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  Concurrency control

  In information technology and computer science, especially in the fields of computer programming, operating systems, multiprocessors, and databases, concurrency control ensures that correct results for concurrent operations are generated, while getting those results as quickly as possible. Computer systems, both software and hardware, consist of modules, or components. Each component is designed to operate correctly, i.e., to obey or to meet certain consistency rules. When components that operate concurrently interact by messaging or by sharing accessed data (in memory or storage), a certain component's consistency may be violated by another component. The general area of concurrency control provides rules, methods, design methodologies, and theories to maintain the consistency of components operating concurrently while interacting, and thus the consistency and correctness of the whole system. Introducing concurrency control into a system means applying operation constraints which typically result in some performance reduction. Operation consistency and correctness should be achieved with as good as possible efficiency, without reducing performance below reasonable levels. Concurrency control can require significant additional complexity and overhead in a concurrent algorithm compared to the simpler sequential algorithm. For example, a failure in concurrency control can result in data corruption from torn read or write operations. == Concurrency control in databases == Comments: This section is applicable to all transactional systems, i.e., to all systems that use database transactions (atomic transactions; e.g., transactional objects in Systems management and in networks of smartphones which typically implement private, dedicated database systems), not only general-purpose database management systems (DBMSs). DBMSs need to deal also with concurrency control issues not typical just to database transactions but rather to operating systems in general. These issues (e.g., see Concurrency control in operating systems below) are out of the scope of this section. Concurrency control in Database management systems (DBMS; e.g., Bernstein et al. 1987, Weikum and Vossen 2001), other transactional objects, and related distributed applications (e.g., Grid computing and Cloud computing) ensures that database transactions are performed concurrently without violating the data integrity of the respective databases. Thus concurrency control is an essential element for correctness in any system where two database transactions or more, executed with time overlap, can access the same data, e.g., virtually in any general-purpose database system. Consequently, a vast body of related research has been accumulated since database systems emerged in the early 1970s. A well established concurrency control theory for database systems is outlined in the references mentioned above: serializability theory, which allows to effectively design and analyze concurrency control methods and mechanisms. An alternative theory for concurrency control of atomic transactions over abstract data types is presented in (Lynch et al. 1993), and not utilized below. This theory is more refined, complex, with a wider scope, and has been less utilized in the Database literature than the classical theory above. Each theory has its pros and cons, emphasis and insight. To some extent they are complementary, and their merging may be useful. To ensure correctness, a DBMS usually guarantees that only serializable transaction schedules are generated, unless serializability is intentionally relaxed to increase performance, but only in cases where application correctness is not harmed. For maintaining correctness in cases of failed (aborted) transactions (which can always happen for many reasons) schedules also need to have the recoverability (from abort) property. A DBMS also guarantees that no effect of committed transactions is lost, and no effect of aborted (rolled back) transactions remains in the related database. Overall transaction characterization is usually summarized by the ACID rules below. As databases have become distributed, or needed to cooperate in distributed environments (e.g., Federated databases in the early 1990, and Cloud computing currently), the effective distribution of concurrency control mechanisms has received special attention. === Database transaction and the ACID rules === The concept of a database transaction (or atomic transaction) has evolved in order to enable both a well understood database system behavior in a faulty environment where crashes can happen any time, and recovery from a crash to a well understood database state. A database transaction is a unit of work, typically encapsulating a number of operations over a database (e.g., reading a database object, writing, acquiring lock, etc.), an abstraction supported in database and also other systems. Each transaction has well defined boundaries in terms of which program/code executions are included in that transaction (determined by the transaction's programmer via special transaction commands). Every database transaction obeys the following rules (by support in the database system; i.e., a database system is designed to guarantee them for the transactions it runs): Atomicity - Either the effects of all or none of its operations remain ("all or nothing" semantics) when a transaction is completed (committed or aborted respectively). In other words, to the outside world a committed transaction appears (by its effects on the database) to be indivisible (atomic), and an aborted transaction does not affect the database at all. Either all the operations are done or none of them are. Consistency - Every transaction must leave the database in a consistent (correct) state, i.e., maintain the predetermined integrity rules of the database (constraints upon and among the database's objects). A transaction must transform a database from one consistent state to another consistent state (however, it is the responsibility of the transaction's programmer to make sure that the transaction itself is correct, i.e., performs correctly what it intends to perform (from the application's point of view) while the predefined integrity rules are enforced by the DBMS). Thus since a database can be normally changed only by transactions, all the database's states are consistent. Isolation - Transactions cannot interfere with each other (as an end result of their executions). Moreover, usually (depending on concurrency control method) the effects of an incomplete transaction are not even visible to another transaction. Providing isolation is the main goal of concurrency control. Durability - Effects of successful (committed) transactions must persist through crashes (typically by recording the transaction's effects and its commit event in a non-volatile memory). The concept of atomic transaction has been extended during the years to what has become Business transactions which actually implement types of Workflow and are not atomic. However also such enhanced transactions typically utilize atomic transactions as components. === Why is concurrency control needed? === If transactions are executed serially, i.e., sequentially with no overlap in time, no transaction concurrency exists. However, if concurrent transactions with interleaving operations are allowed in an uncontrolled manner, some unexpected, undesirable results may occur, such as: The lost update problem: A second transaction writes a second value of a data-item (datum) on top of a first value written by a first concurrent transaction, and the first value is lost to other transactions running concurrently which need, by their precedence, to read the first value. The transactions that have read the wrong value end with incorrect results. The dirty read problem: Transactions read a value written by a transaction that has been later aborted. This value disappears from the database upon abort, and should not have been read by any transaction ("dirty read"). The reading transactions end with incorrect results. The incorrect summary problem: While one transaction takes a summary over the values of all the instances of a repeated data-item, a second transaction updates some instances of that data-item. The resulting summary does not reflect a correct result for any (usually needed for correctness) precedence order between the two transactions (if one is executed before the other), but rather some random result, depending on the timing of the updates, and whether certain update results have been included in the summary or not. Most high-performance transactional systems need to run transactions concurrently to meet their performance requirements. Thus, without concurrency control such systems can neither provide correct results nor maintain their databases consistently. === Concurrency control mechanisms === ==== Categories ==== The main categories of concurrency control mechanis

