Clips is a discontinued mobile video editing software application created by Apple Inc. It was released onto the iOS App Store on April 6, 2017, for free. Initially, it was only available on 64-bit devices running iOS 10.3 or later; as of version 3.1.3, it requires iOS 16.0 or later. Apple describes it as an app for "making and sharing fun videos with text, effects, graphics, and more.". Its final release was on May 9, 2024 before was removed from the App Store on October 10, 2025. == Features == After launching of the app, the user sees the view of the front-facing camera. The app allows the user to create a new clip by tapping on a red record button, or use photos or videos from the device's photo library. Once a clip is recorded, it can be added to a project timeline shown at the bottom of the screen. The user can share their project on social media platforms. The user can also add filters and effects to the project. "Live Titles" (available in several styles) can also be created by dictating to the device.
Content Threat Removal
Content Threat Removal (CTR) is a cybersecurity technology intended to defeat the threat posed by handling digital content in the cyberspace. Unlike other defenses, including antivirus software and sandboxed execution, CTR does not rely on being able to detect threats. Similar to Content Disarm and Reconstruction, CTR is designed to remove the threat without knowing whether it has done so and acts without knowing if data contains a threat or not. Detection strategies work by detecting unsafe content, and then blocking or removing that content. Content that is deemed safe is delivered to its destination. In contrast, Content Threat Removal assumes all data is hostile and delivers none of it to the destination, regardless of whether it is actually hostile. Although no data is delivered, the business information carried by the data is delivered using new data created for the purpose. == Threat == Advanced attacks continuously defeat defenses that are based on detection. These are often referred to as zero-day attacks, because as soon as they are discovered attack detection mechanisms must be updated to identify and neutralize the attack, and until they are, all systems are unprotected. These attacks succeed because attackers find new ways of evading detection. Polymorphic code can be used to evade the detection of known unsafe data and sandbox detection allows attacks to evade dynamic analysis. == Method == A Content Threat Removal defence works by intercepting data on its way to its destination. The business information carried by the data is extracted and the data is discarded. Then entirely new, clean and safe data is built to carry the information to its destination. The effect of building new data to carry the business information is that any unsafe elements of the original data are left behind and discarded. This includes executable data, macros, scripts and malformed data that trigger vulnerabilities in applications. While CTR is a form of content transformation, not all transformations provide a complete defence against the content threat. == Applicability == CTR is applicable to user-to-user traffic, such as email and chat, and machine-to-machine traffic, such as web services. Data transfers can be intercepted by in-line application layer proxies and these can transform the way information content is delivered to remove any threat. CTR works by extracting business information from data and it is not possible to extract information from executable code. This means CTR is not directly applicable to web browsing, since most web pages are code. It can, however, be applied to content that is downloaded from, and uploaded to, websites. Although most web pages cannot be transformed to render them safe, web browsing can be isolated and the remote access protocols used to reach the isolated environment can be subjected to CTR. CTR provides a solution to the problem of stegware. It naturally removes detectable steganography and eliminates symbiotic and permutation steganography through normalisation.
Degree of truth
In classical logic, propositions are typically unambiguously considered as being true or false. For instance, the proposition one is both equal and not equal to itself is regarded as simply false, being contrary to the Law of Noncontradiction; while the proposition one is equal to one is regarded as simply true, by the Law of Identity. However, some mathematicians, computer scientists, and philosophers have been attracted to the idea that a proposition might be more or less true, rather than wholly true or wholly false. Consider this pizza is hot. In mathematics, this idea can be developed in terms of fuzzy logic. In computer science, it has found application in artificial intelligence. In philosophy, the idea has proved particularly appealing in the case of vagueness. Degrees of truth is an important concept in law. The term is an older concept than conditional probability. Instead of determining the objective probability, only a subjective assessment is defined. In adjudicative processes, 'substantive truth' is distinct from 'formal legal truth' which comes in four degrees: hearsay, balance of probabilities, proven beyond reasonable doubt and absolute truth (knowledge reserved unto God).
