Multi-exposure HDR capture

In photography and videography, multi-exposure HDR capture is a technique that creates high dynamic range (HDR) images (or extended dynamic range images) by taking and combining multiple exposures of the same subject matter at different exposures. Combining multiple images in this way results in an image with a greater dynamic range than what would be possible by taking one single image. The technique can also be used to capture video by taking and combining multiple exposures for each frame of the video. The term "HDR" is used frequently to refer to the process of creating HDR images from multiple exposures. Many smartphones have an automated HDR feature that relies on computational imaging techniques to capture and combine multiple exposures. A single image captured by a camera provides a finite range of luminosity inherent to the medium, whether it is a digital sensor or film. Outside this range, tonal information is lost and no features are visible; tones that exceed the range are "burned out" and appear pure white in the brighter areas, while tones that fall below the range are "crushed" and appear pure black in the darker areas. The ratio between the maximum and the minimum tonal values that can be captured in a single image is known as the dynamic range. In photography, dynamic range is measured in exposure value (EV) differences, also known as stops. The human eye's response to light is non-linear: halving the light level does not halve the perceived brightness of a space, it makes it look only slightly dimmer. For most illumination levels, the response is approximately logarithmic. Human eyes adapt fairly rapidly to changes in light levels. HDR can thus produce images that look more like what a human sees when looking at the subject. This technique can be applied to produce images that preserve local contrast for a natural rendering, or exaggerate local contrast for artistic effect. HDR is useful for recording many real-world scenes containing a wider range of brightness than can be captured directly, typically both bright, direct sunlight and deep shadows. Due to the limitations of printing and display contrast, the extended dynamic range of HDR images must be compressed to the range that can be displayed. The method of rendering a high dynamic range image to a standard monitor or printing device is called tone mapping; it reduces the overall contrast of an HDR image to permit display on devices or prints with lower dynamic range. == Benefits == One aim of HDR is to present a similar range of luminance to that experienced through the human visual system. The human eye, through non-linear response, adaptation of the iris, and other methods, adjusts constantly to a broad range of luminance present in the environment. The brain continuously interprets this information so that a viewer can see in a wide range of light conditions. Most cameras are limited to a much narrower range of exposure values within a single image, due to the dynamic range of the capturing medium. With a limited dynamic range, tonal differences can be captured only within a certain range of brightness. Outside of this range, no details can be distinguished: when the tone being captured exceeds the range in bright areas, these tones appear as pure white, and when the tone being captured does not meet the minimum threshold, these tones appear as pure black. Images captured with non-HDR cameras that have a limited exposure range (low dynamic range, LDR), may lose detail in highlights or shadows. Modern CMOS image sensors have improved dynamic range and can often capture a wider range of tones in a single exposure reducing the need to perform multi-exposure HDR. Color film negatives and slides consist of multiple film layers that respond to light differently. Original film (especially negatives versus transparencies or slides) feature a very high dynamic range (in the order of 8 for negatives and 4 to 4.5 for positive transparencies). Multi-exposure HDR is used in photography and also in extreme dynamic range applications such as welding or automotive work. In security cameras the term "wide dynamic range" is used instead of HDR. === Limitations === A fast-moving subject, or camera movement between the multiple exposures, will generate a "ghost" effect or a staggered-blur strobe effect due to the merged images not being identical. Unless the subject is static and the camera mounted on a tripod there may be a tradeoff between extended dynamic range and sharpness. Sudden changes in the lighting conditions (strobed LED light) can also interfere with the desired results, by producing one or more HDR layers that do have the luminosity expected by an automated HDR system, though one might still be able to produce a reasonable HDR image manually in software by rearranging the image layers to merge in order of their actual luminosity. Because of the nonlinearity of some sensors image artifacts can be common. Camera characteristics such as gamma curves, sensor resolution, noise, photometric calibration and color calibration affect resulting high-dynamic-range images. == Process == High-dynamic-range photographs are generally composites of multiple standard dynamic range images, often captured using exposure bracketing. Afterwards, photo manipulation software merges the input files into a single HDR image, which is then also tone mapped in accordance with the limitations of the planned output or display. === Capturing multiple images (exposure bracketing) === Any camera that allows manual exposure control can perform multi-exposure HDR image capture, although one equipped with automatic exposure bracketing (AEB) facilitates the process. Some cameras have an AEB feature that spans a far greater dynamic range than others, from ±0.6 in simpler cameras to ±18 EV in top professional cameras, as of 2020. The exposure value (EV) refers to the amount of light applied to the light-sensitive detector, whether film or digital sensor such as a CCD. An increase or decrease of one stop is defined as a doubling or halving of the amount of light captured. Revealing detail in the darkest of shadows requires an increased EV, while preserving detail in very bright situations requires very low EVs. EV is controlled using one of two photographic controls: varying either the size of the aperture or the exposure time. A set of images with multiple EVs intended for HDR processing should be captured only by altering the exposure time; altering the aperture size also would affect the depth of field and so the resultant multiple images would be quite different, preventing their final combination into a single HDR image. Multi-exposure HDR photography generally is limited to still scenes because any movement between successive images will impede or prevent success in combining them afterward. Also, because the photographer must capture three or more images to obtain the desired luminance range, taking such a full set of images takes extra time. Photographers have developed calculation methods and techniques to partially overcome these problems, but the use of a sturdy tripod is advised to minimize framing differences between exposures. === Merging the images into an HDR image === Tonal information and details from shadow areas can be recovered from images that are deliberately overexposed (i.e., with positive EV compared to the correct scene exposure), while similar tonal information from highlight areas can be recovered from images that are deliberately underexposed (negative EV). The process of selecting and extracting shadow and highlight information from these over/underexposed images and then combining them with image(s) that are exposed correctly for the overall scene is known as exposure fusion. Exposure fusion can be performed manually, relying on the HDR operator's judgment, experience, and training, but usually, fusion is performed automatically by software. === Storing === Information stored in high-dynamic-range images typically corresponds to the physical values of luminance or radiance that can be observed in the real world. This is different from traditional digital images, which represent colors as they should appear on a monitor or a paper print. Therefore, HDR image formats are often called scene-referred, in contrast to traditional digital images, which are device-referred or output-referred. Furthermore, traditional images are usually encoded for the human visual system (maximizing the visual information stored in the fixed number of bits), which is usually called gamma encoding or gamma correction. The values stored for HDR images are often gamma compressed using mathematical functions such as power laws logarithms, or floating point linear values, since fixed-point linear encodings are increasingly inefficient over higher dynamic ranges. HDR images often do not use fixed ranges per color channel, other than traditional images, to represent many more colors over a much wi

