AI Bot Grammar Checker

AI Bot Grammar Checker — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

Blobotics

Blobotics is a term describing research into chemical-based computer processors based on ions rather than electrons. Andrew Adamatzky, a computer scientist at the University of the West of England, Bristol used the term in an article in New Scientist March 28, 2005 [1]. The aim is to create 'liquid logic gates' which would be 'infinitely reconfigurable and self-healing'. The process relies on the Belousov–Zhabotinsky reaction, a repeating cycle of three separate sets of reactions. Such a processor could form the basis of a robot which, using artificial sensors, interact with its surroundings in a way which mimics living creatures. The coining of the term was featured by ABC radio in Australia [2].
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Pushpak Bhattacharyya

Pushpak Bhattacharyya (3 July 1962 – 5 October 2025) was an Indian computer scientist and professor in the Department of Computer Science and Engineering at the IIT Bombay. He served as the Director of the IIT Patna from 2015 to 2021. He was a past President of the Association for Computational Linguistics (2016–17), and held the Vijay and Sita Vashee Chair Professorship at IIT Bombay. Bhattacharyya led the Natural Language Processing (NLP) research group at the Centre for Indian Language Technology (CFILT) at IIT Bombay until his death. At the inauguration of the Nilekani Centre at AI4Bharat, IIT Madras, Nandan Nilekani, Co-founder and Non-Executive Chairman of Infosys, referred to Bhattacharyya as the "Godfather of Indian NLP". == Early life and education == Bhattacharyya was born in Shillong in 1962. He completed his schooling at Jail Road Boys' High School, Shillong. He obtained a B.Tech. in Computer Science from the IIT Kharagpur, followed by an M.Tech. from the IIT Kanpur, and a Ph.D. in Computer Science from IIT Bombay in 1994. == Research == Bhattacharyya’s research areas includes Natural language processing, Artificial intelligence, Machine learning, Psycholinguistics, Eye tracking, and Information retrieval. He made contributions to the development of multilingual lexical databases such as IndoWordNet and other projects related to machine translation and computational linguistics. He authored and co-authored multiple academic works, including Investigations in Computational Sarcasm (with Aditya Joshi), Cognitively Inspired Natural Language Processing: An Investigation Based on Eye Tracking (with Abhijit Mishra), and Machine Translation and Transliteration of Low Resource Related Languages (with Anoop Kunchukuttan). Over his career, Bhattacharyya published more than 350 research papers in journals and conference proceedings and supervised over 300 undergraduate, master’s, and doctoral students. His projects often addressed computational challenges for Indian languages, such as developing wordnets, building translation systems for low-resource languages, and studying cognitive aspects of language processing. He also led government- and industry-funded research initiatives supported by organizations including IBM, Microsoft, Yahoo, and the United Nations. == Death == Bhattacharyya died on 5 October 2025, at the age of 63. == Awards == Patwardhan Award, IIT Bombay, for Technology Development VNMM Award, IIT Roorkee, for Technology Development Fellow, Indian National Academy of Engineering Eminent Engineer Award, Institution of Engineers (India)
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FastText

fastText is a library for learning of word embeddings and text classification created by Facebook's AI Research (FAIR) lab. The model allows one to create an unsupervised learning or supervised learning algorithm for obtaining vector representations for words. Facebook makes available pretrained models for 294 languages. Several papers describe the techniques used by fastText. The GitHub repository was archived on March 19, 2024.
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ROUGE (metric)

ROUGE, or Recall-Oriented Understudy for Gisting Evaluation, is a set of metrics and a software package used for evaluating automatic summarization and machine translation software in natural language processing. The metrics compare an automatically produced summary or translation against a reference or a set of references (human-produced) summary or translation. ROUGE metrics range between 0 and 1, with higher scores indicating higher similarity between the automatically produced summary and the reference. == Metrics == The following five evaluation metrics are available. ROUGE-N: Overlap of n-grams between the system and reference summaries. ROUGE-1 refers to the overlap of unigrams (each word) between the system and reference summaries. ROUGE-2 refers to the overlap of bigrams between the system and reference summaries. ROUGE-L: Longest Common Subsequence (LCS) based statistics. Longest common subsequence problem takes into account sentence-level structure similarity naturally and identifies longest co-occurring in sequence n-grams automatically. ROUGE-W: Weighted LCS-based statistics that favors consecutive LCSes. ROUGE-S: Skip-bigram based co-occurrence statistics. Skip-bigram is any pair of words in their sentence order. ROUGE-SU: Skip-bigram plus unigram-based co-occurrence statistics.
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Lexical Markup Framework

