An intranet is a computer network for sharing information, easier communication, collaboration tools, operational systems, and other computing services within an organization, usually to the exclusion of access by outsiders. The term is used in contrast to public networks, such as the Internet, but uses the same technology based on the Internet protocol suite. An organization-wide intranet can constitute a focal point of internal communication and collaboration, and provide a single starting point to access internal and external resources. In its simplest form, an intranet is established with the technologies for local area networks (LANs) and wide area networks (WANs). Many modern intranets have search engines, user profiles, blogs, mobile apps with notifications, and events planning within their infrastructure. An intranet is sometimes contrasted to an extranet. While an intranet is generally restricted to employees of the organization, extranets may also be accessed by customers, suppliers, or other approved parties. Extranets extend a private network onto the Internet with special provisions for authentication, authorization and accounting (AAA protocol). == Uses == Intranets are increasingly being used to deliver tools, such as for collaboration (to facilitate working in groups and teleconferencing) or corporate directories, sales and customer relationship management, or project management. Intranets are also used as corporate culture-change platforms. For example, a large number of employees using an intranet forum application to host a discussion about key issues could come up with new ideas related to management, productivity, quality, and other corporate issues. In large intranets, website traffic is often similar to public website traffic and can be better understood by using web metrics software to track overall activity. User surveys also improve intranet website effectiveness. Larger businesses allow users within their intranet to access public internet through firewall servers. They have the ability to screen incoming and outgoing messages, keeping security intact. When part of an intranet is made accessible to customers and others outside the business, it becomes part of an extranet. Businesses can send private messages through the public network using special encryption/decryption and other security safeguards to connect one part of their intranet to another. Intranet user-experience, editorial, and technology teams work together to produce in-house sites. Most commonly, intranets are managed by the communications, HR or CIO departments of large organizations, or some combination of these. Because of the scope and variety of content and the number of system interfaces, the intranets of many organizations are much more complex than their respective public websites. Intranets and the use of intranets are growing rapidly. According to the Intranet Design Annual 2007 from Nielsen Norman Group, the number of pages on participants' intranets averaged 200,000 over the years 2001 to 2003 and has grown to an average of 6 million pages over 2005–2007. == Benefits == Intranets can help users locate and view information faster and use applications relevant to their roles and responsibilities. With a web browser interface, users can access data held in any database the organization wants to make available at any time and — subject to security provisions — from anywhere within company workstations, increasing employees' ability to perform their jobs faster, more accurately, and with confidence that they have the right information. It also helps improve services provided to users. Using hypermedia and Web technology, Web publishing allows for the maintenance of and easy access to cumbersome corporate knowledge, such as employee manuals, benefits documents, company policies, business standards, news feeds, and even training, all of which can be accessed throughout a company using common Internet standards (Acrobat files, Flash files, CGI applications). Because each business unit can update the online copy of a document, the most recent version is usually available to employees using the intranet. Intranets are also used as a platform for developing and deploying applications to support business operations and decisions across the internetworked enterprise. Information is easily accessible to all authorised users, enabling collaboration. Being able to communicate in real-time through integrated third-party tools, such as an instant messenger, promotes the sharing of ideas and removes blockages to communication to help boost a business's productivity. Intranets can serve as powerful tools for communicating (such as through chat, email and/or blogs) within a given organization about vertically strategic initiatives that have a global reach throughout said organization. The type of information that can easily be conveyed is the purpose of the initiative and what it is aiming to achieve, who is driving it, results achieved to date, and whom to speak to for more information. By providing this information on the intranet, staff can keep up-to-date with the strategic focus of their organization. For