in the powerset of the set of NFA states are not reachable. == Complexity == Because the DFA states consist of sets of NFA states, an n-state NFA may be converted to a DFA with at most 2n states. For every n, there exist n-state NFAs such that every subset of states is reachable from the initial subset, so that the converted DFA has exactly 2n states, giving Θ(2n) worst-case time complexity. A simple example requiring nearly this many states is the language of strings over the alphabet {0,1} in which there are at least n characters, the nth from last of which is 1. It can be represented by an (n + 1)-state NFA, but it requires 2n DFA states, one for each n-character suffix of the input; cf. picture for n=4. == Applications == Brzozowski's algorithm for DFA minimization uses the powerset construction, twice. It converts the input DFA into an NFA for the reverse language, by reversing all its arrows and exchanging the roles of initial and accepting states, converts the NFA back into a DFA using the powerset construction, and then repeats its process. Its worst-case complexity is exponential, unlike some other known DFA minimization algorithms, but in many examples it performs more quickly than its worst-case complexity would suggest. Safra's construction, which converts a non-deterministic Büchi automaton with n states into a deterministic Muller automaton or into a deterministic Rabin automaton with 2O(n log n) states, uses the powerset construction as part of its machinery.
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The Best Free AI Content Generator for Beginners

Looking for the best AI content generator? An AI content 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 content 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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International Conference on Language Resources and Evaluation

The International Conference on Language Resources and Evaluation is an international conference organised by the ELRA Language Resources Association every other year (on even years) with the support of institutions and organisations involved in Natural language processing. The series of LREC conferences was launched in Granada in 1998. == History of conferences == The survey of the LREC conferences over the period 1998-2013 was presented during the 2014 conference in Reykjavik as a closing session. It appears that the number of papers and signatures is increasing over time. The average number of authors per paper is higher as well. The percentage of new authors is between 68% and 78%. The distribution between male (65%) and female (35%) authors is stable over time. The most frequent technical term is "annotation", then comes "part-of-speech". == The LRE Map == The LRE Map was introduced at LREC 2010 and is now a regular feature of the LREC submission process for both the conference papers and the workshop papers. At the submission stage, the authors are asked to provide some basic information about all the resources (in a broad sense, i.e. including tools, standards and evaluation packages), either used or created, described in their papers. All these descriptors are then gathered in a global matrix called the LRE Map. This feature has been extended to several other conferences.
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AI Pair Programmers Reviews: What Actually Works in 2026

Curious about the best AI pair programmer? An AI pair programmer is software that uses machine learning to help you get more done — it combines speed, accuracy, and an interface that just works. Hands-on testing shows real-world results vary, so a short free trial is the smartest way to decide. Whether you are a beginner or a pro, the right AI pair programmer 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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AI Background Removers: Free vs Paid (2026)

Looking for the best AI background remover? An AI background remover 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 background remover 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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Apertium