Vehicle infrastructure integration
The Vehicle Infrastructure Integration (VII), also known as "Connected Roadways" or "vehicle-to-everything" (V2X) technology, is a United States Department of Transportation initiative that aims to improve road safety by developing technology that connects road vehicles with their environment. This development draws on several disciplines, including transport engineering, electrical engineering, automotive engineering, telematics, and computer science. Although VII specifically covers road transport, similar technologies are under development for other modes of transport. For example, airplanes may use ground-based beacons for automated guidance, allowing the autopilot to fly the plane without human intervention. == Goals == The goal of VII is to establish a communication link between vehicles (via On-Board Equipment, or OBE) and roadside infrastructure (via Roadside Equipment, or RSE) to enhance the safety, efficiency, and convenience of transportation systems. Two potential approaches are the widespread deployment of a dedicated short-range communications (DSRC) link on the 5.9GHz band, and cellular communication (C-V2X). Either of these methods would allow vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I) communication. The initiative has three priorities: Stakeholder evaluation and acceptance of the business model and its deployment schedule, Validation of the technology, with a focus on communications systems, in relation to deployment costs, and Creation of legal structures and policies, especially concerning digital privacy, to improve the system's long-term potential for success. === Safety === Current automotive safety technology relies primarily on vehicle-based radar, lidar, and sonar systems. This technology allows, for instance, a potential reduction in rear-end collisions by monitoring obstacles in front of or behind the vehicle and automatically applying the brakes when necessary. This technology, however, is limited by the sensing range of vehicle-based radar, particularly in angled and left-turn collisions, such as a motorist losing control of the vehicle during an impending head-on collision. The rear-end collisions addressed by current technology are generally less severe than angled, left-turn, or head-on collisions. VII promotes the development of a direct communication link between road vehicles and all other vehicles nearby, allowing for the exchange of information on vehicle speed and orientation or driver awareness and intent. This real-time exchange of information may enable more effective automated emergency maneuvers, such as steering, decelerating, or braking. In addition to nearby vehicle awareness, VII promotes a communication link between vehicles and roadway infrastructure. Such a link may allow for improved real-time traffic information, better queue management, and feedback to vehicles. Existing implementations of VII use vehicle-based sensors that can recognize and respond to roadway markings or signs, automatically adjusting vehicle parameters to follow the recognized instructions. However, this information may also be acquired via roadside beacons or stored in a centralized database accessible to all vehicles. === Efficiency === With a VII system in place, vehicles will be linked together. The headway between vehicles may therefore be reduced so that there is less empty space on the road, increasing the available capacity per lane. More capacity per lane will in turn imply fewer lanes in general, possibly satisfying the community's concerns about the impact of roadway widening. VII will enable precise traffic-signal coordination by tracking vehicle platoons and will benefit from accurate timing by drawing on real-time traffic data covering volume, density, and turning movements. Real-time traffic data can also be used in the design of new roadways or modification of existing systems as the data could be used to provide accurate origin-destination studies and turning-movement counts for uses in transportation forecasting and traffic operations. Such technology would also lead to improvements for transport engineers to address problems whilst reducing the cost of obtaining and compiling data. Tolling is another prospect for VII technology as it could enable roadways to be automatically tolled. Data could be collectively transmitted to road users for in-vehicle display, outlining the lowest cost, shortest distance, and/or fastest route to a destination on the basis of real-time conditions. === Existing applications === To some extent, results along these lines have been achieved in trials performed around the globe, making use of GPS, mobile phone signals, and vehicle registration plates. GPS is becoming standard in many new high-end vehicles and is an option on most new low- and mid-range vehicles. In addition, many users also have mobile phones that transmit trackable signals (and may also be GPS-enabled). Mobile phones can already be traced for purposes of emergency response. GPS and mobile phone tracking, however, do not provide fully reliable data. Furthermore, integrating mobile phones in vehicles may be prohibitively difficult. Data from mobile phones, though useful, might even increase risks to motorists as they tend to look at their phones rather than concentrate on their driving. Automatic registration plate recognition can provide large