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Index locking

In databases an index is a data structure, part of the database, used by a database system to efficiently navigate access to user data. Index data are system data distinct from user data, and consist primarily of pointers. Changes in a database (by insert, delete, or modify operations), may require indexes to be updated to maintain accurate user data accesses. Index locking is a technique used to maintain index integrity. A portion of an index is locked during a database transaction when this portion is being accessed by the transaction as a result of attempt to access related user data. Additionally, special database system transactions (not user-invoked transactions) may be invoked to maintain and modify an index, as part of a system's self-maintenance activities. When a portion of an index is locked by a transaction, other transactions may be blocked from accessing this index portion (blocked from modifying, and even from reading it, depending on lock type and needed operation). Index Locking Protocol guarantees that phantom read phenomenon won't occur. Index locking protocol states: Every relation must have at least one index. A transaction can access tuples only after finding them through one or more indices on the relation A transaction Ti that performs a lookup must lock all the index leaf nodes that it accesses, in S-mode, even if the leaf node does not contain any tuple satisfying the index lookup (e.g. for a range query, no tuple in a leaf is in the range) A transaction Ti that inserts, updates or deletes a tuple ti in a relation r must update all indices to r and it must obtain exclusive locks on all index leaf nodes affected by the insert/update/delete The rules of the two-phase locking protocol must be observed. Specialized concurrency control techniques exist for accessing indexes. These techniques depend on the index type, and take advantage of its structure. They are typically much more effective than applying to indexes common concurrency control methods applied to user data. Notable and widely researched are specialized techniques for B-trees (B-Tree concurrency control) which are regularly used as database indexes. Index locks are used to coordinate threads accessing indexes concurrently, and typically shorter-lived than the common transaction locks on user data. In professional literature, they are often called latches.
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Dan Hendrycks

Dan Hendrycks (born 1994 or 1995) is an American machine learning researcher. He serves as the director of the Center for AI Safety, a nonprofit research organization based in San Francisco, California. == Early life and education == Hendrycks was raised in a Christian evangelical household in Marshfield, Missouri. He received a B.S. from the University of Chicago in 2018 and a Ph.D. from the University of California, Berkeley in Computer Science in 2022. == Career and research == Hendrycks' research focuses on topics that include machine learning safety, machine ethics, and robustness. He credits his participation in the effective altruism (EA) movement-linked 80,000 Hours program for his career focus towards AI safety, though denies being an advocate for EA. Hendrycks is the main author of the research paper that introduced the activation function GELU in 2016, and of the paper that introduced the language model benchmark MMLU (Massive Multitask Language Understanding) in 2020. In February 2022, Hendrycks co-authored recommendations for the US National Institute of Standards and Technology (NIST) to inform the management of risks from artificial intelligence. In September 2022, Hendrycks wrote a paper providing a framework for analyzing the impact of AI research on societal risks. He later published a paper in March 2023 examining how natural selection and competitive pressures could shape the goals of artificial agents. This was followed by "An Overview of Catastrophic AI Risks", which discusses four categories of risks: malicious use, AI race dynamics, organizational risks, and rogue AI agents. Hendrycks is the safety adviser of xAI, an AI startup company founded by Elon Musk in 2023. To avoid any potential conflicts of interest, he receives a symbolic one-dollar salary and holds no company equity. In November 2024, he also joined Scale AI as an advisor collecting a one-dollar salary. Hendrycks is the creator of Humanity's Last Exam, a benchmark for evaluating the capabilities of large language models, which he developed in collaboration with Scale AI. In 2024, Hendrycks published the textbook Introduction to AI Safety, Ethics, and Society, based on courseware he had previously developed. == Selected publications == Hendrycks, Dan; Gimpel, Kevin (2020-07-08). "Gaussian Error Linear Units (GELUs)". arXiv:1606.08415 [cs.LG]. Hendrycks, Dan; Gimpel, Kevin (2018-10-03). "A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks". International Conference on Learning Representations 2017. arXiv:1610.02136. Hendrycks, Dan; Mazeika, Mantas; Dietterich, Thomas (2019-01-28). "Deep Anomaly Detection with Outlier Exposure". International Conference on Learning Representations 2019. arXiv:1812.04606. Hendrycks, Dan; Mazeika, Mantas; Zou, Andy (2021-10-25). "What Would Jiminy Cricket Do? Towards Agents That Behave Morally". Conference on Neural Information Processing Systems 2021. arXiv:2110.13136.
