Kialo

Kialo is an online structured debate platform with argument maps in the form of debate trees. It is a collaborative reasoning tool for thoughtful discussion, understanding different points of view, and collaborative decision-making, showing arguments for and against claims underneath user-submitted theses or questions. The deliberative discourse platform is designed to present hundreds of supporting or opposing arguments in a dynamic argument tree and is streamlined for rational civil debate on topics such as philosophical questions, policy deliberations, entertainment, ethics, science questions, and unsolved problems or subjects of disagreement in general. Argument-boxes are structured into hierarchical branches where the root is the main thesis (or theses) of the debate, enabling deliberation and navigable debates between opposing perspectives. A debate is divided into Pro (supporting) and Con (refuting or devaluing) columns where registered users can add arguments and rate the impact on the weight or validity of the parent claim. The arguments are sorted according to the rating average. Its argument tree structure enables detailed scrutiny of claims at all levels of the tree and allows users to for example quickly understand why a decision was made or which of the aggregated arguments swayed it this way. Newcomers can join a debate at any time and look back at the structured discussion history, and then weigh in at the right place with their new argument or their comment on a specific argument. The design presets a structure on debates "that allows participants to easily see, process, and ultimately assess the many facets of competing claims". The word Kialo is Esperanto for "reason". The platform is the world's largest argument mapping and structured debate site. == Overview == Users can comment on every Pro or Con, for example for requesting sources or expansions. Recent activities of a debate are shown in a panel on the right side of the respective debate. Debates can be found through the search or on the Explore page through their descriptions and topic-tags. Mere comments that do not make a constructive point (a self-contained argument backed by reasoning) are not allowed and are picked up by other users and moderators. "Civil language and sensible observations from opposing perspectives" can be seen also in debates about controversial topics. The site by-design incentivizes fair, rigorous, open-minded dialogue. Contributors making claims often also write counterpoints to their own contribution. Claims need to be shorter than 500 characters and can link to external sources. Debate trees can also start off with multiple theses – such as different policy options or hypotheses. Claims can link to related debates or include segments of them. In the discussion tab of each claim, users can make edit proposals (e.g. for accuracy, improving sources, or changing scope), decide if the argument should be moved or copied to another branch, call for archiving a claim, and ask for extra evidence or clarification. Debates can grow large and complex for which a sunburst diagram visualization of the topology of the debate and the search functionality can be useful. Each debate also has a chat-box. In cases where e.g. a "Con" is a point against multiple in the "Pros", users – through moderators – can link these arguments at the respective places to avoid duplication of content and allowing a clean chain for people to understand which points are arguments against each other. Contributions of users are tracked, enabling a board of thought-leaders for every debate. Other gamification elements include a feature to thank users for their contributions. The "Perspectives" feature allows users to see 'Impact' ratings of supporters and opposers of a thesis as well as of the debate's moderators and individual contributors. It thereby enables participants to see a debate from other participants' perspectives and to sort by them. In Kialo Edu, this feature lets teachers view votes for a whole class, individuals, or supporters/opponents of a specific thesis. Users in both versions of Kialo can vote on the overall debate topic as well as on individual claims to express their perspectives or conclusions, with the rationale (i.e. the main causal arguments) why they voted on the veracity of the thesis as they did not being captured. Voting can be done by any registered user while navigating through any debate that has voting enabled or via using the Guided Voting wizard user interface that automatically walks through branches. As of 2021, Kialo doesn't have a mobile app. == Contents == A 2018 report stated the collaborative argument platform hosts more than 10,000 debates in various languages. It also hosts private debates. The website claims that it has over 18,000 public debates as of July 2023, as well as over 1 million votes and over 720,000 claims. Debates can be found via the site's internal search and up to six tags per debate. Preprint studies have scraped public debates on over 1.4K issues with over 130K statements as of October 2019 and 1628 debates, related to over 1120 categories, with 124,312 unique claims as of June 26, 2020. == Kialo Inc. == The site is run by Kialo Inc. It was founded by German-born entrepreneur and London School of Economics and Political Science graduate Errikos Pitsos in August 2017 and is based in Brooklyn and Berlin. According to a 2018 report, the site does not show advertisements and does not sell user's data. The for-profit company was founded in 2011, Pitsos began to develop the concept in 2012 and described various specifics of the system in 2014. In 2018, he stated that they intend to make money by selling the platform to companies as a deliberation and decision-making tool. The site is free to use for the public and in education. According to the site, as of 2023 Kialo.com is a non-revenue generating site with no ads and no reselling of user data. == Applications and adoption == === Adopted applications === Applications of its content or the platform in society include: Teachers and professors, especially in high schools – including the universities Harvard and Princeton, are using Kialo for class discussions and exercises in critical thinking and reasoning, as consolidating understanding of materials covered in recent