quantities of data, but continuously tracking a vehicle through a corridor is a difficult task with existing technology. Today's equipment is designed for data acquisition and functions such as enforcement and tolling, not for returning data to vehicles or motorists for response. GPS will nevertheless be one of the key components in VII systems. == Limitations == === Privacy === VII architecture is designed to prevent identification of individual vehicles, with all data exchange between the vehicle and the system occurring anonymously. Exchanges between the vehicles and third parties such as OEMs and toll collectors will occur, but the network traffic will be sent via encrypted tunnels and will therefore not be decipherable by the VII system. Data sharing with law enforcement or Homeland Security was not included in system design as of 2006. === Technical issues === ==== Coordination ==== A major issue facing the deployment of VII is the problem of how to set up the system initially. The costs associated with installing the technology in vehicles and providing communications and power at every intersection are significant. ==== Maintenance ==== Another factor for consideration in regard to the technology's distribution is how to update and maintain the units. Traffic systems are highly dynamic, with new traffic controls implemented every day and roadways constructed or repaired every year. The vehicle-based option could be updated via the internet (preferably wireless) but may subsequently require all users to have access to internet technology. Alternatively, if receivers were placed in all vehicles and the VII system was primarily located along the roadside, information could be stored in a centralized database. This would allow the agency responsible to issue updates at any time. These would then be disseminated to the roadside units for passing motorists. Operationally, this method is currently considered to provide the greatest effectiveness but at a high cost to the authorities. ==== Security ==== Security of the units is another concern, especially in light of the public acceptance issue. Criminals could tamper, remove, or destroy VII units regardless of whether they are installed inside vehicles or along the roadside. Magnets, electric shocks, and malicious software (viruses, hacking, or jamming) could be used to damage VII systems – regardless of whether units are located inside vehicle or along the roadside. == Recent developments == Much of the current research and experimentation is conducted in the United States where coordination is ensured through the Vehicle Infrastructure Integration Consortium; consisting of automobile manufacturers (Ford, General Motors, Daimler Chrysler, Toyota, Nissan, Honda, Volkswagen, BMW), IT suppliers, U.S. Federal and state transportation departments, and professional associations. Trialing is taking place in Michigan and California. The specific applications now being developed under the U.S. initiative are: Warning drivers of unsafe conditions or imminent collisions. Warning drivers if they are about to run off the road or speed around a curve too fast. Informing system operators of real-time congestion, weather conditions and incidents. Providing operators with information on corridor capacity for real-time management, planning and provision of corridor-wide advisories to drivers. In mid-2007, a VII environment covering some 20 square miles (52 km2) near Detroit was used to test 20 prototype VII applications. Several automobile manufacturers are also conducting their own VII research and triali
Fragile Dreams: Farewell Ruins of the Moon
Fragile Dreams: Farewell Ruins of the Moon (フラジール ～さよなら月の廃墟～, Furajīru: Sayonara Tsuki no Haikyo; known in Japan as Fragile) is an action role-playing game for the Wii developed by Namco Bandai Games in co-operation with Tri-Crescendo. The game was released by Namco Bandai Games in Japan on January 22, 2009. It was later published by Xseed Games in North America on March 16, 2010, and in Europe by Rising Star Games on March 19, 2010, followed by its release in Australia on April 1, 2010. == Gameplay == In Fragile Dreams, the player character, Seto, must traverse the ruins of Tokyo and the surrounding areas, fighting off ghosts that lurk within these ruins. The game's heads-up display includes a mini-map and HP gauge for Seto's location and health, respectively. Seto will fall unconscious if his HP reaches zero, resulting in a game over. The player controls Seto from a third-person perspective with the Wii Remote and Nunchuk. Seto can use his flashlight (controlled by the Wii Remote pointer) to illuminate his surroundings or solve puzzles and interact with the environment. When searching for certain objectives or hidden enemies, pointing Seto's light in their direction picks up and plays their sounds through the Wii Remote's mini speaker. The Wii Nunchuk, meanwhile, directly controls Seto's movement: aside of basic movement, he can crouch to hide and crawl through small spaces. Seto will often come across damaged floors, which require slow movement (and for heavily damaged floors, crouching) to cross without falling through. As Seto, the player can use weapons found throughout the world to fight off ghosts, ranging from slingshots and golf clubs to crossbows and katanas. Each weapon can only take a certain amount of use: once a weapon reaches its limit, it