Roadie Inc. is an American package delivery company for business and private same-day, urgent and scheduled delivery in the United States. The company was founded in 2014 and launched its web and mobile apps in January 2015. As of September 2021, it reported having over 200,000 drivers covering more than 20,000 zip codes. Roadie states it matches gig drivers with deliveries that are directed along the routes they plan to travel. Major customers include The Home Depot, Walmart, Tractor Supply Company, Best Buy and Delta Air Lines. In September 2021, UPS entered into an agreement to acquire Roadie for an undisclosed amount with the transaction expected to be closed in the fourth quarter. == History == Roadie was founded by Marc Gorlin, a co-founder of Kabbage and founder of VerticalOne and Pretty Good Privacy, as a same-day and urgent delivery company in 2014. In January 2015, Roadie launched the first consumer to consumer (C2C) version of its app with a Series A funding round of $10 million. In February, Roadie announced a partnership with Waffle House to designate its restaurants "Roadie Roadhouses", offering a neutral meeting place for drivers and senders. Drivers receive free food and drink through the partnership. In May, late-night host Jimmy Kimmel discussed the Roadie-Waffle House relationship in an opening monologue on Jimmy Kimmel Live!. Roadie's driver network expanded significantly as a result. Roadie closed a Series B round of funding in June, raising $15 million, and its first business to business (B2B) app version launched that November. In 2015, Delta Air Lines signed an agreement with Roadie to deliver mishandled luggage, becoming Roadie’s first enterprise customer. Roadie launched a pilot program with Delta at Daytona Beach International Airport. Since then, the relationship has expanded to include over 70 airports around the United States and a first mile/last mile line haul relationship with Delta Cargo. In 2017, the company signed a deal with The Home Depot, also based in Atlanta, and in February 2019, closed a Series C round of funding. In October 2019, Roadie and Delta Cargo announced a partnership to create a same-day cross-country delivery offering, DASH Door-to-Door, the first of its kind from a U.S. passenger airline. Tractor Supply Company became the first general merchandise retailer to offer same-day delivery from every store in April 2020 through Roadie. In September 2021, UPS entered an agreement to acquire Roadie for an undisclosed amount. The transaction was expected to close in the fourth quarter of 2021. Roadies, which at the time reported having 200,000 operators serving over 20,000 ZIP Codes, was expected to continue operations under its name as a separate company with no transfer of packages between the UPS and Roadies networks. The relationship between the companies goes back several years with UPS being an early investor. Earlier in 2021, UPS had begun a pilot program testing same-day deliveries via Roadies. == Operations == === On-the-way model === Roadie’s app works by connecting drivers with senders, businesses or consumers who have items that need to be delivered. Deliveries within the app are referred to as "Gigs", which Gorlin said was inspired by live music road crews, also known as roadies. A sender creates a Gig on Roadie's web app or via its API. Drivers then review deliveries in their area on their mobile app and may choose to offer to take on individual or groups of deliveries along the same route. Gigs are then assigned to drivers by Roadie's algorithm. According to the company, this model encourages drivers to choose Gigs that align with their planned schedules and routes. Roadie calls this its "on-the-way" delivery model. The go-to-market approach taken by Roadie also differs from its competitors. Rather than launching in major cities and sequentially adding new markets city-by-city, Roadie launched nationwide from its inception. The company relies on retail and airline partners to drive volume of deliveries in individual markets, which in turn builds up a network of drivers in those areas, making it easier for small businesses and consumers to send deliveries as well. This strategy allows Roadie to reach smaller cities and towns in rural or exurban communities, traditionally difficult markets for delivery providers to serve. === Service lines === Roadie’s platform is most popular for same-day, on-demand or scheduled first mile/last mile delivery, especially delivery from stores and warehouses. Some retailers also use it for returns and reverse logistics, moving inventory, and hot shot shipping. Roadie operates 1-hour grocery delivery for Walmart, and delivers perishable food items for others including small, independent retailers. The on-the-way model complements the grocery industry’s just in time model, making last-mile deliveries that do not break the cold chain. === Cross-country same-day delivery === In October 2019, Roadie and Delta Cargo launched DASH Door-to-Door, a 24/7 door-to-door pick-up and delivery service. Roadie handles the first and last mile and Delta manages the line haul via passenger flights. The service launched originally from Atlanta to 55 cities and is an industry-first for a US commercial airline. === Promotion, awards and corporate citizenship === In September 2015, Roadie announced a partnership with Atlanta-based musician Ludacris, to promote the app. Following the devastation caused by flooding in Baton Rouge in 2016, Roadie offered free pickup and delivery for all deliveries traveling to and from the Baton Rouge area. In December 2020, Walmart named Roadie its top delivery partner for "Highest Driver Customer Satisfaction" and "Highest Net Promoter Score", after expanding into general merchandise deliveries as well as grocery that same year.