classes, more useful and engaging learning experiences, for remote/e-learning, for clearing up misconceptions, teaching logical fallacies and rational argumentation, for academic dialogue, teaching media literacy, and for teaching to sufficiently reflect or research before posting online. Like for debaters of the main site, access for schools and universities is free. Kialo Edu is the custom version of Kialo specifically designed for classroom use where debates are private and locked to invited students. Kialo allows teachers to provide feedback to students on their ideas, argument structure, and research quality while it is left to other students to rate the impacts of their peers' arguments. Students can be allowed to contribute anonymously which may be useful for controversial issues as well as for safeguarding privacy in education. Students are or can be encouraged to back up their claims with evidence which can foster digital literacy and research skills. Students and teachers can use it to arrange their thoughts when structuring an essay or project. The site's name was decided on internally using the software. === Prototypical and theoretical applications === Potential, theoretical, prototypical or little-used applications include: Education Improving critical thinking skills of society at large as well as facilitating deep or efficient thinking and deepening research and debates where e.g. discussions are less shallow and the well-known or many arguments have already been made and in many cases aren't unreasonably over- or underrated. Pitsos claimed that "we're training students to be very good test-takers instead of critical thinkers", suggesting teaching people to think things through may be more important or neglected compared to essay writing skills. Many young people and adults are "submerged into a sea of dispersed information", "[b]rowsing and engaging in superficial thinking activities". Kialo could counteract this issue and help people develop good sane reasoning. Academia, R&D and policy Three scholars from three prestigious U.S. universities outlined possible benefits in this domain, including applications beyond higher education such as for academic communication. They suggest the debate platform could be used for structuring the communication of open peer-review by helping those giving feedback to "hone in on[sic] core arguments and pieces of evidence in an even more direct way" than annotated commenting. It could be used to evaluate extracted argument structures and sequences from raw texts, as in a Semantic Web for arguments. Such "argument mining", to which Kialo is the lar

FoundationDB

FoundationDB is a free and open-source multi-model distributed NoSQL database owned by Apple Inc. with a shared-nothing architecture. The product was designed around a "core" database, with additional features supplied in "layers." The core database exposes an ordered key–value store with transactions. The transactions are able to read or write multiple keys stored on any machine in the cluster while fully supporting ACID properties. Transactions are used to implement a variety of data models via layers. The FoundationDB Alpha program began in January 2012 and concluded on March 4, 2013, with their public Beta release. Their 1.0 version was released for general availability on August 20, 2013. On March 24, 2015, it was reported that Apple has acquired the company. A notice on the FoundationDB web site indicated that the company has "evolved" its mission and would no longer offer downloads of the software. On April 19, 2018, Apple open sourced the software, releasing it under the Apache 2.0 license. == Main features == The main features of FoundationDB include the following: Ordered key–value store In addition to supporting standard key-based reads and writes, the ordering property enables range reads that can efficiently scan large swaths of data. Transactions Transaction processing employs multiversion concurrency control for reads and optimistic concurrency for writes. Transactions can span multiple keys stored on multiple machines. ACID properties FoundationDB guarantees serializable isolation and strong durability via redundant storage on disk before transactions are considered committed. Layers Layers map new data models, APIs, and query languages to the FoundationDB core. They employ FoundationDB's ability to update multiple data elements in a single transaction, ensuring consistency. An example is their SQL layer. Commodity clusters FoundationDB is designed for deployment on distributed clusters of commodity hardware running Linux. Replication FoundationDB stores each piece of data on multiple machines according to a configurable replication factor. Triple replication is the recommended mode for clusters of 5 or more machines. Scalability FoundationDB is designed to support horizontal scaling through the addition of machines to a cluster while automatically handling data replication and partitioning. Systems supported FoundationDB supports packages for Linux, Windows, and macOS. The Linux version supports production clusters, while the Windows and macOS versions support local operation for development purposes. Configurations on Amazon EC2 are also supported. Programming language bindings FoundationDB supports language bindings for Python, Go, Ruby, Node.js, Java, PHP, and C, all of which are made available with the product. == Design limitations == The design of FoundationDB results in several limitations: Long transactions FoundationDB does not support transactions running over five seconds. Large transactions Transaction size cannot exceed 10 MB of total written keys and values. Large keys and values Keys cannot exceed 10 kB in size. Values cannot exceed 100 kB in size. == History == FoundationDB, headquartered in Vienna, Virginia, was started in 2009 by Nick Lavezzo, Dave Rosenthal, and Dave Scherer, drawing on their experience in executive and technology roles at their previous company, Visual Sciences. In March 2015 the FoundationDB Community site was updated to state that the company had changed directions and would no longer be offering downloads of its product. The company was acquired by Apple Inc., which was confirmed March 25, 2015. On April 19, 2018, Apple open sourced the software, releasing it under the Apache 2.0 license.