Language resource management – Lexical markup framework (LMF; ISO 24613), produced by ISO/TC 37, is the ISO standard for natural language processing (NLP) and machine-readable dictionary (MRD) lexicons. The scope is standardization of principles and methods relating to language resources in the contexts of multilingual communication. == Objectives == The goals of LMF are to provide a common model for the creation and use of lexical resources, to manage the exchange of data between and among these resources, and to enable the merging of large number of individual electronic resources to form extensive global electronic resources. Types of individual instantiations of LMF can include monolingual, bilingual or multilingual lexical resources. The same specifications are to be used for both small and large lexicons, for both simple and complex lexicons, for both written and spoken lexical representations. The descriptions range from morphology, syntax, computational semantics to computer-assisted translation. The covered languages are not restricted to European languages but cover all natural languages. The range of targeted NLP applications is not restricted. LMF is able to represent most lexicons, including WordNet, EDR and PAROLE lexicons. == History == In the past, lexicon standardization has been studied and developed by a series of projects like GENELEX, EDR, EAGLES, MULTEXT, PAROLE, SIMPLE and ISLE. Then, the ISO/TC 37 National delegations decided to address standards dedicated to NLP and lexicon representation. The work on LMF started in Summer 2003 by a new work item proposal issued by the US delegation. In Fall 2003, the French delegation issued a technical proposition for a data model dedicated to NLP lexicons. In early 2004, the ISO/TC 37 committee decided to form a common ISO project with Nicoletta Calzolari (CNR-ILC Italy) as convenor and Gil Francopoulo (Tagmatica France) and Monte George (ANSI, United States) as editors. The first step in developing LMF was to design an overall framework based on the general features of existing lexicons and to develop a consistent terminology to describe the components of those lexicons. The next step was the actual design of a comprehensive model that best represented all of the lexicons in detail. A large panel of 60 experts contributed a wide range of requirements for LMF that covered many types of NLP lexicons. The editors of LMF worked closely with the panel of experts to identify the best solutions and reach a consensus on the design of LMF. Special attention was paid to the morphology in order to provide powerful mechanisms for handling problems in several languages that were known as difficult to handle. 13 versions have been written, dispatched (to the National nominated experts), commented and discussed during various ISO technical meetings. After five years of work, including numerous face-to-face meetings and e-mail exchanges, the editors arrived at a coherent UML model. In conclusion, LMF should be considered a synthesis of the state of the art in NLP lexicon field. == Current stage == The ISO number is 24613. The LMF specification has been published officially as an International Standard on 17 November 2008. == As one of the members of the ISO/TC 37 family of standards == The ISO/TC 37 standards are currently elaborated as high level specifications and deal with word segmentation (ISO 24614), annotations (ISO 24611 a.k.a. MAF, ISO 24612 a.k.a. LAF, ISO 24615 a.k.a. SynAF, and ISO 24617-1 a.k.a. SemAF/Time), feature structures (ISO 24610), multimedia containers (ISO 24616 a.k.a. MLIF), and lexicons (ISO 24613). These standards are based on low level specifications dedicated to constants, namely data categories (revision of ISO 12620), language codes (ISO 639), scripts codes (ISO 15924), country codes (ISO 3166) and Unicode (ISO 10646). The two level organization forms a coherent family of standards with the following common and simple rules: the high level specification provides structural elements that are adorned by the standardized constants; the low level specifications provide standardized constants as metadata. == Key standards == The linguistics constants like /feminine/ or /transitive/ are not defined within LMF but are recorded in the Data Category Registry (DCR) that is maintained as a global resource by ISO/TC 37 in compliance with ISO/IEC 11179-3:2003. And these constants are used to adorn the high level structural elements. The LMF specification complies with the modeling principles of Unified Modeling Language (UML) as defined by Object Management Group (OMG). The structure is specified by means of UML class diagrams. The examples are presented by means of UML instance (or object) diagrams. An XML DTD is given in an annex of the LMF document. == Model structure == LMF is composed of the following components: The core package that is the structural skeleton which describes the basic hierarchy of information in a lexical entry. Extensions of the core package which are expressed in a framework that describes the reuse of the core components in conjunction with the additional components required for a specific lexical resource. The extensions are specifically dedicated to morphology, MRD, NLP syntax, NLP semantics, NLP multilingual notations, NLP morphological patterns, multiword expression patterns, and constraint expression patterns. == Example == In the following example, the lexical entry is associated with a lemma clergyman and two inflected forms clergyman and clergymen. The language coding is set for the whole lexical resource. The language value is set for the whole lexicon as shown in the following UML instance diagram. The elements Lexical Resource, Global Information, Lexicon, Lexical Entry, Lemma, and Word Form define the structure of the lexicon. They are specified within the LMF document. On the contrary, languageCoding, language, partOfSpeech, commonNoun, writtenForm, grammaticalNumber, singular, plural are data categories that are taken from the Data Category Registry. These marks adorn the structure. The values ISO 639-3, clergyman, clergymen are plain character strings. The value eng is taken from the list of languages as defined by ISO 639-3. With some additional information like dtdVersion and feat, the same data can be expressed by the following XML fragment: This example is rather simple, while LMF can represent much more complex linguistic descriptions the XML tagging is correspondingly complex. == Selected publications about LMF == The first publication about the LMF specification as it has been ratified by ISO (this paper became (in 2015) the 9th most cited paper within the Language Resources and Evaluation conferences from LREC papers): Language Resources and Evaluation LREC-2006/Genoa: Gil Francopoulo, Monte George, Nicoletta Calzolari, Monica Monachini, Nuria Bel, Mandy Pet, Claudia Soria: Lexical Markup Framework (LMF) About semantic representation: Gesellschaft für linguistische Datenverarbeitung GLDV-2007/Tübingen: Gil Francopoulo, Nuria Bel, Monte George Nicoletta Calzolari, Monica Monachini, Mandy Pet, Claudia Soria: Lexical Markup Framework ISO standard for semantic information in NLP lexicons About African languages: Traitement Automatique des langues naturelles, Marseille, 2014: Mouhamadou Khoule, Mouhamad Ndiankho Thiam, El Hadj Mamadou Nguer: Toward the establishment of a LMF-based Wolof language lexicon (Vers la mise en place d'un lexique basé sur LMF pour la langue wolof) [in French] About Asian languages: Lexicography, Journal of ASIALEX, Springer 2014: Lexical Markup Framework: Gil Francopoulo, Chu-Ren Huang: An ISO Standard for Electronic Lexicons and its Implications for Asian Languages DOI 10.1007/s40607-014-0006-z About European languages: COLING 2010: Verena Henrich, Erhard Hinrichs: Standardizing Wordnets in the ISO Standard LMF: Wordnet-LMF for GermaNet EACL 2012: Judith Eckle-Kohler, Iryna Gurevych: Subcat-LMF: Fleshing out a standardized format for subcategorization frame interoperability EACL 2012: Iryna Gurevych, Judith Eckle-Kohler, Silvana Hartmann, Michael Matuschek, Christian M Meyer, Christian Wirth: UBY - A Large-Scale Unified Lexical-Semantic Resource Based on LMF. About Semitic languages: Journal of Natural Language Engineering, Cambridge University Press (to appear in Spring 2015): Aida Khemakhem, Bilel Gargouri, Abdelmajid Ben Hamadou, Gil Francopoulo: ISO Standard Modeling of a large Arabic Dictionary. Proceedings of the seventh Global Wordnet Conference 2014: Nadia B M Karmani, Hsan Soussou, Adel M Alimi: Building a standardized Wordnet in the ISO LMF for aeb language. Proceedings of the workshop: HLT & NLP within Arabic world, LREC 2008: Noureddine Loukil, Kais Haddar, Abdelmajid Ben Hamadou: Towards a syntactic lexicon of Arabic Verbs. Traitement Automatique des Langues Naturelles, Toulouse (in French) 2007: Khemakhem A, Gargouri B, Abdelwahed A, Francopoulo G: Modélisation des paradigmes de fl
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Top 10 AI Coding Assistants Compared (2026)