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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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  Structured support vector machine

  The structured supportvector machine is a machine learning algorithm that generalizes the support vector machine (SVM) classifier. Whereas the SVM classifier supports binary classification, multiclass classification and regression, the structured SVM allows training of a classifier for general structured output labels. As an example, a sample instance might be a natural language sentence, and the output label is an annotated parse tree. Training a classifier consists of showing pairs of correct sample and output label pairs. After training, the structured SVM model allows one to predict for new sample instances the corresponding output label; that is, given a natural language sentence, the classifier can produce the most likely parse tree. == Training == For a set of n {\displaystyle n} training instances ( x i , y i ) ∈ X × Y {\displaystyle ({\boldsymbol {x}}_{i},y_{i})\in {\mathcal {X}}\times {\mathcal {Y}}} , i = 1 , … , n {\displaystyle i=1,\dots ,n} from a sample space X {\displaystyle {\mathcal {X}}} and label space Y {\displaystyle {\mathcal {Y}}} , the structured SVM minimizes the following regularized risk function. min w ‖ w ‖ 2 + C ∑ i = 1 n max y ∈ Y ( 0 , Δ ( y i , y ) + ⟨ w , Ψ ( x i , y ) ⟩ − ⟨ w , Ψ ( x i , y i ) ⟩ ) {\displaystyle {\underset {\boldsymbol {w}}{\min }}\quad \|{\boldsymbol {w}}\|^{2}+C\sum _{i=1}^{n}{\underset {y\in {\mathcal {Y}}}{\max }}\left(0,\Delta (y_{i},y)+\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y)\rangle -\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y_{i})\rangle \right)} The function is convex in w {\displaystyle {\boldsymbol {w}}} because the maximum of a set of affine functions is convex. The function Δ : Y × Y → R + {\displaystyle \Delta :{\mathcal {Y}}\times {\mathcal {Y}}\to \mathbb {R} _{+}} measures a distance in label space and is an arbitrary function (not necessarily a metric) satisfying Δ ( y , z ) ≥ 0 {\displaystyle \Delta (y,z)\geq 0} and Δ ( y , y ) = 0 ∀ y , z ∈ Y {\displaystyle \Delta (y,y)=0\;\;\forall y,z\in {\mathcal {Y}}} . The function Ψ : X × Y → R d {\displaystyle \Psi :{\mathcal {X}}\times {\mathcal {Y}}\to \mathbb {R} ^{d}} is a feature function, extracting some feature vector from a given sample and label. The design of this function depends very much on the application. Because the regularized risk function above is non-differentiable, it is often reformulated in terms of a quadratic program by introducing one slack variable ξ i {\displaystyle \xi _{i}} for each sample, each representing the value of the maximum. The standard structured SVM primal formulation is given as follows. min w , ξ ‖ w ‖ 2 + C ∑ i = 1 n ξ i s.t. ⟨ w , Ψ ( x i , y i ) ⟩ − ⟨ w , Ψ ( x i , y ) ⟩ + ξ i ≥ Δ ( y i , y ) , i = 1 , … , n , ∀ y ∈ Y {\displaystyle {\begin{array}{cl}{\underset {{\boldsymbol {w}},{\boldsymbol {\xi }}}{\min }}&\|{\boldsymbol {w}}\|^{2}+C\sum _{i=1}^{n}\xi _{i}\\{\textrm {s.t.}}&\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y_{i})\rangle -\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y)\rangle +\xi _{i}\geq \Delta (y_{i},y),\qquad i=1,\dots ,n,\quad \forall y\in {\mathcal {Y}}\end{array}}} == Inference == At test time, only a sample x ∈ X {\displaystyle {\boldsymbol {x}}\in {\mathcal {X}}} is known, and a prediction function f : X → Y {\displaystyle f:{\mathcal {X}}\to {\mathcal {Y}}} maps it to a predicted label from the label space Y {\displaystyle {\mathcal {Y}}} . For structured SVMs, given the vector w {\displaystyle {\boldsymbol {w}}} obtained from training, the prediction function is the following. f ( x ) = argmax y ∈ Y ⟨ w , Ψ ( x , y ) ⟩ {\displaystyle f({\boldsymbol {x}})={\underset {y\in {\mathcal {Y}}}{\textrm {argmax}}}\quad \langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}},y)\rangle } Therefore, the maximizer over the label space is the predicted label. Solving for this maximizer is the so-called inference problem and similar to making a maximum a-posteriori (MAP) prediction in probabilistic models. Depending on the structure of the function Ψ {\displaystyle \Psi } , solving for the maximizer can be a hard problem. == Separation == The above quadratic program involves a very large, possibly infinite number of linear inequality constraints. In general, the number of inequalities is too large to be optimized over explicitly. Instead the problem is solved by using delayed constraint generation where only a finite and small subset of the constraints is used. Optimizing over a subset of the constraints enlarges the feasible set and will yield a solution that provides a lower bound on the objective. To test whether the solution w {\displaystyle {\boldsymbol {w}}} violates constraints of the complete set inequalities, a separation problem needs to be solved. As the inequalities decompose over the samples, for each sample ( x i , y i ) {\displaystyle ({\boldsymbol {x}}_{i},y_{i})} the following problem needs to be solved. y n ∗ = argmax y ∈ Y ( Δ ( y i , y ) + ⟨ w , Ψ ( x i , y ) ⟩ − ⟨ w , Ψ ( x i , y i ) ⟩ − ξ i ) {\displaystyle y_{n}^{}={\underset {y\in {\mathcal {Y}}}{\textrm {argmax}}}\left(\Delta (y_{i},y)+\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y)\rangle -\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y_{i})\rangle -\xi _{i}\right)} The right hand side objective to be maximized is composed of the constant − ⟨ w , Ψ ( x i , y i ) ⟩ − ξ i {\displaystyle -\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y_{i})\rangle -\xi _{i}} and a term dependent on the variables optimized over, namely Δ ( y i , y ) + ⟨ w , Ψ ( x i , y ) ⟩ {\displaystyle \Delta (y_{i},y)+\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y)\rangle } . If the achieved right hand side objective is smaller or equal to zero, no violated constraints for this sample exist. If it is strictly larger than zero, the most violated constraint with respect to this sample has been identified. The problem is enlarged by this constraint and resolved. The process continues until no violated inequalities can be identified. If the constants are dropped from the above problem, we obtain the following problem to be solved. y i ∗ = argmax y ∈ Y ( Δ ( y i , y ) + ⟨ w , Ψ ( x i , y ) ⟩ ) {\displaystyle y_{i}^{}={\underset {y\in {\mathcal {Y}}}{\textrm {argmax}}}\left(\Delta (y_{i},y)+\langle {\boldsymbol {w}},\Psi ({\boldsymbol {x}}_{i},y)\rangle \right)} This problem looks very similar to the inference problem. The only difference is the addition of the term Δ ( y i , y ) {\displaystyle \Delta (y_{i},y)} . Most often, it is chosen such that it has a natural decomposition in label space. In that case, the influence of Δ {\displaystyle \Delta } can be encoded into the inference problem and solving for the most violating constraint is equivalent to solving the inference problem.

