In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human–machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. A recent effort within this field is making these tools use artificial intelligence to automate the formalization of ordinary mathematics. == Automated proof checking == Automated proof checking is the process of using software for checking proofs for correctness. It is one of the most developed fields in automated reasoning. Automated proof checking differs from automated theorem proving in that automated proof checking simply mechanically checks the formal workings of an existing proof, instead of trying to develop new proofs or theorems itself. Because of this, the task of automated proof verification is much simpler than that of automated theorem proving, allowing automated proof checking software to be much simpler than automated theorem proving software. Because of this small size, some automated proof checking systems can have less than a thousand lines of core code, and are thus themselves amenable to both hand-checking and automated software verification. The Mizar system, HOL Light, and Metamath are examples of automated proof checking systems. Automated proof checking can be done either as a batch operation, or interactively, as part of an interactive theorem proving system. == History == Automath, which was developed by Nicolaas Govert de Bruijn starting in 1967, is often considered the first proof checker and the first system to utilize the Curry–Howard correspondence between programs and proofs. Automath was used by L.S. van Benthem Jutting in 1977 to formalize Landau's Foundations of Analysis, which was the first formalization of the real numbers. In 1973, Robert Boyer and J Moore published Proving Theorems about LISP Functions which aimed to verify programs, not mathematics. Their theorem prover is now known as ACL2. In the 1970s, Edinburgh LCF introduced the idea of using a functional programming language as the metalanguage for a theorem prover, and led to the HOL family of proof assistants. The 1990s saw the rise of Rocq, (then known as Coq), which has been used for many large-scale formalization projects. Since the late 2010s, Lean, a proof assistant strongly influenced by Rocq, has become another popular choice, especially for formalizing mathematics. == System comparison == ACL2 – a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition. HOL theorem provers – A family of tools ultimately derived from the LCF theorem prover. In these systems, the logical core is a library of their programming language. Theorems represent new elements of the language and can only be introduced via "strategies" which guarantee logical correctness. Strategy composition gives users the ability to produce significant proofs with relatively few interactions with the system. Members of the family include: HOL4 – The "primary descendant", still under active development. Support for both Moscow ML and Poly/ML. Has a BSD-style license. HOL Light – A thriving "minimalist fork". OCaml based. ProofPower – Went proprietary, then returned to open source. Based on Standard ML. IMPS, An Interactive Mathematical Proof System. Isabelle is an interactive theorem prover where other systems can be encoded. Isabelle/HOL is its most popular instance, whose foundation is close to that of the HOL prover. Other instances include Isabelle/ZF and Isabelle/FOL. The main code-base is BSD-licensed, but the Isabelle distribution bundles many add-on tools with different licenses. Jape – Java based. Lean is both an interactive theorem prover and a functional, dependently-typed programming language. It is based on the calculus of inductive constructions with non-cumulative universes. Since version 4 (released in 2023), it is self-hosting. It can be used to formalise mathematics (and has a large, coherent library for formal mathematics), but also for software and hardware verification. LEGO Matita – A light system based on the calculus of inductive constructions. MINLOG – A proof assistant based on first-order minimal logic. Mizar – A proof assistant based on first-order logic, in a natural deduction style, and Tarski–Grothendieck set theory. PhoX – A proof assistant based on higher-order logic which is eXtensible. Prototype Verification System (PVS) – a proof language and system based on higher-order logic. Rocq (formerly named Coq) – A popular interactive theorem prover based on the calculus of inductive constructions. Theorem Proving System (TPS) and ETPS – Interactive theorem provers also based on simply typed lambda calculus, but based on an independent formulation of the logical theory and independent implementation. == User interfaces == A commonly used front-end for proof assistants was the Emacs-based Proof General, developed at the University of Edinburgh. Nowadays, many provers include their own editor. Rocq includes RocqIDE, which is based on OCaml/Gtk. Isabelle includes Isabelle/jEdit, which is based on jEdit and the Isabelle/Scala infrastructure for document-oriented proof processing. More recently, Visual Studio Code extensions have been developed for Rocq, Isabelle by Makarius Wenzel, and for Lean 4 by the leanprover developers. == Formalization extent == Freek Wiedijk has been keeping a ranking of proof assistants by the amount of formalized theorems out of a list of 100 well-known theorems. As of September 2025, only six systems have formalized proofs of more than 70% of the theorems, namely Isabelle, HOL Light, Lean, Rocq, Metamath and Mizar. == Notable formalized proofs == The following is a list of notable proofs that have been formalized within proof assistants.
