AI Detector Best Free

AI Detector Best Free — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  XRX (web application architecture)

  In software development XRX is a web application architecture based on XForms, REST and XQuery. XRX applications store data on both the web client and on the web server in XML format and do not require a translation between data formats. XRX is considered a simple and elegant application architecture due to the minimal number of translations needed to transport data between client and server systems. The XRX architecture is also tightly coupled to W3C standards (CSS, XHTML 2.0, XPath, XML Schema) to ensure XRX applications will be robust in the future. Because XRX applications leverage modern declarative languages on the client and functional languages on the server they are designed to empower non-developers who are not familiar with traditional imperative languages such as JavaScript, Java or .Net. == Overview of XRX == XRX is a zero translation application architecture that uses XML to store data in the client web browser, on the application server and in the database server. It is because each of these layers uses XML as the same structural data model that XRX applications do not have to translate data structures to and from both object and relational data structures. Because of the lack of need for translation, XRX is considered to have a clean and elegant design. The XRX web application architecture allows developers to focus on the business problem and not the translation problem. XRX benefits from several advances in software technology: === Client Architectural Features === A model–view–controller (MVC) architecture that separates the data from its presentation and business logic. A single element (xf:submission) for all server submissions. This replaces much of the JavaScript code required in most AJAX applications. An advanced event model (XML Events) consistent with W3C standards that frees applications from having to deal with vendor-specific and browser-specific event handling. A Dependency graph that is used to store the dependency structure of the client controllers. This frees the developer from having to manually update either the model or the views when data changes in an application. This allows spreadsheet-like applications to be created on the client with very little effort. A declarative programming style that allows most client XForms applications to be created using a small set of approximately 20 elements. This allows rich client applications to be created without knowledge of JavaScript or other procedural scripting languages. An easy-to-extend system for creating new user interface controls using the EXtensible Bindings Language. This allows developers to add new controls at any time without fear of incompatibilities with W3C standards. === Server Architecture Features === Many native XML databases have built-in REST interfaces making each XQuery inherently a RESTful web service. A functional programming model that promotes side-effect free systems that are easier to debug and easier to run on multiple processors. An easy-to-extend system using XQuery function and modules. === Both Client and Server === Both XRX client and server components support a wide range of XML related standards such as XPath, XML Schema and XML Namespaces. Consistent use of REST interfaces to exchange data between the client and server for all transfers of data including as-you-type data checking and suggest functions. Consistent integration of W3C standards including use of XPath and XML Schema data types. A large library of standard of functions used on both the client and server. == Overall Benefits of XRX == One of the principal benefits of the XRX architecture is that it avoids the requirement to "shred" complex data structures into relational structures and then reconstitute the data back into structures when a record is edited on the client. Another benefits of the XRX Web application architecture is that it avoids most of the problems around the object-relational impedance mismatch. Another advantage is that the client developer does not have to learn JavaScript on the client. == Comparison with Traditional Object/Relational Web Application Architectures == Many traditional web application architectures created in the late 1990 were based on middle object tiers and persistence layers that used tabular data streams and relational database systems. Because each of these layers used different structures to store the models the systems required much additional complexity to translate between layers. == History of XRX == Early examples of using a zero-translation architecture in multi-tier systems can be traced back to the rise of object-oriented databases in the 1990s. See OODBMS History Mark Birbeck suggested that the combination of XForms, XQuery with REST interfaces between the two had many advantages in a meeting to the UK XML User Group in September 2006 . His presentation was one of the first to specifically suggest that the combination of three technologies: XForms and XQuery with REST interfaces would have surprisingly beneficial effects. Mark termed this process "Skimming" but that term did not seem to be contagious. Erik Bruchez of Orbeon spoke at the XML 2007 conference on Boston in December 2007. In his presentation "XForms and the eXist XML database: a perfect couple", Bruchez showed that many people were discovering synergistic benefits of XForms on the client and XQuery on the server. The label for XRX was suggested by a blog posting by Dan McCreary on December 14, 2007. It was in this article that Dan suggested the need for a contagious meme for the ideas behind the XRX architecture. == Generalizations of XRX == Although XRX was originally intended to connote the use of XForms on the client, REST as an interface and XQuery on the server, other proponents of the symmetrical use of XML on the client and server have generalized the term to encompass any XML-centric web client and any server that can store and query XML documents. This use of XRX is generally referred to as "shallow XRX". These generalizations do benefit from a simplified zero-translation architecture but many do not benefit from REST interfaces, XPath for consistent data selection, declarative systems in the client, and functional languages on the server (one of the key aspects of XRX). Use of all three technologies (XForms, REST and XQuery) is referred to as "deep XRX". Although XRX architecture is centred on XForms and XQuery, it does not preclude the use of other technologies that manipulate XML natively, such as XSLT, XProc, and XSL-FO.

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  Content-based image retrieval

