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List of publications in data science

This is a list of publications in data science, generally organized by order of use in a data analysis workflow. See the list of publications in statistics for more research-based and fundamental publications; while this list is more applied, business oriented, and cross-disciplinary. General article inclusion criteria are: Papers from notable practitioners or notable professors, either with a Wikipedia page or reference to their notability Common knowledge all data professionals should know, with references validating this claim Highly cited applied statistics and machine learning publications Discussion-facilitating papers on the field of data science as a whole (for example, the Attention Is All You Need paper is arguably a landmark paper that can be added here, but it is specific to generative artificial intelligence, not for all practitioners of data) Some reasons why a particular publication might be regarded as important: Topic creator – A publication that created a new topic Breakthrough – A publication that changed scientific knowledge significantly Influence – A publication which has significantly influenced the world or has had a massive impact on the teaching of data science. When possible, a reference is used to validate the inclusion of the publication in this list. == History == Statistical Modeling: The Two Cultures (with comments and a rejoinder by the author) Author: Leo Breiman Publication data: Online version: https://projecteuclid.org/journals/statistical-science/volume-16/issue-3/Statistical-Modeling--The-Two-Cultures-with-comments-and-a/10.1214/ss/1009213726.pdf Description: Describes two cultures of statistics, one using a parsimonious and generative stochastic model, while the other is an algorithmic model with no known mechanism for how the data is generated. Breiman argues that while statistics has traditionally favored using the stochastic model, there is value in expanding the methods that statisticians can use to study phenomenon. Importance: Influence on the philosophies of statisticians right before the increased use of machine learning and deep learning methods. In a 20-year retrospective on this article, "Breiman's words are perhaps more relevant than ever". Notable statisticians at the time wrote opinion pieces about the publication. Although overall critical of the publication, David Cox writes that the publication "contains enough truth and exposes enough weaknesses to be thought-provoking." Bradley Efron commented that this publication is a "stimulating paper". Emanuel Parzen also comments about this publication that "Breiman alerts us to systematic blunders (leading to wrong conclusions) that have been committed applying current statistical practice of data modeling". Data Scientist: The Sexiest Job of the 21st Century Author: Thomas H. Davenport and DJ Patil Publication data: Online version: hbr.org/2022/07/is-data-scientist-still-the-sexiest-job-of-the-21st-century Description: Describes the new role at companies that is coined "Data scientist", what they do, how an organization might recruit one to their organization, and how to work with one effectively. Importance: This publication has been an influence on the data community as mentioned near the time it was published in 2012 by institutions like IEEE Spectrum, but also mentioned nearly a decade later asking the same question the title poses. In a retrospective response to their own publication 10 years earlier, authors Davenport and Patil have reflected that the role of a data scientist has "become better institutionalized, the scope of the job has been redefined, the technology it relies on has made huge strides, and the importance of non-technical expertise, such as ethics and change management, has grown". 50 Years of Data Science Author: David Donoho Publication data: Online version: https://www.tandfonline.com/doi/full/10.1080/10618600.2017.1384734 Description: Retrospective discussion paper on the history and origins of data science, with a number of commentary from notable statisticians. Importance: This has been described as "the first in the field to present such a comprehensive and in-depth survey and overview", and helps to define the field that has many definitions. The Composable Data Management System Manifesto Author: Pedro Pedreira, Orri Erling, Konstantinos Karanasos, Scott Schneider, Wes McKinney, Satya R Valluri, Mohamed Zait, Jacques Nadeau Publication data: Online version: https://www.vldb.org/pvldb/vol16/p2679-pedreira.pdf Description: The vision paper advocating for a paradigm shift in how data management systems are designed using standard, composable, interoperable tools rather than siloed software tools. Importance: A paradigm shifting view on how future data science software tools should be designed for more efficient workflows, the principles of which "will be especially crucial for addressing fragmentation, improving interoperability, and promoting user-centricity as data ecosystems grow increasingly complex". == Data collection and organization == Tidy Data Author: Hadley Wickham Publication data: Online version: https://www.jstatsoft.org/article/view/v059i10/ https://vita.had.co.nz/papers/tidy-data.pdf Description: Describes a framework for data cleaning that is summarized in the quote, "each variable is a column, each observation is a row, and each type of observational unit is a table". This allows a standard data structure for which data analysis tools can be consistently built around. Importance: Cited over 1,500 times, this effort for tidy data has been described by David Donoho as having "more impact on today's practice of data analysis than many highly regarded theoretical statistics articles". In the context of data visualization, this publication is said to support "efficient exploration and prototyping because variables can be assigned different roles in the plot without modifying anything about the original dataset". Data Organization in Spreadsheets Author: Karl W. Broman and Kara H. Woo Publication data: Online version: https://www.tandfonline.com/doi/full/10.1080/00031305.2017.1375989 Description: This article offers practical recommendations for organizing data in spreadsheets, like Microsoft Excel and Google Sheets, to reduce errors and lower the barrier for later analyses due to limitations in spreadsheets or quirks in the software. Importance: Influences teaching both data and non-data practitioners to create more analysis-friendly spreadsheets, and has been described to outline "spreadsheet best practices". == Data visualizations == Quantitative Graphics in Statistics: A Brief History Author: James R. Beniger and Dorothy L. Robyn Publication data: Online version: https://www.jstor.org/stable/2683467 Description: Outlines history and evolution of quantitative graphics in statistics, going through spatial organization (17th and 18th centuries), discrete comparison (18th and 19th centuries), continuous distribution (19th century), and multivariate distribution and correlation (late 19th and 20th centuries). Importance: Helps put into perspective for learning data practitioners the recency of graphics that are used. A later publication "Graphical Methods in Statistics" by Stephen Fienberg in 1979 writes that his publication "owes much to the work of Beniger and Robyn". == Practice == Data Science for Business Author: Foster Provost and Tom Fawcett Publication data: Online version: N/A Description: Broadly outlines principles of data science and data-analytic thinking for businesses. Importance: Cited over 3,000 times, it is "highly recommended for students" but also it is also recommended due to its "relevance to senior management leaders who want to build and lead a team of data scientists and implement data science in solving complex business problems". == Tooling == Hidden Technical Debt in Machine Learning Systems Author: D. Sculley, Gary Holy, Daniel Golovin, Eugene Davydov, Todd Phillips, Dietmar Ebner, Vinay Chaudhary, Michael Young, Jean-François Crespo, Dan Dennison Publication data: Online version: https://proceedings.neurips.cc/paper_files/paper/2015/file/86df7dcfd896fcaf2674f757a2463eba-Paper.pdf Description: This paper argues that it is "dangerous to think of [complex machine learning] quick wins as coming for free" and overviews risk factors to account for when implementing a machine learning system. Importance: All authors worked for Google, article is cited over 2,000 times, and helped practitioners thinking about quickly implementing a machine learning tool without understanding the long-term maintenance of the tool. A few useful things to know about machine learning Author: Pedro Domingos Publication data: Online version: https://dl.acm.org/doi/10.1145/2347736.2347755 https://homes.cs.washington.edu/~pedrod/papers/cacm12.pdf Description: The purpose of this paper is to distill inaccessible "folk knowledge" to effectively implement machine learning projects because "machin
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Bernhard Schölkopf