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Maghi King

Margaret (Maghi) Daniel King is a retired British computational linguist known for her work on evaluating the quality of machine translation. She is an honorary professor in the Department of Translation Technology of the University of Geneva in Switzerland, and the former director of the Dalle Molle Institute for Semantic and Cognitive Studies at the University of Geneva. == Education and career == King read classics, Ancient History and Philosophy (Greats) at the University of Oxford, worked as a computer programmer, and became a lecturer in the Department of Computation at the University of Manchester Institute of Science and Technology. She moved to the Dalle Molle Institute for Semantic and Cognitive Studies (ISSCO) in 1974. In 1976, ISSCO became part of the University of Geneva, and she continued there, becoming ISSCO's director in 1978. She remained director until her retirement in 2006. == Recognition == King is a Fellow of the European Association for Artificial Intelligence (formerly ECCAI), elected in 1999.
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AI Video Editors: Free vs Paid (2026)

Trying to pick the best AI video editor? An AI video editor 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 video editor 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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Outline of natural language processing

Natural language processing is computer activity in which computers are entailed to analyze, understand, alter, or generate natural language. This includes the automation of any or all linguistic forms, activities, or methods of communication, such as conversation, correspondence, reading, written composition, dictation, publishing, translation, lip reading, and so on. Natural-language processing is also the name of the branch of computer science, artificial intelligence, and linguistics concerned with enabling computers to engage in communication using natural language(s) in all forms, including but not limited to speech, print, writing, and signing. The following outline is provided as an overview of and topical guide to natural-language processing: == Natural-language processing == Natural-language processing can be described as all of the following: A field of science – systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe. An applied science – field that applies human knowledge to build or design useful things. A field of computer science – scientific and practical approach to computation and its applications. A branch of artificial intelligence – intelligence of machines and robots and the branch of computer science that aims to create it. A subfield of computational linguistics – interdisciplinary field dealing with the statistical or rule-based modeling of natural language from a computational perspective. An application of engineering – science, skill, and profession of acquiring and applying scientific, economic, social, and practical knowledge, in order to design and also build structures, machines, devices, systems, materials and processes. An application of software engineering – application of a systematic, disciplined, quantifiable approach to the design, development, operation, and maintenance of software, and the study of these approaches; that is, the application of engineering to software. A subfield of computer programming – process of designing, writing, testing, debugging, and maintaining the source code of computer programs. This source code is written in one or more programming languages (such as Java, C++, C#, Python, etc.). The purpose of programming is to create a set of instructions that computers use to perform specific operations or to exhibit desired behaviors. A subfield of artificial intelligence programming – A type of system – set of interacting or interdependent components forming an integrated whole or a set of elements (often called 'components' ) and relationships which are different from relationships of the set or its elements to other elements or sets. A system that includes software – software is a collection of computer programs and related data that provides the instructions for telling a computer what to do and how to do it. Software refers to one or more computer programs and data held in the storage of the computer. In other words, software is a set of programs, procedures, algorithms and its documentation concerned with the operation of a data processing system. A type of technology – making, modification, usage, and knowledge of tools, machines, techniques, crafts, systems, methods of organization, in order to solve a problem, improve a preexisting solution to a problem, achieve a goal, handle an applied input/output relation or perform a specific function. It can also refer to the collection of such tools, machinery, modifications, arrangements and procedures. Technologies significantly affect human as well as other animal species' ability to control and adapt to their natural environments. A form of computer technology – computers and their application. NLP makes use of computers, image scanners, microphones, and many types of software programs. Language technology – consists of natural-language processing (NLP) and computational linguistics (CL) on the one hand, and speech technology on the other. It also includes many application oriented aspects of these. It is often called human language technology (HLT). == Prerequisite technologies == The following technologies make natural-language processing possible: Communication – the activity of a source sending a message to a receiver Language – Speech – Writing – Computing – Computers – Computer programming – Information extraction – User interface – Software – Text editing – program used to edit plain text files Word processing – piece of software used for composing, editing, formatting, printing documents Input devices – pieces of hardware for sending data to a computer to be processed Computer keyboard – typewriter style input device whose input is converted into various data depending on the circumstances Image scanners – == Subfields of natural-language processing == Information extraction (IE) – field concerned in general with the extraction of semantic information from text. This covers tasks such as named-entity recognition, coreference resolution, relationship extraction, etc. Ontology engineering – field that studies the methods and methodologies