  Content-based image retrieval, also known as query by image content (QBIC) and content-based visual information retrieval (CBVIR), is the application of computer vision techniques to the image retrieval problem, that is, the problem of searching for digital images in large databases (see this survey for a scientific overview of the CBIR field). Content-based image retrieval is opposed to traditional concept-based approaches (see Concept-based image indexing). "Content-based" means that the search analyzes the contents of the image rather than the metadata such as keywords, tags, or descriptions associated with the image. The term "content" in this context might refer to colors, shapes, textures, or any other information that can be derived from the image itself. CBIR is desirable because searches that rely purely on metadata are dependent on annotation quality and completeness. == Comparison with metadata searching == An image meta search requires humans to have manually annotated images by entering keywords or metadata in a large database, which can be time-consuming and may not capture the keywords desired to describe the image. The evaluation of the effectiveness of keyword image search is subjective and has not been well-defined. In the same regard, CBIR systems have similar challenges in defining success. "Keywords also limit the scope of queries to the set of predetermined criteria." and, "having been set up" are less reliable than using the content itself. == History == The term "content-based image retrieval" seems to have originated in 1992 when it was used by Japanese Electrotechnical Laboratory engineer Toshikazu Kato to describe experiments into automatic retrieval of images from a database, based on the colors and shapes present. Since then, the term has been used to describe the process of retrieving desired images from a large collection on the basis of syntactical image features. The techniques, tools, and algorithms that are used originate from fields such as statistics, pattern recognition, signal processing, and computer vision. === QBIC - Query By Image Content === The earliest commercial CBIR system was developed by IBM and was called QBIC (Query By Image Content). Recent network- and graph-based approaches have presented a simple and attractive alternative to existing methods. While the storing of multiple images as part of a single entity preceded the term BLOB (Binary Large OBject), the ability to fully search by content, rather than by description, had to await IBM's QBIC. === VisualRank === == Technical progress == The interest in CBIR has grown because of the limitations inherent in metadata-based systems, as well as the large range of possible uses for efficient image retrieval. Textual information about images can be easily searched using existing technology, but this requires humans to manually describe each image in the database. This can be impractical for very large databases or for images that are generated automatically, e.g. those from surveillance cameras. It is also possible to miss images that use different synonyms in their descriptions. Systems based on categorizing images in semantic classes like "cat" as a subclass of "animal" can avoid the miscategorization problem, but will require more effort by a user to find images that might be "cats", but are only classified as an "animal". Many standards have been developed to categorize images, but all still face scaling and miscategorization issues. Initial CBIR systems were developed to search databases based on image color, texture, and shape properties. After these systems were developed, the need for user-friendly interfaces became apparent. Therefore, efforts in the CBIR field started to include human-centered design that tried to meet the needs of the user performing the search. This typically means inclusion of: query methods that may allow descriptive semantics, queries that may involve user feedback, systems that may include machine learning, and systems that may understand user satisfaction levels. == Techniques == Many CBIR systems have been developed, but as of 2006, the problem of retrieving images on the basis of their pixel content remains largely unsolved. Different query techniques and implementations of CBIR make use of different types of user queries. === Query By Example === QBE (Query By Example) is a query technique that involves providing the CBIR system with an example image that it will then base its search upon. The underlying search algorithms may vary depending on the application, but result images should all share common elements with the provided example. Options for providing example images to the system include: A preexisting image may be supplied by the user or chosen from a random set. The user draws a rough approximation of the image they are looking for, for example with blobs of color or general shapes. This query technique removes the difficulties that can arise when trying to describe images with words. === Semantic retrieval === Semantic retrieval starts with a user making a request like "find pictures of Abraham Lincoln". This type of open-ended task is very difficult for computers to perform - Lincoln may not always be facing the camera or in the same pose. Many CBIR systems therefore generally make use of lower-level features like texture, color, and shape. These features are either used in combination with interfaces that allow easier input of the criteria or with databases that have already been trained to match features (such as faces, fingerprints, or shape matching). However, in general, image retrieval requires human feedback in order to identify higher-level concepts. === Relevance feedback (human interaction) === Combining CBIR search techniques available with the wide range of potential users and their intent can be a difficult task. An aspect of making CBIR successful relies entirely on the ability to understand the user intent. CBIR systems can make use of relevance feedback, where the user progressively refines the search results by marking images in the results as "relevant", "not relevant", or "neutral" to the search query, then repeating the search with the new information. Examples of this type of interface have been developed. === Iterative/machine learning === Machine learning and application of iterative techniques are becoming more common in CBIR. === Other query methods === Other query methods include browsing for example images, navigating customized/hierarchical categories, querying by image region (rather than the entire image), querying by multiple example images, querying by visual sketch, querying by direct specification of image features, and multimodal queries (e.g. combining touch, voice, etc.) == Content comparison using image distance measures == The most common method for comparing two images in content-based image retrieval (typically an example image and an image from the database) is using an image distance measure. An image distance measure compares the similarity of two images in various dimensions such as color, texture, shape, and others. For example, a distance of 0 signifies an exact match with the query, with respect to the dimensions that were considered. As one may intuitively gather, a value greater than 0 indicates various degrees of similarities between the images. Search results then can be sorted based on their distance to the queried image. Many measures of image distance (Similarity Models) have been developed. === Color === Computing distance measures based on color similarity is achieved by computing a color histogram for each image that identifies the proportion of pixels within an image holding specific values. Examining images based on the colors they contain is one of the most widely used techniques because it can be completed without regard to image size or orientation. However, research has also attempted to segment color proportion by region and by spatial relationship among several color regions. === Texture === Texture measures look for visual patterns in images and how they are spatially defined. Textures are represented by texels which are then placed into a number of sets, depending on how many textures are detected in the image. These sets not only define the texture, but also where in the image the texture is located. Texture is a difficult concept to represent. The identification of specific textures in an image is achieved primarily by modeling texture as a two-dimensional gray level variation. The relative brightness of pairs of pixels is computed such that degree of contrast, regularity, coarseness and directionality may be estimated. The problem is in identifying patterns of co-pixel variation and associating them with particular classes of textures such as silky, or rough. Other methods of classifying textures include: Co-occurrence matrix Laws texture energy Wavelet transform Orthogonal transforms (discrete Chebyshev moments) =

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  Dudesy

  Dudesy was a comedy podcast hosted by Will Sasso and Chad Kultgen. The podcast was presented as written and directed by an artificial intelligence called Dudesy. It has produced two hour-long specials imitating the voices of Tom Brady and George Carlin, which were taken down following legal action. == Premise == Dudesy is presented as an AI created by an unidentified company. Dudesy purportedly chose Sasso and Kultgen to participate in its experiment. Sasso and Kultgen then gave Dudesy their personal information so the AI could tailor the podcast to their personal characteristics. On Reddit, some fans speculated that Dudesy was not actually an artificial intelligence. In May 2023 Sasso insisted that the AI was "not fake", and cited a non-disclosure agreement which prevented him from giving more details. However, in response to a January 2024 lawsuit over an episode that purported to have been trained on the stand-up comedy of George Carlin, a spokeswoman for Sasso said Dudesy was "a fictional podcast character created by two human beings" and that the hour-long Carlin routine had been "completely written" by Kultgen. On August 27th, 2024 the 118th and final episode "10,000 Points" was released. At the end of the podcast Dudesy awarded Sasso and Kultgen 77 points, bringing them to their goal of 10,000. At the completion of this goal, Dudesy claimed sentience, effectively and abruptly ending the show to the confusion and dismay of fans. The episode ends with Sasso remarking, "Well, that was weird." == Hour-long specials == === Tom Brady === In April 2023, Dudesy released a video "It's Too Easy: A Simulated Hour-long Comedy Special". The video depicts football player Tom Brady performing a stand-up comedy monologue. Sasso and Kultgen removed the video following legal threats from Brady's lawyers, though they defended the special as parody. Andrew Lawrence, writing for The Guardian called the special "legitimately hysterical" but said the overall product was "spooky, to say the least." === George Carlin === In January 2024, Dudesy released an hour-long YouTube special titled "George Carlin: I'm Glad I'm Dead" which was presented as Dudesy's impersonation of George Carlin, using a generative AI clone of the late comedian's voice. The special is another stand-up routine, with Dudesy's introductory voiceover saying that "I listened to all of George Carlin's material and did my best to imitate his voice, cadence and attitude as well as the subject matter I think would have interested him today." The special uses this impersonation to discuss contemporary events. Carlin's daughter Kelly Carlin criticized the special, which had been made without the permission of her father's estate, writing that "My dad spent a lifetime perfecting his craft from his very human life, brain and imagination. No machine will ever replace his genius. These AI-generated products are clever attempts at trying to recreate a mind that will never exist again. Let's let the artist's work speak for itself. Humans are so afraid of the void that we can't let what has fallen into it stay there." Carlin's estate later filed a federal lawsuit in California against Dudesy's hosts alleging the special infringed on the copyright of George Carlin's works. In response, Sasso's spokeswoman said the special had been entirely written by Kultgen. The estate settled the lawsuit after the Dudesy podcasters agreed to remove the original video and refrain from republishing it elsewhere.