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  Brian D. Ripley

  Brian David Ripley FRSE (born 29 April 1952) is a British statistician. From 1990, he was professor of applied statistics at the University of Oxford and also a professorial fellow at St Peter's College. He retired August 2014 due to ill health. == Biography == Ripley has made contributions to the fields of spatial statistics and pattern recognition. His work on artificial neural networks in the 1990s helped to bring aspects of machine learning and data mining to the attention of statistical audiences. He emphasised the value of robust statistics in his books Pattern Recognition and Neural Networks and Modern Applied Statistics with S. Ripley helped develop the S-PLUS programming language and its open source derivative R. He co-authored two books based on S, S Programming and Modern Applied Statistics with S. Since mid-1997 he is a member of the "R Core Team" and from 2000 to 2021 he was one of the most active committers to the R core. The package MASS is one of only fifteen "recommended packages" for R (with June 2024 more than 20,900). He was educated at the University of Cambridge, where he was awarded both the Smith's Prize (at the time awarded to the best graduate essay writer who had been undergraduate at Cambridge in that cohort) and the Rollo Davidson Prize. The university also awarded him the Adams Prize in 1987 for an essay entitled Statistical Inference for Spatial Processes, later published as a book. He served on the faculty of Imperial College, London from 1976 until 1983, at which point he moved to the University of Strathclyde. == Authored books == Ripley, B. D. (1981) Spatial Statistics. Wiley, 252pp. ISBN 0-471-08367-4. Ripley, B. D. (1983) Stochastic Simulation. Wiley, ISBN 0-471-81884-4. Ripley, B. D. (1988). Statistical Inference for Spatial Processes. Cambridge University Press. ISBN 0-521-35234-7. Ripley, B. D. (1996) Pattern Recognition and Neural Networks. Cambridge University Press. 403 pages. ISBN 0-521-46086-7. Venables, W. N. and Ripley, B. D. (2000) S Programming. Springer, 264pp. ISBN 978-0-387-98966-2. Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S (Fourth Edition; previous editions published as Modern Applied Statistics with S-PLUS in 1994, 1997 & 1999). Springer, 462pp. ISBN 978-0-387-95457-8.

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  MetaMask

  MetaMask is a software cryptocurrency wallet developed by ConsenSys for interacting with the Ethereum blockchain and other EVM-compatible networks. It enables users to manage Ethereum accounts and connect to decentralized applications (dApps) via a browser extension or mobile app. As of early 2026, MetaMask reports over 100 million users worldwide. == Overview == MetaMask allows users to store and manage private keys, send and receive Ethereum-based cryptocurrencies and tokens (including ERC-20 and ERC-721 standards), broadcast transactions, and interact with dApps. dApps connect to the wallet via JavaScript interfaces, prompting users to approve signatures or transactions. The wallet features MetaMask Swaps, an in-app token swap aggregator sourcing liquidity from multiple decentralized exchanges (DEXs), with a service fee of 0.875%. In 2025, MetaMask introduced the MetaMask Rewards program (initially mobile-only), where users earn points for activities such as swaps, bridging, and referrals. Season 1 (October 2025 – January 2026) distributed over $30 million in Linea tokens and other perks to participants. == History == MetaMask launched in 2016 as open-source software under the MIT license. It initially supported browser extensions for Chrome and Firefox. Mobile versions were in closed beta from 2019 and publicly released for iOS and Android in September 2020. In August 2020, the license changed to a custom proprietary one. MetaMask Swaps launched on desktop in October 2020 and on mobile in March 2021. The Rewards program launched in late 2025 with Linea integration. == Criticism == MetaMask has faced criticism over privacy, including default analytics settings that share some user data (which can be disabled). Its reliance on Infura (acquired by ConsenSys in 2019) has raised concerns about centralization in Ethereum infrastructure. The wallet regularly issues warnings about phishing scams and fake airdrops impersonating MetaMask.