Construction of t-norms

In mathematics, t-norms are a special kind of binary operations on the real unit interval [0, 1]. Various constructions of t-norms, either by explicit definition or by transformation from previously known functions, provide a plenitude of examples and classes of t-norms. This is important, e.g., for finding counter-examples or supplying t-norms with particular properties for use in engineering applications of fuzzy logic. The main ways of construction of t-norms include using generators, defining parametric classes of t-norms, rotations, or ordinal sums of t-norms. Relevant background can be found in the article on t-norms. == Generators of t-norms == The method of constructing t-norms by generators consists in using a unary function (generator) to transform some known binary function (most often, addition or multiplication) into a t-norm. In order to allow using non-bijective generators, which do not have the inverse function, the following notion of pseudo-inverse function is employed: Let f: [a, b] → [c, d] be a monotone function between two closed subintervals of extended real line. The pseudo-inverse function to f is the function f (−1): [c, d] → [a, b] defined as f ( − 1 ) ( y ) = { sup { x ∈ [ a , b ] ∣ f ( x ) < y } for f non-decreasing sup { x ∈ [ a , b ] ∣ f ( x ) > y } for f non-increasing. {\displaystyle f^{(-1)}(y)={\begin{cases}\sup\{x\in [a,b]\mid f(x)y\}&{\text{for }}f{\text{ non-increasing.}}\end{cases}}} === Additive generators === The construction of t-norms by additive generators is based on the following theorem: Let f: [0, 1] → [0, +∞] be a strictly decreasing function such that f(1) = 0 and f(x) + f(y) is in the range of f or in [f(0+), +∞] for all x, y in [0, 1]. Then the function T: [0, 1]2 → [0, 1] defined as T(x, y) = f (-1)(f(x) + f(y)) is a t-norm. Alternatively, one may avoid using the notion of pseudo-inverse function by having T ( x , y ) = f − 1 ( min ( f ( 0 + ) , f ( x ) + f ( y ) ) ) {\displaystyle T(x,y)=f^{-1}\left(\min \left(f(0^{+}),f(x)+f(y)\right)\right)} . The corresponding residuum can then be expressed as ( x ⇒ y ) = f − 1 ( max ( 0 , f ( y ) − f ( x ) ) ) {\displaystyle (x\Rightarrow y)=f^{-1}\left(\max \left(0,f(y)-f(x)\right)\right)} . And the biresiduum as ( x ⇔ y ) = f − 1 ( | f ( x ) − f ( y ) | ) {\displaystyle (x\Leftrightarrow y)=f^{-1}\left(\left|f(x)-f(y)\right|\right)} . If a t-norm T results from the latter construction by a function f which is right-continuous in 0, then f is called an additive generator of T. Examples: The function f(x) = 1 – x for x in [0, 1] is an additive generator of the Łukasiewicz t-norm. The function f defined as f(x) = –log(x) if 0 < x ≤ 1 and f(0) = +∞ is an additive generator of the product t-norm. The function f defined as f(x) = 2 – x if 0 ≤ x < 1 and f(1) = 0 is an additive generator of the drastic t-norm. Basic properties of additive generators are summarized by the following theorem: Let f: [0, 1] → [0, +∞] be an additive generator of a t-norm T. Then: T is an Archimedean t-norm. T is continuous if and only if f is continuous. T is strictly monotone if and only if f(0) = +∞. Each element of (0, 1) is a nilpotent element of T if and only if f(0) < +∞. The multiple of f by a positive constant is also an additive generator of T. T has no non-trivial idempotents. (Consequently, e.g., the minimum