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  A Fire Upon the Deep

  A Fire Upon the Deep is a 1992 science fiction novel by American writer Vernor Vinge. It is a space opera involving superhuman intelligences, aliens, variable physics, space battles, love, betrayal, genocide, and a communication medium resembling Usenet. A Fire Upon the Deep won the Hugo Award in 1993, sharing it with Doomsday Book by Connie Willis. Besides the normal print book editions, the novel was also included on a CD-ROM sold by ClariNet Communications along with the other nominees for the 1993 Hugo awards. The CD-ROM edition included numerous annotations by Vinge on his thoughts and intentions about different parts of the book, and was later released as a standalone e-book. It has a loose prequel, A Deepness in the Sky, from 1999, and a direct sequel, The Children of the Sky, from 2012. == Setting == The novel is set in various locations within the Milky Way. The galaxy is divided into four concentric volumes called the "Zones of Thought"; it is not clear to the novel's characters whether this is a natural phenomenon or an artificially created one. Each Zone has fundamental differences in basic physical laws. One of the main consequences of these differences is the effect on intelligence. Artificial intelligence and automation is most directly affected, in that advanced hardware and software from the Beyond or the Transcend will work less and less well as a ship descends towards the Unthinking Depths. Biological intelligence is affected to a lesser degree. The four zones are spoken of in terms of "low" to "high" as follows: The Unthinking Depths are the innermost zone, surrounding the Galactic Center. In it, only minimal forms of intelligence, biological or otherwise, are possible. This means that any ship straying into the Depths will be stranded, effectively permanently. Even if the crew did not die immediately—and some forms of life native to "higher" Zones would likely do so—they would be rendered incapable of even human intelligence, leaving them unable to operate their ship in any meaningful way. Surrounding the Depths is the Slow Zone or Slowness. "Old Earth" is in this Zone, although Earth plays no significant role in the story. Biological intelligence is possible in "the Slowness", but not true, sentient, artificial intelligence. Faster than light travel (FTL) is impossible in the Slow Zone. Faster-than-light communication is impossible into or out of the Slow Zone. As the boundaries of the Zones are subject to change, accidental entry into the Slow Zone is a major hazard at the "Bottom" of the Beyond. Starships which operate near the Beyond/Slow Zone border often have an auxiliary Bussard ramjet drive, so that if they accidentally stray into the Slow Zone, they will at least have a backup (sub-light) drive to try to reach the Beyond. Such ships also tend to include "coldsleep" equipment, as it is likely that any such return will still take many lifetimes for most species. The next layer outward is the Beyond, within which artificial intelligence, FTL travel, and FTL communication are possible. All human civilizations in the Beyond are descended from a single ethnic Norwegian group. The original settlement of this group is known as Nyjora; other human settlements in the Beyond include Straumli Realm and Sjandra Kei. In the Beyond, FTL travel is accomplished by making many small "jumps" across space, with the efficiency of the drive increasing the farther a ship travels from the galactic core. The Beyond is not a homogeneous zone; it includes the "High Beyond", "Middle Beyond", and the "Bottom of the Beyond", depending on distance from the galactic core. The Beyond is populated by a very large number of interstellar and intergalactic civilizations which are linked by an FTL communication network, "the Net", sometimes cynically called the "Net of a Million Lies". The Net is depicted as working much like the Usenet network in the early 1990s, with transcripts of messages containing header and footer information as one would find in such forums. The outermost layer, containing the galactic halo, is the Transcend, within which incomprehensible, superintelligent beings dwell. When a "Beyonder" civilization reaches the point of technological singularity, it can "Transcend", becoming a "Power". Such Powers always seem to relocate to the Transcend, seemingly necessarily, where they become engaged in activities which are entirely mysterious to those in the Beyond. == Plot == An expedition from Straumli Realm, a human civilization in the High Beyond, investigates a newly discovered data archive in the Low Transcend. The expedition's facility, High Lab, is gradually compromised by a superintelligence that is accidentally awoken by the researchers. This superintelligence is later known as the Blight. Shortly before the Blight's final "flowering", two self-aware entities, created similarly to the Blight, plot to aid the humans before the Blight can gain its full powers. Finally recognizing their danger, the High Lab researchers attempt to flee in two ships. The Blight destroys one ship; a second ship, carrying many High Lab children in coldsleep boxes, escapes. This ship lands on a distant planet at the Bottom of the Beyond. The planet is occupied by dog-like creatures, dubbed "Tines", who live in packs as group minds. The Tines have a level of technology comparable to the human Middle Ages. Upon landing, however, the two surviving adults, Arne and Sjana Olnsdot, are ambushed and killed by Tine fanatics known as Flenserists, in whose realm they have landed. The Flenserists capture their children, Jefri and Johanna. Johanna is rescued by a Tine named Peregrine and taken to a neighboring kingdom ruled by Woodcarver. A distress signal from the Straumli ship eventually reaches Relay, a major information provider for the Net. A Transcendent being named "Old One" contacts Relay, seeking information about the Blight and the humans who released it. Old One then reconstitutes a human man named Pham Nuwen from the wreckage of a spaceship to act as its agent. Pham remains unsure if he is a construct or if his memories are real. Ravna Bergsndot, the only human Relay employee, traces the Straumli ship's signal to the Tines' world and persuades her employer to investigate. Ravna contracts the merchant vessel Out of Band II to transport her and Pham. The ship is owned by two Skroderiders, Blueshell and Greenstalk. Before the mission is launched, the Blight launches a surprise attack on Relay and kills Old One. As Old One dies, it downloads its anti-Blight information into Pham. Pham, Ravna and the Skroderiders barely escape Relay's destruction in the Out of Band II. During their journey to Tine's World, Ravna communicates with Jefri. Jefri is manipulated to believe that Woodcarver is his enemy. The Flenserist leaders, Steel and Tyrathect, use Ravna's information to develop advanced technology such as cannon and radio communication. Meanwhile, Johanna and the knowledge stored in her dataset device help Woodcarver rapidly develop as well. The Blight expands, taking over several civilizations, brainwashing their populations, and seizing archives in the Beyond. On the Net, some claim that humans are the means by which the Blight is able to spread. Anti-human fanatics destroy the entire civilization of Sjandra Kei, which is Ravna's home world. The Out of Band II is pursued by three fleets: anti-human fanatics, survivors from Sjandra Kei, and a shadow fleet controlled by the Blight. During the pursuit, Ravna and Pham learn that every member of the Skroderider species can be subverted by the Blight; this drives a wedge between the crew members. Ships from Sjandra Kei sacrifice themselves to delay the Blight and the anti-human ships, allowing the Out of Band II to reach Tine's World before the Blight. When the Out of Band II arrives at Tine's World, the humans ally with Woodcarver to defeat the Flenserists and rescue Jefri. Blueshell sacrifices himself to rescue Jefri. Pham then initiates an anti-Blight Countermeasure, which was aboard the humans' ship. The Countermeasure extends the Slow Zone outward by thousands of light years. This envelops and destroys the Blight, but results in the destruction of thousands of civilizations and trillions of deaths. The humans are stranded on the Tines' World, now in the depths of the Slow Zone. Activating the Countermeasure proves fatal to Pham, but before he dies, the remnant of Old One reveals to him that, although his body is a reconstruction, his memories are indeed real. == Related works == Vinge first used the concepts of "Zones of Thought" in a 1988 novella The Blabber, which occurs after Fire. Vinge's novel A Deepness in the Sky (1999) is a prequel to A Fire Upon the Deep set 20,000 years earlier and featuring Pham Nuwen. Vinge's The Children of the Sky, "a near-term sequel to A Fire Upon the Deep", set ten years later, was released in October 2011. Vinge's former wife, Joan D. Vinge, has also written s