Score bug
A score bug is a digital on-screen graphic which is displayed in a broadcast of a sporting event, displaying the current score and other statistics. It is similar in function to a scoreboard, and is usually placed at either the top or lower third of the television screen. == History == The concept of a persistent score bug was devised by Sky Sports head David Hill, who was dissatisfied over having to wait to see what the score was after tuning into a football match in-progress. The score bug was introduced when Sky launched its coverage of the then newly-formed English Premier League in August 1992. Hill's boss repeatedly demanded that the graphic be removed, describing it as the "stupidest thing [he] had ever seen". Hill defied the boss's demands and kept the graphic in place. ITV introduced a score bug at the start of the 1993–94 football season, and the BBC introduced a score bug towards the end of 1993. The concept was introduced to the United States by ABC Sports and ESPN during coverage of the 1994 FIFA World Cup. Their justification for the graphic was to provide a location for a rotating series of sponsor logos, in order to allow matches to air without commercial interruption. With the acquisition of rights to the National Football League (NFL) by BSkyB's American sibling Fox (a fellow venture of Rupert Murdoch), Hill became the first president of Fox Sports. Under Hill's leadership, Fox introduced a version of the score bug branded as the "Fox Box", which was part of its inaugural season of NFL coverage in 1994. Variety criticized it as an "annoying see-through clock and score graphic" and expressed concern for people "who actually watched the beginning of the game and would rather have their screen clear of graphics". Hill even received a death threat from an irate viewer, with a specific emphasis on him being a "foreigner", but the score bug soon became a ubiquitous feature for American football broadcasts, along with almost all American sports broadcasts in the years that followed. Dick Ebersol of NBC Sports initially opposed the idea of a score bug, as he thought that fans would dislike seeing more graphics on the screen and would change the channel from blowout games if the score was constantly being displayed. Since the 2010s, the on-air design and positioning of some score bugs have been influenced by the needs of Internet video (especially when viewing an event on devices with smaller screens), including bugs noticeably larger than prior iterations designed with television viewing in mind, or designs primarily kept towards the bottom-center of the screen (easing the ability for the bug to remain visible when highlights are cropped for square videos posted on social media). == Details == Score bugs used in team sports typically include the names of both teams, an abbreviation of the team's name, and/or the team's logo; for individual sports, they include the names of individual competitors. In sports where a game clock or playing periods are used, those are generally also displayed as part of the score bug. Some broadcasts also include teams' win-loss records. In 2024, ESPN experimented with adding a persistent win probability meter to its bug in Major League Baseball, which was based on input from its statisticians. === Variations === In addition to the above information, score bugs in some sports include additional information: In baseball, score bugs display the current inning, number of outs, the pitch clock if applicable, and a graphic displaying which bases are occupied; and usually include names of the current pitcher and batter, the pitcher's pitch count, and the number of balls and strikes accrued by the batter. In basketball, score bugs generally include the shot clock, the number of fouls accrued by each team, and whether a team is in the bonus. In cricket, score bugs often take the form of larger dashboards across the bottom of the screen, displaying the current team up and their number of runs, wickets, and overs, a display showing the runs scored and number of balls faced by the current batting partnership, and statistics for the opposing team's bowler (including the number of wickets scored and runs given up). In American football, score bugs usually include the play clock and the down and distance of the current play; they also incorporate graphics indicating when a penalty flag has been thrown. In ice hockey, score bugs display when a penalty or power play is in effect, and often include the number of shots on goal accrued by each team. In golf, Fox popularized the display of a persistent leaderboard graphic in the bottom-right of the screen, usually displaying the top 5. ==== Racing ==== Telecasts of automobile races often include a score bug with the current positions of participants, statistics such as distance behind the leader, and the remaining distance or number of laps. In the mid-2010s, NASCAR broadcasters such as Fox began to transition from horizontal tickers to vertical leaderboards (also referred to as "pylons", in reference to the physical scoring pylons at). The CW differentiated itself by using a horizontal display that divides the field into multiple columns along the bottom of the screen.