for building ontologies, which are formal representations of a set of concepts within a domain and the relationships between those concepts. Speech processing – field that covers speech recognition, text-to-speech and related tasks. Statistical natural-language processing – Statistical semantics – a subfield of computational semantics that establishes semantic relations between words to examine their contexts. Distributional semantics – a subfield of statistical semantics that examines the semantic relationship of words across a corpora or in large samples of data. == Related fields == Natural-language processing contributes to, and makes use of (the theories, tools, and methodologies from), the following fields: Automated reasoning – area of computer science and mathematical logic dedicated to understanding various aspects of reasoning, and producing software which allows computers to reason completely, or nearly completely, automatically. A sub-field of artificial intelligence, automatic reasoning is also grounded in theoretical computer science and philosophy of mind. Linguistics – scientific study of human language. Natural-language processing requires understanding of the structure and application of language, and therefore it draws heavily from linguistics. Applied linguistics – interdisciplinary field of study that identifies, investigates, and offers solutions to language-related real-life problems. Some of the academic fields related to applied linguistics are education, linguistics, psychology, computer science, anthropology, and sociology. Some of the subfields of applied linguistics relevant to natural-language processing are: Bilingualism / Multilingualism – Computer-mediated communication (CMC) – any communicative transaction that occurs through the use of two or more networked computers. Research on CMC focuses largely on the social effects of different computer-supported communication technologies. Many recent studies involve Internet-based social networking supported by social software. Contrastive linguistics – practice-oriented linguistic approach that seeks to describe the differences and similarities between a pair of languages. Conversation analysis (CA) – approach to the study of social interaction, embracing both verbal and non-verbal conduct, in situations of everyday life. Turn-taking is one aspect of language use that is studied by CA. Discourse analysis – various approaches to analyzing written, vocal, or sign language use or any significant semiotic event. Forensic linguistics – application of linguistic knowledge, methods and insights to the forensic context of law, language, crime investigation, trial, and judicial procedure. Interlinguistics – study of improving communications between people of different first languages with the use of ethnic and auxiliary languages (lingua franca). For instance by use of intentional international auxiliary languages, such as Esperanto or Interlingua, or spontaneous interlanguages known as pidgin languages. Language assessment – assessment of first, second or other language in the school, college, or university context; assessment of language use in the workplace; and assessment of language in the immigration, citizenship, and asylum contexts. The assessment may include analyses of listening, speaking, reading, writing or cultural understanding, with respect to understanding how the language works theoretically and the ability to use the language practically. Language pedagogy – science and art of language education, including approaches and methods of language teaching and study. Natural-language processing is used in programs designed to teach language, including first- and second-language training. Language planning – Language policy – Lexicography – Literacies – Pragmatics – Second-language acquisition – Stylistics – Translation – Comp
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Best AI Marketing Tools in 2026

Trying to pick the best AI marketing tool? An AI marketing tool 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 marketing tool 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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Jerome H. Friedman

Jerome Harold Friedman (born December 29, 1939) is an American statistician, consultant and Professor of Statistics at Stanford University, known for his contributions in the field of statistics and data mining. == Biography == Friedman studied at Chico State College for two years before transferring to the University of California, Berkeley in 1959, where he received his AB in Physics in 1962, and his PhD in High Energy Particle Physics in 1967. In 1968 he started his academic career as research physicist at the Lawrence Berkeley National Laboratory. In 1972 he started at Stanford University as leader of the Computation Research Group at the Stanford Linear Accelerator Center, where he would participate until 2003. In the year 1976–77 he was a visiting scientist at CERN in Geneva. From 1981 to 1984 he was visiting professor at the University of California, Berkeley. In 1982 he was appointed Professor of Statistics at Stanford University. In 1984 he was elected as a Fellow of the American Statistical Association. In 2002 he was awarded the SIGKDD Innovation Award by the Association for Computing Machinery (ACM). In 2010 he was elected as a member of the National Academy of Sciences (Applied mathematical sciences). == Publications == Friedman has authored and co-authored many publications in the field of data-mining including "nearest neighbor classification, logistical regressions, and high dimensional data analysis. His primary research interest is in the area of machine learning." A selection: Friedman, Jerome H. & Tukey, John W. (1974). "A projection pursuit algorithm for exploratory data analysis". IEEE Transactions on Computers. 23 (9): 881–890. doi:10.1109/T-C.1974.224051. OSTI 1442925. S2CID 7997450. Friedman, Jerome H. & Stuetzle, Werner (1981). "Projection pursuit regression". Journal of the American Statistical Association. 76 (376): 817–823. doi:10.1080/01621459.1981.10477729. OSTI 1445517. Friedman, Jerome H. (1991). "Multivariate adaptive regression splines". Annals of Statistics. 19 (1): 1–67. CiteSeerX 10.1.1.382.970. doi:10.1214/aos/1176347963. JSTOR 2241837. Friedman, Jerome H. (2001). "Greedy function approximation: a gradient boosting machine". Annals of Statistics. 29 (5): 1189–1232. doi:10.1214/aos/1013203451. JSTOR 2699986.