example, when Nestlé had a number of food processing plants in Scandinavia, their central support system had to deal with a number of queries every day. When Nestlé decided to invest in an intranet, they quickly realized the savings. Gerry McGovern says that the savings from the reduction in query calls was substantially greater than the investment in the intranet. Users can view information and data via a web browser rather than maintaining physical documents such as procedure manuals, internal phone list and requisition forms. This can potentially save the business money on printing, duplicating documents, and the environment, as well as document maintenance overhead. For example, the HRM company PeopleSoft "derived significant cost savings by shifting HR processes to the intranet". McGovern goes on to say the manual cost of enrolling in benefits was found to be US$109.48 per enrollment. "Shifting this process to the intranet reduced the cost per enrollment to $21.79; a saving of 80 percent". Another company that saved money on expense reports was Cisco. "In 1996, Cisco processed 54,000 reports and the amount of dollars processed was USD19 million". Many companies dictate computer specifications which, in turn, may allow Intranet developers to write applications that only have to work on one browser such that there are no cross-browser compatibility issues. Being able to specifically address one's "viewer" is a great advantage. Since intranets are user-specific (requiring database/network authentication prior to access), users know exactly who they are interfacing with and can personalize their intranet based on role (job title, department) or individual ("Congratulations Jane, on your 3rd year with our company!"). Since "involvement in decision making" is one of the main drivers of employee engagement, offering tools (like forums or surveys) that foster peer-to-peer collaboration and employee participation can make employees feel more valued and involved. == Planning and creation == Most organizations devote considerable resources into the planning and implementation of their intranet as it is of strategic importance to the organization's success. Some of the planning would include topics such as determining the purpose and goals of the intranet, identifying persons or departments responsible for implementation and management and devising functional plans, page layouts and designs. The appropriate staff would also ensure that implementation schedules and phase-out of existing systems were organized, while defining and implementing security of the intranet and ensuring it lies within legal boundaries and other constraints. In order to produce a high-value end product, systems planners should determine the level of interactivity (e.g. wikis, on-line forms) desired. Planners may also consider whether the input of new data and updating of existing data is to be centrally controlled or devolve. These decisions sit alongside to the hardware and software considerations (like content management systems), participation issues (like good taste, harassment, confidentiality), and features to be supported. Intranets are often static sites; they are a shared drive, serving up centrally stored documents alongside internal articles or communications (often one-way communication). By leveraging firms which specialise in 'social' intranets, organisations are beginning to think of how their intranets can become a 'communication hub' for their entire team. The actual implementation would include steps such as securing senior management support and funding, conducting a business requirement analysis and identifying users' information needs. From the technical perspective, there would need to be a coordinated installation of the web server and user access netw
Digital image processing
Digital image processing is the use of a digital computer to process digital images through an algorithm. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing. It allows a much wider range of algorithms to be applied to the input data and can avoid problems such as the build-up of noise and distortion during processing. Since images are defined over two dimensions (perhaps more), digital image processing may be modeled in the form of multidimensional systems. The generation and development of digital image processing are mainly affected by three factors: first, the development of computers; second, the development of mathematics (especially the creation and improvement of discrete mathematics theory); and third, the demand for a wide range of applications in environment, agriculture, military, industry and medical science has increased. == History == Many of the techniques of digital image processing, or digital picture processing as it often was called, were developed in the 1960s, at Bell Laboratories, the Jet Propulsion Laboratory, Massachusetts Institute of Technology, University of Maryland, and a few other research facilities, with application to satellite imagery, wire-photo standards conversion, medical imaging, videophone, character recognition, and photograph enhancement. The purpose of early image processing