ZeroPC

ZeroPC was a commercial webtop developed by ZeroDesktop, Inc. located in San Mateo, California. ZeroPC has been called a personal cloud OS. It mimicked the look, feel and functionality of the desktop environment of a real operating system. The software was launched in September 2011 through Disrupt SF 2011 event and recently selected to the finalist of SXSW 2012 in Innovative Web Technology category. ZeroPC is web-based and required a Java applet to operate bundled productivity tool Thinkfree. The web applications found on ZeroPC are built on Java in the back end. Features included drag-and-drop functionality, cloud dashboard and personal cloud storage meta services. ZeroPC belonged to a category of services that intended to turn the Web into a full-fledged platform by using Web services as a foundation along with presentation technologies that replicated the experience of desktop applications for users. ZeroPC aggregates content so users can easily access, transfer and share whatever content they want, using a web browser from any device. Its meta-cloud layer supports Dropbox, Box, SugarSync, OneDrive, 4Shared, Google Drive, Evernote, Picasa, Flickr, Instagram, Facebook, Twitter, and Photobucket. ZeroPC Cloud OS platform also provides extensive APIs for iOS and Android App developers. Some of the features found on ZeroPC are: File sharing, Webmail, Cloud Content Navigator, Instant messenger, Sticky Note, Audio/Video Player and Office productivity applications. ZeroPC 2.0 platform ran on AWS for free and paid users. Its platform is licensable to Telco and ISV for commercial purpose. Their clients are SFR, SK Telecom, Hancom and others. As of June 1, 2017, ZeroPC's servers were switched off completely, and ZeroPC is no longer in service since its parent company, NComputing, had launched Virtual Desktop Service in the cloud (AWS) to public. == Browser and Platform Compatibility == The ZeroPC web desktop was compatible with Mac OS X and Microsoft Windows platforms. It is certified to operate on Safari 6.0, Firefox 15.0.1, Google Chrome 22.0.1229.79 m and Internet Explorer 8 and 9. The ZeroPC front end user interface executes entirely within a web browser (see above) and uses HTML, some features of HTML5, JavaScript, AJAX and an optional Java plug-in. == Security == All communication between the ZeroPC front end user interface and the ZeroPC back end servers is encrypted using SSL (HTTPS) protocol. Furthermore, any content stored in the ZeroPC server-side repository is also encrypted using 256-bit Advanced Encryption Standard (AES-256) by Amazon S3 on AWS. ZeroPC users could connect their ZeroPC profile to other storage services such as Dropbox and Box. This connection allows the ZeroPC user to fully manage their content stored in these other storage services. To establish the connection ZeroPC rigorously adhered to the Oauth implementation provided by the target storage service. Upon completion of the Oauth process, ZeroPC stores the relevant access token in the user's profile. This token, along with all other sensitive password related data was encrypted using AES 256-bit key size. == Implementations == As noted above, the ZeroPC platform was hosted on Amazon Web Services infrastructure and is available to the general consumer. A user was allowed to sign up by selecting one of three account plans including a no-cost option. The ZeroPC could also be white-labeled for organizations wishing to provide this functionality to their own users. The white-label options include managed hosting on Amazon Web Services infrastructure and also installation within the organization's IT infrastructure. == User Access Points == The ZeroPC infrastructure provided user access to content and features in several different ways. As described in this article the user can access their information by signing into the ZeroPC web desktop. Additionally, ZeroPC offers native applications designed to run on popular mobile devices including smartphones and tablets. == Leadership == ZeroPC was founded by Chief Executive Officer, Young Song, an entrepreneur who previously founded NComputing, a $60 million venture-backed company. He also co-founded eMachines, Inc., a low-cost computer brand (later acquired by Gateway).