Bernhard Schölkopf (born 20 February 1968) is a German computer scientist known for his work in machine learning, especially on kernel methods and causality. He is a director at the Max Planck Institute for Intelligent Systems in Tübingen, Germany, where he heads the Department of Empirical Inference. He is also an affiliated professor at ETH Zürich, honorary professor at the University of Tübingen and Technische Universität Berlin, and chairman of the European Laboratory for Learning and Intelligent Systems (ELLIS). == Research == === Kernel methods === Schölkopf developed SVM methods achieving world record performance on the MNIST pattern recognition benchmark at the time. With the introduction of kernel PCA, Schölkopf and coauthors argued that SVMs are a special case of a much larger class of methods, and all algorithms that can be expressed in terms of dot products can be generalized to a nonlinear setting by means of what is known as reproducing kernels. Another significant observation was that the data on which the kernel is defined need not be vectorial, as long as the kernel Gram matrix is positive definite. Both insights together led to the foundation of the field of kernel methods, encompassing SVMs and many other algorithms. Kernel methods are now textbook knowledge and one of the major machine learning paradigms in research and applications. Developing kernel PCA, Schölkopf extended it to extract invariant features and to design invariant kernels and showed how to view other major dimensionality reduction methods such as LLE and Isomap as special cases. In further work with Alex Smola and others, he extended the SVM method to regression and classification with pre-specified sparsity and quantile/support estimation. He proved a representer theorem implying that SVMs, kernel PCA, and most other kernel algorithms, regularized by a norm in a reproducing kernel Hilbert space, have solutions taking the form of kernel expansions on the training data, thus reducing an infinite dimensional optimization problem to a finite dimensional one. He co-developed kernel embeddings of distributions methods to represent probability distributions in Hilbert Spaces, with links to Fraunhofer diffraction as well as applications to independence testing. === Causality === Starting in 2005, Schölkopf turned his attention to causal inference. Causal mechanisms in the world give rise to statistical dependencies as epiphenomena, but only the latter are exploited by popular machine learning algorithms. Knowledge about causal structures and mechanisms is useful by letting us predict not only future data coming from the same source, but also the effect of interventions in a system, and by facilitating transfer of detected regularities to new situations. Schölkopf and co-workers addressed (and in certain settings solved) the problem of causal discovery for the two-variable setting and connected causality to Kolmogorov complexity. Around 2010, Schölkopf began to explore how to use causality for machine learning, exploiting assumptions of independence of mechanisms and invariance. His early work on causal learning was exposed to a wider machine learning audience during his Posner lecture at NeurIPS 2011, as well as in a keynote talk at ICML 2017. He assayed how to exploit underlying causal structures in order to make machine learning methods more robust with respect to distribution shifts and systematic errors, the latter leading to the discovery of a number of new exoplanets including K2-18b, which was subsequently found to contain water vapour in its atmosphere, a first for an exoplanet in the habitable zone. == Education and employment == Schölkopf studied mathematics, physics, and philosophy in Tübingen and London. He was supported by the Studienstiftung and won the Lionel Cooper Memorial Prize for the best M.Sc. in Mathematics at the University of London. He completed a Diplom in Physics, and then moved to Bell Labs in New Jersey, where he worked with Vladimir Vapnik, who became co-adviser of his PhD thesis at TU Berlin (with Stefan Jähnichen). His thesis, defended in 1997, won the annual award of the German Informatics Association. In 2001, following positions in Berlin, Cambridge and New York, he founded the Department for Empirical Inference at the Max Planck Institute for Biological Cybernetics, which grew into a leading center for research in machine learning. In 2011, he became founding director at the Max Planck Institute for Intelligent Systems. With Alex Smola, Schölkopf co-founded the series of Machine Learning Summer Schools. He also co-founded a Cambridge-Tübingen PhD Programme and the Max Planck-ETH Center for Learning Systems. In 2016, he co-founded the Cyber Valley research consortium. He participated in the IEEE Global Initiative on "Ethically Aligned Design". Schölkopf is co-editor-in-Chief of the Journal of Machine Learning Research, a journal he helped found, being part of a mass resignation of the editorial board of Machine Learning (journal). He is among the world’s most cited computer scientists. Alumni of his lab include Ulrike von Luxburg, Carl Rasmussen, Matthias Hein, Arthur Gretton, Gunnar Rätsch, Matthias Bethge, Stefanie Jegelka, Jason Weston, Olivier Bousquet, Olivier Chapelle, Joaquin Quinonero-Candela, and Sebastian Nowozin. As of late 2023, Schölkopf is also a scientific advisor to French research group Kyutai which is being funded by Xavier Niel, Rodolphe Saadé, Eric Schmidt, and others. == Awards and recognition == Schölkopf’s awards include the Royal Society Milner Award and, shared with Isabelle Guyon and Vladimir Vapnik, the BBVA Foundation Frontiers of Knowledge Award in the Information and Communication Technologies category. He was the first scientist working in Europe to receive this award. He was elected a Fellow of the Royal Society in 2026.
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Best AI Bug Finders in 2026