Shopping for the best AI coding assistant? An AI coding assistant 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 coding assistant 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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Isolation forest

Isolation forest is an unsupervised learning algorithm for anomaly detection that works on the principle of isolating anomalies, instead of the most common techniques of profiling normal points. In statistics, an anomaly (a.k.a. outlier) is an observation or event that deviates so much from other events to arouse suspicion it was generated by a different mean. For example, the graph in Fig.1 represents ingress traffic to a web server, expressed as the number of requests in 3-hours intervals, for a period of one month. It is quite evident by simply looking at the picture that some points (marked with a red circle) are unusually high, to the point of inducing suspect that the web server might have been under attack at that time. On the other hand, the flat segment indicated by the red arrow also seems unusual and might possibly be a sign that the server was down during that time period. Anomalies in a big dataset may follow very complicated patterns, which are difficult to detect "by eye" in the great majority of cases. This is the reason why the field of anomaly detection is well suited for the application of machine learning techniques. The most common techniques employed for anomaly detection are based on the construction of a profile of what is "normal": anomalies are reported as those instances in the dataset that do not conform to the normal profile. Isolation Forest uses a different approach: instead of trying to build a model of normal instances, it explicitly isolates anomalous points in the dataset. The main advantage of this approach is the possibility of exploiting sampling techniques to an extent that is not allowed to the profile-based methods, creating a very fast algorithm with a low memory demand. == History == The Isolation Forest (iForest) algorithm was initially proposed by Fei Tony Liu, Kai Ming Ting and Zhi-Hua Zhou in 2008. The authors took advantage of two quantitative properties of anomalous data points in a sample, that is: they are the minority consisting of fewer instances and they have attribute-values that are very different from those of normal instances Since anomalies are typically few and very different from the other points in the sample, they must be easier to "isolate" compared to normal points. On the basis of this principle, Isolation Forest builds an ensemble of "Isolation Trees" (iTrees) for the data set and marks as anomalies the points that have short average path lengths on the iTrees. In a later paper, published in 2012 the same authors described a set of experiments to prove that iForest: has a low linear time complexity and a small memory requirement is able to deal with high dimensional data with irrelevant attributes can be trained with or without anomalies in the training set can provide detection results with different levels of granularity without re-training In 2013 Zhiguo Ding and Minrui Fei proposed a framework based on iForest to resolve the problem of detecting anomalies in streaming data. More application of iForest to streaming data are described in papers by Swee Chuan Tan et al., G. A. Susto et al. and Yu Weng et al. One of the main problems of the application of iForest to anomaly detection was not with the model itself, but rather in the way the "anomaly score" was computed. This problem was highlighted by Sahand Hariri, Matias Carrasco Kind and Robert J. Brunner in a 2018 paper, wherein they proposed an improved iForest model named Extended Isolation Forest (EIF). In the same paper the authors describe the improvements made to the original model and how they are able to enhance the consistency and reliability of the anomaly score produced for a given data point. == Algorithm == At the basis of the Isolation Forest algorithm there is the tendency of anomalous instances in a dataset to be easier to separate from the rest of the sample (isolate), compared to normal points. In order to isolate a data point the algorithm recursively generates partitions on the sample by randomly selecting an attribute and then randomly selecting a split value for the attribute, between the minimum and maximum values allowed for that attribute. An example of random partitioning in a 2D dataset of normally distributed points is given in Fig. 2 for a non-anomalous point and Fig. 3 for a point that's more likely to be an anomaly. It is apparent from the pictures how anomalies require fewer random partitions to be isolated, compared to normal points. From a mathematical point of view, recursive partitioning can be represented by a tree structure named Isolation Tree, while the number of partitions required to isolate a point can be interpreted as the length of the path, within the tree, to reach a terminating node starting from the root. For example, the path length of point xi in Fig. 2 is greater than the path length of xj in Fig. 3. More formally, let X = { x1, ..., xn } be a set of d-dimensional points and X' ⊂ X a subset of X. An Isolation Tree (iTree) is defined as a data structure with the following properties: for each node T in the Tree, T is either an external-node with no child, or an internal-node with one "test" and exactly two daughter nodes (Tl, Tr) a test at node T consists of an attribute q and a split value p such that the test q < p determines the traversal of a data point to either Tl or Tr. In order to build an iTree, the algorithm recursively divides X' by randomly selecting an attribute q and a split value p, until either (i) the node has only one instance or (ii) all data at the node have the same values. When the iTree is fully grown, each point in X is isolated at one of the external nodes. Intuitively, the anomalous points are those (easier to isolate, hence) with the smaller path length in the tree, where the path length h(xi) of point x i ∈ X {\displaystyle x_{i}\in X} is defined as the number of edges xi traverses from the root node to get to an external node. A probabilistic explanation of iTree is provided in the iForest original paper. == Properties of Isolation Forest == Sub-sampling: since iForest does not need to isolate all of normal instances, it can frequently ignore the big majority of the training sample. As a consequence, iForest works very well when the sampling size is kept small, a property that is in contrast with the great majority of existing methods, where large sampling size is usually desirable. Swamping: when normal instances are too close to anomalies, the number of partitions required to separate anomalies increases, a phenomena known as swamping, which makes it more difficult for iForest to discriminate between anomalies and normal points. One of the main reasons for swamping is the presence of too many data for the purpose of anomaly detection, which implies one possible solution to the problem is sub-sampling. Since iForest respond very well to sub-sampling in terms of performance, the reduction of the number of points in the sample is also a good way to reduce the effect of swamping. Masking: when the number of anomalies is high it is possible that some of those aggregate in a dense and large cluster, making it more difficult to separate the single anomalies and, in turn, to detect such points as anomalous. Similarly to swamping, this phenomena (known as "masking") is also more likely when the number of points in the sample is big, and can be alleviated through sub-sampling. High Dimensional Data: one of the main limitation to standard, distance-based methods is their inefficiency in dealing with high dimensional datasets:. The main reason for that is, in a high dimensional space every point is equally sparse, so using a distance-based measure of separation is pretty ineffective. Unfortunately, high-dimensional data also affects the detection performance of iForest, but the performance can be vastly improved by adding a features selection test like Kurtosis to reduce the dimensionality of the sample space. Normal Instances Only: iForest performs well even if the training set does not contain any anomalous point, the reason being that iForest describes data distributions in such a way that high values of the path length h(xi) correspond to the presence of data points. As a consequence, the presence of anomalies is pretty irrelevant to iForest's detection performance. == Anomaly Detection with Isolation Forest == Anomaly detection with Isolation Forest is a process composed of two main stages: in the first stage, a training dataset is used to build iTrees as described in previous sections. in the second stage, each instance in test set is passed through the iTrees build in the previous stage, and a proper "anomaly score" is assigned to the instance using the algorithm described below Once all the instances in the test set have been assigned an anomaly score, it is possible to mark as "anomaly" any point whose score is greater than a predefined threshold, which depends on the domain the analysis is being applied to. === Anomaly Score === Th
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Google matrix