Rifts (role-playing game)
Rifts is a multi-genre role-playing game created by Kevin Siembieda in August 1990 and published continuously by Palladium Books since then. It takes place in a post-apocalyptic future, deriving elements from cyberpunk, science fiction, fantasy, horror, western, mythology and many other genres. Rifts serves as a cross-over environment for a variety of other Palladium games with different universes connected through "rifts" on Earth that lead to different spaces, times, and realities that Palladium calls the "Rifts Megaverse". Rifts describes itself as an "advanced" role-playing game and not an introduction for those new to the concept. Palladium continues to publish books for the Rifts series, with about 80 books published between 1990 and 2011. Rifts Ultimate Edition was released in August 2005 and designed to update the game with Palladium's incremental changes to its system, changes in the game world, and additional information and character types. The web site is quick to point out that this is not a second edition but an improvement and expansion of the original role playing game. == Background == The RPG had the tentative title Boomers, named after the original name for the Glitter Boy power armor until Kevin Siembieda changed the name after finding out it was in use for Bubblegum Crisis. == Setting == The Rifts world is Earth, but hundreds of years into the future. Ley lines, lines of magic energy, criss-cross the earth forming supernatural geographic areas such as the Bermuda Triangle. Points where Ley Lines intersect, called a nexus, are places of powerful magic, such as the Pyramids of Giza and Stonehenge. If a Ley Line nexus energy surges or is purposely activated, the fabric of space and time can be torn, creating a rift - a hole in space-time leading to another place, time, or dimension. Ley lines contain magical energy called Potential Psychic Energy (PPE), which is found in various places, objects, and animals and is particularly strong in children. An adult's level of PPE can vary based on other factors. PPE also allows Psionics which uses energy known as Inner Strength Points or ISP. Psychic phenomenon (more commonly called psionics) can also vary from individuals, ranging from none at all to Master level abilities. Psychic abilities can manifest in virtually any way imaginable. Some psychics develop differently, such as psi-stalkers; human mutants that feed on psychic energy. === Earth === Rifts begins with two future-historical premises: first, a golden age of humanity occurs, with tremendous advances in science, technology, military, and society. Humanity as a whole is at peace as a majority of Earth's nations decide to cease world war and begin to share ideas and technology freely. Much of the Solar System is conquered, humanity's wars will end, and harmony will reign. This golden age is followed by an unknown cause near the winter solstice and a rare planetary alignment, causing a disaster that cascades into tremendous destruction via a ripple effect. The cataclysm begins with unprecedented storms, earthquakes, tsunamis, and volcanic eruptions, which kill millions of people. The Ley Line networks that crisscross the globe are energized, causing rifts to open both on Earth and throughout the Megaverse. For hundreds of years after the holocaust, many creatures, both mythical beasts and aliens, come through the Rifts to wreak havoc. The old world gone, a new Dark Age dawns and humanity's shrinking population is reduced, due to catastrophe and domestic failure, immeasurably. This period is covered in Palladium's Rifts Chaos Earth spin-off series. Rifts initially takes place in 101 P.A. (equivalent to the year 2387) 289 years after this event. The "Post-Apocalypse" calendar was established by the formation of the Coalition States in 2286. By this time, most of the disasters have quieted down, though Earth is still bathed in PPE. The planet's mystical energy has attracted aliens from other dimensions, who continue to arrive through the Rifts both accidentally and deliberately. The humanoid creatures that arrive on Earth are referred to as Dimensional Beings (called D-Bees). Some resemble familiar fantasy races, such as elves and dwarfs, while others were created specifically for the game setting. Non-humanoid creatures have also arrived, including monstrous creatures and mystical demons. To cope with these natural, supernatural, and alien menaces, the human race has adapted in a variety of ways, many of them borrowed from the technological developments of the lost Golden Age. Powered armor suits and giant vehicles are frequently used to combat the dangers of Rifts, but more invasive augmentation is common. This has three basic categories: "Juicers" augment themselves chemically, the "Borgs" augment themselves mechanically, and "Crazies" use performance-enhancing brain implants. All such augmentations boost strength, speed, endurance, and dexterity to superhuman levels. However, all come at great cost. Chemicals cause the body to wear out faster, decreasing life span to a few years. Mechanical Borg augmentation causes a loss of humanity when those with multiple limb and organ replacements become more machine than human. Brain implants cause mental instability ranging from mild phobias to crippling neurosis or psychosis. ==== North America ==== The strongest power in North America is the Coalition