Abiquo Enterprise Edition
Abiquo Hybrid Cloud Management Platform is a web-based cloud computing software platform developed by Abiquo. Written entirely in Java, it is used to build, integrate and manage public and private clouds in homogeneous environments. Users can deploy and manage servers, storage system and network and virtual devices. It also supports LDAP integration. == Hypervisors == Abiquo supports five hypervisor systems. VMware ESXi Microsoft Hyper-V Citrix XenServer Oracle VM Server for x86 KVM From version 3.1, it also supports multiple public cloud providers: Amazon AWS Rackspace Google Compute Engine HP Cloud ElasticHosts DigitalOcean Abiquo version 3.2 added: Microsoft Azure Abiquo version 3.4 added: Support for Docker hosts, adding multi-tenant networking, storage management and private registry management for Docker SoftLayer CloudSigma Later versions continued to add features including autoscaling on any cloud, integration to VMware NSX and OpenStack Neutron for software defined networking, guest config with cloud-init and integrated monitoring driving guest automation. == Storage services == Abiquo supports any vendor for hypervisor storage, and also supports tiered storage pools, enabling storage-as-a-service from specific vendors and technologies including: NFS Generic iSCSI NetApp Nexenta == SAAS version == In April 2014 Abiquo launched Abiquo anyCloud, a SAAS version of the Abiquo Hybrid Cloud Platform software. This version lets users manage public cloud resources from: Amazon AWS Microsoft Azure IBM SoftLayer DigitalOcean Rackspace Open Cloud (an OpenStack cloud) HP Public Cloud (an OpenStack cloud) Google Compute Engine ElasticHosts Additional security and process features include workflow, to have an enterprise administrator electronically sign off on changes, an audit trail of activity and the ability to share cloud accounts among and enterprise team in a secure way. == Reviews and awards == Finalist for the 2015 Cloud Awards Finalist for the 2015 UK Cloud Awards in the category Cloud Management Product of the Year EMA Radar for Private Cloud platforms 2013 Global Telecoms Business Innovation Summit and Awards 2013 (with Interoute) EuroCloud UK Awards
Vagueness
In linguistics and philosophy, a vague predicate is one which gives rise to borderline cases. For example, the English adjective "tall" is vague since it is not clearly true or false for someone of middling height. By contrast, the word "prime" is not vague since every number is definitively either prime or not. Vagueness is commonly diagnosed by a predicate's ability to give rise to the sorites paradox. Vagueness is separate from ambiguity, in which an expression has multiple denotations. For instance the word "bank" is ambiguous since it can refer either to a river bank or to a financial institution, but there are no borderline cases between both interpretations. Vagueness is a major topic of research in philosophical logic, where it serves as a potential challenge to classical logic. Work in formal semantics has sought to provide a compositional semantics for vague expressions in natural language. Work in philosophy of language has addressed implications of vagueness for the theory of meaning, while metaphysicists have considered whether reality itself is vague. == Importance == The concept of vagueness has philosophical importance. Suppose one wants to come up with a definition of "right" in the moral sense. One wants a definition to cover actions that are clearly right and exclude actions that are clearly wrong, but what does one do with the borderline cases? Surely, there are such cases. Some philosophers say that one should try to come up with a definition that is itself unclear on just those cases. Others say that one has an interest in making his or her definitions more precise than ordinary language, or his or her ordinary concepts, themselves allow; they recommend one advances precising definitions. === In law === Vagueness is also a problem which arises in law, and in some cases, judges have to arbitrate regarding whether a borderline case does, or does not, satisfy a given vague concept. Examples include disability (how much loss of vision is required before one is legally blind?), human life (at what point from conception to birth is one a legal human being, protected for instance by laws against murder?), adulthood (most familiarly reflected in legal ages for driving, drinking, voting, consensual sex, etc.), race (how to classify someone of mixed racial heritage), etc. Even such apparently unambiguous concepts such as biological sex can be subject to vagueness problems, not just from transsexuals' gender transitions but also