t-norm has no additive generator.) === Multiplicative generators === The isomorphism between addition on [0, +∞] and multiplication on [0, 1] by the logarithm and the exponential function allow two-way transformations between additive and multiplicative generators of a t-norm. If f is an additive generator of a t-norm T, then the function h: [0, 1] → [0, 1] defined as h(x) = e−f (x) is a multiplicative generator of T, that is, a function h such that h is strictly increasing h(1) = 1 h(x) · h(y) is in the range of h or equal to 0 or h(0+) for all x, y in [0, 1] h is right-continuous in 0 T(x, y) = h (−1)(h(x) · h(y)). Vice versa, if h is a multiplicative generator of T, then f: [0, 1] → [0, +∞] defined by f(x) = −log(h(x)) is an additive generator of T. == Parametric classes of t-norms == Many families of related t-norms can be defined by an explicit formula depending on a parameter p. This section lists the best known parameterized families of t-norms. The following definitions will be used in the list: A family of t-norms Tp parameterized by p is increasing if Tp(x, y) ≤ Tq(x, y) for all x, y in [0, 1] whenever p ≤ q (similarly for decreasing and strictly increasing or decreasing). A family of t-norms Tp is continuous with respect to the parameter p if lim p → p 0 T p = T p 0 {\displaystyle \lim _{p\to p_{0}}T_{p}=T_{p_{0}}} for all values p0 of the parameter. === Schweizer–Sklar t-norms === The family of Schweizer–Sklar t-norms, introduced by Berthold Schweizer and Abe Sklar in the early 1960s, is given by the parametric definition T p S S ( x , y ) = { T min ( x , y ) if p = − ∞ ( x p + y p − 1 ) 1 / p if − ∞ < p < 0 T p r o d ( x , y ) if p = 0 ( max ( 0 , x p + y p − 1 ) ) 1 / p if 0 < p < + ∞ T D ( x , y ) if p = + ∞ . {\displaystyle T_{p}^{\mathrm {SS} }(x,y)={\begin{cases}T_{\min }(x,y)&{\text{if }}p=-\infty \\(x^{p}+y^{p}-1)^{1/p}&{\text{if }}-\infty −∞ Continuous if and only if p < +∞ Strict if and only if −∞ < p ≤ 0 (for p = −1 it is the Hamacher product) Nilpotent if and only if 0 < p < +∞ (for p = 1 it is the Łukasiewicz t-norm). The family is strictly decreasing for p ≥ 0 and continuous with respect to p in [−∞, +∞]. An additive generator for T p S S {\displaystyle T_{p}^{\mathrm {SS} }} for −∞ < p < +∞ is f p S S ( x ) = { − log ⁡ x if p = 0 1 − x p p otherwise. {\displaystyle f_{p}^{\mathrm {SS} }(x)={\begin{cases}-\log x&{\text{if }}p=0\\{\frac {1-x^{p}}{p}}&{\text{otherwise.}}\end{cases}}} === Hamacher t-norms === The family of Hamacher t-norms, introduced by Horst Hamacher in the late 1970s, is given by the following parametric definition for 0 ≤ p ≤ +∞: T p H ( x , y ) = { T D ( x , y ) if p = + ∞ 0 if p = x = y = 0 x y p + ( 1 − p ) ( x + y − x y ) otherwise. {\displaystyle T_{p}^{\mathrm {H} }(x,y)={\begin{cases}T_{\mathrm {D} }(x,y)&{\text{if }}p=+\infty \\0&{\text{if }}p=x=y=0\\{\frac {xy}{p+(1-p)(x+y-xy)}}&{\text{otherwise.}}\end{cases}}} The t-norm T 0 H {\displaystyle T_{0}^{\mathrm {H} }} is called the Hamacher product. Hamacher t-norms are the only t-norms which are rational functions. The Hamacher t-norm T p H {\displaystyle T_{p}^{\mathrm {H} }} is strict if and only if p < +∞ (for p = 1 it is the product t-norm). The family is strictly decreasing and continuous with respect to p. An additive generator of T p H {\displaystyle T_{p}^{\mathrm {H} }} for p < +∞ is f p H ( x ) = { 1 − x x if p = 0 log ⁡ p + ( 1 − p ) x x otherwise. {\displaystyle f_{p}^{\mathrm {H} }(x)={\begin{cases}{\frac {1-x}{x}}&{\text{if }}p=0\\\log {\frac {p+(1-p)x}{x}}&{\text{otherwise.