will break after battle. The player can also find other usable and collectable items in the field, marked with fireflies. The player can only save their game by resting at small fire pits scattered throughout the world: used fire pits are marked with a bonfire. The player can also examine and identify Mystery Items, organize their inventory, as well as after encountering the Merchant, buy and sell items. As stated by the producer of the game, Kentarō Kawashima, Fragile Dreams is not strictly a survival horror: rather, its story focuses on human drama. In Fragile Dreams, aside of the main story, the player can find and examine objects and graffiti throughout the world. Objects called memory items (ranging from origami and stones to cell phones and books) hold the memories of their former owners (only accessible at bonfires), while the graffiti contains messages only seen by pointing at them in first-person. By examining these messages, the player can piece together hints to the game's backstory. == Story == === Setting and characters === Fragile Dreams is set in a post-apocalyptic version of Earth in the near-future. Almost all the world's population has vanished, leaving the surviving buildings and structures abandoned. The game is set in and near the ruins of Tokyo, Japan, where the event that nearly wiped out humanity may have originated. The protagonist, Seto, is a 15-year-old boy who searches the world for other living humans. He encounters Ren, a silver-haired girl who often leaves behind large, cryptic drawings. Other characters include: Sai, the ghost of a young woman; Crow, a mischievous and straightforward amnesiac boy; Personal Frame (P.F.), a portable computer who loves having conversations more than anything else; Chiyo, the ghost of a little girl; and the Merchant, a mysterious yet merry man who trades various goods. The game's host of enemies mainly consist of ghosts, but also include humanoid robots and security proxies. The main antagonist, Shin, is the AI of a scientist who considers speech to be an inferior means of communication. Various memory items include a greater set of characters, each giving hints to the game's backstory. === Plot === At the end of Seto's fifteenth summer, his grandfather dies. Seto buries him in front of their home, an old observatory, and that from then on he became "truly alone". At night, he searches for anything the old man had left for him and discovers a letter, along with a strange blue stone in a locket. Suddenly, a mask-like ghost appears and attacks Seto. After driving the creature off, Seto reads the old man's letter, who tells him to "reach a tall red tower" east of the observatory, where he might find other survivors. After departing for the tower, Seto reaches an old subway entrance in the Azabudai district and finds Ren sitting on a collapsed pillar, singing to the stars. He accidentally startles her and the frightened Ren flees into the subway station: getting over the shock of meeting another person, Seto follows her. While searching the station, he discovers a Personal Frame, who guides him towards Ren. Unfortunately, just as they reach the exit, P.F.'s battery dies out: Seto buries the device, keeping a screw from it in his locket. From the underground, Seto finds himself at an abandoned amusement park and encounters Crow, who steals Seto's locket. After a long chase across the park and another encounter with the masked ghost, Crow returns Seto's locket and directs him to a hotel nearby, where he saw a girl who might know something about Ren. Crow also gives Seto his skull ring to keep in his locket and kisses him. At the hotel, Seto encounters Sai and fights the masked ghost again. After laying to rest the spirit of an old woman named Chiyo, the two discover Ren's drawings by a sewer. Returning to the underground, Seto and Sai find themselves at a hydropower dam. While searching for Ren, Seto discovers that Crow is actually a robot, but his battery begins to fail and Seto mourns for him as he "die[s]". Finally, they encounter Ren in a cell: although glad to see him again, Ren runs off after Shin calls. Sai explains to Seto that most of humanity died because of a "human empathy expansion project" called Glass Cage. The project was meant to make human thoughts transparent, meaning that no one would need words to communicate. However, after Glass Cage activated, people who went to sleep never woke up again. Sai reveals that she was Glass Cage's first catalyst: this time, Shin intends to use Ren as the catalyst. After exiting the dam, a demolition crane attempts to destroy it. Hearing both Shin's and the masked ghost's voices from the crane — saying, "Any threat to the project must be eliminated." — the player realizes both are manifestations of Glass Cage. After Seto destroys the crane, Sai leads him to the facility where Ren was taken to. Entering the laboratory, Seto and Sai are confronted by Shin, who coldly dismisses Sai's attempts at reasoning with him and is adamant about proceeding with his plans. As they traverse the laboratory, they overhear a voice announcing "Glass Cage Launch Preparations Complete", strengthening their resolve to save Ren. Making it into the room where Ren is being held, Shin tells them of his intention to use Glass Cage to "obliterate corporeal beings". After Seto defeats him, Shin disappears and Seto releases Ren from the device holding