In search of the best AI bug finder? An AI bug finder 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 bug finder 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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European Association for Machine Translation

The European Association for Machine Translation is the European branch of the International Association for Machine Translation Archived 2010-06-24 at the Wayback Machine. It is a non-profit organisation and organises conferences and workshops on the subject of machine translation. It was registered in 1991 in Switzerland and is the only organisation of its type in Europe.
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Bin picking

Bin picking (also referred to as random bin picking) is a core problem in computer vision and robotics. The goal is to have a robot with sensors and cameras attached to it pick-up known objects with random poses out of a bin using a suction gripper, parallel gripper, or other kind of robot end effector. Early work on bin picking made use of Photometric Stereo in recovering the shapes of objects and to determine their orientation in space. Amazon previously held a competition focused on bin picking referred to as the "Amazon Picking Challenge", which was held from 2015 to 2017. The challenge tasked entrants with building their own robot hardware and software that could attempt simplified versions of the general task of picking and stowing items on shelves. The robots were scored by how many items were picked and stowed in a fixed amount of time. The first Amazon Robotics challenge was won by a team from TU Berlin in 2015, followed by a team from TU Delft and the Dutch company "Fizyr" in 2016. The last Amazon Robotics Challenge was won by the Australian Centre for Robotic Vision at Queensland University of Technology with their robot named Cartman. The Amazon Robotics/Picking Challenge was discontinued following the 2017 competition. Although there can be some overlap, bin picking is distinct from "each picking" and the bin packing problem.
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How to Choose an AI Sales Assistant

In search of the best AI sales assistant? An AI sales assistant 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 sales assistant 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 Voice Assistants Reviews: What Actually Works in 2026

In search of the best AI voice assistant? An AI voice assistant 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 voice assistant 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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Ann Copestake

Ann Alicia Copestake is professor of computational linguistics and head of the Department of Computer Science and Technology at the University of Cambridge and a fellow of Wolfson College, Cambridge. == Education == Copestake was educated at the University of Cambridge where she was awarded a Bachelor of Arts degree in Natural Sciences. After two years working for Unilever Research she completed the Cambridge Diploma in Computer Science. She went on to study at the University of Sussex where she was awarded a PhD in 1992 for research on lexical semantics supervised by Gerald Gazdar. == Career and research == Copestake started doing research in Natural language processing and Computational Linguistics at the University of Cambridge in 1985. Since then she has been a visiting researcher at Xerox PARC (1993/4) and the University of Stuttgart (1994/5). From July 1994 to October 2000 she worked at the Center for the Study of Language and Information (CSLI) at Stanford University, as a Senior Researcher. Copestake was appointed a University Lecturer at Cambridge in October 2000. In the UK, her research has been funded by the Engineering and Physical Sciences Research Council (EPSRC) and Arts and Humanities Research Council (AHRC). According to Google Scholar and Scopus her most cited publications include papers on minimal recursion semantics, multiword expressions, polysemy, named-entity recognition and feature structure grammars.
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Google Mobile Services

Google Mobile Services (GMS) is a collection of proprietary applications and application programming interfaces (APIs) services from Google that are typically pre-installed on the majority of Android devices, such as smartphones, tablets, and smart TVs. GMS is not a part of the Android Open Source Project (AOSP), which means an Android manufacturer needs to obtain a license from Google in order to legally pre-install GMS on an Android device. This license is provided by Google without any licensing fees except in the EU. == Core applications == The following are core applications that are part of Google Mobile Services: Google Search Google Chrome YouTube Google Play Google Drive Gmail Google Meet Google Maps Google Photos Google TV YouTube Music === Historically === Google+ Google Hangouts Google Wallet Google Play Magazines Google Play Music Google Play Movies & TV Google Duo == Reception, competitors, and regulators == === FairSearch === Numerous European firms filed a complaint to the European Commission stating that Google had manipulated their power and dominance within the market to push their Services to be used by phone manufacturers. The firms were joined under the name FairSearch, and the main firms included were Microsoft, Expedia, TripAdvisor, Nokia and Oracle. FairSearch's major problem with Google's practices was that they believed Google were forcing phone manufacturers to use their Mobile Services. They claimed Google managed this by asking these manufacturers to sign a contract stating that they must preinstall specific Google Mobile Services, such as Maps, Search and YouTube, in order to get the latest version of Android. Google swiftly responded stating that they "continue to work co-operatively with the European Commission". === Aptoide === The third-party Android app store Aptoide also filed an EU competition complaint against Google once again stating that they are misusing their power within the market. Aptoide alleged that Google was blocking third-party app stores from being on Google Play, as well as blocking Google Chrome from downloading any third-party apps and app stores. As of June 2014, Google had not responded to these allegations. === Abuse of Android dominance === In May 2019, Umar Javeed, Sukarma Thapar, Aaqib Javeed vs. Google LLC & Ors. the Competition Commission of India ordered an antitrust probe against Google for abusing its dominant position with Android to block market rivals. In Prima Facie opinion the commission held that mandatory pre-installation of the entire Google Mobile Services (GMS) suite, under Mobile Application Distribution Agreements (MADA), amounts to the imposition of unfair conditions on the device manufacturers. === EU antitrust ruling === On July 18, 2018, the European Commission fined Google €4.34 billion for breaching EU antitrust rules which resulted in a change of licensing policy for the GMS in the EU. A new paid licensing agreement for smartphones and tablets shipped into the EEA was created. The change is that the GMS is now decoupled from the base Android and will be offered under a separate paid licensing agreement. === Privacy policy === At the same time, Google faced problems with various European data protection agencies, most notably In the United Kingdom and France. The problem they faced was that they had a set of 60 rules merged into one, which allowed Google to "track users more closely". Google once again came out and stated that their new policies still abide by European Union laws. === Android distributions without Google Mobile Services === After surveillance and privacy concerns, several custom android distributions have been implemented, such as GrapheneOS, LineageOS, CalyxOS, iodéOS or /e/OS, and they come either without any GMS installed by default or with microG, that adds a compatibility layer.
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Weighted automaton