Apertium is a free/open-source rule-based machine translation platform. It is free software and released under the terms of the GNU General Public License. == Overview == Apertium is a transfer-based machine translation system, which uses finite state transducers for all of its lexical transformations, and Constraint Grammar taggers as well as hidden Markov models or Perceptrons for part-of-speech tagging / word category disambiguation. A structural transfer component is responsible for word movement and agreement; most Apertium language pairs up until now have used "chunking" or shallow transfer rules, though newer pairs use (possibly recursive) rules defined in a Context-free grammar. Many existing machine translation systems available at present are commercial or use proprietary technologies, which makes them very hard to adapt to new usages. Apertium code and data is free software and uses a language-independent specification, to allow for the ease of contributing to Apertium, more efficient development, and enhancing the project's overall growth. At present (December 2020), Apertium has released 51 stable language pairs, delivering fast translation with reasonably intelligible results (errors are easily corrected). Being an open-source project, Apertium provides tools for potential developers to build their own language pair and contribute to the project. == History == Apertium originated as one of the machine translation engines in the project OpenTrad, which was funded by the Spanish government, and developed by the Transducens research group at the Universitat d'Alacant. It was originally designed to translate between closely related languages, although it has recently been expanded to treat more divergent language pairs. To create a new machine translation system, one just has to develop linguistic data (dictionaries, rules) in well-specified XML formats. Language data developed for it (in collaboration with the Universidade de Vigo, the Universitat Politècnica de Catalunya and the Universitat Pompeu Fabra) currently support (in stable version) the Arabic, Aragonese, Asturian, Basque, Belarusian, Breton, Bulgarian, Catalan, Crimean Tatar, Danish, English, Esperanto, French, Galician, Hindi, Icelandic, Indonesian, Italian, Kazakh, Macedonian, Malaysian, Maltese, Northern Sami, Norwegian (Bokmål and Nynorsk), Occitan, Polish, Portuguese, Romanian, Russian, Sardinian, Serbo-Croatian, Silesian, Slovene, Spanish, Swedish, Tatar, Ukrainian, Urdu, and Welsh languages. A full list is available below. Several companies are also involved in the development of Apertium, including Prompsit Language Engineering, Imaxin Software and Eleka Ingeniaritza Linguistikoa. The project has taken part in the 2009, 2010, 2011, 2012, 2013 and 2014 editions of Google Summer of Code and the 2010, 2011, 2012, 2013, 2014, 2015, 2016 and 2017 editions of Google Code-In. == Translation methodology == This is an overall, step-by-step view how Apertium works. The diagram displays the steps that Apertium takes to translate a source-language text (the text we want to translate) into a target-language text (the translated text). Source language text is passed into Apertium for translation. The deformatter removes formatting markup (HTML, RTF, etc.) that should be kept in place but not translated. The morphological analyser segments the text (expanding elisions, marking set phrases, etc.), and looks up segments in the language dictionaries, returning dictionary forms and tags for all matches. In pairs that involve agglutinative morphology, including a number of Turkic languages, a Helsinki Finite State Transducer (HFST) is used. Otherwise, an Apertium-specific finite state transducer system called lttoolbox, is used. The morphological disambiguator (the morphological analyser and the morphological disambiguator together form the part of speech tagger) resolves ambiguous segments (i.e., when there is more than one match) by choosing one match. Apertium uses Constraint Grammar rules (with the vislcg3 parser) for most of its language pairs. Retokenisation uses a finite state transducer to match sequences of lexical units and may reorder or translate tags (often used for translating idiomatic expressions into something that more approaches the target language grammar) Lexical transfer looks up disambiguated source-language basewords to find their target-language equivalents (i.e., mapping source language to target language). For lexical transfer, Apertium uses an XML-based dictionary format called bidix. Lexical selection chooses between alternative translations when the source text word has alternative meanings. Apertium uses a specific XML-based technology, apertium-lex-tools, to perform lexical selection. Structural transfer (i.e., it is an XML format that allows writing complex structural transfer rules) can consist of one-step chunking transfer, three-step chunking transfer or a CFG-based transfer module. The chunking modules flag grammatical differences between the source language and target language (e.g. gender or number agreement) by creating a sequence of chunks containing markers for this. They then reorder or modify chunks in order to produce a grammatical translation in the target-language. The newer CFG-based module matches input sequences into possible parse trees, selecting the best-ranking one and applying transformation rules on the tree. The morphological generator uses the tags to deliver the correct target language surface form. The morphological generator is a morphological transducer, just like the morphological analyser. A morphological transducer both analyses and generates forms. The post-generator makes any necessary orthographic changes due to the contact of words (e.g. elisions). The reformatter replaces formatting markup (HTML, RTF, etc.) that was removed by the deformatter in the first step. Apertium delivers the target-language translation. == Supported languages == As of June 2026, the following 108 pairs and 51 languages and languages varieties are supported by Apertium.
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Neural network Gaussian process