was to improve the quality of the image. In image processing, the input is a low-quality image, and the output is an image with improved quality. Common image processing includes image enhancement, restoration, encoding, and compression. The first successful application was the American Jet Propulsion Laboratory (JPL). They used image processing techniques such as geometric correction, gradation transformation, noise removal, etc. on the thousands of lunar photos sent back by the Space Detector Ranger 7 in 1964, taking into account the position of the Sun and the environment of the Moon. The impact of the successful mapping of the Moon's surface map by the computer has been a success. Later, more complex image processing was performed on the nearly 100,000 photos sent back by the spacecraft, so that the topographic map, color map and panoramic mosaic of the Moon were obtained, which achieved extraordinary results and laid a solid foundation for human landing on the Moon. The cost of processing was fairly high, however, with the computing equipment of that era. That changed in the 1970s, when digital image processing proliferated as cheaper computers and dedicated hardware became available. This led to images being processed in real-time, for some dedicated problems such as television standards conversion. As general-purpose computers became faster, they started to take over the role of dedicated hardware for all but the most specialized and computer-intensive operations. With the fast computers and signal processors available in the 2000s, digital image processing has become the most common form of image processing, and is generally used because it is not only the most versatile method, but also the cheapest. === Image sensors === The basis for modern image sensors is metal–oxide–semiconductor (MOS) technology, invented at Bell Labs between 1955 and 1960, This led to the development of digital semiconductor image sensors, including the charge-coupled device (CCD) and later the CMOS sensor. The charge-coupled device was invented by Willard S. Boyle and George E. Smith at Bell Labs in 1969. While researching MOS technology, they realized that an electric charge was the analogy of the magnetic bubble and that it could be stored on a tiny MOS capacitor. As it was fairly straightforward to fabricate a series of MOS capacitors in a row, they connected a suitable voltage to them so that the charge could be stepped along from one to the next. The CCD is a semiconductor circuit that was later used in the first digital video cameras for television broadcasting. The NMOS active-pixel sensor (APS) was invented by Olympus in Japan during the mid-1980s. This was enabled by advances in MOS semiconductor device fabrication, with MOSFET scaling reaching smaller micron and then sub-micron levels. The NMOS APS was fabricated by Tsutomu Nakamura's team at Olympus in 1985. The CMOS active-pixel sensor (CMOS sensor) was later developed by Eric Fossum's team at the NASA Jet Propulsion Laboratory in 1993. By 2007, sales of CMOS sensors had surpassed CCD sensors. MOS image sensors are widely used in optical mouse technology. The first optical mouse, invented by Richard F. Lyon at Xerox in 1980, used a 5 μm NMOS integrated circuit sensor chip. Since the first commercial optical mouse, the IntelliMouse introduced in 1999, most optical mouse devices use CMOS sensors. === Image compression === An important development in digital image compression technology was the discrete cosine transform (DCT), a lossy compression technique first proposed by Nasir Ahmed in 1972. DCT compression became the basis for JPEG, which was introduced by the Joint Photographic Experts Group in 1992. JPEG compresses images down to much smaller file sizes, and has become the most widely used image file format on the Internet. Its highly efficient DCT compression algorithm was largely responsible for the wide proliferation of digital images and digital photos, with several billion JPEG images produced every day as of 2015. Medical imaging techniques produce very large amounts of data, especially from CT, MRI and PET modalities. As a result, storage and communications of electronic image data are prohibitive without the use of compression. JPEG 2000 image compression is used by the DICOM standard for storage and transmission of medical images. The cost and feasibility of accessing large image data sets over low or various bandwidths are further addressed by use of another DICOM standard, called JPIP, to enable efficient streaming of the JPEG 2000 compressed image data. === Digital signal processor (DSP) === Electronic signal processing was revolutionized by the wide adoption of MOS technology in the 1970s. MOS integrated circuit technology was the basis for the first single-chip microprocessors and microcontrollers in the early 1970s, and then the first single-chip digital signal processor (DSP) chips in the late 1970s. DSP chips have since been widely used in digital image processing. The discrete cosine transform (DCT) image compression algorithm has been widely implemented in DSP chips, with many companies developing DSP chips based