A Google matrix is a particular stochastic matrix that is used by Google's PageRank algorithm. The matrix represents a graph with edges representing links between pages. The PageRank of each page can then be generated iteratively from the Google matrix using the power method. However, in order for the power method to converge, the matrix must be stochastic, irreducible and aperiodic. == Adjacency matrix A and Markov matrix S == In order to generate the Google matrix G, we must first generate an adjacency matrix A which represents the relations between pages or nodes. Assuming there are N pages, we can fill out A by doing the following: A matrix element A i , j {\displaystyle A_{i,j}} is filled with 1 if node j {\displaystyle j} has a link to node i {\displaystyle i} , and 0 otherwise; this is the adjacency matrix of links. A related matrix S corresponding to the transitions in a Markov chain of given network is constructed from A by dividing the elements of column "j" by a number of k j = Σ i = 1 N A i , j {\displaystyle k_{j}=\Sigma _{i=1}^{N}A_{i,j}} where k j {\displaystyle k_{j}} is the total number of outgoing links from node j to all other nodes. The columns having zero matrix elements, corresponding to dangling nodes, are replaced by a constant value 1/N. Such a procedure adds a link from every sink, dangling state a {\displaystyle a} to every other node. Now by the construction the sum of all elements in any column of matrix S is equal to unity. In this way the matrix S is mathematically well defined and it belongs to the class of Markov chains and the class of Perron-Frobenius operators. That makes S suitable for the PageRank algorithm. == Construction of Google matrix G == Then the final Google matrix G can be expressed via S as: G i j = α S i j + ( 1 − α ) 1 N ( 1 ) {\displaystyle G_{ij}=\alpha S_{ij}+(1-\alpha ){\frac {1}{N}}\;\;\;\;\;\;\;\;\;\;\;(1)} By the construction the sum of all non-negative elements inside each matrix column is equal to unity. The numerical coefficient α {\displaystyle \alpha } is known as a damping factor. Usually S is a sparse matrix and for modern directed networks it has only about ten nonzero elements in a line or column, thus only about 10N multiplications are needed to multiply a vector by matrix G. == Examples of Google matrix == An example of the matrix S {\displaystyle S} construction via Eq.(1) within a simple network is given in the article CheiRank. For the actual matrix, Google uses a damping factor α {\displaystyle \alpha } around 0.85. The term ( 1 − α ) {\displaystyle (1-\alpha )} gives a surfer probability to jump randomly on any page. The matrix G {\displaystyle G} belongs to the class of Perron-Frobenius operators of Markov chains. The examples of Google matrix structure are shown in Fig.1 for Wikipedia articles hyperlink network in 2009 at small scale and in Fig.2 for University of Cambridge network in 2006 at large scale. == Spectrum and eigenstates of G matrix == For 0 < α < 1 {\displaystyle 0<\alpha <1} there is only one maximal eigenvalue λ = 1 {\displaystyle \lambda =1} with the corresponding right eigenvector which has non-negative elements P i {\displaystyle P_{i}} which can be viewed as stationary probability distribution. These probabilities ordered by their decreasing values give the PageRank vector P i {\displaystyle P_{i}} with the PageRank K i {\displaystyle K_{i}} used by Google search to rank webpages. Usually one has for the World Wide Web that P ∝ 1 / K β {\displaystyle P\propto 1/K^{\beta }} with β ≈ 0.9 {\displaystyle \beta \approx 0.9} . The number of nodes with a given PageRank value scales as N P ∝ 1 / P ν {\displaystyle N_{P}\propto 1/P^{\nu }} with the exponent ν = 1 + 1 / β ≈ 2.1 {\displaystyle \nu =1+1/\beta \approx 2.1} . The left eigenvector at λ = 1 {\displaystyle \lambda =1} has constant matrix elements. With 0 < α {\displaystyle 0<\alpha } all eigenvalues move as λ i → α λ i {\displaystyle \lambda _{i}\rightarrow \alpha \lambda _{i}} except the maximal eigenvalue λ = 1 {\displaystyle \lambda =1} , which remains unchanged. The PageRank vector varies with α {\displaystyle \alpha } but other eigenvectors with λ i < 1 {\displaystyle \lambda _{i}<1} remain unchanged due to their orthogonality to the constant left vector at λ = 1 {\displaystyle \lambda =1} . The gap between λ = 1 {\displaystyle \lambda =1} and other eigenvalue being 1 − α ≈ 0.15 {\displaystyle 1-\alpha \approx 0.15} gives a rapid convergence of a random initial vector to the PageRank approximately after 50 multiplications on G {\displaystyle G} matrix. At α = 1 {\displaystyle \alpha =1} the matrix G {\displaystyle G} has generally many degenerate eigenvalues λ = 1 {\displaystyle \lambda =1} (see e.g. [6]). Examples of the eigenvalue spectrum of the Google matrix of various directed networks is shown in Fig.3 from and Fig.4 from. The Google matrix can be also constructed for the Ulam networks generated by the Ulam method [8] for dynamical maps. The spectral properties of such matrices are discussed in [9,10,11,12,13,15]. In a number of cases the spectrum is described by the fractal Weyl law [10,12]. The Google matrix can be constructed also for other directed networks, e.g. for the procedure call network of the Linux Kernel software introduced in [15]. In this case the spectrum of λ {\displaystyle \lambda } is described by the fractal Weyl law with the fractal dimension d ≈ 1.3 {\displaystyle d\approx 1.3} (see Fig.5 from ). Numerical analysis shows that the eigenstates of matrix G {\displaystyle G} are localized (see Fig.6 from ). Arnoldi iteration method allows to compute many eigenvalues and eigenvectors for matrices of rather large size [13]. Other examples of G {\displaystyle G} matrix include the Google matrix of brain [17] and business process management [18], see also. Applications of Google matrix analysis to DNA sequences is described in [20]. Such a Google matrix approach allows also to analyze entanglement of cultures via ranking of multilingual Wikipedia articles abouts persons [21] == Historical notes == The Google matrix with damping factor was described by Sergey Brin and Larry Page in 1998 [22], see also articles on PageRank history [23], [24].
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Imieliński–Lipski algebra