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  Diane Litman

  Diane Litman is an American professor of computer science at the University of Pittsburgh. She also jointly holds the positions of senior scientist with the Learning Research and Development Center and faculty with the Intelligent Systems department. Litman is noted for her work in the areas of artificial intelligence, computational linguistics, knowledge representation and reasoning, natural language processing, and user modeling. == Education == Litman did her undergraduate studies at the College of William and Mary and her master's and PhD degrees at the University of Rochester. == Career == Before joining the University of Pittsburgh, she was an assistant professor at Columbia University. She additionally held the position of a research scientist in the Artificial Intelligence Principles Research Department Laboratory at AT&T Labs. Litman has held the position of Chair of the North American Chapter of the Association for Computational Linguistics two times, elected twice for the position, whose tenure lasts four years. She is also a distinguished member of the executive committee of the Association for Computational Linguistics, and a member of the editorial boards of Computational Linguistics and User Modeling and User-Adapted Interaction. She has also held the position of Leverhulme Professor at the University of Edinburgh. Litman was the keynote speaker at the Speech and Language Technology in Education 2013 symposium, the 2006 SIGdial Meeting on Discourse and Dialogue, and at the 2008 Symposium of the Annual Meeting of the Society for the Study of Artificial Intelligence and Simulation of Behaviour. She also sits on the board of the several interest groups, including the International Speech Communication Association's Special Interest Group on Speech and Language Technology in Education. Litman has served as chair, organizer, and a senior member of numerous committees of peer-reviewed scientific journals. == Awards and recognition == She has also co-authored numerous award-winning papers and was awarded senior member status by the Association for the Advancement of Artificial Intelligence in 2011, an award designed to honor those who have "achieved significant accomplishments within the field of artificial intelligence."

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  How to Choose an AI Chatbot

  Looking for the best AI chatbot? An AI chatbot 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 chatbot 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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  Monica S. Lam

  Monica Sin-Ling Lam is an American computer scientist. She is a professor in the Computer Science Department at Stanford University. == Education == Monica Lam received a B.Sc. from University of British Columbia in 1980 and a Ph.D. in computer science from Carnegie Mellon University in 1987. == Career == Lam joined the faculty of Computer Science at Stanford University in 1988. She has contributed to the research of a wide range of computer systems topics including compilers, program analysis, operating systems, security, computer architecture, and high-performance computing. More recently, she is working in natural language processing, and virtual assistants with an emphasis on privacy protection. She is the faculty director of the Open Virtual Assistant Lab, which organized the first workshop for the World Wide Voice Web. The lab developed the open-source Almond voice assistant, which is sponsored by the National Science Foundation. Almond received Popular Science's Best of What's New award in 2019. Previously, Lam led the SUIF (Stanford University Intermediate Format) Compiler project, which produced a widely used compiler infrastructure known for its locality optimizations and interprocedural parallelization. Many of the compiler techniques she developed have been adopted by industry. Her other research projects included the architecture and compiler for the CMU Warp machine, a systolic array of VLIW processors, and the Stanford DASH distributed shared memory machine. In 1998, she took a sabbatical leave from Stanford to help start Tensilica Inc., a company that specializes in configurable processor cores. In another research project, her program analysis group developed a collection of tools for improving software security and reliability. They developed the first scalable context-sensitive inclusion-based pointer analysis and a freely available tool called BDDBDDB, that allows programmers to express context-sensitive analyses simply by writing Datalog queries. Other tools developed include Griffin, static and dynamic analysis for finding security vulnerabilities in Web applications such as SQL injection, a static and dynamic program query language called QL, a static memory leak detector called Clouseau, a dynamic buffer overrun detector called CRED, and a dynamic error diagnosis tool called DIDUCE. In the Collective project, her research group and she developed the concept of a livePC: subscribers of the livePC will automatically run the latest of the published PC virtual images with each reboot. This approach allows computers to be managed scalably and securely. In 2005, the group started a company called MokaFive to transfer the technology to industry. She also directed the MobiSocial laboratory at Stanford, as part of the Programmable Open Mobile Internet 2020 initiative. Lam is also the cofounder of Omlet, which launched in 2014. Omlet is the first product from MobiSocial. Omlet is an open, decentralized social networking tool, based on an extensible chat platform. Lam chaired the ACM SIGPLAN Programming Languages Design and Implementation Conference in 2000, served on the Editorial Board of ACM Transactions on Computer Systems and numerous program committees for conferences on languages and compilers (PLDI, POPL), operating systems (SOSP), and computer architecture (ASPLOS, ISCA). == Awards and honors == National Academy of Engineering member, 2019 University of British Columbia Computer Science 50th Anniversary Research Award, 2018 Fellow of the ACM, 2007 ACM Programming Language Design and Implementation Best Paper Award in 2004 ACM SIGSOFT Distinguished Paper Award in 2002 ACM Most Influential Programming Language Design and Implementation Paper Award in 2001 NSF Young Investigator award in 1992 Two of her papers were recognized in "20 Years of PLDI--a Selection (1979-1999)" One of her papers was recognized in the "25 Years of the International Symposia on Computer Architecture", 1988. == Selected works == Compilers: Principles, Techniques and Tools (2d Ed) (2006) (the "Dragon Book") by Alfred V. Aho, Monica S. Lam, Ravi Sethi, and Jeffrey D. Ullman (ISBN 0-321-48681-1) A Systolic Array Optimizing Compiler (1989) (ISBN 0-89838-300-5) Monica Lam, Dissertation