from certain genetic conditions which can give an individual mixed male and female biological traits (see intersex). In the common law system, vagueness is a possible legal defence against by-laws and other regulations. The legal principle is that delegated power cannot be used more broadly than the delegator intended. Therefore, a regulation may not be so vague as to regulate areas beyond what the law allows. Any such regulation would be "void for vagueness" and unenforceable. This principle is sometimes used to strike down municipal by-laws that forbid "explicit" or "objectionable" contents from being sold in a certain city; courts often find such expressions to be too vague, giving municipal inspectors discretion beyond what the law allows. In the US this is known as the vagueness doctrine and in Europe as the principle of legal certainty. === In science === Many scientific concepts are of necessity vague, for instance species in biology cannot be precisely defined, owing to unclear cases such as ring species. Nonetheless, the concept of species can be clearly applied in the vast majority of cases. As this example illustrates, to say that a definition is "vague" is not necessarily a criticism. Consider those animals in Alaska that are the result of breeding huskies and wolves: are they dogs? It is not clear: they are borderline cases of dogs. This means one's ordinary concept of doghood is not clear enough to let us rule conclusively in this case. == Approaches == The philosophical question of what the best theoretical treatment of vagueness is—which is closely related to the problem of the paradox of the heap, a.k.a. sorites paradox—has been the subject of much philosophical debate. === Fuzzy logic === One theoretical approach is that of fuzzy logic, developed by American mathematician Lotfi Zadeh. Fuzzy logic proposes a gradual transition between "perfect falsity", for example, the statement "Bill Clinton is bald", to "perfect truth", for, say, "Patrick Stewart is bald". In ordinary logics, there are only two truth-values: "true" and "false". The fuzzy perspective differs by introducing an infinite number of truth-values along a spectrum between perfect truth and perfect falsity. Perfect truth may be represented by "1", and perfect falsity by "0". Borderline cases are thought of as having a "truth-value" anywhere between 0 and 1 (for example, 0.6). Advocates of the fuzzy logic approach have included K. F. Machina (1976) and Dorothy Edgington (1993). === Supervaluationism === Another theoretical approach is known as "supervaluationism". This approach has been defended by Kit Fine and Rosanna Keefe. Fine argues that borderline applications of vague predicates are neither true nor false, but rather are instances of "truth value gaps". He defends an interesting and sophisticated system of vague semantics, based on the notion that a vague predicate might be "made precise" in many alternative ways. This system has the consequence that borderline cases of vague terms yield statements that are neither true, nor false. Given a supervaluationist semantics, one can define the predicate "supertrue" as meaning "true on all precisifications". This predicate will not change the semantics of atomic statements (e.g. "Frank is bald", where Frank is a borderline case of baldness), but does have consequences for logically complex statements. In particular, the tautologies of sentential logic, such as "Frank is bald or Frank is not bald", will turn out to be supertrue, since on any precisification of baldness, either "Frank is bald" or "Frank is not bald" will be true. Since the presence of borderline cases seems to threaten principles like this one (excluded middle), the fact that supervaluationism can "rescue" them is seen as a virtue. === Subvaluationism === Subvaluationism is the logical dual of supervaluationism, and has been defended by Dominic Hyde (2008) and Pablo Cobreros (2011). Whereas the supervaluationist characterises truth as 'supertruth', the subvaluationist characterises truth as 'subtruth', or "true on at least some precisifications". Subvaluationism proposes that borderline applications of vague terms are both true and false. It thus has "truth-value gluts". According to this theory, a vague statement is true if it is true on at least one precisification and false if it is false under at least one precisification. If a vague statement comes out true under one precisification and false under another, it is both true and false. Subvaluationism ultimately amounts to the claim that vagueness is a truly contradictory phenomenon. Of a borderline case of "bald man" it would be both true and false to say that he is bald, and both