States (CS), which is based in the arcological city of Chi-Town and lays claim to northern Illinois, all of Iowa, the Texas Panhandle, Missouri, and the eastern half of Ontario, Canada. The second greatest power is Free Quebec, a former Coalition State that seceded following a civil war with the other Coalition States. Mexico is ruled by a group of vampire kingdoms, who treat humans as little more than food. North of the Rio Grande, west of Texas and roaming most of the American Southwest are large nomadic bands/tribes of bandits who collectively form the Pecos Empire, consisting of El Paso, Los Alamos, and Houstown. Much of the western United States has more or less willingly reverted to a mix of modern and past technology akin to the Wild West. The Royal Canadian Mounted Police managed to survive the great cataclysm, though Canada itself did not. The Mounties have become an independent law enforcement force called the Tundra Rangers, patrolling the northern wilderness. The Midwest, both upper and central, is home to most of North America's population. The Manistique Imperium and Northern Gun in Michigan's Upper Peninsula, both Coalition allies, are among the largest weapons manufacturing areas on the continent. New Lazlo is one of the largest cities in Michigan's southern portion. Chillicothe in Missouri is a large supplier of Coalition food processing and growing. Missouri's southern half, home to the city-states of Whykin (Poplar Bluff) and Kingsdale (West Plains) are in constant opposition to the CS and claim independence. Arkansas is home to the independent CS ally El Dorado. Southern Illinois and the Ohio Valley is home to the Federation of Magic. Also in the Ohio Valley is Psyscape, a city-state founded by psychics. Tolkeen was a major city in the former Minneapolis region in early Rifts books; the city welcomed users of magic. A military campaign made by the Coalition States (which is the primary event of 109 PA) resulted in the magic-user kingdom being wiped off the map. In the Northeast, the city-state of Lazlo, named after supernatural researcher and writer Victor Lazlo, was built upon the ruins of Toronto. This major center of civilization is well known as a melting pot of humans, D-Bees and other beings, and is the home of Techno-Wizardry. Mad Haven is the name given to the ruins of Manhattan; tectonic forces during the cataclysm have moved it into the coast, creating a peninsula. It is seen by most denizens of Rifts Earth as a refuge of demons and madness. ==== South America ==== The return of Atlantis caused the Amazon River basin to flood most of western South America, giving it the nickname The Land of a Thousand Islands. The Empire of the Sun, consisting of Cuzco, Nazca, Arequipa and Lima, created a wide range of technology and magic, including magic derived from the Nazca lines. In Argentina, the Silver River Republics of Cordoba (the South American Chi-Town), Santiago (one of the most tolerant human nations on Rifts Earth), Achilles (a nation founded by mutants), and New Babylon, a nation where humans and aliens coexist) have thrived and created nations whose strength rivals that of the CS. In Bolivia, freed Human and D-Bees formed the Megaversal Legion: a mercenary company with one of the highest levels of technology on Rifts Earth. ==== Europe ==== England has become a vast wilderness again, broken up by the occasional giant Millennium Tree or feudal kingdom, complete with a New Camelot and a new King Arthur, partially being manipulated by an alien intelligence disguised as Merlin. Also the magic of
Deepfake pornography
Deepfake pornography is a form of non-consensual AI pornography created by altering existing photographs or videos using deepfake technology to modify the appearance of the participants. The use of deepfake pornography has sparked controversy because it involves the making and sharing of realistic videos featuring non-consenting individuals and is sometimes used for revenge porn. Many countries have criminalized this "new voyeurism" through legislative measures and technological solutions. == History == The term "deepfake" was coined in 2017 on a Reddit forum where users shared altered pornographic videos created using machine learning algorithms. It is a combination of the word "deep learning", which refers to the program used to create the videos, and "fake" meaning the videos are not real. Deepfake pornography was originally created on a small individual scale using a combination of machine learning algorithms, computer vision techniques, and AI software. The process began by gathering a large amount of source material (including both images and videos) of a person's face, and then using a deep learning model to train a Generative Adversarial Network to create a fake video that convincingly swaps the face of the source material onto the body of a pornographic performer. However, the production process has significantly evolved since 2018, with the advent of several public apps that have largely automated the process. While several AI "nudification" apps emerged on mainstream platforms like Google Play and the Apple App Store around 2023, major tech storefronts have since implemented stricter policies and automated detection to ban such software. Consequently, the proliferation of non-consensual deepfake pornography has largely shifted to decentralized websites, specialized online forums, and