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  FMLLR

  In signal processing, Feature space Maximum Likelihood Linear Regression (fMLLR) is a global feature transform that are typically applied in a speaker adaptive way, where fMLLR transforms acoustic features to speaker adapted features by a multiplication operation with a transformation matrix. In some literature, fMLLR is also known as the Constrained Maximum Likelihood Linear Regression (cMLLR). == Overview == fMLLR transformations are trained in a maximum likelihood sense on adaptation data. These transformations may be estimated in many ways, but only maximum likelihood (ML) estimation is considered in fMLLR. The fMLLR transformation is trained on a particular set of adaptation data, such that it maximizes the likelihood of that adaptation data given a current model-set. This technique is a widely used approach for speaker adaptation in HMM-based speech recognition. Later research also shows that fMLLR is an excellent acoustic feature for DNN/HMM hybrid speech recognition models. The advantage of fMLLR includes the following: the adaptation process can be performed within a pre-processing phase, and is independent of the ASR training and decoding process. this type of adapted feature can be applied to deep neural networks (DNN) to replace traditionally used mel-spectrogram in end-to-end speech recognition models. fMLLR's speaker adaptation process leads to a significant performance boost for ASR models, hence outperforming other transform or features like MFCCs (Mel-Frequency Cepstral Coefficients) and FBANKs (Filter bank) coefficients. fMLLR features can be efficiently realized with speech toolkits like Kaldi. Major problem and disadvantage of fMLLR: when the amount of adaptation data is limited, the transformation matrices tends to easily overfit the given data. == Computing fMLLR transform == Feature transform of fMLLR can be easily computed with the open source speech tool Kaldi, the Kaldi script uses the standard estimation scheme described in Appendix B of the original paper, in particular the section Appendix B.1 "Direct method over rows". In the Kaldi formulation, fMLLR is an affine feature transform of the form x {\displaystyle x} → A {\displaystyle A} x {\displaystyle x} + b {\displaystyle +b} , which can be written in the form x {\displaystyle x} →W x ^ {\displaystyle {\hat {x}}} , where x ^ {\displaystyle {\hat {x}}} = [ x 1 ] {\displaystyle {\begin{bmatrix}x\\1\end{bmatrix}}} is the acoustic feature x {\displaystyle x} with a 1 appended. Note that this differs from some of the literature where the 1 comes first as x ^ {\displaystyle {\hat {x}}} = [ 1 x ] {\displaystyle {\begin{bmatrix}1\\x\end{bmatrix}}} . The sufficient statistics stored are: K = ∑ t , j , m γ j , m ( t ) Σ j m − 1 μ j m x ( t ) + {\displaystyle K=\sum _{t,j,m}\gamma _{j,m}(t)\textstyle \Sigma _{jm}^{-1}\mu _{jm}x(t)^{+}\displaystyle } where Σ j m − 1 {\displaystyle \textstyle \Sigma _{jm}^{-1}\displaystyle } is the inverse co-variance matrix. And for 0 ≤ i ≤ D {\displaystyle 0\leq i\leq D} where D {\displaystyle D} is the feature dimension: G ( i ) = ∑ t , j , m γ j , m ( t ) ( 1 σ j , m 2 ( i ) ) x ( t ) + x ( t ) + T {\displaystyle G^{(i)}=\sum _{t,j,m}\gamma _{j,m}(t)\left({\frac {1}{\sigma _{j,m}^{2}(i)}}\right)x(t)^{+}x(t)^{+T}\displaystyle } For a thorough review that explains fMLLR and the commonly used estimation techniques, see the original paper "Maximum likelihood linear transformations for HMM-based speech recognition ". Note that the Kaldi script that performs the feature transforms of fMLLR differs with by using a column of the inverse in place of the cofactor row. In other words, the factor of the determinant is ignored, as it does not affect the transform result and can causes potential danger of numerical underflow or overflow. == Comparing with other features or transforms == Experiment result shows that by using the fMLLR feature in speech recognition, constant improvement is gained over other acoustic features on various commonly used benchmark datasets (TIMIT, LibriSpeech, etc). In particular, fMLLR features outperform MFCCs and FBANKs coefficients, which is mainly due to the speaker adaptation process that fMLLR performs. In, phoneme error rate (PER, %) is reported for the test set of TIMIT with various neural architectures: As expected, fMLLR features outperform MFCCs and FBANKs coefficients despite the use of different model architecture. Where MLP (multi-layer perceptron) serves as a simple baseline, on the other hand RNN, LSTM, and GRU are all well known recurrent models. The Li-GRU architecture is based on a single gate and thus saves 33% of the computations over a standard GRU model, Li-GRU thus effectively address the gradient vanishing problem of recurrent models. As a result, the best performance is obtained with the Li-GRU model on fMLLR features. == Extract fMLLR features with Kaldi == fMLLR can be extracted as reported in the s5 recipe of Kaldi. Kaldi scripts can certainly extract fMLLR features on different dataset, below are the basic example steps to extract fMLLR features from the open source speech corpora Librispeech. Note that the instructions below are for the subsets train-clean-100,train-clean-360,dev-clean, and test-clean, but they can be easily extended to support the other sets dev-other, test-other, and train-other-500. These instruction are based on the codes provided in this GitHub repository, which contains Kaldi recipes on the LibriSpeech corpora to execute the fMLLR feature extraction process, replace the files under $KALDI_ROOT/egs/librispeech/s5/ with the files in the repository. Install Kaldi. Install Kaldiio. If running on a single machine, change the following lines in $KALDI_ROOT/egs/librispeech/s5/cmd.sh to replace queue.pl to run.pl: Change the data path in run.sh to your LibriSpeech data path, the directory LibriSpeech/ should be under that path. For example: Install flac with: sudo apt-get install flac Run the Kaldi recipe run.sh for LibriSpeech at least until Stage 13 (included), for simplicity you can use the modified run.sh. Copy exp/tri4b/trans. files into exp/tri4b/decode_tgsmall_train_clean_/ with the following command: Compute the fMLLR features by running the following script, the script can also be downloaded here: Compute alignments using: Apply CMVN and dump the fMLLR features to new .ark files, the script can also be downloaded here: Use the Python script to convert Kaldi generated .ark features to .npy for your own dataloader, an example Python script is provided:

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  Yejin Choi

  Yejin Choi (Korean: 최예진; born 1977) is the Dieter Schwarz Foundation Professor and Senior Fellow at the Department of Computer Science at Stanford University and the Stanford Institute for Human-Centered Artificial Intelligence (HAI) respectively. Her research considers natural language processing and computer vision. == Early life and education == Choi is from South Korea. She attended Seoul National University. After earning a bachelor's degree in Computer Science, Choi moved to the United States, where she joined Cornell University as a graduate student. There she worked with Claire Cardie on natural language processing. After earning her doctorate, Choi joined Stony Brook University as an Assistant Professor of Computer Science. At Stony Brook University Choi developed a statistical technique to identify fake hotel reviews. == Research and career == In 2018 Choi joined the Allen Institute for AI. Her research looks to endow computers with a statistical understanding of written language. She became interested in neural networks and their application in artificial intelligence. She started to assemble a knowledge base that became known as the atlas of machine commonsense (ATOMIC). By the time she had finished the creation of ATOMIC, the language model generative Pre-trained Transformer 2 (GPT-2) had been released. ATOMIC does not make use of linguistic rules, but combines the representations of different languages within a neural network. In 2020, Choi was endowed with the Brett Helsel Professorship, which she held until she became Chair of Computer Science in 2023. She has since made use of Commonsense Transformers (COMET) with Good old fashioned artificial intelligence (GOFAI). The approach combines symbolic reasoning and neural networks. She has developed computational models that can detect biases in language that work against people from underrepresented groups. For example, one study demonstrated that female film characters are portrayed as less powerful than their male counterparts. In 2023, Choi became The Wissner-Slivka Chair of Computer Science. Choi is also a scientific advisor to French research group Kyutai which is being funded by Xavier Niel, Rodolphe Saadé, Eric Schmidt, and others. In 2025, Stanford HAI announced the appointment of Choi as senior fellow and the Dieter Schwarz Foundation HAI Professor and Professor of Computer Science at Stanford University. == Awards and honours == 2013 International Conference on Computer Vision Marr Prize 2016 Institute of Electrical and Electronics Engineers AI One to Watch 2017 Facebook ParlAI Research Award 2018 Anita Borg Early Career Award 2020 Association for the Advancement of Artificial Intelligence Outstanding Paper Award 2021 Conference on Neural Information Processing Systems Outstanding Paper Award 2021 Association for Computational Linguistics Test-of-time Paper Award 2021 Conference on Computer Vision and Pattern Recognition Longuet-Higgins Prize 2022 North American Chapter of the Association for Computational Linguistics Best Paper Award 2022 International Conference on Machine Learning Outstanding Paper Award 2022 MacArthur Fellowship 2023 Association for Computational Linguistics Best Paper Award 2023 TIME100 Archived 2024-12-27 at the Wayback Machine AI 2023 2023 Empirical Methods in Natural Language Processing Outstanding Paper Award 2025 Association for Computational Linguistics Outstanding Paper Award 2025 Association for Computational Linguistics Best Demo Paper Award 2025 TIME100 AI 2025 == Select publications == Ott, Myle; Choi, Yejin; Cardie, Claire; Hancock, Jeffrey T. (2011). "Finding Deceptive Opinion Spam by Any Stretch of the Imagination". Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies. Portland, Oregon, USA: Association for Computational Linguistics: 309–319. arXiv:1107.4557. Bibcode:2011arXiv1107.4557O. ISBN 9781932432879. S2CID 2510724. Kulkarni, Girish; Premraj, Visruth; Ordonez, Vicente; Dhar, Sagnik; Li, Siming; Choi, Yejin; Berg, Alexander C.; Berg, Tamara L. (2013). "BabyTalk: Understanding and Generating Simple Image Descriptions". IEEE Transactions on Pattern Analysis and Machine Intelligence. 35 (12): 2891–2903. Bibcode:2013ITPAM..35.2891K. CiteSeerX 10.1.1.225.5228. doi:10.1109/TPAMI.2012.162. ISSN 1939-3539. PMID 22848128. Choi, Yejin; Cardie, Claire; Riloff, Ellen; Patwardhan, Siddharth (2005). "Identifying sources of opinions with conditional random fields and extraction patterns". Proceedings of the conference on Human Language Technology and Empirical Methods in Natural Language Processing - HLT '05. Morristown, NJ, USA: Association for Computational Linguistics. pp. 355–362. doi:10.3115/1220575.1220620.

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  Best AI Content Generators 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. This guide breaks down the top picks, their pros and cons, and who each one is best for.

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  Amebis

  Amebis from Kamnik is a company in Slovenia in the field of language technologies. The company has published several electronic dictionaries and encyclopedic dictionaries (e.g. ASP (32) dictionaries) and developed spell checkers, grammar checker Besana, hyphenators and lemmatizers for Slovene, Serbian and Albanian languages. The company maintains and edits the largest Slovenian dictionary portal Termania, which contains more than 135 dictionaries. The most used terminological dictionary on Termania is the Slovenian medical dictionary. In co-operation with company Alpineon and the Jožef Stefan Institute they have developed a speech synthesizer and screen reader Govorec (Speaker). They have also provided technical support for the largest text corpus of Slovene, called FidaPLUS, Fran and Franček. Amebis also developed the system of machine translation Amebis Presis, which incorporates the Slovenian language. On 11 October 2023 Amebis received award of the Father Stanislav Škrabec Foundation for special achievements in Slovene linguistics.

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  H2O (software)

  H2O is an open-source, in-memory, distributed machine learning and predictive analytics platform developed by the company H2O.ai (previously 0xdata). The software uses a distributed architecture for parallel processing on standard hardware. It supports algorithms for large-scale data analysis and model deployment. H2O is primarily used by data scientists and developers for statistical modeling and data-driven decision-making. The platform is designed to handle in-memory computations across a distributed computing environment. It offers implementations for numerous statistical and machine learning algorithms, which are accessible through various programming interfaces. The software is released under the Apache License 2.0. == Functionality and features == H2O provides a suite of supervised and unsupervised machine learning algorithms. Its core functions include: Supervised learning: algorithms in the field of statistics, data mining and machine learning such as generalized linear models, random forests, gradient boosting and deep learning are implemented for classification and regression tasks. Unsupervised learning: including K-Means clustering and principal component analysis. Automated machine learning: a features designed to automate the processes of model selection, tuning, and ensemble creation. The software can ingest data from various sources, including the Hadoop Distributed File System, Amazon S3, SQL databases, as well as local file systems. It operates natively on Apache Spark clusters through Sparkling Water. Proponents claim that improved performance is achieved compared to other analysis tools. The software is distributed free of charge, under a business model based on the development of individual applications and support. == Architecture == H2O is primarily written in Java. It uses a distributed architecture that allows the platform to cluster nodes for parallel processing and in-memory storage of data and models. Users interact with the H2O platform through several primary interfaces: Programming language interfaces: APIs are provided for the R and Python programming languages, and various Apache offerings (Apache Hadoop and Spark, as well as Maven). H2O Flow: a graphical web-based interactive computational environment that functions as a notebook interface for data exploration, model building, and scripting. REST-API: allows for integration with other applications and frameworks such as Microsoft Excel or RStudio. With the H2O Machine Learning Integration Nodes, KNIME offers algorithmic workflows. While the algorithm executes, approximate results are displayed, so that users can track the progress and intervene if needed. == History, influences, and extensions == The software project was initiated by the company 0xdata, which later changed its name to H2O.ai. The three Stanford professors Stephen P. Boyd, Robert Tibshirani and Trevor Hastie form a panel that advises H2O on scientific issues. Since its inception, H2O provides open-source machine learning libraries for enterprise use. The core H2O platform is often complemented by offerings from H2O.ai, such as H2O Driverless AI. == Reception == H2O is referenced in peer-reviewed literature regarding automated machine learning (AutoML). The platform has been categorized as a "Leader" and a "Strong Performer" in industry reports by Forrester Research. H2O (the open-source platform) and the associated commercial platform Driverless AI have been recurring winners of InfoWorld's most prestigious awards, including both the Best of Open Source Software ("Bossies") and the Technology of the Year awards.