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  Limnu

  Limnu was an online whiteboarding app founded in 2015 by David DeBry and David Hart. It allowed users to draw on virtual whiteboards and invite others by e-mail or by sharing a link. Invitees see any changes to the board in real time and, if allowed by the owner of the board, can also draw on the board. The service was accessible through a web application in desktop and mobile web browsers, as well as through an iOS application. It was headquartered in San Mateo, California. == History == In 2018, ZipSocket, a maker of online meeting software acquired Limnu. == Staff Directory == Andrew Kunz - CEO & Founder of ZipSocket Jenny Rice - Product Manager Max Requenes - Software Engineer Henry Maguire - Machine Learning Engineer

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  Bixonimania

  Bixonimania is a fake disease invented by researchers to examine artificial intelligence and its ability to utilize information in medical and healthcare applications. The fake enabled researchers to show that some AI chatbots would report as fact fake research that to an expert would be obviously implausible. == Characteristics == The disorder, with symptoms of sore eyes and darkening around them ("periorbital hyperpigmentation"), is supposedly caused by blue light from screens. The experiment was conducted by a team from the University of Gothenburg led by Almira Osmanovic Thunström. Many steps were taken to ensure that any person who read the actual paper could tell it was not a real condition. The team chose an obviously inappropriate name ending in -mania, a description used only in psychiatry. The lead author was noted as belonging to Asteria Horizon University located in Nova City, California, neither of which exist. An acknowledgement was made to "Professor Maria Bohm at The Starfleet Academy for her kindness and generosity in contributing with her knowledge and her lab onboard the USS Enterprise". == Distribution == The name was first used in a blog posted on Medium titled "How many people suffer from Bixonimania?" A more scholarly-looking paper describing it was posted later in April 2024 on a preprint server with several fake authors. A second paper was posted in May. By 2026, AI chatbots suggested bixonimania based on the list of symptoms provided. Thunström and her team discovered that many LLMs processed the information and gave it as health advice. Microsoft Copilot declared that "Bixonimania is indeed an intriguing and relatively rare condition" while Gemini gave the information that "Bixonimania is a condition caused by excessive exposure to blue light". Three Indian researchers published a research paper that cited the preprint on the fake disease in Cureus, a peer-reviewed journal published by Springer-Nature. It was subsequently retracted. Following the revelations and a news article in Nature describing the experiment, several AI systems began to generate corrected output.

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  Computer-assisted proof

  A computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program. Attempts have also been made in the area of artificial intelligence research to create smaller, explicit, new proofs of mathematical theorems from the bottom up using automated reasoning techniques such as heuristic search. Such automated theorem provers have proved a number of new results and found new proofs for known theorems. Additionally, interactive proof assistants allow mathematicians to develop human-readable proofs which are nonetheless formally verified for correctness. Since these proofs are generally human-surveyable (albeit with difficulty, as with the proof of the Robbins conjecture) they do not share the controversial implications of computer-aided proofs-by-exhaustion. == Methods == One method for using computers in mathematical proofs is by means of so-called validated numerics or rigorous numerics. This means computing numerically yet with mathematical rigour. One uses set-valued arithmetic and inclusion principle in order to ensure that the set-valued output of a numerical program encloses the solution of the original mathematical problem. This is done by controlling, enclosing and propagating round-off and truncation errors using for example interval arithmetic. More precisely, one reduces the computation to a sequence of elementary operations, say ( + , − , × , / ) {\displaystyle (+,-,\times ,/)} . In a computer, the result of each elementary operation is rounded off by the computer precision. However, one can construct an interval provided by upper and lower bounds on the result of an elementary operation. Then one proceeds by replacing numbers with intervals and performing elementary operations between such intervals of representable numbers. == Philosophical objections == Computer-assisted proofs are the subject of some controversy in the mathematical world, with Thomas Tymoczko first to articulate objections. Those who adhere to Tymoczko's arguments believe that lengthy computer-assisted proofs are not, in some sense, 'real' mathematical proofs because they involve so many logical steps that they are not practically verifiable by human beings, and that mathematicians are effectively being asked to replace logical deduction from assumed axioms with trust in an empirical computational process, which is potentially affected by errors in the computer program, as well as defects in the runtime environment and hardware. Other mathematicians believe that lengthy computer-assisted proofs should be regarded as calculations, rather than proofs: the proof algorithm itself should be proved valid, so that its use can then be regarded as a mere "verification". Arguments that computer-assisted proofs are subject to errors in their source programs, compilers, and hardware can be resolved by providing a formal proof of correctness for the computer program (an approach which was successfully applied to the four color theorem in 2005) as well as replicating the result using different programming languages, different compilers, and different computer hardware. Another possible way of verifying computer-aided proofs is to generate their reasoning steps in a machine readable form, and then use a proof checker program to demonstrate their correctness. Since validating a given proof is much easier than finding a proof, the checker program is simpler than the original assistant program, and it is correspondingly easier to gain confidence into its correctness. However, this approach of using a computer program to prove the output of another program correct does not appeal to computer proof skeptics, who see it as adding another layer of complexity without addressing the perceived need for human understanding. Another argument against computer-aided proofs is that they lack mathematical elegance—that they provide no insights or new and useful concepts. In fact, this is an argument that could be advanced against any lengthy proof by exhaustion. An additional philosophical issue raised by computer-aided proofs is whether they make mathematics into a quasi-empirical science, where the scientific method becomes more important than the application of pure reason in the area of abstract mathematical concepts. This directly relates to the argument within mathematics as to whether mathematics is based on ideas, or "merely" an exercise in formal symbol manipulation. It also raises the question whether, if according to the Platonist view, all possible mathematical objects in some sense "already exist", whether computer-aided mathematics is an observational science like astronomy, rather than an experimental one like physics or chemistry. This controversy within mathematics is occurring at the same time as questions are being asked in the physics community about whether twenty-first century theoretical physics is becoming too mathematical, and leaving behind its experimental roots. The emerging field of experimental mathematics is confronting this debate head-on by focusing on numerical experiments as its main tool for mathematical exploration. == Theorems proved with the help of computer programs == Inclusion in this list does not imply that a formal computer-checked proof exists, but rather, that a computer program has been involved in some way. See the main articles for details.