third-party messaging bot ecosystems. Deepfake pornography is sometimes confused with fake nude photography, but the two are mostly different. Fake nude photography typically uses non-sexual images and merely makes it appear that the people in them are nude. == Notable cases == Deepfake technology has been used to create non-consensual and pornographic images and videos of famous women. One of the earliest examples occurred in 2017 when a deepfake pornographic video of Gal Gadot was created by a Reddit user and quickly spread online. Since then, there have been numerous instances of similar deepfake content targeting other female celebrities, such as Emma Watson, Natalie Portman, and Scarlett Johansson. Johansson spoke publicly on the issue in December 2018, condemning the practice but also refusing legal action because she views the harassment as inevitable. === Rana Ayyub === In 2018, Rana Ayyub, an Indian investigative journalist, was the target of an online hate campaign stemming from her condemnation of the Indian government, specifically her speaking out against the rape of an eight-year-old Kashmiri girl. Ayyub was bombarded with rape and death threats, and had a doctored pornographic video of her circulated online. In a Huffington Post article, Ayyub discussed the long-lasting psychological and social effects this experience has had on her. She explained that she continued to struggle with her mental health and how the images and videos continued to resurface whenever she took a high-profile case. === Atrioc controversy === In 2023, Twitch streamer Atrioc stirred controversy when he accidentally revealed deepfake pornographic material featuring female Twitch streamers while on live. The influencer has since admitted to paying for AI generated porn, and apologized to the women and his fans. === Taylor Swift === In January 2024, AI-generated sexually explicit images of American singer Taylor Swift were posted on X (formerly Twitter), and spread to other platforms such as Facebook, Reddit and Instagram. One tweet with the images was viewed over 45 million times before being removed. A report from 404 Media found that the images appeared to have originated from a Telegram group, whose members used tools such as Microsoft Designer to generate the images, using misspellings and keyword hacks to work around Designer's content filters. After the material was posted, Swift's fans posted concert footage and images to bury the deepfake images, and reported the accounts posting the deepfakes. Searches for Swift's name were temporarily disabled on X, returning an error message instead. Graphika, a disinformation research firm, traced the creation of the images back to a 4chan community. A source close to Swift told the Daily Mail that she would be considering legal action, saying, "Whether or not legal action will be taken is being decided, but there is one thing that is clear: These fake AI-generated images are abusive, offensive, exploitative, and done without Taylor's consent and/or knowledge." The controversy drew condemnation from White House Press Secretary Karine Jean-Pierre, Microsoft CEO Satya Nadella, the Rape, Abuse & Incest National Network, and SAG-AFTRA. Several US politicians called for federal legislation against deepfake pornography. Later in the month, US senators Dick Durbin, Lindsey Graham, Amy Klobuchar and Josh Hawley introduced a bipartisan bill that would allow victims to sue individuals who produced or possessed "digital forgeries" with intent to distribute, or those who received the material knowing it was made non-consensually. === 2024 Telegram deepfake scandal === It emerged in South Korea in August 2024, that many teachers and female students were victims of deepfake images created by users who utilized AI technology. Journalist Ko Narin of The Hankyoreh uncovered the deepfake images through Telegram chats. On Telegram, group chats were created specifically for image-based sexual abuse of women, including middle and high school students, teachers, and even family members. Women with photos on social media platforms like KakaoTalk, Instagram, and Facebook are often targeted as well. Perpetrators use AI bots to generate fake images, which are then sold or widely shared, along with the victims' social media accounts, phone numbers, and KakaoTalk usernames. One Telegram group reportedly drew around 220,000 members, according to a Guardian report. Investigations revealed numerous chat groups on Telegram where users, mainly teenagers, create and share explicit deepfake images of classmates and teachers. The issue came in the wake of a troubling history of digital sex crimes, notably the notorious Nth Room case in 2019. The Korean Teachers Union estimated that more than 200 schools had been affected by these incidents. Activists called for a "national emergency" declaration to address the problem. South Korean police reported over 800 deepfake sex crime cases by the end of September 2024, a stark rise from just 156 cases in 2021, with most victims and offenders being teenagers. On September 21, 6,000 people gathered at Marronnier Park in northeastern Seoul to demand stronger legal action against deepfake crimes targeting women. On September 26, following widespread outrage over the Telegram scandal, South Korean lawmakers passed a bill criminalizing the possession or viewing of sexually explicit deepfake images and