true and false to say that he is not bald. === Epistemicist view === A fourth approach, known as "the epistemicist view", has been defended by Timothy Williamson (1994), R. A. Sorensen (1988) and (2001), and Nicholas Rescher (2009). They maintain that vague predicates do, in fact, draw sharp boundaries, but that one cannot know where these boundaries lie. One's confusion about whether some vague word does or does not apply in a borderline case is due to one's ignorance. For example, in the epistemicist view, there is a fact of the matter, for every person, about whether that person is old or not old; some people are ignorant of this fact. === As a property of objects === One possibility is that one's words and concepts are perfectly precise, but that objects themselves are vague. Consider Peter Unger's example of a cloud (from his famous 1980 paper, "The Problem of the Many"): it is not clear where the boundary of a cloud lies; for any given bit of water vapor, one can ask whether it is part of the cloud or not, and for many such bits, one will not know how to answer. Hence, perhaps such a term as 'cloud' is not itself vague, but rather precisely denotes a vague object. This strategy has occasionally been poorly received; most notably, in Gareth Evans' short paper "Can There Be Vague Objects?" (1978), wherein an argument is examined which appears to show that vague identity-statements are impossible (i.e., result in logical incoherence). David Lewis explains that the reader is intended to conclude, with Evans, that—since there clearly are, in fact, meaningful vague identities—any purported proof to the contrary cannot be right; and as the proof relies upon the premise that vague terms precisely denote vague objects, but fails under the view that vague terms reflect a merel
ICAART
The International Conference on Agents and Artificial Intelligence (ICAART) is a meeting point for researchers (among others) with interest in the areas of Agents and Artificial Intelligence. There are 2 tracks in ICAART, one related to Agents and Distributed AI in general and the other one focused in topics related to Intelligent Systems and Computational Intelligence. The conference program is composed of several different kind of sessions like technical sessions, poster sessions, keynote lectures, tutorials, special sessions, doctoral consortiums, panels and industrial tracks. The papers presented in the conference are made available at the SCITEPRESS digital library, published in the conference proceedings and some of the best papers are invited to a post-publication with Springer. ICAART's first edition was in 2009 counting with several keynote speakers like Marco Dorigo, Edward H. Shortliffe and Eduard Hovy. Since then, the conference had several other invited speakers like Katia Sycara, Nick Jennings, Robert Kowalski, Boi Faltings and Tim Finin. Bart Selman is one of the names confirmed for the next edition of this conference. Since 2012 the conference is held in conjunction with 2 other conferences: the International Conference on Operations Research and Enterprise Systems (ICORES) and the International Conference on Pattern Recognition Applications and Methods (ICPRAM). == Areas == === Agents === Agent communication languages Cooperation and Coordination Distributed Problem Solving Economic Agent Models Emotional Intelligence Group Decision Making Intelligent Auctions and Markets Mobile Agents Multi-agent systems Negotiation and Interaction Protocols Nep News Detection Agent Models and Architectures Physical Agents at Work Privacy, Safety and Security Programming Environments and Languages Robot and Multi-Robot Systems Self Organizing Systems Semantic Web Simulation Swarm Intelligence Task Planning and Execution Transparency and Ethical Issues Agent-Oriented Software Engineering Web Intelligence Agent Platforms and Interoperability Autonomous systems Cloud Computing and Its Impact Cognitive robotics Collective Intelligence Conversational Agents === Artificial intelligence === AI and Creativity Deep Learning Evolutionary Computing Fuzzy Systems Hybrid Intelligent Systems Industrial Applications of AI Intelligence and Cybersecurity Intelligent User Interfaces Knowledge Representation and Reasoning Knowledge-Based Systems Ambient Intelligence Machine learning Model-Based Reasoning Natural Language Processing Neural Networks Ontologies Planning and Scheduling Social Network Analysis Soft Computing State Space Search Bayesian Networks Uncertainty in AI Vision and Perception Visualization Big Data Case-Based Reasoning Cognitive Systems Constraint Satisfaction Data Mining Data Science == Editions == === ICAART 2023 – Lisbon, Portugal === === ICAART 