Is an AI Sales Assistant Worth It in 2026?

Shopping for the best AI sales assistant? An AI sales assistant is software that uses machine learning to help you get more done — it keeps getting smarter as the underlying models improve. Pricing, accuracy, and the size of the model behind the tool are the three factors that most affect daily usefulness. Whether you are a beginner or a pro, the right AI sales assistant slots into your workflow and pays for itself fast. Below we compare features, pricing, and real output so you can choose with confidence.

AI Website Builders: Free vs Paid (2026)

Looking for the best AI website builder? An AI website builder 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 website builder 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.

Karen Spärck Jones

Karen Ida Boalth Spärck Jones (26 August 1935 – 4 April 2007) was a self-taught programmer and a pioneering British computer and information scientist responsible for the concept of inverse document frequency (IDF), a technology that underlies most modern search engines. She was an advocate for women in computer science, her slogan being, "Computing is too important to be left to men." In 2019, The New York Times published her belated obituary in its series Overlooked, calling her "a pioneer of computer science for work combining statistics and linguistics, and an advocate for women in the field." From 2008, to recognise her achievements in the fields of information retrieval (IR) and natural language processing (NLP), the Karen Spärck Jones Award is awarded annually to a recipient for outstanding research in one or both of her fields. == Early life and education == Karen Ida Boalth Spärck Jones was born in Huddersfield, Yorkshire, England. Her parents were Alfred Owen Jones, a chemistry lecturer, and Ida Spärck, a Norwegian who worked for the Norwegian government while in exile in London during World War II. Spärck Jones was educated at a grammar school in Huddersfield and then from 1953 to 1956 at Girton College, Cambridge, studying history, with an additional final year in Moral Sciences (philosophy). While at Cambridge, Spärck Jones joined the organisation known as the Cambridge Language Research Unit (CLRU) and met the head of CLRU Margaret Masterman, who would inspire her to go into computer science. While working at the CLRU, Spärck Jones began pursuing her PhD. At the time of submission, her PhD thesis was cast aside as uninspired and lacking original thought, but was later published in its entirety as a book. She briefly became a school teacher before moving into computer science. Spärck Jones married fellow Cambridge computer scientist Roger Needham in 1958. Spärck Jones's mother, Ida Spärck, had fled Norway on one of the last boats out after the German invasion in April 1940, going on to serve the Norwegian government in exile in London throughout the war. This background of displacement and resilience shaped the household in which Spärck Jones grew up. She later kept her mother's Norwegian surname professionally after marrying, stating that "it maintains a permanent existence of your own." Spärck Jones described her entry into computing as almost accidental. She had been working as a schoolteacher when she began visiting the CLRU out of curiosity about her husband's work. It was Margaret Masterman — whom she later described as "a very strange and interesting woman" — who offered her a research position and drew her fully into the field. == Career == Spärck Jones worked at the Cambridge Language Research Unit from the late 1950s, then at Cambridge University Computer Laboratory from 1974 until her retirement in 2002. From 1999, she held the post of Professor of Computers and Information. She had been given a permanent position only in 1993, and earlier in her career had been employed on a series of short-term contracts. She continued to work in the Computer Laboratory until shortly before her death. Her publications include nine books and numerous papers. A full list of her publications is available from the Cambridge Computer Laboratory. Spärck Jones' main research interests, since the late 1950s, were natural language processing and information retrieval. In 1964, Spärck Jones published "Synonymy and Semantic Classification", which is now seen as a foundational paper in the field of natural language processing. One of her most important contributions was the concept of inverse document frequency (IDF) weighting in information retrieval, which she introduced in a 1972 paper. IDF is used in most search engines today, usually as part of the term frequency–inverse document frequency (TF–IDF) weighting scheme. In the 1980s, Spärck Jones began her work on early speech recognition systems. In 1982 she became involved in the Alvey Programme which was an initiative to motivate more computer science research across the