A Neural Network Gaussian Process (NNGP) is a Gaussian process (GP) obtained as the limit of a certain type of sequence of neural networks. Specifically, a wide variety of network architectures converges to a GP in the infinitely wide limit, in the sense of distribution. The concept constitutes an intensional definition, i.e., a NNGP is just a GP, but distinguished by how it is obtained. == Motivation == Bayesian networks are a modeling tool for assigning probabilities to events, and thereby characterizing the uncertainty in a model's predictions. Deep learning and artificial neural networks are approaches used in machine learning to build computational models which learn from training examples. Bayesian neural networks merge these fields. They are a type of neural network whose parameters and predictions are both probabilistic. While standard neural networks often assign high confidence even to incorrect predictions, Bayesian neural networks can more accurately evaluate how likely their predictions are to be correct. Computation in artificial neural networks is usually organized into sequential layers of artificial neurons. The number of neurons in a layer is called the layer width. When we consider a sequence of Bayesian neural networks with increasingly wide layers (see figure), they converge in distribution to a NNGP. This large width limit is of practical interest, since the networks often improve as layers get wider. And the process may give a closed form way to evaluate networks. NNGPs also appears in several other contexts: It describes the distribution over predictions made by wide non-Bayesian artificial neural networks after random initialization of their parameters, but before training; it appears as a term in neural tangent kernel prediction equations; it is used in deep information propagation to characterize whether hyperparameters and architectures will be trainable. It is related to other large width limits of neural networks. === Scope === The first correspondence result had been established in the 1995 PhD thesis of Radford M. Neal, then supervised by Geoffrey Hinton at University of Toronto. Neal cites David J. C. MacKay as inspiration, who worked in Bayesian learning. Today the correspondence is proven for: Single hidden layer Bayesian neural networks; deep fully connected networks as the number of units per layer is taken to infinity; convolutional neural networks as the number of channels is taken to infinity; transformer networks as the number of attention heads is taken to infinity; recurrent networks as the number of units is taken to infinity. In fact, this NNGP correspondence holds for almost any architecture: Generally, if an architecture can be expressed solely via matrix multiplication and coordinatewise nonlinearities (i.e., a tensor program), then it has an infinite-width GP. This in particular includes all feedforward or recurrent neural networks composed of multilayer perceptron, recurrent neural networks (e.g., LSTMs, GRUs), (nD or graph) convolution, pooling, skip connection, attention, batch normalization, and/or layer normalization. === Illustration === Every setting of a neural network's parameters θ {\displaystyle \theta } corresponds to a specific function computed by the neural network. A prior distribution p ( θ ) {\displaystyle p(\theta )} over neural network parameters therefore corresponds to a prior distribution over functions computed by the network. As neural networks are made infinitely wide, this distribution over functions converges to a Gaussian process for many architectures. The notation used in this section is the same as the notation used below to derive the correspondence between NNGPs and fully connected networks, and more details can be found there. The figure to the right plots the one-dimensional outputs z L ( ⋅ ; θ ) {\displaystyle z^{L}(\cdot ;\theta )} of a neural network for two inputs x {\displaystyle x} and x ∗ {\displaystyle x^{}} against each other. The black dots show the function computed by the neural network on these inputs for random draws of the parameters from p ( θ ) {\displaystyle p(\theta )} . The red lines are iso-probability contours for the joint distribution over network outputs z L ( x ; θ ) {\displaystyle z^{L}(x;\theta )} and z L ( x ∗ ; θ ) {\displaystyle z^{L}(x^{};\theta )} induced by p ( θ ) {\displaystyle p(\theta )} . This is the distribution in function space corresponding to the distribution p ( θ ) {\displaystyle p(\theta )} in parameter space, and the black dots are samples from this distribution. For infinitely wide neural networks, since the distribution over functions computed by the neural network is a Gaussian process, the joint distribution over network outputs is a multivariate Gaussian for any finite set of network inputs. == Discussion == === Infinitely wide fully connected network === This section expands on the correspondence between infinitely wide neural networks and Gaussian processes for the specific case of a fully connected architecture. It provides a proof sketch outlining why the correspondence holds, and introduces the specific functional form of the NNGP for fully connected networks. The proof sketch closely follows the approach by Novak and coauthors. ==== Network architecture specification ==== Consider a fully connected artificial neural network with inputs x {\displaystyle x} , parameters θ {\displaystyle \theta } consisting of weights W l {\displaystyle W^{l}} and biases b l {\displaystyle b^{l}} for each layer l {\displaystyle l} in the network, pre-activations (pre-nonlinearity) z l {\displaystyle z^{l}} , activations (post-nonlinearity) y l {\displaystyle y^{l}} , pointwise nonlinearity ϕ ( ⋅ ) {\displaystyle \phi (\cdot )} , and layer widths n l {\displaystyle n^{l}} . For simplicity, the width n L + 1 {\displaystyle n^{L+1}} of the readout vector z L {\displaystyle z^{L}} is taken to be 1. The parameters of this network have a prior distribution p ( θ ) {\displaystyle p(\theta )} , which consists of an isotropic Gaussian for each weight and bias, with the variance of the weights scaled inversely with layer width. This network is illustrated in the figure to the right, and described by the following set of equations: x ≡ input y l ( x ) = { x l = 0 ϕ ( z l − 1 ( x ) ) l > 0 z i l ( x ) = ∑ j W i j l y j l ( x ) + b i l W i j l ∼ N ( 0 , σ w 2 n l ) b i l ∼ N ( 0 , σ b 2 ) ϕ ( ⋅ ) ≡ nonlinearity y l ( x ) , z l − 1 ( x ) ∈ R n l × 1 n L + 1 = 1 θ = { W 0 , b 0 , … , W L , b L } {\displaystyle {\begin{aligned}x&\equiv {\text{input}}\\y^{l}(x)&=\left\{{\begin{array}{lcl}x&&l=0\\\phi \left(z^{l-1}(x)\right)&&l>0\end{array}}\right.\\z_{i}^{l}(x)&=\sum _{j}W_{ij}^{l}y_{j}^{l}(x)+b_{i}^{l}\\W_{ij}^{l}&\sim {\mathcal {N}}\left(0,{\frac {\sigma _{w}^{2}}{n^{l}}}\right)\\b_{i}^{l}&\sim {\mathcal {N}}\left(0,\sigma _{b}^{2}\right)\\\phi (\cdot )&\equiv {\text{nonlinearity}}\\y^{l}(x),z^{l-1}(x)&\in \mathbb {R} ^{n^{l}\times 1}\\n^{L+1}&=1\\\theta &=\left\{W^{0},b^{0},\dots ,W^{L},b^{L}\right\}\end{aligned}}} ==== ==== z l | y l {\displaystyle z^{l}|y^{l}} is a Gaussian process We first observe that the pre-activations z l {\displaystyle z^{l}} are described by a Gaussian process conditioned on the preceding activations y l {\displaystyle y^{l}} . This result holds even at finite width. Each pre-activation z i l {\displaystyle z_{i}^{l}} is a weighted sum of Gaussian random variables, corresponding to the weights W i j l {\displaystyle W_{ij}^{l}} and biases b i l {\displaystyle b_{i}^{l}} , where the coefficients for each of those Gaussian variables are the preceding activations y j l {\displaystyle y_{j}^{l}} . Because they are a weighted sum of zero-mean Gaussians, the z i l {\displaystyle z_{i}^{l}} are themselves zero-mean Gaussians (conditioned on the coefficients y j l {\displaystyle y_{j}^{l}} ). Since the z l {\displaystyle z^{l}} are jointly Gaussian for any set of y l {\displaystyle y^{l}} , they are described by a Gaussian process conditioned on the preceding activations y l {\displaystyle y^{l}} . The covariance or kernel of this Gaussian process depends on the weight and bias variances σ w 2 {\displaystyle \sigma _{w}^{2}} and σ b 2 {\displaystyle \sigma _{b}^{2}} , as well as the second moment matrix K l {\displaystyle K^{l}} of the preceding activations y l {\displaystyle y^{l}} , z i l ∣ y l ∼ G P ( 0 , σ w 2 K l + σ b 2 ) K l ( x , x ′ ) = 1 n l ∑ i y i l ( x ) y i l ( x ′ ) {\displaystyle {\begin{aligned}z_{i}^{l}\mid y^{l}&\sim {\mathcal {GP}}\left(0,\sigma _{w}^{2}K^{l}+\sigma _{b}^{2}\right)\\K^{l}(x,x')&={\frac {1}{n^{l}}}\sum _{i}y_{i}^{l}(x)y_{i}^{l}(x')\end{aligned}}} The effect of the weight scale σ w 2 {\displaystyle \sigma _{w}^{2}} is to rescale the contribution to the covariance matrix from K l {\displaystyle K^{l}} , while the bias is shared for all inputs, and so σ b 2 {\displaystyle \sigma _{b}^{2}} makes the z i l {\displaystyle z_{i}^{l}} for different datapoints more similar and
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Extended affix grammar