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  In search of the best AI text-to-image tool? An AI text-to-image 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 text-to-image 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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  State complexity

  State complexity is an area of theoretical computer science dealing with the size of abstract automata, such as different kinds of finite automata. The classical result in the area is that simulating an n {\displaystyle n} -state nondeterministic finite automaton by a deterministic finite automaton requires exactly 2 n {\displaystyle 2^{n}} states in the worst case. == Transformation between variants of finite automata == Finite automata can be deterministic and nondeterministic, one-way (DFA, NFA) and two-way (2DFA, 2NFA). Other related classes are unambiguous (UFA), self-verifying (SVFA) and alternating (AFA) finite automata. These automata can also be two-way (2UFA, 2SVFA, 2AFA). All these machines can accept exactly the regular languages. However, the size of different types of automata necessary to accept the same language (measured in the number of their states) may be different. For any two types of finite automata, the state complexity tradeoff between them is an integer function f {\displaystyle f} where f ( n ) {\displaystyle f(n)} is the least number of states in automata of the second type sufficient to recognize every language recognized by an n {\displaystyle n} -state automaton of the first type. The following results are known. NFA to DFA: 2 n {\displaystyle 2^{n}} states. This is the subset construction by Rabin and Scott, proved optimal by Lupanov. UFA to DFA: 2 n {\displaystyle 2^{n}} states, see Leung, An earlier lower bound by Schmidt was smaller. NFA to UFA: 2 n − 1 {\displaystyle 2^{n}-1} states, see Leung. There was an earlier smaller lower bound by Schmidt. SVFA to DFA: Θ ( 3 n / 3 ) {\displaystyle \Theta (3^{n/3})} states, see Jirásková and Pighizzini 2DFA to DFA: n ( n n − ( n − 1 ) n ) {\displaystyle n(n^{n}-(n-1)^{n})} states, see Kapoutsis. Earlier construction by Shepherdson used more states, and an earlier lower bound by Moore was smaller. 2DFA to NFA: ( 2 n n + 1 ) = O ( 4 n n ) {\displaystyle {\binom {2n}{n+1}}=O({\frac {4^{n}}{\sqrt {n}}})} , see Kapoutsis. Earlier construction by Birget used more states. 2NFA to NFA: ( 2 n n + 1 ) {\displaystyle {\binom {2n}{n+1}}} , see Kapoutsis. 2NFA to NFA accepting the complement: O ( 4 n ) {\displaystyle O(4^{n})} states, see Vardi. AFA to DFA: 2 2 n {\displaystyle 2^{2^{n}}} states, see Chandra, Kozen and Stockmeyer. AFA to NFA: 2 n {\displaystyle 2^{n}} states, see Fellah, Jürgensen and Yu. 2AFA to DFA: 2 n 2 n {\displaystyle 2^{n2^{n}}} , see Ladner, Lipton and Stockmeyer. 2AFA to NFA: 2 Θ ( n log ⁡ n ) {\displaystyle 2^{\Theta (n\log n)}} , see Geffert and Okhotin. === The 2DFA vs. 2NFA problem and logarithmic space === It is an open problem whether all 2NFAs can be converted to 2DFAs with polynomially many states, i.e. whether there is a polynomial p ( n ) {\displaystyle p(n)} such that for every n {\displaystyle n} -state 2NFA there exists a p ( n ) {\displaystyle p(n)} -state 2DFA. The problem was raised by Sakoda and Sipser, who compared it to the P vs. NP problem in the computational complexity theory. Berman and Lingas discovered a formal relation between this problem and the L vs. NL open problem. This relation was further elaborated by Kapoutsis. == State complexity of operations for finite automata == Given a binary regularity-preserving operation on languages ∘ {\displaystyle \circ } and a family of automata X (DFA, NFA, etc.), the state complexity of ∘ {\displaystyle \circ } is an integer function f ( m , n ) {\displaystyle f(m,n)} such that for each m-state X-automaton A and n-state X-automaton B there is an f ( m , n ) {\displaystyle f(m,n)} -state X-automaton for L ( A ) ∘ L ( B ) {\displaystyle L(A)\circ L(B)} , and for all integers m, n there is an m-state X-automaton A and an n-state X-automaton B such that every X-automaton for L ( A ) ∘ L ( B ) {\displaystyle L(A)\circ L(B)} must have at least f ( m , n ) {\displaystyle f(m,n)} states. Analogous definition applies for operations with any number of arguments. The first results on state complexity of operations for DFAs were published by Maslov and by Yu, Zhuang and Salomaa. Holzer and Kutrib pioneered the state complexity of operations on NFA. The known results for basic operations are listed below. === Union === If language L 1 {\displaystyle L_{1}} requires m states and language L 2 {\displaystyle L_{2}} requires n states, how many states does L 1 ∪ L 2 {\displaystyle L_{1}\cup L_{2}} require? DFA: m n {\displaystyle mn} states, see Maslov and Yu, Zhuang and Salomaa. NFA: m + n + 1 {\displaystyle m+n+1} states, see Holzer and Kutrib. UFA: at least min ( n , m ) Ω ( log ⁡ ( min ( n , m ) ) ) {\displaystyle \min(n,m)^{\Omega (\log(\min(n,m)))}} ; between m n + m + n {\displaystyle mn+m+n} and m + n m 2 0.79 m {\displaystyle m+nm2^{0.79m}} states, see Jirásek, Jirásková and Šebej. SVFA: m n {\displaystyle mn} states, see Jirásek, Jirásková and Szabari. 