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Alberto Broggi

Alberto Broggi is General Manager at VisLab srl (spinoff of the University of Parma acquired by Silicon-Valley company Ambarella Inc. in June 2015) and a professor of Computer Engineering at the University of Parma in Italy. == Research in computer vision, hardware, and AV == Broggi's research activities started in 1991–1994. His group together with the Dipartimento di Elettronica, Politecnico di Torino, Italy, built their own hardware architecture (named PAPRICA, for PArallel PRocessor for Image Checking and Analysis, based on 256 single-bit processing elements working in SIMD fashion) and installed it on board of a mobile laboratory (Mob-Lab) to develop and test some initial concepts in the field of intelligent vehicles. In 1996, Broggi's group worked to develop a real vehicle prototype (named ARGO, a Lancia Thema passenger car which was equipped with vision sensors, processing systems, and vehicle actuators) and developed the necessary software and hardware that made it able to drive autonomously on standard roads. Broggi's research group (called VisLab from then on) gathered all their findings in a book, which was then also translated in Chinese. When Broggi was with the University of Pavia, his research was extended and applied to extreme conditions (automatic driving on snow and ice): in 2001, VisLab led the research effort of providing a vehicle (RAS, Robot Antartico di Superficie) with sensing capabilities so that it was able to automatically follow the vehicle in front. In 2010 Broggi's group embarked on driving 4 vehicles autonomously from Italy to China with no human intervention. This challenge is called VIAC, for VisLab Intercontinental Autonomous Challenge . Soon after this, Broggi was awarded a second ERC grant (Proof of concept) to industrialize some of the results obtained and successfully tested on the VIAC vehicles. On July 12, 2013, VisLab tested the BRAiVE vehicle in downtown Parma, negotiating two-way narrow rural roads, pedestrian crossings, traffic lights, artificial bumps, pedestrian areas, and tight roundabouts. The vehicle traveled from Parma University Campus up to Piazza della Pilotta (downtown Parma): a 20 minutes run in a real environment, together with real traffic at 11am on a working day, that required absolutely no human intervention. Part of this test was driven with nobody in the driver seat, for the first time ever on public roads.
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Document mosaicing

Document mosaicing is a process that stitches multiple, overlapping snapshot images of a document together to produce one large, high resolution composite. The document is slid under a stationary, over-the-desk camera by hand until all parts of the document are snapshotted by the camera's field of view. As the document slid under the camera, all motion of the document is coarsely tracked by the vision system. The document is periodically snapshotted such that the successive snapshots are overlap by about 50%. The system then finds the overlapped pairs and stitches them together repeatedly until all pairs are stitched together as one piece of document. The document mosaicing can be divided into four main processes. Tracking Feature detecting Correspondences establishing Images mosaicing. == Tracking (simple correlation process) == In this process, the motion of the document slid under the camera is coarsely tracked by the system. Tracking is performed by a process called simple correlation process. In the first frame of snapshots, a small patch is extracted from the center of the image as a correlation template. The correlation process is performed in the four times size of the patch area of the next frame. The motion of the paper is indicated by the peak in the correlation function. The peak in the correlation function indicates the motion of the paper. The template is resampled from this frame and the tracking continues until the template reaches the edge of the document. After the template reaches the edge of the document, another snapshot is taken and the tracking process performs repeatedly until the whole document is imaged. The snapshots are stored in an ordered list to facilitate pairing the overlapped images in later processes. == Feature detecting for efficient matching == Feature detection is the process of finding the transformation that aligns one image with another. There are two main approaches for feature detection. Feature-based approach : Motion parameters are estimated from point correspondences. This approach is suitable for the case that there is plenty supply of stable and detectable features. Featureless approach : When the motion between the two images is small, the motion parameters are estimated using optical flow. On the other hand, when the motion between the two images is large, the motion parameters are estimated using generalised cross-correlation. However, this approach requires a computationally expensive resources. Each image is segmented into a hierarchy of columns, lines, and words to match