on DCT technology. DCTs are widely used for encoding, decoding, video coding, audio coding, multiplexing, control signals, signaling, analog-to-digital conversion, formatting luminance and color differences, and color formats such as YUV444 and YUV411. DCTs are also used for encoding operations such as motion estimation, motion compensation, inter-frame prediction, quantization, perceptual weighting, entropy encoding, variable encoding, and motion vectors, and decoding operations such as the inverse operation between different color formats (YIQ, YUV and RGB) for display purposes. DCTs are also commonly used for high-definition television (HDTV) encoder/decoder chips. == Tasks == Digital image processing allows the use of much more complex algorithms, and hence, can offer both more sophisticated performance at simple tasks, and the implementation of methods which would be impossible by analogue means. In particular, digital image processing is a concrete application of, and a practical technology based on: Classification Feature extraction Multi-scale signal analysis Pattern recognition Projection Some techniques that are used in digital image processing include: Anisotropic diffusion Hidden Markov models Image editing Image restoration Independent component analysis Linear filtering Neural networks Partial differential equations Pixelation Point feature matching Principal components analysis Self-organizing maps Wavelets == Digital image transformations == === Filtering === Digital filters are used to blur and sharpen digital images. Filtering can be performed by: convolution with specifically designed kernels (filter array) in the spatial domain masking specific frequency regions in the frequency (Fourier) domain The following examples show both methods: ==== Image padding in Fourier domain filtering ==== Images are typically padded before being transformed to the Fourier space, the highpass filtered images below illustrate the consequences of different padding techniques: Notice that the highpass filter shows extra edges when zero padded compared to the repeated edge padding. ==== Filtering code examples ==== MATLAB example for spatial domain highpass filtering. === Affine transformations === Affine transformations enable basic image transformations including scale, rotate, translate, mirror and shear as is shown in the following examples: To apply the affine
Nathalie Japkowicz
Nathalie Japkowicz is a Canadian computer scientist specializing in machine learning. She is a professor and department chair of computer science at the American University College of Arts and Sciences. == Life == Nathalie Japkowicz completed a B.Sc. at McGill University in 1988. She earned an M.Sc. from the University of Toronto in 1990. She completed a Ph.D. at Rutgers University in 1999. Her dissertation was titled Concept-learning in the absence of counter-examples: an autoassociation-based approach to classification. Stephen José Hanson and Casimir Alexander Kulikowski were her doctoral advisors. Japkowicz worked at the University of Ottawa in the school of electrical engineering and computer science. She was the lead of its laboratory for research on machine learning for defense security. From 2003 to 2005, Japkowicz was the secretary of the Canadian Artificial Intelligence Association (CAIAC). She was CAIAC vice president from 2009 to 2014 and president from 2013 to 2015, and part-president from 2015 to 2017. Japkowicz is a professor and department chair of computer science at the American University College of Arts and Sciences. She researches artificial intelligence, machine learning, data mining, and big data analysis. == Selected works == Gao, Yong; Japkowicz, Nathalie, eds. (2009). Advances in Artificial Intelligence: 22nd Canadian Conference on Artificial Intelligence, Canadian AI 2009 Kelowna, Canada, May 25–27, 2009 Proceedings. Lecture Notes in Computer Science. Vol. 5549. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-642-01818-3. ISBN 978-3-642-01817-6. S2CID 27083226. Japkowicz, Nathalie; Shah, Mohak (2011). Evaluating Learning Algorithms: A Classification Perspective (1 ed.). Cambridge University Press. doi:10.1017/cbo9780511921803. ISBN 978-0-511-92180-3. Japkowicz, Nathalie; Matwin, Stan, eds. (2015). Discovery Science: 18th International Conference, DS 2015, Banff, AB, Canada, October 4–6, 2015. Proceedings. Lecture Notes in Computer Science. Vol. 9356. Cham: Springer International Publishing. doi:10.1007/978-3-319-24282-8. ISBN 978-3-319-24281-1. S2CID 1302223. Japkowicz, Nathalie; Stefanowski, Jerzy, eds. (2016). Big Data Analysis: New Algorithms for a New Society. Studies in Big Data. Vol. 16. Cham: Springer International Publishing. doi:10.1007/978-3-319-26989-4. ISBN 978-3-319-26987-0. Ceci, Michelangelo; Japkowicz, Nathalie; Liu, Jiming; Papadopoulos, George A.; Raś, Zbigniew W., eds. (2018). Foundations of Intelligent Systems: 24th International Symposium, ISMIS 2018, Limassol, Cyprus, October 29–31, 2018, Proceedings. Lecture Notes in Computer Science. Vol. 11177. Cham: Springer International Publishing. doi:10.1007/978-3-030-01851-1. ISBN 978-3-030-01850-4. S2CID 53038780.