In database theory, Imieliński–Lipski algebra is an extension of relational algebra onto tables with different types of null values. It is used to operate on relations with incomplete information. Imieliński–Lipski algebras are defined to satisfy precise conditions for semantically meaningful extension of the usual relational operators, such as projection, selection, union, and join, from operators on relations to operators on relations with various kinds of "null values". These conditions require that the system be safe in the sense that no incorrect conclusion is derivable by using a specified subset F of the relational operators; and that it be complete in the sense that all valid conclusions expressible by relational expressions using operators in F are in fact derivable in this system. For example, it is well known that the three-valued logic approach to deal with null values, supported treatment of nulls values by SQL is not complete, see Ullman book. To show this, let T be: Take SQL query Q SQL query Q will return empty set (no results) under 3-valued semantics currently adopted by all variants of SQL. This is the case because in SQL, NULL is never equal to any constant – in this case, neither to “Spring” nor “Fall” nor “Winter” (if there is Winter semester in this school). NULL='Spring' will evaluate to MAYBE and so will NULL='Fall'. The disjunction MAYBE OR MAYBE evaluates to MAYBE (not TRUE). Thus Igor will not be part of the answer (and of course neither will Rohit). But Igor should be returned as the answer. Indeed, regardless what semester Igor took the Networks class (no matter what was the unknown value of NULL), the selection condition will be true. This “Igor” will be missed by SQL and the SQL answer would be incomplete according to completeness requirements specified in Tomasz Imieliński, Witold Lipski, 'Incomplete Information in Relational Databases'. It is also argued there that 3-valued logic (TRUE, FALSE, MAYBE) can never provide guarantee of complete answer for tables with incomplete information. Three algebras which satisfy conditions of safety and completeness are defined as Imielinski–Lipski algebras: the Codd-Tables algebra, the V-tables algebra and the Conditional tables (C-tables) algebra. == Codd-tables algebra == Codd-tables algebra is based on the usual Codd's single NULL values. The table T above is an example of Codd-table. Codd-table algebra supports projection and positive selections only. It is also demonstrated in [IL84 that it is not possible to correctly extend more relational operators over Codd-Tables. For example, such basic operation as join is not extendable over Codd-tables. It is not possible to define selections with Boolean conditions involving negation and preserve completeness. For example, queries like the above query Q cannot be supported. In order to be able to extend more relational operators, more expressive form of null value representation is needed in tables which are called V-table. == V-tables algebra == V-tables algebra is based on many different ("marked") null values or variables allowed to appear in a table. V-tables allow to show that a value may be unknown but the same for different tuples. For example, in the table below Gaurav and Igor order the same (but unknown) beer in two unknown bars (which may, or may not be different – but remain unknown). Gaurav and Jane frequent the same unknown bar (Y1). Thus, instead one NULL value, we use indexed variables, or Skolem constants . V-tables algebra is shown to correctly support projection, positive selection (with no negation occurring in the selection condition), union, and renaming of attributes, which allows for processing arbitrary conjunctive queries. A very desirable property enjoyed by the V-table algebra is that all relational operators on tables are performed in exactly the same way as in the case of the usual relations. === Conditional tables (c-tables) algebra === Example of conditional table (c-table) is shown below. It has additional column “con” which is a Boolean condition involving variables, null values – same as in V-tables. over the following table c-table Conditional tables algebra, mainly of theoretical interest, supports projection, selection, union, join, and renaming. Under closed-world assumption, it can also handle the operator of difference, thus it can support all relational operators. == History == Imieliński–Lipski algebras were introduced by Tomasz Imieliński and Witold Lipski Jr. in Incomplete Information in Relational Databases.
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François Chollet

François Chollet (French: [fʁɑ̃swa ʃoˈlɛ]; born 20 October 1989) is a French software engineer, artificial intelligence (AI) researcher, and former Senior Staff Engineer at Google. Chollet is the creator of the Keras deep-learning library released in 2015. His research focuses on computer vision, the application of machine learning to formal reasoning, abstraction, and how to achieve greater generality in artificial intelligence (AGI). == Education and career == In 2012, Chollet graduated with a Diplôme d'Ingénieur (Master of Engineering) from ENSTA Paris, a school of the Polytechnic Institute of Paris. In 2015, Chollet started working at Google shortly after releasing Keras. In 2019, he published the Abstraction and Reasoning Corpus for Artificial General Intelligence (ARC-AGI) benchmark, which measures the ability of AI systems to solve novel reasoning problems. In 2024, Chollet launched ARC Prize, a US$1 million competition to solve the ARC-AGI benchmark. He left Google in November 2024 after more than 9 years with the company to found with Zapier co-founder Mike Knoop a new startup focused on developing AGI with program synthesis. In early 2025, Chollet announced the expansion of ARC Prize into a full-fledged non-profit foundation, to further the mission of guiding and accelerating research progress towards artificial general intelligence. == Books and publications == Chollet's research papers in artificial intelligence have been published at major conferences in the field, including the Conference on Computer Vision and Pattern Recognition (CVPR), the Conference on Neural Information Processing Systems (NeurIPS), and the International Conference on Learning Representations (ICLR). Chollet is the author of Xception: Deep Learning with Depthwise Separable Convolutions, which is among the top ten most cited papers in CVPR proceedings at more than 18,000 citations. Chollet is the author of the book Deep Learning with Python, which sold over 100,000 copies, and the co-author with Tomasz Kalinowski of Deep Learning With R. == Awards == On December 1, 2021, Chollet won the Global Swiss AI Award for breakthroughs in AI. In September 2024, Chollet was named by TIME as one of the 100 most influential people in AI.
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Stephen Muggleton