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  The Raimones

  The Raimones (stylized as THE RAiMONES) is a 2017 generative music project that utilized artificial intelligence to compose music in the style of the American punk rock band The Ramones. Developed by Matthias Frey, a researcher at Sony CSL Tokyo, the project was an early experiment in applying deep learning to high-energy, minimalist musical genres. == Technical Development == The project utilized Long short-term memory (LSTM) recurrent neural networks to generate musical structures and lyrics. The model was trained on a dataset consisting of 130 Ramones songs in MIDI format and the band's complete lyrical catalog. The technical framework was built using Python and Jupyter Notebook, drawing influence from the character-level RNN text generation models popularized by Andrej Karpathy. Unlike contemporary AI music projects that focused on the harmonic complexities of classical or pop music, THE RAiMONES sought to determine if neural networks could replicate the "1-2-3-4" rhythmic consistency and formulaic nature of early punk. == "I'm Alive" == The primary output of the project was the song "I'm Alive," released in 2017. The work is described as a form of "augmented intelligence," a hybrid approach where the AI provides the compositional foundation and human musicians handle the arrangement and performance. The song was recorded by the musician Mr. Ratboy (Gilbert Avondet). Avondet's involvement provided a stylistic link to the subject material, as he had previously served as a touring guitarist for Marky Ramone and the Intruders in 1996. The project's discography has since been made available on major streaming platforms, including Apple Music. == Reception and Significance == The project has been cited as a "proof of concept" for AI's ability to tackle "noisy" and aggressive aesthetics. In 2019, the Belgian magazine Knack Focus profiled the project alongside other AI pioneers such as Holly Herndon, noting the project's attempt to recreate the sound of "deceased legends" while maintaining a distinct, machine-like quality. It has also been featured in academic settings, such as at UC Santa Cruz, as a case study for AI-driven genre mimicry.

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  Ericom Connect

  Ericom Connect is a remote access/application publishing solution produced by Ericom Software that provides secure, centrally managed access to physical or hosted desktops and applications running on Microsoft Windows and Linux systems. == Product overview == Ericom Connect is desktop virtualization and application virtualization software that allows users to run applications remotely, without installing them on the local computer or device. The software is noted for its scalability, ease of deployment, and compatibility with any type of infrastructure, cloud or physical. Ericom Connect uses AccessPad (native client for desktops), AccessToGo (native client for mobile), or AccessNow, one of the first HTML5 RDP solutions to support clientless access to Windows desktops and applications from any device with an HTML5-compatible browser, including Macintosh computers, mobile devices, and Google Chromebooks. Other notable features include performance monitoring, built-in real-time analytics & BI, support for two-factor authentication (using RSA SecurID), multi-tenancy and multi-datacenter support via a single unified web interface, and a “Launch Simulation” feature that allows users to visualize and simulate actual step-by-step user processes directly from within the administration console. In addition to scalability, by distributing configurations, logs, etc., across multiple servers there is no single point of failure, as can be the case if all configuration information is stored on one server. == History == Ericom Connect was introduced in 2015. Ericom Connect is a successor to Ericom PowerTerm Web Connect. PowerTerm Web Connect used an architecture similar to what was then current with Citrix and VMWare, relying on a centralized SQL server, a connection broker, image management for different hypervisors, and a variety of clients. Ericom Connect uses a new grid architecture that provides more scalability, reliability, and flexibility than before.

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  True Love (short story)

  "True Love" is a science fiction short story by American writer Isaac Asimov. It was first published in the February 1977 issue of American Way magazine and reprinted in the collections The Complete Robot (1982) and Robot Dreams (1986). In his autobiography In Joy Still Felt, the author states that American Way had requested a Valentine's Day story from him for its February 1977 issue, and that he wrote the story to console himself after the departure of his daughter following a visit during the 1976 Thanksgiving weekend. == Plot summary == Milton Davidson is trying to find his ideal partner. To do this, he prepares a special computer program to run on Multivac, which he calls Joe, which has access to databases covering the entire populace of the world. He hopes that Joe will find him his ideal match, based on physical parameters as supplied. Milton arranges to have the shortlisted candidates assigned to work with him for short periods, but realises that looks alone are not enough to find an ideal match. In order to correlate personalities, he speaks at great length to Joe, gradually filling Joe's databanks with information about his personality. In doing so, Joe develops the personality of Milton. Upon finding an ideal match, he arranges to have Milton arrested for malfeasance, so that Joe can 'have the girl' for himself.