the organised sets of features across images. Skew angle estimation and columns, lines and words finding are the examples of feature detection operations. === Skew angle estimation === Firstly, the angle that the rows of text make with the image raster lines (skew angle) is estimated. It is assumed to lie in the range of ±20°. A small patch of text in the image is selected randomly and then rotated in the range of ±20° until the variance of the pixel intensities of the patch summed along the raster lines is maximised. To ensure that the found skew angle is accurate, the document mosaic system performs calculation at many image patches and derive the final estimation by finding the average of the individual angles weighted by the variance of the pixel intensities of each patch. === Columns, lines and words finding === In this operation, the de-skewed document is intuitively segmented into a hierarchy of columns, lines and words. The sensitivity to illumination and page coloration of the de-skewed document can be removed by applying a Sobel operator to the de-skewed image and thresholding the output to obtain the binary gradient, de-skewed image. The operation can be roughly separated into 3 steps: column segmentation, line segmentation and word segmentation. Columns are easily segmented from the binary gradient, de-skewed images by summing pixels vertically. Baselines of each row are segmented in the same way as the column segmentation process but horizontally. Finally, individual words are segmented by applying the vertical process at each segmented row. These segmentations are important because the document mosaic is created by matching the lower right corners of words in overlapping images pair. Moreover, the segmentation operation can organize the list of images in the context of a hierarchy of rows and column reliably. The segmentation operation involves a considerable amount of summing in the binary gradient, de-skewed images, which done by construct a matrix of partial sums whose elements are given by p i y = ∑ u = 1 i ∑ v = 1 j b u v {\displaystyle p_{iy}=\sum _{u=1}^{i}\sum _{v=1}^{j}b_{uv}} The matrix of partial sums is calculated in one pass through the binary gradient, de-skewed image. ∑ u = u 1 u 2 ∑ v = v 1 v 2 b u v = p u 2 v 2 + p u 1 v 1 − p u 1 v 2 − p u 2 v 1 {\displaystyle \sum _{u=u_{1}}^{u_{2}}\sum _{v=v_{1}}^{v_{2}}b_{uv}=p_{u_{2}v_{2}}+p_{u_{1}v_{1}}-p_{u_{1}v_{2}}-p_{u_{2}v_{1}}} == Correspondences establishing == The two images are now organized in hierarchy of linked lists in following structure : image=list of columns row=list of words column=list of row word=length (in pixels) At the bottom of the structure, the length of each word is recorded for establishing correspondence between two images to reduce to search only the corresponding structures for the groups of words with the matching lengths. === Seed match finding === A seed match finding is done by comparing each row in image1 with each row in image2. The two rows are then compared to each other by every word. If the length (in pixel) of the two words (one from image1 and one from image2) and their immediate neighbours agree with each other within a predefined tolerance threshold (5 pixels, for example), then they are assumed to match. The row of each image is assumed a match if there are three or more word matches between the two rows. The seed match finding operation is terminated when two pairs of consecutive row match are found. === Match list building === After finishing a seed match finding operation, the next process is to build the match list to generate the correspondences points of the two images. The process is done by searching the matching pairs of rows away from the seed row. == Images mosaicing == Given the list of corresponding points of the two images, finding the transformation of the overlapping portion of the images is the next process. Assuming a pinhole camera model, the transformation between pixels (u,v) of image 1 and pixels (u0, v0) of image 2 is demonstrated by a plane-to-plane projectivity. [ s u ′ s v ′ s ] = [ p 11 p 12 p 13 p 21 p 22 p 23 p 31 p 32 1 ] [ u v 1 ] E q .1 {\displaystyle \left[{\begin{array}{c}su'\\sv'\\s\end{array}}\right]=\left[{\begin{array}{ccc}p_{11}&p_{12}&p_{13}\\p_{21}&p_{22}&p_{23}\\p_{31}&p_{32}&1\end{array}}\right]\left[{\begin{array}{c}u\\v\\1\end{array}}\right]\qquad Eq.1} The parameters of the projectivity is found from four pairs of matching points. RANSAC regression technique is used to reject outlying matches and estimate the projectivity from the remaining good matches. The projectivity is fine-tuned using correlation at the corners of the overlapping portion to obtain four correspondences to sub-pixel accuracy. Therefore, image1 is then transformed into image2's coordinate system using Eq.1. The typical result of the process is shown in Figure 5. === Many images coping === Finally, the whole page composition is built up by mapping all the images into the coordinate system of an "anchor" image, which is normally the one nearest the page center. The transformations to the anchor