2020 – Valletta, Malta === === ICAART 2019 – Prague, Czech Republic === Proceedings - Proceedings of the 11th International Conference on Web Information Systems and Technologies - Volume 1. ISBN 978-989-758-350-6 Proceedings - Proceedings of the 11th International Conference on Web Information Systems and Technologies - Volume 2. ISBN 978-989-758-350-6 === ICAART 2018 – Funchal, Madeira, Portugal === Proceedings - Proceedings of the 10th International Conference on Web Information Systems and Technologies - Volume 1. ISBN 978-989-758-275-2 Proceedings - Proceedings of the 10th International Conference on Web Information Systems and Technologies - Volume 2. ISBN 978-989-758-275-2 === ICAART 2017 – Porto, Portugal === Proceedings - Proceedings of the 9th International Conference on Web Information Systems and Technologies - Volume 1. ISBN 978-989-758-219-6 Proceedings - Proceedings of the 9th International Conference on Web Information Systems and Technologies - Volume 2. ISBN 978-989-758-220-2 === ICAART 2016 – Rome, Italy === Proceedings - Proceedings of the 8th International Conference on Web Information Systems and Technologies - Volume 1. ISBN 978-989-758-172-4 Proceedings - Proceedings of the 8th International Conference on Web Information Systems and Technologies - Volume 2. ISBN 978-989-758-172-4 === ICAART 2015 – Lisbon, Portugal === Proceedings - Proceedings of the 7th International Conference on Web Information Systems and Technologies - Volume 1. ISBN 978-989-758-073-4 Proceedings - Proceedings of the 7th International Conference on Web Information Systems and Technologies - Volume 2. ISBN 978-989-758-074-1 === ICAART 2014 – ESEO, Angers, Loire Valley, France === Proceedings - Proceedings of the 6th International Conference on Web Information Systems and Technologies - Volume 1. ISBN 978-989-758-015-4 Proceedings - Proceedings of the 6th International Conference on Web Information Systems and Technologies - Volume 2. ISBN 978-989-758-016-1 === ICAART 2013 – Barcelona, Spain === Proceedings - Proceedings of the 5th International Conference on Web Information Systems and Technologies - Volume 1. ISBN 978-989-8565-38-9 Proceedings - Proceedings of the 5th International Conference on Web Information Systems and Technologies - Volume 2. ISBN 978-989-8565-39-6 === ICAART 2012 – Vilamoura, Algarve, Portugal === Proceedings - Proceedings of the 4th International Conference on Web Information Systems and Technologies - Volume 1. ISBN 978-989-8425-95-9 Proceedings - Proceedings of the 4th International Conference on Web Information Systems and Technologies - Volume 2. ISBN 978-989-8425-96-6 === ICAART 2011 – Rome, Italy === Proceedings - Proceedings of the 3rd International Conference on Web Information Systems and Technologies - Volume 1. ISBN 978-989-8425-40-9 Proceedings - Proceedings of the 3rd International Conference on Web Information Systems and Technologies - Volume 2. ISBN 978-989-8425-41-6 === ICAART 2010 – Valencia, Spain === Proceedings - Proceedings of the 2nd International Conference on Web Information Systems and Technologies - Volume 1. ISBN 978-989-674-021-4 Proceedings - Proceedings of the 2nd International Conference on Web Information Systems and Technologies - Volume 2. ISBN 978-989-674-022-1 === ICAART 2009 – Porto, Portugal === Proceedings - Proceedings of the 1st International Conference on Web Information Systems and Technologies. ISBN 978-989-8111-66-1
Statistical semantics
In linguistics, statistical semantics applies the methods of statistics to the problem of determining the meaning of words or phrases, ideally through unsupervised learning, to a degree of precision at least sufficient for the purpose of information retrieval. == History == The term statistical semantics was first used by Warren Weaver in his well-known paper on machine translation. He argued that word-sense disambiguation for machine translation should be based on the co-occurrence frequency of the context words near a given target word. The underlying assumption that "a word is characterized by the company it keeps" was advocated by J. R. Firth. This assumption is known in linguistics as the distributional hypothesis. Emile Delavenay defined statistical semantics as the "statistical study of the meanings of words and their frequency and order of recurrence". "Furnas et al. 1983" is frequently cited as a foundational contribution to statistical semantics. An early success in the field was latent semantic analysis. == Applications == Research in statistical semantics has resulted in a wide variety of algorithms that use the distributional hypothesis to discover many aspects of semantics, by applying statistical techniques to large