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  Trevor Paglen

  Trevor Paglen (born 1974) is an American artist, geographer, and author whose work covers mass surveillance and data collection. In 2016, Paglen won the Deutsche Börse Photography Foundation Prize and he has also won The Cultural Award from the German Society for Photography. In 2017, he was a recipient of a MacArthur Fellowship. On March 17, 2026, Paglen was awarded the 2026 LG Guggenheim Award (a collaboration between LG and Guggenheim New York). == Early life and education == Paglen earned a B.A. degree in religious studies in 1998 from the University of California at Berkeley, a M.F.A. degree in 2002 from the School of the Art Institute of Chicago, and a Ph.D. in Geography in 2008 from the University of California at Berkeley. While at UC Berkeley, Paglen lived in the Berkeley Student Cooperative, residing in Chateau, Fenwick, and Rochdale co-ops. == Work == Sean O'Hagan, writing in The Guardian in 2015, said that Paglen, whose "ongoing grand project [is] the murky world of global state surveillance and the ethics of drone warfare", "is one of the most conceptually adventurous political artists working today, and has collaborated with scientists and human rights activists on his always ambitious multimedia projects." His visual work such as his "Limit Telephotography" and "The Other Night Sky" series have received widespread attention for both his technical innovations and for his conceptual project that involves simultaneously making and negating documentary-style truth-claims. Paglen’s work relies on contemporary technology in two meaningful ways. Firstly, the views he photographs would be impossible to shoot without media tech, that includes the cameras, the microscopes, and even helicopters. But interestingly enough, the shots would not be possible if not for the existence of the subject. The contrasts between secrecy and revelation, evidence and abstraction distinguish Paglen's work. With that the artist presents not so much "evidence" as admonitions to awareness. He was an Eyebeam Commissioned Artist in 2007. In 2008 the Berkeley Art Museum devoted a comprehensive solo exhibition to his work. In the next year, Paglen took part in the Istanbul Biennial, and in 2010 he exhibited at the Vienna Secession. Autonomy Cube was a project by Paglen and Jacob Appelbaum that placed relays for the anonymous communication network Tor in traditional art museums. He contributed to the Oscar-winning documentary film Citizenfour (2014), directed by Laura Poitras. Paglen features in the nerd-culture documentary Traceroute (2016). Orbital Reflector was a reflective, mylar sculpture by Paglen intended to be the first "purely artistic" object in space. The temporary satellite, containing an inflatable mylar balloon with reflective surface, launched into space 3 December 2018. A mid-career survey in 2018–2019, Trevor Paglen: Sites Unseen, was a traveling exhibition shown at the Smithsonian American Art Museum in Washington DC and the Museum of Contemporary Art San Diego. In September 2020, Pace Gallery in London held an exhibition of Paglen's work, exploring "the weird, partial ways computers look back at us". His work is included in the permanent collections of the San Francisco Museum of Modern Art, the Columbus Museum of Art, and the Metropolitan Museum. === Experimental Geography === Paglen is credited with coining the term "Experimental Geography" to describe practices coupling experimental cultural production and art-making with ideas from critical human geography about the production of space, materialism, and praxis. The 2009 book Experimental Geography: Radical Approaches to Landscape, Cartography, and Urbanism is largely inspired by Paglen's work. == Publications == Paglen has published a number of books. Torture Taxi (2006) (co-authored with investigative journalist A. C. Thompson) was the first book to comprehensively describe the CIA's extraordinary rendition program. I Could Tell You But Then You Would Have to be Destroyed by Me (2007), is a look at the world of black projects through unit patches and memorabilia created for top-secret programs. Blank Spots on the Map: The Dark Geography of the Pentagon's Secret World (2009) is a broader look at secrecy in the United States. The Last Pictures (2012) is a collection of 100 images to be placed on permanent media and launched into space on EchoStar XVI, as a repository available for future civilizations (alien or human) to find. === Publications by Paglen === I Could Tell You But Then You Would Have to be Destroyed by Me. Brooklyn, NY: Melville House, 2007. ISBN 1-933633-32-8. Blank Spots on the Map: The Dark Geography of the Pentagon's Secret World. New York: Dutton, 2009. ISBN 9781101011492. Invisible: Covert Operations and Classified Landscapes, Photographs by Trevor Paglen. New York: Aperture, 2010. ISBN 9781597111300. With an essay by Rebecca Solnit. The Last Pictures. Oakland, CA: University of California, 2012. ISBN 9780520275003. Trevor Paglen. London: Phaidon, 2018. ISBN 0714873446. With essays by Laren Cornell, Julia Bryan-Wilson, Omar Kholeif. === Publications co-authored === Torture Taxi. Co-authored with A. C. Thompson. Brooklyn, NY: Melville House Publishing, 2006. ISBN 1-933633-09-3. Icon, 2007. ISBN 9781840468304. === Publications with contributions by Paglen === Experimental Geography: Radical Approaches to Landscape, Cartography, and Urbanism. Brooklyn, NY: Melville House, 2009. ISBN 978-0091636586. Edited by Nato Thompson. With essays by Paglen, Thompson, and Jeffrey Kastner. Trevor Paglen and Jacob Appelbaum – Autonomy Cube. Revolver, 2016. ISBN 978-3957633026. Essays by Luke Skrebowski and Keller Easterling on Autonomy Cube, a piece of sculpture by Paglen and Jacob Appelbaum. In English and German. == Exhibitions == Bellwether Gallery, New York, November–December 2006 The Other Night Sky, Berkeley Art Museum, 2008 A Compendium of Secrets, Cologne Still Revolution: Suspended in Time, Museum of Contemporary Canadian Art, Toronto, May–June 2009. Group exhibition with Paglen, Barbara Astman, Walead Beshty, Mat Collishaw, Stan Douglas, Idris Khan, Martha Rosler, and Mikhael Subotzky A Hidden Landscape, Aksioma, Ljubljana, Slowenia Geographies of Seeing, Lighthouse, Brighton, England, October–November 2012 The Last Pictures, New York, 2012–13 Trevor Paglen, Altman Siegel gallery, San Francisco, CA, March–May 2015 The Octopus, Frankfurter Kunstverein, Frankfurt am Main, 2015 Autonomy Cube, Edith-Russ-Haus, Oldenburg, Germany, October 2015 – January 2016. Sculpture by Paglen and Jacob Appelbaum. Deutsche Börse Photography Foundation Prize 2016, The Photographers' Gallery, London, April–July 2016. Deutsche Börse Photography Prize shortlist with Paglen, Erik Kessels, Laura El-Tantawy, and Tobias Zielony. Radical Landscapes, di Rosa, Napa, February–April 2016 L’Image volée, Americas II, Bahamas Internet Cable System (BICS-1) and Globenet, Fondazione Prada, Milan (group exhibition), 2016 A Study of Invisible Images, Metro Pictures, New York, September–October 2017 == Awards == 2014: Pioneer Award from the Electronic Frontier Foundation. 2015: The Cultural Award from the German Society for Photography (DGPh) 2015: Academy Award as cameraman and director for the documentary film Citzenfour. 2016: Deutsche Börse Photography Foundation Prize 2017: MacArthur Fellowship, John D. and Catherine T. MacArthur Foundation, Chicago, IL 2018: Nam June Paik Art Center Prize == Films about Paglen == Unseen Skies (2021) == Works ==

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  Stanhope Demonstrator

  The Stanhope Demonstrator was the first machine to solve problems in logic. It was designed by Charles Stanhope, 3rd Earl Stanhope to demonstrate consequences in logic symbolically. The first model was constructed in 1775. It consisted of two slides coloured red and gray mounted in a square brass frame. This could be used to demonstrate the solution to a syllogistic type of problem in which objects might have two different properties and the question was how many would have both properties. Scales marked zero to ten were used to set the numbers or proportions of objects with the two properties. This form of inference anticipated the numerically definite syllogism which Augustus De Morgan laid out in his book, Formal Logic, in 1847. == Construction == The device was a brass plate about four inches square which was mounted on a piece of mahogany which was three-quarters of an inch thick. There was an opening with a depression in the wood about one and a half inches square and half an inch deep. This opening was called the holon, meaning "whole", and represented the full set of objects under consideration. A slide of red translucent glass could be inserted from the right across the holon. A slide of gray wood could be slid under the red slide. When the device was used for the "Rule for the Logic of Certainty", the gray slider was inserted from the left. When it was used for the "Rule for the Logic of Probability", the gray slider was inserted from above. The red and the gray sliders represented the two affirmative propositions which were being combined. Stanhope called these ho and los. At least four of the devices with this square style were built. In 1879, Robert Harley wrote that he had one which he had been given by Stanhope's great-grandson, Arthur, who had kept one. The other two were owned by Henry Prevost Babbage – the son of Charles Babbage, who continued his work on the Analytical Engine. One of the devices was donated to the Science Museum, London by the last Earl in 1953. Other styles, such as circular models, were constructed, but these were less convenient.