Stephen H. Muggleton (born 6 December 1959, son of Louis Muggleton) is Professor of Machine Learning and Head of the Computational Bioinformatics Laboratory at Imperial College London. == Education == Muggleton received his Bachelor of Science degree in computer science (1982) and Doctor of Philosophy in artificial intelligence (1986) supervised by Donald Michie at the University of Edinburgh. == Career == Following his PhD, Muggleton went on to work as a postdoctoral research associate at the Turing Institute in Glasgow (1987–1991) and later an EPSRC Advanced Research Fellow at Oxford University Computing Laboratory (OUCL) (1992–1997) where he founded the Machine Learning Group. In 1997 he moved to the University of York and in 2001 to Imperial College London. From 2025, Muggleton has joined Nanjing University as a full-time professor. == Research == Muggleton's research interests are primarily in Artificial intelligence. From 1997 to 2001 he held the Chair of Machine Learning at the University of York and from 2001 to 2006 the EPSRC Chair of Computational Bioinformatics at Imperial College in London. Since 2013 he holds the Syngenta/Royal Academy of Engineering Research Chair as well as the post of Director of Modelling for the Imperial College Centre for Integrated Systems Biology. He is known for founding the field of Inductive logic programming. In this field he has made contributions to theory introducing predicate invention, inverse entailment and stochastic logic programs. He has also played a role in systems development where he was instrumental in the systems Duce, Cigol, Golem, Progol and Metagol and applications – especially biological prediction tasks. He worked on a Robot Scientist together with Ross D. King that is capable of combining Inductive Logic Programming with active learning. His present work concentrates on the development of Meta-Interpretive Learning, a new form of Inductive Logic Programming which supports predicate invention and learning of recursive programs.
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The Best Free AI Virtual Assistant for Beginners

Comparing the best AI virtual assistant? An AI virtual assistant is software that uses machine learning to help you get more done — it lowers the barrier so anyone can produce professional output. Privacy matters too: check whether your data trains the model and whether a no-log or enterprise tier is available. Whether you are a beginner or a pro, the right AI virtual assistant 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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ChatGPT