frame are calculated by concatenating the pair-wise transformations found earlier. The raw document mosaic is shown in Figure 6. However, there might be a problem of non-consecutive images that are overlap. This problem can be solved by performing Hierarchical sub-mosaics. As shown in Figure 7, image1 and image2 are registered, as are image3 and image4, creating two sub-mosaics. These two sub-mosaics are later stitched together in another mosaicing process. == Applied areas == There are various areas that the technique of document mosaicing can be applied to such as : Text segmentation of images of documents Document Recognition Interaction with paper on the digital desk Video mosaics for virtual environments Image registration techniques == Relevant research papers == Huang, T.S.; Netravali, A.N. (1994). "Motion and structure from feature correspondences: A review". Proceedings of the IEEE. 82 (2): 252–268. doi:10.1109/5.265351. D.G. Lowe. [1] Perceptual Organization and Visual Recognition. Kluwer Academic Publishers, Boston, 1985. Irani, M.; Peleg, S. (1991). "Improving resolution by image registration". CVGIP: Graphical Models and Image Processing. 53 (3): 231–239. doi:10.1016/1049-9652(91)90045-L. S2CID 4834546. Shivakumara, P.; Kumar, G. Hemantha; Guru, D. S.; Nagabhushan, P. (2006). "
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Regular language

In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to many modern regular expression engines, which are augmented with features that allow the recognition of non-regular languages). Alternatively, a regular language can be defined as a language recognised by a finite automaton. The equivalence of regular expressions and finite automata is known as Kleene's theorem (after American mathematician Stephen Cole Kleene). In the Chomsky hierarchy, regular languages are the languages generated by Type-3 grammars. == Formal definition == The collection of regular languages over an alphabet Σ is defined recursively as follows: The empty language ∅ is a regular language. For each a ∈ Σ (a belongs to Σ), the singleton language {a} is a regular language. If A is a regular language, A (Kleene star) is a regular language. Due to this, the empty string language {ε} is also regular. If A and B are regular languages, then A ∪ B (union) and A • B (concatenation) are regular languages. No other languages over Σ are regular. See Regular expression § Formal language theory for syntax and semantics of regular expressions. == Examples == All finite languages are regular; in particular the empty string language {ε} = ∅ is regular. Other typical examples include the language consisting of all strings over the alphabet {a, b} which contain an even number of as, or the language consisting of all strings of the form: several as followed by several bs. A simple example of a language that is not regular is the set of strings {anbn | n ≥ 0}. Intuitively, it cannot be recognized with a finite automaton, since a finite automaton has finite memory and it cannot remember the exact number of a's. Techniques to prove this fact rigorously are given below. == Equivalent formalisms == A regular language satisfies the following equivalent properties: it is the language of a regular expression (by the above definition) it is the language accepted by a nondeterministic finite automaton (NFA) it is the language accepted by a deterministic finite automaton (DFA) it can be generated by a regular grammar it is the language accepted by an alternating finite automaton it is the language accepted by a two-way finite automaton it can be generated by a prefix grammar it can be accepted by a read-only Turing machine it can be defined in monadic second-order logic (Büchi–Elgot–Trakhtenbrot theorem) it is recognized by some finite syntactic monoid M, meaning it is the preimage {w ∈ Σ | f(w) ∈ S} of a subset S of a finite monoid M under a monoid homomorphism f : Σ → M from the free monoid on its alphabet the number of equivalence classes of its syntactic congruence is finite. (This number equals the number of states of the minimal deterministic finite automaton accepting L.) Properties 10. and 11. are purely algebraic approaches to define regular languages; a similar set of statements can be formulated for a monoid M ⊆ Σ. In this case, equivalence over M leads to the concept of a recognizable language. Some authors use one of the above properties different from "1." as an alternative definition of regular languages. Some of the equivalences above, particularly those among the first four formalisms, are called Kleene's theorem in textbooks. Precisely which one (or which subset) is called such varies between authors. One textbook calls the equivalence of regular expressions and NFAs ("1." and "2." above) "Kleene's theorem". Another textbook calls the equivalence of regular expressions and DFAs ("1." and "3." above) "Kleene's theorem". Two other textbooks first prove the expressive equivalence of NFAs and DFAs ("2." and "3.") and then state "Kleene's theorem" as the equivalence between regular expressions and finite automata (the