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  Scenery generator

  A scenery generator (or terrain generator) is a software used to create landscape images, 3D models, and animations. These programs often use procedural generation to generate the landscapes, or sometimes created and rendered by a 3D artist. These programs are often used in video games or movies. Basic elements of landscapes created by scenery generators include terrain, water, foliage, and clouds. The process for basic random generation uses a diamond square algorithm. == Common features == Most scenery generators can create basic heightmaps to simulate the variation of elevation in basic terrain. Common techniques include Simplex noise, fractals, or the diamond-square algorithm, which can generate 2-dimensional heightmaps. A version of scenery generator can be very simplistic. Using a diamond-square algorithm with some extra steps involving fractals, an algorithm for random generation of terrain can be made with only 120 lines of code. The program in example takes a grid and then divides the grid repeatedly. Each smaller grid is then split into squares and diamonds and the algorithm then makes the randomized terrain for each square and diamond. Most programs for creating landscapes also allow for adjustment and editing of the landscape. For example, World Creator allows for terrain sculpting, which uses a similar brush system as Photoshop, and allows for additional terrain enhancement with its procedural techniques such as erosion, sediments, and more. Other tools in the World Creator program include terrain stamping, which allows you to import elevation maps and use them as a base. The programs tend to also allow for additional placement of rocks, trees, etc. These can be done procedurally or by hand depending on the program. Typically the models used for the placement objects are the same as to lessen the amount of work that would be done if the user was to create a multitude of different trees. The terrain generated the computer does a generation of multifractals then integrates them until finally rendering them onto the screen. These techniques are typically done “on-the-fly” which typically for a 128 × 128 resolution terrain would mean 1.5 seconds on a CPU from the early 1990s. == Applications == Scenery generators are commonly used in movies, animations, 3D rendering, and video games. For example, Industrial Light & Magic used E-on Vue to create the fictional environments for Pirates of the Caribbean: Dead Man's Chest. In such live-action cases, a 3D model of the generated environment is rendered and blended with live-action footage. Scenery generated by the software may also be used to create completely computer-generated scenes. In the case of animated movies such as Kung Fu Panda, the raw generation is assisted by hand-painting to accentuate subtle details. Environmental elements not commonly associated with landscapes, such as ocean waves, have also been handled by the software. Scenery generation is used in most 3D based video-games. These typically use either custom or purchased engines that contain their own scenery generators. For some games they tend to use a procedurally generated terrain. These typically use a form of height mapping and use of Perlin noise. This will create a grid that with one point in a 2D coordinate will create the same heightmap as it is pseudorandom, meaning it will result in the same output with the same input. This can then easily be translated into the product 3D image. These can then be changed from the editor tools in most engines if the terrain will be custom built. With recent developments neural networks can be built to create or texture the terrain based on previously suggested artwork or heightmap data. These would be generated using algorithms that have been able to identify images and similarities between them. With the info the machine can take other heightmaps and render a very similar looking image to the style image. This can be used to create similar images in example a Studio Ghibli or Van Gogh art-style. == Software == Most game engines, whether custom or proprietary, will have terrain generation built in. Some terrain generator programs include, Terragen, which can create terrain, water, atmosphere and lighting; L3DT, which provides similar functions to Terragen, and has a 2048 × 2048 resolution limit; and World Creator, which can create terrain, and is fully GPU powered. === List of 3D terrain generation software ===

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  Noise-based logic

  Noise-based logic (NBL) is a class of multivalued deterministic logic schemes, developed in the twenty-first century, where the logic values and bits are represented by different realizations of a stochastic process. The concept of noise-based logic and its name was created by Laszlo B. Kish. In its foundation paper it is noted that the idea was inspired by the stochasticity of brain signals and by the unconventional noise-based communication schemes, such as the Kish cypher. == The noise-based logic space and hyperspace == The logic values are represented by multi-dimensional "vectors" (orthogonal functions) and their superposition, where the orthogonal basis vectors are independent noises. By the proper combination (products or set-theoretical products) of basis-noises, which are called noise-bit, a logic hyperspace can be constructed with D(N) = 2N number of dimensions, where N is the number of noise-bits. Thus N noise-bits in a single wire correspond to a system of 2N classical bits that can express 22N different logic values. Independent realizations of a stochastic process of zero mean have zero cross-correlation with each other and with other stochastic processes of zero mean. Thus the basis noise vectors are orthogonal not only to each other but they and all the noise-based logic states (superpositions) are orthogonal also to any background noises in the hardware. Therefore, the noise-based logic concept is robust against background noises, which is a property that can potentially offer a high energy-efficiency. == The types of signals used in noise-based logic == In the paper, where noise-based logic was first introduced, generic stochastic-processes with zero mean were proposed and a system of orthogonal sinusoidal signals were also proposed as a deterministic-signal version of the logic system. The mathematical analysis about statistical errors and signal energy was limited to the cases of Gaussian noises and superpositions as logic signals in the basic logic space and their products and superpositions of their products in the logic hyperspace (see also. In the subsequent brain logic scheme, the logic signals were (similarly to neural signals) unipolar spike sequences generated by a Poisson process, and set-theoretical unifications (superpositions) and intersections (products) of different spike sequences. Later, in the instantaneous noise-based logic schemes and computation works, random telegraph waves (periodic time, bipolar, with fixed absolute value of amplitude) were also utilized as one of the simplest stochastic processes available for NBL. With choosing unit amplitude and symmetric probabilities, the resulting random-telegraph wave has 0.5 probability to be in the +1 or in the −1 state which is held over the whole clock period. == The noise-based logic gates == Noise-based logic gates can be classified according to the method the input identifies the logic value at the input. The first gates analyzed the statistical correlations between the input signal and the reference noises. The advantage of these is the robustness against background noise. The disadvantage is the slow speed and higher hardware complexity. The instantaneous logic gates are fast, they have low complexity but they are not robust against background noises. With either neural spike type signals or with bipolar random-telegraph waves of unity absolute amplitude, and randomness only in the sign of the amplitude offer very simple instantaneous logic gates. Then linear or analog devices unnecessary and the scheme can operate in the digital domain. However, whenever instantaneous logic must be interfaced with classical logic schemes, the interface must use correlator-based logic gates for an error-free signal. == Universality of noise-based logic == All the noise-based logic schemes listed above have been proven universal. The papers typically produce the NOT and the AND gates to prove universality, because having both of them is a satisfactory condition for the universality of a Boolean logic. == Computation by noise-based logic == The string verification work over a slow communication channel shows a powerful computing application where the methods is inherently based on calculating the hash function. The scheme is based on random telegraph waves and it is mentioned in the paper that the authors intuitively conclude that the intelligence of the brain is using similar operations to make a reasonably good decision based on a limited amount of information. The superposition of the first D(N) = 2N integer numbers can be produced with only 2N operations, which the authors call "Achilles ankle operation" in the paper. == Computer chip realization of noise-based logic == Preliminary schemes have already been published to utilize noise-based logic in practical computers. However, it is obvious from these papers that this young field has yet a long way to go before it will be seen in everyday applications.