In theoretical computer science and formal language theory, a weighted automaton or weighted finite-state machine is a generalization of a finite-state machine in which the edges have weights, for example real numbers or integers. Finite-state machines are only capable of answering decision problems; they take as input a string and produce a Boolean output, i.e. either "accept" or "reject". In contrast, weighted automata produce a quantitative output, for example a count of how many answers are possible on a given input string, or a probability of how likely the input string is according to a probability distribution. They are one of the simplest studied models of quantitative automata. The definition of a weighted automaton is generally given over an arbitrary semiring R {\displaystyle R} , an abstract set with an addition operation + {\displaystyle +} and a multiplication operation × {\displaystyle \times } . The automaton consists of a finite set of states, a finite input alphabet of characters Σ {\displaystyle \Sigma } and edges which are labeled with both a character in Σ {\displaystyle \Sigma } and a weight in R {\displaystyle R} . The weight of any path in the automaton is defined to be the product of weights along the path, and the weight of a string is the sum of the weights of all paths which are labeled with that string. The weighted automaton thus defines a function from Σ ∗ {\displaystyle \Sigma ^{}} to R {\displaystyle R} . Weighted automata generalize deterministic finite automata (DFAs) and nondeterministic finite automata (NFAs), which correspond to weighted automata over the Boolean semiring, where addition is logical disjunction and multiplication is logical conjunction. In the DFA case, there is only one accepting path for any input string, so disjunction is not applied. When the weights are real numbers and the outgoing weights for each state add to one, weighted automata can be considered a probabilistic model and are also known as probabilistic automata. These machines define a probability distribution over all strings, and are related to other probabilistic models such as Markov decision processes and Markov chains. Weighted automata have applications in natural language processing where they are used to assign weights to words and sentences, as well as in image compression. They were first introduced by Marcel-Paul Schützenberger in his 1961 paper On the definition of a family of automata. Since their introduction, many extensions have been proposed, for example nested weighted automata, cost register automata, and weighted finite-state transducers. Researchers have studied weighted automata from the perspective of learning a machine from its input-output behavior (see computational learning theory) and studying decidability questions. == Definition == A commutative semiring (or rig) is a set R equipped with two distinguished elements 0 ≠ 1 {\displaystyle 0\neq 1} and addition and multiplication operations ⊕ {\displaystyle \oplus } and ⊗ {\displaystyle \otimes } such that ⊕ {\displaystyle \oplus } is commutative and associative with identity 0 {\displaystyle 0} , ⊗ {\displaystyle \otimes } is commutative and associative with identity 1 {\displaystyle 1} , ⊗ {\displaystyle \otimes } distributes over ⊕ {\displaystyle \oplus } , and 0 is an absorbing element for ⊗ {\displaystyle \otimes } . A weighted automaton over R {\displaystyle R} is a tuple A = ( Q , Σ , Δ , I , F ) {\displaystyle {\mathcal {A}}=(Q,\Sigma ,\Delta ,I,F)} where: Q {\displaystyle Q} is a finite set of states. Σ {\displaystyle \Sigma } is a finite alphabet. Δ ⊆ Q × Σ × R × Q {\displaystyle \Delta \subseteq Q\times \Sigma \times R\times Q} is a finite set of transitions ( q , σ , w , q ′ ) {\displaystyle (q,\sigma ,w,q')} , where σ {\displaystyle \sigma } is called a character and w {\displaystyle w} is called a weight. I : Q → R {\displaystyle I:Q\to R} is an initial weight function. F : Q → R {\displaystyle F:Q\to R} is a final weight