ChatGPT is a generative artificial intelligence chatbot developed by OpenAI. Originally released in November 2022, the product uses large language models—specifically generative pre-trained transformers (GPTs)—to generate text, speech, and images in response to user prompts. ChatGPT accelerated the AI boom, an ongoing period marked by rapid investment and public attention toward the field of artificial intelligence (AI). OpenAI operates the service on a freemium model. Users can interact with ChatGPT through text, audio, and image prompts. ChatGPT was quickly adopted, reaching 100 million monthly active users two months after its release and 900 million weekly active users in February 2026. It has been lauded for its potential to transform numerous professional fields, and has instigated public debate about the nature of creativity and the future of knowledge work. The chatbot has also been criticized for its limitations and potential for unethical use. It can generate plausible-sounding but incorrect or nonsensical answers, known as hallucinations. Biases in its training data have been reflected in its responses. The chatbot can facilitate academic dishonesty, generate misinformation, and create malicious code. The ethics of its development, particularly the use of copyrighted content as training data, have also drawn controversy. == Features == ChatGPT is a chatbot and AI assistant built on large language model (LLM) technology. It is designed to generate human-like text and can carry out a wide variety of tasks. These include, among many others, writing and debugging computer programs, composing music, scripts, fairy tales, and essays, answering questions (sometimes at a level exceeding that of an average human test-taker), and generating business concepts. ChatGPT is frequently used for translation and summarization tasks, and can simulate interactive environments such as a Linux terminal, a multi-user chat room, or simple text-based games such as tic-tac-toe. Users interact with ChatGPT through conversations which consist of text, audio, and image inputs and outputs. The user's inputs to these conversations are referred to as prompts. An optional "Memory" feature allows users to tell ChatGPT to memorize specific information. Another option allows ChatGPT to recall old conversations. GPT-based moderation classifiers are used to reduce the risk of harmful outputs being presented to users. In March 2023, OpenAI added support for plugins for ChatGPT. This includes both plugins made by OpenAI, such as web browsing and code interpretation, and external plugins from developers such as Expedia, OpenTable, and Zapier. From October to December 2024, ChatGPT Search was deployed. It allows ChatGPT to search the web in an attempt to make more accurate and up-to-date responses. It increased OpenAI's direct competition with major search engines. OpenAI allows businesses to tailor how their content appears in the ChatGPT Search results and influence what sources are used. In December 2024, OpenAI launched a new feature allowing users to call ChatGPT with a telephone for up to 15 minutes per month for free. In September 2025, OpenAI added a feature called Pulse, which generates a daily analysis of a user's chats and connected apps such as Gmail and Google Calendar. In October 2025, OpenAI launched ChatGPT Atlas, a browser integrating the ChatGPT assistant directly into web navigation, to compete with existing browsers such as Google Chrome. It has an additional feature called "agentic mode" that allows it to take online actions for the user. === Paid tier === ChatGPT was initially free to the public and remains free in a limited capacity. In February 2023, OpenAI launched a premium service, ChatGPT Plus, that costs US$20 per month. What was offered on the paid plan versus the free tier changed as OpenAI has continued to update ChatGPT, and a Pro tier at $200/mo was introduced in December 2024. The Pro launch coincided with the release of the o1 model. In August 2025, ChatGPT Go was offered in India for ₹399 per month. The plan has higher limits than the free version. === Mobile apps === In May-July 2023, OpenAI began offering ChatGPT iOS and Android apps. ChatGPT can also power Android's assistant. An app for Windows launched on the Microsoft Store on October 15, 2024. === Languages === OpenAI met Icelandic President Guðni Th. Jóhannesson in 2022. In 2023, OpenAI worked with a team of 40 Icelandic volunteers to fine-tune ChatGPT's Icelandic conversation skills as a part of Iceland's attempts to preserve the Icelandic language. ChatGPT (based on GPT-4) was better able to translate Japanese to English when compared to Bing, Bard, and DeepL Translator in 2023. In December 2023, the Albanian government decided to use ChatGPT for the rapid translation of European Union documents and the analysis of required changes needed for Albania's accession to the EU. Several studies have shown that ChatGPT can outperform Google Translate in some mainstream translation tasks. However, as of 2024, no machine translation services match human expert performance. In August 