2DFA: between m + n {\displaystyle m+n} and 4 m + n + 4 {\displaystyle 4m+n+4} states, see Kunc and Okhotin. 2NFA: m + n {\displaystyle m+n} states, see Kunc and Okhotin. === Intersection === How many states does L 1 ∩ L 2 {\displaystyle L_{1}\cap L_{2}} require? DFA: m n {\displaystyle mn} states, see Maslov and Yu, Zhuang and Salomaa. NFA: m n {\displaystyle mn} states, see Holzer and Kutrib. UFA: m n {\displaystyle mn} states, see Jirásek, Jirásková and Šebej. SVFA: m n {\displaystyle mn} states, see Jirásek, Jirásková and Szabari. 2DFA: between m + n {\displaystyle m+n} and m + n + 1 {\displaystyle m+n+1} states, see Kunc and Okhotin. 2NFA: between m + n {\displaystyle m+n} and m + n + 1 {\displaystyle m+n+1} states, see Kunc and Okhotin. === Complementation === If language L requires n states then how many states does its complement require? DFA: n {\displaystyle n} states, by exchanging accepting and rejecting states. NFA: 2 n {\displaystyle 2^{n}} states, see Birget. or Jirásková UFA: at least n Ω ~ ( log ⁡ n ) {\displaystyle n^{{\tilde {\Omega }}(\log n)}} states, see Göös, Kiefer and Yuan, (this follows an earlier bound by Raskin); and at most n + 1 ⋅ 2 0.5 n {\displaystyle {\sqrt {n+1}}\cdot 2^{0.5n}} states, see Indzhev and Kiefer. SVFA: n {\displaystyle n} states, by exchanging accepting and rejecting states. 2DFA: at least n {\displaystyle n} and at most 4 n {\displaystyle 4n} states, see Geffert, Mereghetti and Pighizzini. === Concatenation === How many states does L 1 L 2 = { w 1 w 2 ∣ w 1 ∈ L 1 , w 2 ∈ L 2 } {\displaystyle L_{1}L_{2}=\{w_{1}w_{2}\mid w_{1}\in L_{1},w_{2}\in L_{2}\}} require? DFA: m ⋅ 2 n − 2 n − 1 {\displaystyle m\cdot 2^{n}-2^{n-1}} states, see Maslov and Yu, Zhuang and Salomaa. NFA: m + n {\displaystyle m+n} states, see Holzer and Kutrib. UFA: 3 4 2 m + n − 1 {\displaystyle {\frac {3}{4}}2^{m+n}-1} states, see Jirásek, Jirásková and Šebej. SVFA: Θ ( 3 n / 3 2 m ) {\displaystyle \Theta (3^{n/3}2^{m})} states, see Jirásek, Jirásková and Szabari. 2DFA: at least 2 Ω ( n ) log ⁡ m {\displaystyle {\frac {2^{\Omega (n)}}{\log m}}} and at most 2 m m + 1 ⋅ 2 n n + 1 {\displaystyle 2m^{m+1}\cdot 2^{n^{n+1}}} states, see Jirásková and Okhotin. === Kleene star === DFA: 3 4 2 n {\displaystyle {\frac {3}{4}}2^{n}} states, see Maslov and Yu, Zhuang and Salomaa. NFA: n + 1 {\displaystyle n+1} states, see Holzer and Kutrib. UFA: 3 4 2 n {\displaystyle {\frac {3}{4}}2^{n}} states, see Jirásek, Jirásková and Šebej. SVFA: 3 4 2 n {\displaystyle {\frac {3}{4}}2^{n}} states, see Jirásek, Jirásková and Szabari. 2DFA: at least 1 n 2 n 2 − 1 {\displaystyle {\frac {1}{n}}2^{{\frac {n}{2}}-1}} and at most 2 O ( n n + 1 ) {\displaystyle 2^{O(n^{n+1})}} states, see Jirásková and Okhotin. === Reversal === DFA: 2 n {\displaystyle 2^{n}} states, see Mirkin, Leiss, and Yu, Zhuang and Salomaa. NFA: n + 1 {\displaystyle n+1} states, see Holzer and Kutrib. UFA: n {\displaystyle n} states. SVFA: 2 n + 1 {\displaystyle 2n+1} states, see Jirásek, Jirásková and Szabari. 2DFA: between n + 1 {\displaystyle n+1} and n + 2 {\displaystyle n+2} states, see Jirásková and Okhotin. == Finite automata over a unary alphabet == State complexity of finite automata with a one-letter (unary) alphabet, pioneered by Chrobak, is different from the multi-letter case. Let g ( n ) = e Θ ( n ln ⁡ n ) {\displaystyle g(n)=e^{\Theta ({\sqrt {n\ln n}})}} be Landau's function. === Transformation between models === For a one-letter alphabet, transformations between different types of finite automata are sometimes more efficient than in the general case. NFA to DFA: g ( n ) + O ( n 2 ) {\displaystyle g(n)+O(n^{2})} states, see Chrobak. 2DFA to DFA: g ( n ) + O ( n ) {\displaystyle g(n)+O(n)} states, see Chrobak and Kunc and Okhotin. 2NFA to DFA: O ( g ( n ) ) {\displaystyle O(g(n))} states, see Mereghetti and Pighizzini. and Geffert, Mereghetti and Pighizzini. NFA to 2DFA: at most O ( n 2 ) {\displaystyle O(n^{2})} states, see Chrobak. 2NFA to 2DFA: at most n O ( log ⁡ n ) {\displaystyle n^{O(\log n)}} states, proved by implementing the method of Savitch's theorem, see

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