function. A path on input w ∈ Σ ∗ {\displaystyle w\in \Sigma ^{}} is a finite path in the graph, where the concatenation of the character labels equals w {\displaystyle w} . The weight of the path q 0 , q 1 , … , q n {\displaystyle q_{0},q_{1},\ldots ,q_{n}} is the product ( ⊗ {\displaystyle \otimes } ) of the weights along the path, additionally multiplied by the initial and final weights I ( q 0 ) ⊗ F ( q n ) {\displaystyle I(q_{0})\otimes F(q_{n})} . The weight of the word w {\displaystyle w} is the sum ( ⊕ {\displaystyle \oplus } ) of the weights of all paths on input w {\displaystyle w} (or 0 if there are no accepting paths). In this way the machine defines a function [ [ A ] ] : Σ ∗ → R {\displaystyle [\![{\mathcal {A}}]\!]:\Sigma ^{}\to R} . == Ambiguity and determinism == Since Δ {\displaystyle \Delta } is a set of transitions, weighted automata allow multiple transitions (or paths) on a single input string. Therefore a weighted automaton can be considered analogous to a nondeterministic finite automaton (NFA). As is the case with NFAs, restrictions of weighted automata are considered that correspond to the concepts of deterministic finite automaton and unambiguous finite automaton (deterministic weighted automata and unambiguous weighted automata, respectively). First, a preliminary definition: the underlying NFA of A {\displaystyle {\mathcal {A}}} is an NFA formed by removing all transitions with weight 0 {\displaystyle 0} and then erasing all of the weights on the transitions Δ {\displaystyle \Delta } , so that the new transition set lies in Q × Σ × Q {\displaystyle Q\times \Sigma \times Q} . The initial states and final states are the set of states q {\displaystyle q} such that I ( q ) ≠ 0 {\displaystyle I(q)\neq 0} and F ( q ) ≠ 0 {\displaystyle F(q)\neq 0} , respectively. A weighted automaton is deterministic if the underlying NFA is deterministic and unambiguous if the underlying NFA is unambiguous. Every deterministic weighted automaton is unambiguous. In both the deterministic and unambiguous cases, there is always at most one accepting path, so the ⊕ {\displaystyle \oplus } operation is never applied and can be omitted from the definition. == Variations == The requirement that there is a zero element for ⊕ {\displaystyle \oplus } is sometimes omitted; in this case the machine defines a partial function from Σ ∗ {\displaystyle \Sigma ^{}} to R {\displaystyle R} rather than a total function. It is possible to extend the definition to allow epsilon transitions ( q , ϵ , w , q ′ ) {\displaystyle (q,\epsilon ,w,q')} , where ϵ {\displaystyle \epsilon } is the empty string. In this case, one must then require that there are no cycles of epsilon transitions. This does not increase the expressiveness of weighted automata. If epsilon transitions are allowed, the initial weights and final weights can be replaced by initial and final sets of states without loss of expressiveness. Some authors omit the initial and final weight functions I {\displaystyle I} and F {\displaystyle F} . Instead, I {\displaystyle I} and F {\displaystyle F} are replaced by a set of initial and final states. If epsilon transitions are not present, this technically decreases expressiveness as it forces [ [ A ] ] ( ε ) {\displaystyle [\![{\mathcal {A}}]\!](\varepsilon )} to depend only on the number of states that are both initial and final. The transition function can be given as a matrix Δ σ ∈ R Q × Q {\displaystyle \Delta _{\sigma }\in R^{Q\times Q}} with entries in R {\displaystyle R} for each σ {\displaystyle \sigma } , rather than a set of transitions. The entry of the matrix at ( q , q ′ ) {\displaystyle (q,q')} is the sum of all transitions labeled ( q , σ , q ′ ) {\displaystyle (q,\sigma ,q')} . Some authors restrict to specific semirings, such as N {\displaystyle \mathbb {N} } or Z {\displaystyle \mathbb {Z} } , particularly when studying decidability results.
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Best AI Paragraph Rewriters in 2026