corpora: Measuring the similarity in word meanings Measuring the similarity in word relations Modeling similarity-based generalization Discovering words with a given relation Classifying relations between words Extracting keywords from documents Measuring the cohesiveness of text Discovering the different senses of words Distinguishing the different senses of words Subcognitive aspects of words Distinguishing praise from criticism == Related fields == Statistical semantics focuses on the meanings of common words and the relations between common words, unlike text mining, which tends to focus on whole documents, document collections, or named entities (names of people, places, and organizations). Statistical semantics is a subfield of computational semantics, which is in turn a subfield of computational linguistics and natural language processing. Many of the applications of statistical semantics (listed above) can also be addressed by lexicon-based algorithms, instead of the corpus-based algorithms of statistical semantics. One advantage of corpus-based algorithms is that they are typically not as labour-intensive as lexicon-based algorithms. Another advantage is that they are usually easier to adapt to new languages or noisier new text types from e.g. social media than lexicon-based algorithms are. However, the best performance on an application is often achieved by combining the two approaches.
Contextual AI
Contextual AI is an enterprise software company based in Mountain View, California. It develops a platform for building specialized Retrieval-Augmented Generation (RAG) agents for enterprise use. The company was founded in 2023 by Douwe Kiela and Amanpreet Singh, both former AI researchers at Facebook AI Research (FAIR) and Hugging Face. Douwe Kiela previously led the Meta research team that introduced the Retrieval-Augmented Generation (RAG) approach in 2020. Contextual AI focuses on enterprise generative AI applications using RAG 2.0 technology, with deployments primarily in the technology, banking, finance and media sectors. == History == In June 2023, Contextual AI announced it had raised $20 million in a seed funding round led by Bain Capital Ventures (BCV), with participation from Lightspeed Venture Partners, Greycroft, SV Angel, and several angel investors. In August 2024, the company raised $80 million in a Series A funding round led by Greycroft, with participation from previous investors including Bain Capital Ventures, Lightspeed, and Conviction Partners. The round also included new backers such as Bezos Expeditions, NVentures (Nvidia), HSBC Ventures, and Snowflake Ventures. == Features == Retrieval-Augmented Generation (RAG) is an artificial intelligence framework that integrates information retrieval with text generation to improve the performance of large language models (LLMs) on complex, knowledge-intensive tasks. It was introduced in 2020 by researchers at Meta AI, including Douwe Kiela, Patrick Lewis and others, in their paper Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks. RAG enables language models to access and incorporate external information, such as proprietary databases or real-time web content, at query time, instead of relying solely on pre-trained, internal, static knowledge. This architecture addresses common limitations of standard LLMs, including hallucination, outdated information, and lack of attribution to source materials. RAG systems retrieve relevant context through a variety of techniques - including vector search, keyword search, text-to-SQL - and feeds this context into the language model to generate responses. The approach improves factual accuracy, supports domain-specific customization, enables citation of sources, and allows for more updated information without retraining the model itself. General Availability. In January 2025, Contextual AI announced the general availability of its enterprise platform for building specialized RAG agents. Early adopters included Qualcomm, which used the platform for their Customer Engineering team needs. Grounded Language Model. In March 2025, the company introduced a Grounded Language Model (GLM) for factual accuracy in enterprise AI applications. Reranker. In March 2025, Contextual AI released an instruction-following reranker that allows users to influence the ranking of retrieved documents through natural language instructions, such as prioritizing recent files, specific formats, or content from designated sources. == Applications == Contextual AI's platform has been adopted across a range of industries, including finance, technology, media and professional services. Clients include Fortune 500 companies such as Qualcomm and HSBC.