country. == Significance of inverse document frequency == At the time Spärck Jones was working, most computer scientists were focused on making people adapt to machines — learning precise codes and commands to retrieve information. Spärck Jones was working in the opposite direction: teaching computers to understand human language as it is actually used. Her 1972 paper introduced the concept of inverse document frequency (IDF) by observing that not all words carry equal informational value. A word like "the" appears in virtually every document and tells a retrieval system almost nothing about what any specific document is about. A rare word like "photosynthesis," by contrast, is highly specific and informative. IDF assigns each word a statistical weight based on how rarely it occurs across a document collection — the rarer the word, the higher its weight. When combined with term frequency (TF), which measures how often a word appears within a single document, the resulting TF–IDF score gives every word a relevance rating that can be used to rank documents in response to a search query. By 2007, Spärck Jones noted that "pretty much every web engine uses those principles." Her colleague John Tait remarked that "a lot of the stuff she was working on until five or ten years ago seemed like mad nonsense, and now we take it for granted." The 1972 paper remains among the most cited works in information retrieval research, with over 4,500 citations recorded in Google Scholar at the time of her death. The conceptual foundation of TF–IDF — that word meaning is statistical and contextual — has also informed later developments in machine learning and natural language processing, including transformer-based language models such as BERT. == Impact on artificial intelligence == Even though Spärck Jones' views on artificial intelligence (AI) were rather pessimistic in regard to the perceived limitations of AI in information retrieval, her work in natural language processing, information retrieval, and introducing the concept of inverse document frequency (IDF) contributed to the future technological development of AI. Her statistical and ranking methods shifted the direction of the development of AI towards being more expandable and led by data. Her work had a more indirect and conceptual impact on AI, compared to the current and direct impact it has had on search engines. == Gender and advocacy == Spärck Jones spent the majority of her career at Cambridge on short-term contracts without permanent employment, a situation she attributed directly to gender. In her 2001 IEEE oral history interview she stated that Cambridge was "in many ways not user-friendly, in the sense of women-friendly." She was frequently the only woman present in professional meetings throughout her career. She channelled this experience into active advocacy. She was a founding member of the women@cl network at Cambridge's Computer Laboratory, worked on outreach programmes aimed at encouraging girls into computing, and became widely known for her slogan: "Computing is too important to be left to men." She was the first woman ever to receive the BCS Lovelace Medal. === Honours and awards === These include: Gerard Salton Award (1988) Elected a Fellow of Association for the Advancement of Artificial Intelligence (AAAI) in 1993 President of the Association for Computational Linguistics (ACL) in 1994 Honorary degree of Doctor of Science from The City University in 1997. Elected a Fellow of the British Academy (FBA), where she also served as Vice-President in 2000–2002 Fellow of European Association for Artificial Intelligence (ECCAI) Association for Information Science and Technology (ASIS&T) Award of Merit (2002) Association for Computational Linguistics (ACL) Lifetime Achievement Award (2004) ACM - AAAI Allen Newell Award (2006) BCS Lovelace Medal (2007) Association for Computing Machinery (ACM) Women's Group Athena Award (2007) == Death and legacy == Spärck Jones died on 4 April 2007, due to cancer at the age of 71. In 2008, the BCS Information Retrieval Specialist Group (BCS IRSG) in conjunction with the British Computer Society established an annual Karen Spärck Jones Award in her honour, to encourage and promote research that advances understanding of Natural Language Processing or Information Retrieval. The Karen Spärck Jones lecture sponsored by BCS recognises the contribution that women have made to computing. In August 2017, the University of Huddersfield renamed one of its campus buildings in her honour. Formerly known as Canalside West, the Spärck Jones building houses the University's School of Computing and Engineering. When Spärck Jones died in 2007, The Times did not publish an obituary for her, despite having published one for her husband Roger Needham in 2003. In 2019, The New York Times included her in its Overlooked series under the title "Ove