her. Their reunion is cut short as Sai tells them that the backup system has "finished copying her psyche to the AI", allowing Glass Cage to proceed. Ren reveals Shin has escaped to the top of the Tokyo Tower and Seto asks Ren to wait at the base of the tower and for Sai to accompany her. On his way up the tower, Seto hears the voices of P.F., Chiyo and Crow wishing him luck. He confronts and defeats Shin a second time, who reveals his motivations: he had secretly used himself as the first test subject of the human empathy expansion project and gained the ability to hear the thoughts of those around him. Despite his initial belief in the project as a way for humans to empathize with one another, all he heard around him was "jealousy and contempt" and he soon grew disillusioned with the world as even his parents turned against him. Believing no person loved him, Shin wants to put an end to humanity. His words meet with a vehement response from Sai, as she tells him that she loves him, having developed those feelings while she was the catalyst and all she ever wanted was to be part of his life. Hearing this, Shin finds peace, tossing the AI mainframe away so Glass Cage can never be reactivated and vanishes together with Sai, hand-in-hand, after thanking Seto. Descending from the tower, Seto finally learns Ren's name and they resolve to look for other survivors together. == Development == Fragile Dreams was developed by the team at Namco Bandai Games. Director and producer Kentarō Kawashima came up with the concept for the game in 2003, before the Wii console was revealed. When the Wii was unveiled, it became the obvious choice as the game's platform as the Wii remote could be used to control the flashlight. Kawashima wrote the main scenario for the title, w
CPT Corporation
CPT Corporation was founded in 1971 by Dean Scheff in Minneapolis, Minnesota, with co-founders James Wienhold and Richard Eichhorn. CPT first designed, manufactured, and marketed the CPT 4200, a dual-cassette-tape machine that controlled a modified IBM Selectric typewriter to support text editing and word processing. The CPT 4200 was followed in 1976 by the CPT VM (Visual Memory), a partial-page display-screen dual-cassette-tape unit, and shortly thereafter by the CPT 8000, a full-page display dual-diskette desktop microcomputer that drove stand-alone daisy wheel printers. Subsequent products included (1) variants on the 8000 series; (2) the CPT 6000 series, which had a lower capacity, smaller screen, and was less expensive; (3) the CPT 9000 series, which had a larger capacity and could run IBM personal computer software; (4) the CPT Phoenix series, which had a graphical capabilities; (5) CPT PT, a software-only reduced version that ran on IBM personal computers and clones; and (6) other related products. The CPT logo—originally three letters chosen to sound well together—began to be taken as an acronym for "cassette powered typewriting," and subsequently for "computer processed text," and numerous other variants. Major competition was IBM, Wang, Lanier, Xerox, and other word processing vendors. CPT Corporation was fifth in size among Minnesota-based top high-tech companies, after 3M, Honeywell, Control Data, and Medtronic. Corporate revenues grew to approximately a quarter-billion dollars per year in the mid-1980s, then declined with the proliferation of personal computers. CPT ultimately ceased major manufacturing late in the 20th century. == Selected products == === Cassette based === The CPT 4200 was a dual-cassette-tape unit with a small built-in keyboard that controlled a modified IBM Selectric typewriter. Keystrokes entered on the typewriter appeared on the paper as they were recorded on the output cassette, which formed a magnetic replica of the characters printed on the page. That output cassette could later be used as an input cassette, where it would be played back to the typewriter along with new keystrokes to accomplish text editing. The keyboard of the CPT 4200 had action keys for "skip", "read" and "stop", mode keys for "word", "line", "paragraph," and "page." Pressing "read" transferred a word, line, paragraph, or page (depending on which mode key had been selected) from the input tape to both the typewriter and the output tape. Line boundaries (aka printer margins) recorded on the input tape were ignored or retained depending on whether or not the "adjust" key had been selected. Alternatively, pressing "skip" moved past the corresponding amount of text on the input tape without sending it to the typewriter or to the output tape. The Selectric's keyboard was active for any new typing, which would appear on the paper and transferred to the output tape. Thus a document was edited by reading back those parts of the text to be retained and skipping those parts to be discarded, with new typing added from the Selectric's keyboard. Price: approx. $5000, 1980-era values. The CPT Communicator was an add-on to the CPT 4200 that allowed data to be transferred from one text-editing machine to another, or between a text-editing machine and a remote computer, via phone lines. Price: not available. === Microprocessor based === ==== CPT 8000 series ==== The CPT 8000 was the company's first microcomputer product, exhibited in spring of 1976. It was a self-contained desktop machine with two 8-inch floppy diskette drives, a movable keyboard, and a full-page