Best AI Paragraph Rewriters in 2026
In search of the best AI paragraph rewriter? An AI paragraph rewriter is software that uses machine learning to help you get more done — it turns a rough idea into a polished result in seconds. When choosing one, weigh output quality, pricing, export formats, and how well it fits the tools you already use. Whether you are a beginner or a pro, the right AI paragraph rewriter slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.
Dan Roth
Dan Roth (Hebrew: דן רוט) is the Eduardo D. Glandt Distinguished Professor of Computer and Information Science at the University of Pennsylvania and the Chief AI Scientist at Oracle. Until June 2024 Roth was a VP and distinguished scientist at AWS AI. In his role at AWS, Roth led over the last three years the scientific effort behind the first-generation Generative AI products from AWS, including Titan Models, Amazon Q efforts, and Bedrock, from inception until they became generally available. Roth got his B.A. summa cum laude in mathematics from the Technion, Israel, and his Ph.D. in computer science from Harvard University in 1995. He taught at the University of Illinois at Urbana-Champaign from 1998 to 2017 before moving to the University of Pennsylvania. == Professional career == Roth is a Fellow of the American Association for the Advancement of Science (AAAS), the Association for Computing Machinery (ACM), the Association for the Advancement of Artificial Intelligence (AAAI), and the Association of Computational Linguistics (ACL). Roth’s research focuses on the computational foundations of intelligent behavior. He develops theories and systems pertaining to intelligent behavior using a unified methodology, at the heart of which is the idea that learning has a central role in intelligence. His work centers around the study of machine learning and inference methods to facilitate natural language understanding. In doing that he has pursued several interrelated lines of work that span multiple aspects of this problem - from fundamental questions in learning and inference and how they interact, to the study of a range of natural language processing (NLP) problems and developing advanced machine learning based tools for natural language applications. Roth has made seminal contribution to the fusion of Learning and Reasoning, Machine Learning with weak, incidental supervision, and to machine learning and inference approaches to natural language understanding. He has written the first paper on zero-shot learning in natural language processing, a 2008 paper by Chang, Ratinov, Roth, and Srikumar that was published at AAAI’08, but the name given to the learning paradigm there was dataless classification. Roth has worked on probabilistic reasoning (including its complexity and probabilistic lifted inference ), Constrained Conditional Models (ILP formulations of NLP problems) and constraints-driven learning, part-based (constellation) methods in object recognition, response based Learning, He has developed NLP and Information extraction tools that are being used broadly by researchers and commercially, including NER, coreference resolution, wikification, SRL, and ESL text correction. Roth is a co-founder of NexLP, Inc., a startup that applies natural language processing and machine learning in the legal and compliance domains. In 2020, NexLP was acquired by Reveal, Inc., an e-discovery software company. He is currently on the scientific advisory board of the Allen Institute for AI.