Type-2 fuzzy sets and systems
Type-2 fuzzy sets and systems generalize standard type-1 fuzzy sets and systems so that more uncertainty can be handled. From the beginning of fuzzy sets, criticism was made about the fact that the membership function of a type-1 fuzzy set has no uncertainty associated with it, something that seems to contradict the word fuzzy, since that word has the connotation of much uncertainty. So, what does one do when there is uncertainty about the value of the membership function? The answer to this question was provided in 1975 by the inventor of fuzzy sets, Lotfi A. Zadeh, when he proposed more sophisticated kinds of fuzzy sets, the first of which he called a "type-2 fuzzy set". A type-2 fuzzy set lets us incorporate uncertainty about the membership function into fuzzy set theory, and is a way to address the above criticism of type-1 fuzzy sets head-on. And, if there is no uncertainty, then a type-2 fuzzy set reduces to a type-1 fuzzy set, which is analogous to probability reducing to determinism when unpredictability vanishes. Type1 fuzzy systems are working with a fixed membership function, while in type-2 fuzzy systems the membership function is fluctuating. A fuzzy set determines how input values are converted into fuzzy variables. == Overview == In order to symbolically distinguish between a type-1 fuzzy set and a type-2 fuzzy set, a tilde symbol is put over the symbol for the fuzzy set; so, A denotes a type-1 fuzzy set, whereas à denotes the comparable type-2 fuzzy set. When the latter is done, the resulting type-2 fuzzy set is called a "general type-2 fuzzy set" (to distinguish it from the special interval type-2 fuzzy set). Zadeh didn't stop with type-2 fuzzy sets, because in that 1976 paper he also generalized all of this to type-n fuzzy sets. The present article focuses only on type-2 fuzzy sets because they are the next step in the logical progression from type-1 to type-n fuzzy sets, where n = 1, 2, ... . Although some researchers are beginning to explore higher than type-2 fuzzy sets, as of early 2009, this work is in its infancy. The membership function of a general type-2 fuzzy set, Ã, is three-dimensional (Fig. 1), where the third dimension is the value of the membership function at each point on its two-dimensional domain that is called its "footprint of uncertainty"(FOU). For an interval type-2 fuzzy set that third-dimension value is the same (e.g., 1) everywhere, which means that no new information is contained in the third dimension of an interval type-2 fuzzy set. So, for such a set, the third dimension is ignored, and only the FOU is used to describe it. It is for this reason that an interval type-2 fuzzy set is sometimes called a first-order uncertainty fuzzy set model, whereas a general type-2 fuzzy set (with its useful third-dimension) is sometimes referred to as a second-order uncertainty fuzzy set model. The FOU represents the blurring of a type-1 membership function, and is completely described by its two bounding functions (Fig. 2), a lower membership function (LMF) and an upper membership function (UMF), both of which are type-1 fuzzy sets! Consequently, it is possible to use type-1 fuzzy set mathematics to characterize and work with interval type-2 fuzzy sets. This means that engineers and scientists who already know type-1 fuzzy sets will not have to invest a lot of time learning about general type-2 fuzzy set mathematics in order to understand and use interval type-2 fuzzy sets. Work on type-2 fuzzy sets languished during the 1980s and early-to-mid 1990s, although a small number of articles were published about them. People were still trying to figure out what to do with type-1 fuzzy sets, so even though Zadeh proposed type-2 fuzzy sets in 1976, the time was not right for researchers to drop what they were doing with type-1 fuzzy sets to focus on type-2 fuzzy sets. This changed in the latter part of the 1990s as a result of Jerry Mendel and his student's works on type-2 fuzzy sets and systems. Since then, more researchers around the world are writing articles about type-2 fuzzy sets and systems. == Interval type-2 fuzzy sets == Interval type-2 fuzzy sets have received the most attention because the mathematics that is needed for such sets—primarily Interval arithmetic—is much simpler than the mathematics that is needed for general type-2 fuzzy sets. The literature about interval type-2 fuzzy sets is large, whereas the literature about general type-2 fuzzy sets is much