NAPLPS

NAPLPS (North American Presentation Layer Protocol Syntax) is a graphics language for use originally with videotex and teletext services. NAPLPS was developed from the Telidon system developed in Canada, with a small number of additions from AT&T Corporation. The basics of NAPLPS were later used as the basis for several other microcomputer-based graphics systems. == History == The Canadian Communications Research Centre (CRC), based in Ottawa, had been working on various graphics systems since the late 1960s, much of it led by Herb Bown. Through the 1970s they turned their attention to building out a system of "picture description instructions", which encoded graphics commands as a text stream. Graphics were encoded as a series of instructions (graphics primitives) each represented by a single ASCII character. Graphic coordinates were encoded in multiple 6-bit strings of XY coordinate data, flagged to place them in the printable ASCII range so that they could be transmitted with conventional text transmission techniques. ASCII SI/SO characters were used to differentiate the text from graphic portions of a transmitted "page". These instructions were decoded by separate programs to produce graphics output, on a plotter for instance. Other work produced a fully interactive version. In 1975, the CRC gave a contract to Norpak to develop an interactive graphics terminal that could decode the instructions and display them on a color display. During this period, a number of companies were developing the first teletext systems, notably the BBC's Ceefax system. Ceefax encoded character data into the lines in the vertical blanking interval of normal television signals where they could not be seen on-screen, and then used a buffer and decoder in the user's television to convert these into "pages" of text on the display. The Independent Broadcasting Authority quickly introduced their own ORACLE system, and the two organizations subsequently agreed to use a single standard, the "Broadcast Teletext Specification". This later became World System Teletext. At about the same time, other organizations were developing videotex systems, similar to teletext except they used modems to transmit their data instead of television signals. This was potentially slower and used up a telephone line, but had the major advantage of allowing the user to transmit data back to the sender. The UK's General Post Office developed a system using the Ceefax/ORACLE standard, launching it as Prestel, while France prepared the first steps for its ultimately very successful Minitel system, using a rival display standard called Antiope. By 1977, the Norpak system was running, and from this work the CRC decided to create their own teletext/videotext system. Unlike the systems being rolled out in Europe, the CRC decided from the start that the system should be able to run on any combination of communications links. For instance, it could use the vertical blanking interval to send data to the user, and a modem to return selections to the servers. It could be used in a one-way or two-way system. In teletext mode, character codes were sent to users' televisions by encoding them as dot patterns in the vertical blanking interval of the video signal. Various technical "tweaks" and details of the NTSC signals used by North American televisions allowed the downstream videotex channel to increase to 600 bit/s, about twice that used in the European systems. In videotext mode, Bell 202 modems were typical, offering a 1,200 bit/s download rate. A set top box attached to the TV decoded these signals back into text and graphics pages, which the user could select among. The system was publicly launched as Telidon on August 15, 1978. Compared to the European standards, the CRC system was faster, bi-directional, and offered real graphics as opposed to simple character graphics. The downside of the system was that it required much more advanced decoders, typically featuring Zilog Z80 or Motorola 6809 processors with RGB and/or RF output. The Innovation, Science and Economic Development Canada (then Department of Communications) launched a four-year plan to fund public roll-outs of the technology in an effort to spur the development of a commercial Telidon system. AT&T Corporation was so impressed by Telidon that they decided to join the project. They added a number of useful extensions, notably the ability to define original graphics commands (macro) and character sets (DRCS). They also tabled algorithms for proportionally spaced text, which greatly improved the quality of the displayed pages. A joint CSA/ANSI working group (X3L2.1) revised the specifications, which were submitted for standardization. In 1983, they became CSA T500 and ANSI X3.110, or NAPLPS. The data encoding system was also standardized as the NABTS (North American Broadcast Teletext Specification) protocol. Business models for Telidon services were poorly developed. Unlike the UK, where teletext was supported by one of only two large companies whose whole revenue model was based on a read-only medium (television), in North America Telidon was being offered by companies who worked on a subscriber basis. == One-way systems == Telidon-based teletext was tested in a few North American trials in the early 1980s — CBC IRIS, TVOntario, MTS-sponsored Project IDA, to name a few. NAPLPS was also part of the NABTS teletext standard, for the encoding and display of teletext pages. In the late 1980s and early 1990s, affiliates of the regional sports network group SportsChannel ran a service called Sports Plus Network, which ran sports news and scores while SportsChannel was not otherwise on the air. The screens, which frequently featured team logos or likenesses of players in addition to text, were drawn entirely with NAPLPS graphics and resembled the loading of Prodigy pages over a modem, though slightly faster. == Two-way systems == Various two-way systems using NAPLPS appeared in North America in the early 1980s. The biggest North American examples were Knight Ridder's Viewtron (based in Miami) and the Los Angeles Times' Gateway service (based in Orange County). Both used the Sceptre NAPLPS terminal from AT&T. The Sceptre contained a slow modem that connected over the consumer's telephone line to host computers. The Sceptre was expensive whether purchased or rented. Despite huge investments by their parent companies, neither Viewtron nor Gateway lasted into the second half of the decade. Another system, Keyfax, was developed by Keycom Electronic Publishing, a joint venture of Honeywell, Centel (since acquired by Sprint) and Field Enterprises, then-owner of the Chicago Sun-Times newspaper. Keyfax had originally been a WST teletext service, broadcast overnights on Field's Chicago television station WFLD-32 and through the VBI of both WFLD and national superstation WTBS; the decision was made to convert Keyfax into a subscription service, using a proprietary NAPLPS terminal device in a last-ditch effort to save the service. It did not work and Keyfax had ceased operations by the end of 1986. Other early-1980s NAPLPS technology was deployed in Canada, both as a way for rural Canadians to get news and weather information and as the platform for touchscreen information kiosks. In Vancouver these were featured at Expo 86. The kiosks became ubiquitous in Toronto under the name Teleguide, and were deployed in many shopping centres and at major tourist attractions. The latter city was the North American nexus of NAPLPS and the home of Norpak, the most successful of NAPLPS-oriented developers. Norpak created and sold hardware and software for NAPLPS development and display. TVOntario also developed NAPLPS content creation software. London, Ontario - based Cableshare used NAPLPS as the basis of touch-screen information kiosks for shopping malls, the flagship of which was deployed at Toronto's Eaton Centre. The system relied on an 8085-based microcomputer which drove several NAPLPS terminals fitted with touch screens, all communicating via Datapac to a back end database. The system offered news, weather and sports information along with shopping mall guides and coupons. Cableshare also developed and sold a leading NAPLPS page creation utility called the "Picture Painter." In the late 1980s, Tribune Media Services (TMS) and the Associated Press operated a cable television channel called AP News Plus that provided NAPLPS-based news screens to cable television subscribers in many U.S. cities. The news pages were created and edited by TMS staffers working on an Atex editing system in Orlando, Florida, and sent by satellite to NAPLPS decoder devices located at the local cable television companies. Among the firms providing technology to TMS and the Associated Press for the AP News Plus channel was Minneapolis-based Electronic Publishers Inc. (1985–1988). In 1981, two amateur radio operators (VE3FTT and VE3GQW) received special permission from the Canad