VOCEDplus
VOCEDplus is a free international research database about tertiary education, maintained and developed by staff at the c (NCVER) in Adelaide, South Australia. The focus of the database content is the relation of post-compulsory education and training to workforce needs, skills development, and social inclusion. == Structure == The content of the VOCEDplus database encompasses vocational education and training (VET), higher education, lifelong learning, informal learning, VET in schools, adult and community education, apprenticeships/traineeships, international education, providers of education and training, and workforce development. It is international in scope and contains over 84,000 English language records, many with links to full text documents. VOCEDplus contains extensive Australian materials and includes a wide range of international information, covering outcomes of tertiary education in the shape of published research, practice, policy, and statistics. Entries are included for the following types of publications: reports; annual reports; papers; discussion papers; occasional papers; working papers; books; book chapters; conference papers; conference proceedings; journals; journal articles; policy documents; published statistics; theses; podcasts; and teaching and training materials. Each database entry contains standard bibliographic information and an abstract. Many entries include full text access via the publisher's website or a digitised copy. == History == === 1989-1997 === In the early years VOCEDplus was known as VOCED. The original database was produced by a network of clearinghouses across Australia with the aim of sharing activities in the technical and further education (TAFE) sector. VOCED was produced in hardcopy and an electronic version was distributed on diskette. === 1997-2001 === 1997 - the first web version of VOCED was made available from the National Centre for Vocational Education Research (NCVER) organisational website 1998 - a major project to upgrade the database and expand its international coverage commenced 2001 - creation of VOCED's own website 2001 - VOCED endorsed as the UNESCO international database for technical and vocational education and training (TVET) research information === 2001-2009 === Many changes to the database and website occurred during this period with a focus on continuous improvement to meet the needs of users and utilise emerging technologies. 2006 - materials produced for two adult literacy and learning programs funded by the Australian Department of Education, Employment and Workplace Relations (DEEWR) - the Workplace English Language and Learning (WELL) Programme and the Adult Literacy National Project (ALNP) included in VOCED 2007 - the Australian clearinghouse network transferred most of the hardcopy collections to NCVER, to form a centralised repository of resources 2009 - materials produced by Reframing the Future (RTF) a vocational education and training workforce development initiative of the Australian, State and Territory Governments included in VOCED === 2009-2014 === A major rebuild of the database and website was undertaken during this period to take advantage of the potential of new technologies to provide improved services and incorporate Web 2.0 technologies (RSS feeds, and share and bookmark tools). 2009 - scope expanded to more fully encompass the higher education sector 2011 - launch of VOCEDplus with the name change representing the enhanced features and extended focus 2012 - a major retrospective digitisation project commenced and by the end of the 2012-2013 financial year a total of 9,328 publications (593,534 pages/microfiche frames) had been digitised, ensuring these publications are available electronically for free === 2014-2019 === A number of significant curated content products were released during this period. 2015 - release of a refreshed look to adopt the new NCVER branding plus a number of search enhancements (Guided search, Expert search, and Glossary search) were added 2015 - first in the series of 'Focus on...' pages released 2016 - launch of the 'Pod Network', a convenient and efficient platform that allows instant access to research and a multitude of resources on a range of subjects 2017 - completion of the 'Pod Network', consisting of 20 Pods (on broad subjects including Apprenticeships and traineeships, Foundation skills, Teaching and learning, Career development, and Students) and 74 Podlets (on narrow topics including Online learning, Social media, VET in schools, STEM skills, and Adult literacy) 2018 - launch of the 'Timeline of Australian VET Policy Initiatives' and the 'VET Knowledge Bank' which contains a suite of products capturing Australia's diverse, complex and ever-changing VET system 2019 - after an internal review, a refreshed, streamlined version of the 'Pod Network' was released, consisting of 13 Pods and 20 Podlets 2019 - launch of the 'VET Practitioner Resource' which contains a range of information to support VET practitioners in their work and is organised into three sections: (1) Teaching, training and assessment: standards, guidance, research and good practice resources to inform daily work; (2) Practitioners as researchers: information for undertaking practitioner-led research; and (3) The VET workforce: information about VET teachers and trainers, and the professional development needs of the VET workforce 2019 - VOCEDplus celebrated 30 years of providing information to the tertiary education sector and the homepage was refreshed to make it more modern and easier to use === 2020- === VOCEDplus continued to be accessible throughout the COVID-19 pandemic. 2020-2021 - the VET Knowledge Bank added a dedicated page, 'COVID-19 announcements', that showcases the measures introduced by the Australian, state and territory governments to mitigate the impact of the pandemic and promote economic recovery 2020-2024 - published research about the effects of the pandemic on education and training, providers, students, labour markets, employment and employees was collected and made permanently available in the database 2024 - VOCEDplus celebrated 35 years of providing information to the tertiary education sector. The homepage was refreshed and a number of enhancements and new features were implemented including a new My Profile feature, improvements to My Selection, accessible search history and saved searches, enhanced search functionality, and improved navigation.
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