smaller. Both kinds of fuzzy sets are being actively researched by an ever-growing number of researchers around the world and have resulted in successful employment in a variety of domains such as robot control. Formally, the following have already been worked out for interval type-2 fuzzy sets: Fuzzy set operations: union, intersection and complement Centroid (a very widely used operation by practitioners of such sets, and also an important uncertainty measure for them) Other uncertainty measures [fuzziness, cardinality, variance and skewness and uncertainty bounds Similarity Subsethood Embedded fuzzy sets Fuzzy set ranking Fuzzy rule ranking and selection Type-reduction methods Firing intervals for an interval type-2 fuzzy logic system Fuzzy weighted average Linguistic weighted average Synthesizing an FOU from data that are collected from a group of subject == Interval type-2 fuzzy logic systems == Type-2 fuzzy sets are finding very wide applicability in rule-based fuzzy logic systems (FLSs) because they let uncertainties be modeled by them whereas such uncertainties cannot be modeled by type-1 fuzzy sets. A block diagram of a type-2 FLS is depicted in Fig. 3. This kind of FLS is used in fuzzy logic control, fuzzy logic signal processing, rule-based classification, etc., and is sometimes referred to as a function approximation application of fuzzy sets, because the FLS is designed to minimize an error function. The following discussions, about the four components in Fig. 3 rule-based FLS, are given for an interval type-2 FLS, because to-date they are the most popular kind of type-2 FLS; however, most of the discussions are also applicable for a general type-2 FLS. Rules, that are either provided by subject experts or are extracted from numerical data, are expressed as a collection of IF-THEN statements, e.g., IF temperature is moderate and pressure is high, then rotate the valve a bit to the right. Fuzzy sets are associated with the terms that appear in the antecedents (IF-part) or consequents (THEN-part) of rules, and with the inputs to and the outputs of the FLS. Membership functions are used to describe these fuzzy sets, and in a type-1 FLS they are all type-1 fuzzy sets, whereas in an interval type-2 FLS at least one membership function is an interval type-2 fuzzy set. An interval type-2 FLS lets any one or all of the following kinds of uncertainties be quantified: Words that are used in antecedents and consequents of rules—because words can mean different things to different people. Uncertain consequents—because when rules are obtained from a group of experts, consequents will often be different for the same rule, i.e. the experts will not necessarily be in agreement. Membership function parameters—because when those parameters are optimized using uncertain (noisy) training data, the parameters become uncertain. Noisy measurements—because very often it is such measurements that activate the FLS. In Fig. 3, measured (crisp) inputs are first transformed into fuzzy sets in the Fuzzifier block because it is fuzzy sets and not numbers that activate the rules which are described in terms of fuzzy sets and not numbers. Three kinds of fuzzifiers are possible in an interval type-2 FLS. When measurements are: Perfect, they are modeled as a crisp set; Noisy, but the noise is stationary, they are modeled as a type-1 fuzzy set; and, Noisy, but the noise is non-stationary, they are modeled as an interval type-2 fuzzy set (this latter kind of fuzzification cannot be done in a type-1 FLS). In Fig. 3, after measurements are fuzzified, the resulting input fuzzy sets are mapped into fuzzy output sets by the Inference block. This is accomplished by first quantifying each rule using fuzzy set theory, and by then using the mathematics of fuzzy sets to establish the output of each rule, with the help of an inference mechanism. If there are M rules then the fuzzy input sets to the Inference block will activate only a subset of those rules, where the subset contains at least one rule and usually way fewer than M rules. The inference is done one rule at a time. So, at the output of the Inference block, there will be one or more fired-rule fuzzy output sets. In most engineering applications of an FLS, a number (and not a fuzzy set) is needed as its final output, e.g., the consequent of the rule given above is "Rotate the valve a bit to the right." No automatic valve will know what this means because "a bit to the right" is a linguistic expression, and a valv