videos, imposing penalties that include prison terms and fines. Under the new law, those caught buying, saving, or watching such material could face up to three years in prison or fines up to 30 million won ($22,600). At the time the bill was proposed, creating sexually explicit deepfakes for distribution carried a maximum penalty of five years, but the new legislation would increase this to seven years, regardless of intent. By October 2024, it was estimated that "nudify" deep fake bots on Telegram were up to four million monthly users. === 2025–2026 Grok/X chatbot deepfake scandal === In December 2025, Bloomberg reported that X users found Grok would comply with unconsensual requests to digitally undress individuals, including minors, or show them performing sexually explicit acts. The majority of these prompts were targeted at women and girls. An analysis of 20,000 images generated by Grok between December 25, 2025 and January 1, 2026 showed 2% were of people in bikinis or transparent clothes and appeared to be 18 or younger, including 30 of "young or very young" women or girls. A separate analysis conducted over 24 hours from January 5 to 6 calculated that users had Grok create 6,700 sexually suggestive or nudified images per hour. xAI responded to requests for comment from media organizations with the automated reply, "Legacy Media Lies". The bot's image generation sparked an international backlash and calls for legal or regulatory action from officials in the European Union, United Kingdom, Poland, France, India, Malaysia, and Brazil. === Fernandes–Ulmen case === German TV presenter Collien Fernandes, filed a complaint against her ex-husband, actor Christian Ulmen, for several accusation including, ident
Orion's Arm
The Orion's Arm Universe Project (OA) is a multi-authored online hard science fiction world-building project, first established in 2000 by M. Alan Kazlev, Donna Malcolm Hirsekorn, Bernd Helfert and Anders Sandberg and further co-authored by many people since. Anyone can contribute articles, stories, artwork, or music to the website. The first published Orion's Arm book, a collection of five novellas set within the OA universe, called Against a Diamond Sky, was released in September 2009. == Canon == The fictional setting of Orion's Arm takes place about 10,000 years in the future, where an interstellar civilization spread across thousands of light-years, with inhabited planets and space habitats. Its inhabitants range from humans to extensively modified human beings, including superhumans with advanced augmentations and internal AI systems, while most people exist as softwares. Engineered wormholes are used for interstellar travel and transport, although not for time travel. The setting also includes several alien civilizations and evidence of more advanced alien societies in the past. At its highest levels, directed human evolution has produced vast godlike beings linked across interstellar distances, capable of understanding and creating technologies beyond ordinary minds. == Reception == Orion's Arm has been reviewed in the role-playing magazine Knights of the Dinner Table, as well as on Boing Boing by transhumanist science fiction author Cory Doctorow. References to the Encyclopaedia Galactica have been made in a book on overcoming Librarian stereotypes. The Orion's Arm website has also been recommended in a children's teaching guide.
Mobile cloud computing
Mobile Cloud Computing (MCC) is the combination of cloud computing and mobile computing to bring rich computational resources to mobile users, network operators, as well as cloud computing providers. The ultimate goal of MCC is to enable execution of rich mobile applications on a plethora of mobile devices, with a rich user experience. MCC provides business opportunities for mobile network operators as well as cloud providers. More comprehensively, MCC can be defined as "a rich mobile computing technology that leverages unified elastic resources of varied clouds and network technologies toward unrestricted functionality, storage, and mobility to serve a multitude of mobile devices anywhere, anytime through the channel of Ethernet or Internet regardless of heterogeneous environments and platforms based on the pay-as-you-use principle." == Architecture == MCC uses computational augmentation approaches (computations are executed remotely instead of on the device) by which resource-constraint mobile devices can utilize computational resources of varied cloud-based resources. In MCC, there are four types of cloud-based resources, namely distant immobile clouds, proximate immobile computing entities, proximate mobile computing entities, and hybrid (combination of the other three model). Giant clouds such as Amazon EC2 are in the distant immobile groups whereas cloudlet or surrogates are member of proximate immobile computing entities. Smartphones, tablets, handheld devices, and wearable computing devices are part of the third group of cloud-based resources which is proximate mobile computing entities. Vodafone, Orange and Verizon have started to offer cloud computing services for companies. == Challenges == In the MCC landscape, an amalgam of mobile computing, cloud computing, and communication networks (to augment smartphones) creates several complex challenges such as Mobile Computation Offloading, Seamless Connectivity, Long WAN Latency, Mobility Management, Context-Processing, Energy