In search of the best AI paragraph rewriter? An AI paragraph rewriter 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 paragraph rewriter 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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Top 10 AI Analytics Tools Compared (2026)

Shopping for the best AI analytics tool? An AI analytics tool 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 analytics 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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Colour banding

Colour banding is a subtle form of posterisation in digital images, caused by the colour of each pixel being rounded to the nearest of the digital colour levels. While posterisation is often done for artistic effect, colour banding is an undesired artefact. In 24-bit colour modes, 8 bits per channel is usually considered sufficient to render images in Rec. 709 or sRGB. However the eye can see the difference between the colour levels, especially when there is a sharp border between two large areas of adjacent colour levels. This will happen with gradual gradients (like sunsets, dawns or clear blue skies), and also when blurring an image a large amount. Colour banding is more noticeable with fewer bits per pixel (BPP) at 16–256 colours (4–8 BPP), where there are fewer shades with a larger difference between them. The appearance of colour banding is exaggerated by the Mach bands effect. Possible solutions include the introduction of dithering and increasing the number of bits per colour channel. Because the banding comes from limitations in the presentation of the image, blurring the image does not fix this unless the image BPP is higher than the original.
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AI Customer-support Bots Reviews: What Actually Works in 2026

Looking for the best AI customer-support bot? An AI customer-support bot 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 customer-support bot 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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Co-Büchi automaton

In automata theory, a co-Büchi automaton is a variant of Büchi automaton. The only difference is the accepting condition: a Co-Büchi automaton accepts an infinite word w {\displaystyle w} if there exists a run, such that all the states occurring infinitely often in the run are in the final state set F {\displaystyle F} . In contrast, a Büchi automaton accepts a word w {\displaystyle w} if there exists a run, such that at least one state occurring infinitely often in the final state set F {\displaystyle F} . (Deterministic) Co-Büchi automata are strictly weaker than (nondeterministic) Büchi automata. == Formal definition == Formally, a deterministic co-Büchi automaton is a tuple A = ( Q , Σ , δ , q 0 , F ) {\displaystyle {\mathcal {A}}=(Q,\Sigma ,\delta ,q_{0},F)} that consists of the following components: Q {\displaystyle Q} is a finite set. The elements of Q {\displaystyle Q} are called the states of A {\displaystyle {\mathcal {A}}} . Σ {\displaystyle \Sigma } is a finite set called the alphabet of A {\displaystyle {\mathcal {A}}} . δ : Q × Σ → Q {\displaystyle \delta :Q\times \Sigma \rightarrow Q} is the transition function of A {\displaystyle {\mathcal {A}}} . q 0 {\displaystyle q_{0}} is an element of Q {\displaystyle Q} , called the initial state. F ⊆ Q {\displaystyle F\subseteq Q} is the final state set. A {\displaystyle {\mathcal {A}}} accepts exactly those words w {\displaystyle w} with the run ρ ( w ) {\displaystyle \rho (w)} , in which all of the infinitely often occurring states in ρ ( w ) {\displaystyle \rho (w)} are in F {\displaystyle F} . In a non-deterministic co-Büchi automaton, the transition function δ {\displaystyle \delta } is replaced with a transition relation Δ {\displaystyle \Delta } . The initial state q 0 {\displaystyle q_{0}} is replaced with an initial state set Q 0 {\displaystyle Q_{0}} . Generally, the term co-Büchi automaton refers to the non-deterministic co-Büchi automaton. For more comprehensive formalism see also ω-automaton. == Acceptance Condition == The acceptance condition of a co-Büchi automaton is formally ∃ i ∀ j : j ≥ i ρ ( w j ) ∈ F . {\displaystyle \exists i\forall j:\;j\geq i\quad \rho (w_{j})\in F.} The Büchi acceptance condition is the complement of the co-Büchi acceptance condition: ∀ i ∃ j : j ≥ i ρ ( w j ) ∈ F . {\displaystyle \forall i\exists j:\;j\geq i\quad \rho (w_{j})\in F.} == Properties == Co-Büchi automata are closed under union, intersection, projection and determinization.
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