2024, a representative of the Asia Pacific wing of OpenAI made a visit to Taiwan, during which a demonstration of ChatGPT's Chinese abilities was made. ChatGPT's Mandarin Chinese abilities were lauded, but the ability of the AI to produce content in Mandarin Chinese in a Taiwanese accent was found to be "less than ideal" due to differences between mainland Mandarin Chinese and Taiwanese Mandarin. === GPT Store === In November 2023, OpenAI released GPT Builder, a tool allowing users to customize ChatGPT's behavior for a specific use case. The customized systems are referred to as GPTs. In January 2024, OpenAI launched the GPT Store, a marketplace for GPTs. At launch, OpenAI included more than 3 million GPTs created by GPT Builder users in the GPT Store. === ChatGPT Apps === In September 2025, OpenAI added support for Model Context Protocol (MCP) to ChatGPT apps. When enabled in developer mode, this allows for improved third-party access to ChatGPT tools and servers. === Deep Research === In February 2025, OpenAI released Deep Research, a feature that generates reports based on extensive web searches. It was initially based on the reasoning model o3 and took 5 to 30 minutes per report. === Images === In October 2023, OpenAI's image generation model DALL-E 3 was integrated into ChatGPT. The integration used ChatGPT to write prompts for DALL-E guided by conversations with users. In March 2025, OpenAI updated ChatGPT to generate images using GPT Image instead of DALL-E. One of the most significant improvements was in the generation of text within images, which is especially useful for branded content. However, this ability is noticeably worse in non-Latin alphabets. The model can also generate new images based on existing ones provided in the prompt. These images are generated with C2PA metadata, which can be used to verify that they are AI-generated. OpenAI has emplaced additional safeguards to prevent what the company deems to be harmful image generation. === Agents === In 2025, OpenAI added several features to make ChatGPT more agentic (capable of autonomously performing longer tasks). In January, Operator was released. It was capable of autonomously performing tasks through web browser interactions, including filling forms, placing online orders, scheduling appointments, and other browser-based tasks. It was controlling a software environment inside a virtual machine with limited internet connectivity and with safety restrictions. It struggled with complex user interfaces. In May 2025, OpenAI introduced an agent for coding named Codex. It is capable of writing software, answering codebase questions, running tests, and proposing pull requests. It is based on a fine-tuned version of OpenAI o3. It has two versions, one running in a virtual machine in the cloud, and one where the agent runs in the cloud, but performs actions on a local machine connected via API. In July 2025, OpenAI released ChatGPT agent, an AI agent that can perform multi-step tasks. Like Operator, it controls a virtual computer. It also inherits from Deep Research's ability to gather and summarize significant volumes of information. The user can interrupt tasks or provide additional instructions as needed. In September 2025, OpenAI partnered with Stripe, Inc. to release Agentic Commerce Protocol, enabling purchases through ChatGPT. At launch, the feature was limited to purchases on Etsy from US users with a payment method linked to their OpenAI account. OpenAI takes an undisclosed cut from the merchant's payment. === ChatGPT Health === On January 7, 2026, OpenAI introduced a feature called "ChatGPT Health", whereby ChatGPT can discuss the user's health in a way that is separate from other chats. The feature is not available for users in the United Kingdom, Switzerland, or the European Economic Area, and is available on a waitli
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David Blei

David Meir Blei is a professor in the Statistics and Computer Science departments at Columbia University. Prior to fall 2014 he was an associate professor in the Department of Computer Science at Princeton University. His work is primarily in machine learning. == Research == His research interests include topic models and he was one of the original developers of latent Dirichlet allocation, along with Andrew Ng and Michael I. Jordan. As of June 18, 2020, his publications have been cited 109,821 times, giving him an h-index of 116. == Honors and awards == Blei received the ACM Infosys Foundation Award in 2013. (This award is given to a computer scientist under the age of 45. It has since been renamed the ACM Prize in Computing.) He was named Fellow of ACM "For contributions to the theory and practice of probabilistic topic modeling and Bayesian machine learning" in 2015.
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AI Blog Writers: Free vs Paid (2026)

Shopping for the best AI blog writer? An AI blog writer 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 blog writer 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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