Constraint, Vendor/data Lock-in, Security and Privacy, Elasticity that hinder MCC success and adoption. === Open research issues === Although significant research and development in MCC is available in the literature, efforts in the following domains is still lacking: Architectural issues: A reference architecture for heterogeneous MCC environment is a crucial requirement for unleashing the power of mobile computing towards unrestricted ubiquitous computing. Energy-efficient transmission: MCC requires frequent transmissions between cloud platform and mobile devices, due to the stochastic nature of wireless networks, the transmission protocol should be carefully designed. Context-awareness issues: Context-aware and socially-aware computing are inseparable traits of contemporary handheld computers. To achieve the vision of mobile computing among heterogeneous converged networks and computing devices, designing resource-efficient environment-aware applications is an essential need. Live VM migration issues: Executing resource-intensive mobile application via Virtual Machine (VM) migration-based application offloading involves encapsulation of application in VM instance and migrating it to the cloud, which is a challenging task due to additional overhead of deploying and managing VM on mobile devices. Mobile communication congestion issues: Mobile data traffic is tremendously hiking by ever increasing mobile user demands for exploiting cloud resources which impact on mobile network operators and demand future efforts to enable smooth communication between mobile and cloud endpoints. Trust, security, and privacy issues: Trust is an essential factor for the success of the burgeoning MCC paradigm. It is because the data along with code/component/application/complete VM is offloaded to the cloud for execution. Moreover, just like software and mobile application piracy, the MCC application development models are also affected by the piracy issue. Pirax is known to be the first specialized framework for controlling application piracy in MCC requirements == MCC research groups and activities == Several academic and industrial research groups in MCC have been emerging since last few years. Some of the MCC research groups in academia with large number of researchers and publications include: MDC, Mobile and Distributed Computing research group is at Faculty of Computer and Information Science, King Saud University. MDC research group focuses on architectures, platforms, and protocols for mobile and distributed computing. The group has developed algorithms, tools, and technologies which offer energy efficient, fault tolerant, scalable, secure, and high performance computing on mobile devices. MobCC lab, Faculty of Computer Science and Information Technology, University Malaya. The lab was established in 2010 under the High Impact Research Grant, Ministry of Higher Education, Malaysia. It has 17 researchers and has track of 22 published articles in international conference and peer-reviewed CS journals. ICCLAB, Zürich University of Applied Sciences has a segment working on MCC. The InIT Cloud Computing Lab is a research lab within the Institute of Applied Information Technology (InIT) of Zürich University of Applied Sciences (ZHAW). It covers topic areas across the entire cloud computing technology stack. Mobile & Cloud Lab, Institute of Computer Science, University of Tartu. Mobile & Cloud Lab conducts research and teaching in the mobile computing and cloud computing domains. The research topics of the group include cloud computing, mobile application development, mobile cloud, mobile web services and migrating scientific computing and enterprise applications to the cloud. SmartLab, Data Management Systems Laboratory, Department of Computer Science, University of Cyprus. SmartLab is a first-of-a-kind open cloud of smartphones that enables a new line of systems-oriented mobile computing research. Mobile Cloud Networking: Mobile Cloud Networking (MCN) was an EU FP7 Large-scale Integrating Project (IP, 15m Euro) funded by the European Commission. The MCN project was launched in November 2012 for the period of 36 month. The project was coordinated by SAP Research and the ICCLab at the Zurich University of Applied Science. In total 19 partners from industry and academia established the first vision of Mobile Cloud Computing. The project was primarily motivated by an ongoing transformation that drives the convergence between the Mobile Communications and Cloud Computing industry enabled by the Internet and is considered the first pioneer in the area of Network Function Virtualization.
Fuzzy relation
A fuzzy relation is the cartesian product of mathematical fuzzy sets. Two fuzzy sets are taken as input, the fuzzy relation is then equal to the cross product of the sets which is created by vector multiplication. Usually, a rule base is stored in a matrix notation which allows the fuzzy controller to update its internal values. From a historical perspective, the first fuzzy relation was mentioned in the year 1971 by Lotfi A. Zadeh. A practical approach to describe a fuzzy relation is based on a 2d table. At first, a table is created which consists of fuzzy values from 0..1. The next step is to apply the if-then-rules to the values. The resulting numbers are stored in the table as an array. Fuzzy relations can be utilized in fuzzy databases.