Permutation automaton

In automata theory, a permutation automaton, or pure-group automaton, is a deterministic finite automaton such that each input symbol permutes the set of states. Formally, a deterministic finite automaton A may be defined by the tuple (Q, Σ, δ, q0, F), where Q is the set of states of the automaton, Σ is the set of input symbols, δ is the transition function that takes a state q and an input symbol x to a new state δ(q,x), q0 is the initial state of the automaton, and F is the set of accepting states (also: final states) of the automaton. A is a permutation automaton if and only if, for every two distinct states qi and qj in Q and every input symbol x in Σ, δ(qi,x) ≠ δ(qj,x). A formal language is p-regular (also: a pure-group language) if it is accepted by a permutation automaton. For example, the set of strings of even length forms a p-regular language: it may be accepted by a permutation automaton with two states in which every transition replaces one state by the other. == Applications == The pure-group languages were the first interesting family of regular languages for which the star height problem was proved to be computable. Another mathematical problem on regular languages is the separating words problem, which asks for the size of a smallest deterministic finite automaton that distinguishes between two given words of length at most n – by accepting one word and rejecting the other. The known upper bound in the general case is O ( n 2 / 5 ( log ⁡ n ) 3 / 5 ) {\displaystyle O(n^{2/5}(\log n)^{3/5})} . The problem was later studied for the restriction to permutation automata. In this case, the known upper bound changes to O ( n 1 / 2 ) {\displaystyle O(n^{1/2})} .