vertically oriented CRT display simulating paper with black characters on a white background, for a wysiwyg view of text on paper. It was promoted as familiar and easy to use for those experienced with typewriters. A keyboard with a large set of extra keys made operating the 8000 quite easy even for people without any computer skills or background. IN, OUT, PRINT, OOPS OOPS was changed thinking it was insulting to the buyer to assume they would ever make an error. The CPT 8000 was designed to show a full page of text with a static line showing the margin and tab stops. An additional line would display status or error messages with a times square like display. The times square error and status messages were very well done, "The printer needs a new ribbon" rather than "ERROR 034892". The text page could both smooth pan and scroll by the hardware in the display board and nothing quite like it existed for a very long time. The 8000 ran its own multitasking hardware interrupt-driven operating system but it also ran CP/M quite well. So unlike other companies that sold Wordprocessor only systems, CPT had a system that could run any of the many popular CP/M applications. Using the CP/M OS users could develop Fortran, CBasic, Cobol and other language's programs. The 8000 used Intel's 8080 microprocessor. The display board was bleeding-edge, high-speed logic. The parts available at this time were pushed to their limits to provide the speed needed to display this much text. There were times that batches of parts from one manufacturer simply could not be clocked as fast as the 8000 display required. Memory was initially 64K, but larger boards of 128K were most common then later 256K were offered. The 8080 accessed this additional RAM by running a custom page flipping circuit. The 8000 was originally priced at $8000 and its daisy wheel printer an additional $8000. The model number having been confused with the price at its first appearance at the Hanover fair. An RS-232 serial communication option was available for the 8000 series that allowed the electronic transfer of documents. One very popular use of this was to access the Westlaw system. A tempest approved version of the 8000 was developed that was RF tight with nothing being emitted that could be monitored or spied on. === Storage Systems === ==== CPT WordPak ==== The CPT WordPak series was CPT's first external document storage system that enabled multiple 8000 series workstations to store documents in an electronic filing cabinet. Prior to WordPak, all documents were stored on removable 8-inch floppy diskettes. Sharing documents involved handing off the original disk, or copying the document to a second disk and 'sneaker-net-ing' (walking it over) to the second 8000. But this resulted in two copies of the document, one at each workstation. A circuit board with a proprietary cable connector was installed in the 8000/6000 family of "workstations" and connected to the WordPak by a multi-conductor cable. WordPak 1 consisted of a single Shugart Associates SA4000 14"-diameter hard disk with a capacity of 30 megabytes. WordPak 2 added a 2nd drive for a total of 60 megabytes. ==== CPT SRS 45 ==== The CPT SRS 45 was what would now be called a server (quite likely the first of its kind) but in practice was much more. It was maybe the worlds easiest networking shared resource system. It combined a ZIP drive for backup and hard disk(s) that would be shared simultaneously by up to eight CPT machines that had the PC AT bus. The primary person responsible for its development was Bill Davidson whose wife Cheryl was responsible for bringing up CP/M, MP/M and other Digital Research products running on the Phoenix. The brilliance of the system were the networking cards that plugged into the individual machines. These used the 55AA installable driver of the IBM BIOS to simply add the zip and hard disk drives to each computers drives list. So a system that started with floppy drives A and B and a C hard disk on the machine would have the SRS 45 drives added as drives D (E, F depending on the number of hard disk) and Z for the zip drive. Sharing (avoiding writing to the same file at the same time) was handled by simply assigning parts of the drives for individuals and other directories for shared use. No "driver" software was needed. You simply plugged in the networking card and your machine had additional drives that were internal to the SRS45. This approach was far ahead of its time and sadly never recognized for its brilliance. The SRS45 as were all CPT machines not just dedicated Word Processors. === Personal-computer based === ==== CPT PT software ==== CPT PT was a reduced a version of the software that ran under MS-DOS as an application on IBM PC compatible computers. The corporation intended it as a bridge to allow data to flow in and out of personal computer packages, as well as providing a personal-computer word processing application for those familiar with standalone CPT equipment or who preferred the CPT style of dual-window text editing. Price: approx. $200, 1980-era values. ==== CPT Genius Display ==== The Genius display was a stand-alone, vertically-oriented (portrait) configuration monochrome grey-scale CRT monitor unit and an IBM PC form factor display card to allow high-resolution, full-page text & graphics on IBM PC compatible computers.
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