In computer science, extended affix grammars (EAGs) are a formal grammar formalism for describing the context free and context sensitive syntax of language, both natural language and programming languages. EAGs are a member of the family of two-level grammars; more specifically, a restriction of Van Wijngaarden grammars with the specific purpose of making parsing feasible. Like Van Wijngaarden grammars, EAGs have hyperrules that form a context-free grammar except in that their nonterminals may have arguments, known as affixes, the possible values of which are supplied by another context-free grammar, the metarules. EAGs were introduced and studied by D.A. Watt in 1974; recognizers were developed at the University of Nijmegen between 1985 and 1995. The EAG compiler developed there will generate either a recogniser, a transducer, a translator, or a syntax directed editor for a language described in the EAG formalism. The formalism is quite similar to Prolog, to the extent that it borrowed its cut operator. EAGs have been used to write grammars of natural languages such as English, Spanish, and Hungarian. The aim was to verify the grammars by making them parse corpora of text (corpus linguistics); hence, parsing had to be sufficiently practical. However, the parse tree explosion problem that ambiguities in natural language tend to produce in this type of approach is worsened for EAGs because each choice of affix value may produce a separate parse, even when several different values are equivalent. The remedy proposed was to switch to the much simpler Affix Grammar over a Finite Lattice (AGFL) instead, in which metagrammars can only produce simple finite languages.
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