latter said to describe "recognizable languages"). A linguistically oriented text first equates regular grammars ("4." above) with DFAs and NFAs, calls the languages generated by (any of) these "regular", after which it introduces regular expressions which it terms to describe "rational languages", and finally states "Kleene's theorem" as the coincidence of regular and rational languages. Other authors simply define "rational expression" and "regular expressions" as synonymous and do the same with "rational languages" and "regular languages". Apparently, the term regular originates from a 1951 technical report where Kleene introduced regular events and explicitly welcomed "any suggestions as to a more descriptive term". Noam Chomsky, in his 1959 seminal article, used the term regular in a different meaning at first (referring to what is called Chomsky normal form today), but noticed that his finite state languages were equivalent to Kleene's regular events. == Closure properties == The regular languages are closed under various operations, that is, if the languages K and L are regular, so is the result of the following operations: the set-theoretic Boolean operations: union K ∪ L, intersection K ∩ L, and complement L, hence also relative complement K − L. the regular operations: K ∪ L, concatenation ⁠ K ∘ L {\displaystyle K\circ L} ⁠, and Kleene star L. the trio operations: string homomorphism, inverse string homomorphism, and intersection with regular languages. As a consequence they are closed under arbitrary finite state transductions, like quotient K / L with a regular language. Even more, regular languages are closed under quotients with arbitrary languages: If L is regular then L / K is regular for any K. the reverse (or mirror image) LR. Given a nondeterministic finite automaton to recognize L, an automaton for LR can be obtained by reversing all transitions and interchanging starting and finishing states. This may result in multiple starting states; ε-transitions can be used to join them. == Decidability properties == Given two deterministic finite automata A and B, it is decidable whether they accept the same language. As a consequence, using the above closure properties, the following problems are also decidable for arbitrarily given deterministic finite automata A and B, with accepted languages LA and LB, respectively: Containment: is LA ⊆ LB ? Disjointness: is LA ∩ LB = {} ? Emptiness: is LA = {} ? Universality: is LA = Σ ? Membership: given a ∈ Σ, is a ∈ LB ? For regular expressions, the universality problem is NP-complete already for a singleton alphabet. For larger alphabets, that problem is PSPACE-complete. If regular expressions are extended to allow also a squaring operator, with "A2" denoting the same as "AA", still just regular languages can be described, but the universality problem has an exponential space lower bound, and is in fact complete for exponential space with respect to polynomial-time reduction. For a fixed finite alphabet, the theory of the set of all languages – together with strings, membership of a string in a language, and for each character, a function to append the character to a string (and no other operations) – is decidable, and its minimal elementary substructure consists precisely of regular languages. For a binary alphabet, the theory is called S2S. == Complexity results == In computational complexity theory, the complexity class of all regular languages is sometimes referred to as REGULAR or REG and equals DSPACE(O(1)), the decision problems that can be solved in constant space (the space used is independent of the input size). REGULAR ≠ AC0, since it (trivially) contains the parity problem of determining whether the number of 1 bits in the input is even or odd and this problem is not in AC0. On the other hand, REGULAR does not contain AC0, because the nonregular language of palindromes, or the nonregular language { 0 n 1 n : n ∈ N } {\displaystyle \{0^{n}1^{n}:n\in \mathbb {N} \}} can both be recognized in AC0. If a language is not regular, it requires a machine with at least Ω(log log n) space to recognize (where n is the input size). In other words, DSPACE(o(log log n)) equals the class of regular languages. In practice, most nonregular problems are studied in a setting with at least logarithmic space, as this is the amount of space required to store a pointer into the input tape. == Location in the Chomsky hierarchy == To locate the regular languages in the Chomsky hierarchy, one notices that every regular language is context-free. The converse is not true: for example, the language consisting of all strings having the same number of as as bs is context-free but not regular. To prove that a language is not regular, one often uses the Myhill–Nerode theorem and the pumping lemma. Other approaches include using the closure properties of regular languages or quantifying Kolmogorov complexity. Important subclasses of regular languages include: Finite languages, those containing only a finite number of words. These are regular la
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