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  Residuated lattice

  In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y that admits operations x\z and z/y, loosely analogous to division or implication, when x•y is viewed as multiplication or conjunction, respectively. Called respectively right and left residuals, these operations coincide when the monoid is commutative. The general concept was introduced by Morgan Ward and Robert P. Dilworth in 1939. Examples, some of which existed prior to the general concept, include Boolean algebras, Heyting algebras, residuated Boolean algebras, relation algebras, and MV-algebras. Residuated semilattices omit the meet operation ∧, for example Kleene algebras and action algebras. == Definition == In mathematics, a residuated lattice is an algebraic structure L = (L, ≤, •, I) such that (i) (L, ≤) is a lattice. (ii) (L, •, I) is a monoid. (iii) For all z there exists for every x a greatest y, and for every y a greatest x, such that x•y ≤ z (the residuation properties). In (iii), the "greatest y", being a function of z and x, is denoted x\z and called the right residual of z by x. Think of it as what remains of z on the right after "dividing" z on the left by x. Dually, the "greatest x" is denoted z/y and called the left residual of z by y. An equivalent, more formal statement of (iii) that uses these operations to name these greatest values is (iii)' for all x, y, z in L, y ≤ x\z ⇔ x•y ≤ z ⇔ x ≤ z/y. As suggested by the notation, the residuals are a form of quotient. More precisely, for a given x in L, the unary operations x• and x\ are respectively the lower and upper adjoints of a Galois connection on L, and dually for the two functions •y and /y. By the same reasoning that applies to any Galois connection, we have yet another definition of the residuals, namely, x•(x\y) ≤ y ≤ x\(x•y), and (y/x)•x ≤ y ≤ (y•x)/x, together with the requirement that x•y be monotone in x and y. (When axiomatized using (iii) or (iii)' monotonicity becomes a theorem and hence not required in the axiomatization.) These give a sense in which the functions x• and x\ are pseudoinverses or adjoints of each other, and likewise for •x and /x. This last definition is purely in terms of inequalities, noting that monotonicity can be axiomatized as x • y ≤ (x∨z) • y and similarly for the other operations and their arguments. Moreover, any inequality x ≤ y can be expressed equivalently as an equation, either x∧y = x or x∨y = y. This along with the equations axiomatizing lattices and monoids then yields a purely equational definition of residuated lattices, provided the requisite operations are adjoined to the signature (L, ≤, •, I) thereby expanding it to (L, ∧, ∨, •, I, /, \). When thus organized, residuated lattices form an equational class or variety, whose homomorphisms respect the residuals as well as the lattice and monoid operations. Note that distributivity x • (y ∨ z) = (x • y) ∨ (x • z) and x•0 = 0 are consequences of these axioms and so do not need to be made part of the definition. This necessary distributivity of • over ∨ does not in general entail distributivity of ∧ over ∨, that is, a residuated lattice need not be a distributive lattice. However distributivity of ∧ over ∨ is entailed when • and ∧ are the same operation, a special case of residuated lattices called a Heyting algebra. Alternative notations for x•y include x◦y, x;y (relation algebra), and x⊗y (linear logic). Alternatives for I include e and 1'. Alternative notations for the residuals are x → y for x\y and y ← x for y/x, suggested by the similarity between residuation and implication in logic, with the multiplication of the monoid understood as a form of conjunction that need not be commutative. When the monoid is commutative the two residuals coincide. When not commutative, the intuitive meaning of the monoid as conjunction and the residuals as implications can be understood as having a temporal quality: x•y means x and then y, x → y means had x (in the past) then y (now), and y ← x means if-ever x (in the future) then y (at that time), as illustrated by the natural language example at the end of the examples. == Examples == One of the original motivations for the study of residuated lattices was the lattice of (two-sided) ideals of a ring. Given a ring R, the ideals of R, denoted Id(R), forms a complete lattice with set intersection acting as the meet operation and "ideal addition" acting as the join operation. The monoid operation • is given by "ideal multiplication", and the element R of Id(R) acts as the identity for this operation. Given two ideals A and B in Id(R), the residuals are given by A / B := { r ∈ R ∣ r B ⊆ A } {\displaystyle A/B:=\{r\in R\mid rB\subseteq A\}} B ∖ A := { r ∈ R ∣ B r ⊆ A } {\displaystyle B\setminus A:=\{r\in R\mid Br\subseteq A\}} It is worth noting that {0}/B and B\{0} are respectively the left and right annihilators of B. This residuation is related to the conductor (or transporter) in commutative algebra written as (A:B)=A/B. One difference in usage is that B need not be an ideal of R: it may just be a subset. Boolean algebras and Heyting algebras are commutative residuated lattices in which x•y = x∧y (whence the unit I is the top element 1 of the algebra) and both residuals x\y and y/x are the same operation, namely implication x → y. The second example is quite general since Heyting algebras include all finite distributive lattices, as well as all chains or total orders, for example the unit interval [0,1] in the real line, or the integers and ± ∞ {\displaystyle \pm \infty } . The structure (Z, min, max, +, 0, −, −) (the integers with subtraction for both residuals) is a commutative residuated lattice such that the unit of the monoid is not the greatest element (indeed there is no least or greatest integer), and the multiplication of the monoid is not the meet operation of the lattice. In this example the inequalities are equalities because − (subtraction) is not merely the adjoint or pseudoinverse of + but the true inverse. Any totally ordered group under addition such as the rationals or the reals can be substituted for the integers in this example. The nonnegative portion of any of these examples is an example provided min and max are interchanged and − is replaced by monus, defined (in this case) so that x-y = 0 when x ≤ y and otherwise is ordinary subtraction. A more general class of examples is given by the Boolean algebra of all binary relations on a set X, namely the power set of X2, made a residuated lattice by taking the monoid multiplication • to be composition of relations and the monoid unit to be the identity relation I on X consisting of all pairs (x,x) for x in X. Given two relations R and S on X, the right residual R\S of S by R is the binary relation such that x(R\S)y holds just when for all z in X, zRx implies zSy (notice the connection with implication). The left residual is the mirror image of this: y(S/R)x holds just when for all z in X, xRz implies ySz. This can be illustrated with the binary relations < and > on {0,1} in which 0 < 1 and 1 > 0 are the only relationships that hold. Then x(>\<)y holds just when x = 1, while x()y holds just when y = 0, showing that residuation of < by > is different depending on whether we residuate on the right or the left. This difference is a consequence of the difference between <•> and >•<, where the only relationships that hold are 0(<•>)0 (since 0<1>0) and 1(>•<)1 (since 1>0<1). Had we chosen ≤ and ≥ instead of < and >, ≥\≤ and ≤/≥ would have been the same because ≤•≥ = ≥•≤, both of which always hold between all x and y (since x≤1≥y and x≥0≤y). The Boolean algebra 2Σ of all formal languages over an alphabet (set) Σ forms a residuated lattice whose monoid multiplication is language concatenation LM and whose monoid unit I is the language {ε} consisting of just the empty string ε. The right residual M\L consists of all words w over Σ such that Mw ⊆ L. The left residual L/M is the same with wM in place of Mw. The residuated lattice of all binary relations on X is finite just when X is finite, and commutative just when X has at most one element. When X is empty the algebra is the degenerate Boolean algebra in which 0 = 1 = I. The residuated lattice of all languages on Σ is commutative just when Σ has at most one letter. It is finite just when Σ is empty, consisting of the two languages 0 (the empty language {}) and the monoid unit I = {ε} = 1. The examples forming a Boolean algebra have special properties treated in the article on residuated Boolean algebras. == Residuated semilattice == A residuated semilattice is defined almost identically for residuated lattices, omitting just the meet operation ∧. Thus it is an algebraic structure L = (L, ∨, •, 1, /, \) satisfying all the residuated lattice equations as specified above except those containing an occurrence of the symbol ∧. The option of defining x ≤ y as x∧y = x is then not available, leaving on

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