}}\end{cases}}} === Frank t-norms === The family of Frank t-norms, introduced by M.J. Frank in the late 1970s, is given by the parametric definition for 0 ≤ p ≤ +∞ as follows: T p F ( x , y ) = { T m i n ( x , y ) if p = 0 T p r o d ( x , y ) if p = 1 T L u k ( x , y ) if p = + ∞ log p ⁡ ( 1 + ( p x − 1 ) ( p y − 1 ) p − 1 ) otherwise. {\displaystyle T_{p}^{\mathrm {F} }(x,y)={\begin{cases}T_{\mathrm {min} }(x,y)&{\text{if }}p=0\\T_{\mathrm {prod} }(x,y)&{\text{if }}p=1\\T_{\mathrm {Luk} }(x,y)&{\text{if }}p=+\infty \\\log _{p}\left(1+{\frac {(p^{x}-1)(p^{y}-1)}{p-1}}\right)&{\text{otherwise.}}\end{cases}}} The Frank t-norm T p F {\displaystyle T_{p}^{\mathrm {F} }} is strict if p < +∞. The family is strictly decreasing and continuous with respect to p. An additive generator for T p F {\displaystyle T_{p}^{\mathrm {F} }} is f p F ( x ) = { − log ⁡ x if p = 1 1 − x if p = + ∞ log ⁡ p − 1 p x − 1 otherwise. {\displaystyle f_{p}^{\mathrm {F} }(x)={\begin{cases}-\log x&{\text{if }}p=1\\1-x&{\text{if }}p=+\infty \\\log {\frac {p-1}{p^{x}-1}}&{\text{otherwise.}}\end{cases}}} === Yager t-norms === The family of Yager t-norms, introduced in the early 1980s by Ronald R. Yager, is given for 0 ≤ p ≤ +∞ by T p Y ( x , y ) = { T D ( x , y ) if p = 0 max ( 0 , 1 − ( ( 1 − x ) p + ( 1 − y ) p ) 1 / p ) if 0 < p < + ∞ T m i n ( x , y ) if p = + ∞ {\displaystyle T_{p}^{\mathrm {Y} }(x,y)={\begin{cases}T_{\mathrm {D} }(x,y)&{\text{if }}p=0\\\max \left(0,1-((1-x)^{p}+(1-y)^{p})^{1/p}\right)&{\text{if }}0

Structure mapping engine

In artificial intelligence and cognitive science, the structure mapping engine (SME) is an implementation in software of an algorithm for analogical matching based on the psychological theory of Dedre Gentner. The basis of Gentner's structure-mapping idea is that an analogy is a mapping of knowledge from one domain (the base) into another (the target). The structure-mapping engine is a computer simulation of the analogy and similarity comparisons. The theory is useful because it ignores surface features and finds matches between potentially very different things if they have the same representational structure. For example, SME could determine that a pen is like a sponge because both are involved in dispensing liquid, even though they do this very differently. == Structure mapping theory == Structure mapping theory is based on the systematicity principle, which states that connected knowledge is preferred over independent facts. Therefore, the structure mapping engine should ignore isolated source-target mappings unless they are part of a bigger structure. The SME, the theory goes, should map objects that are related to knowledge that has already been mapped. The theory also requires that mappings be done one-to-one, which means that no part of the source description can map to more than one item in the target and no part of the target description can be mapped to more than one part of the source. The theory also requires that if a match maps subject to target, the arguments of subject and target must also be mapped. If both these conditions are met, the mapping is said to be "structurally consistent." == Concepts in SME == SME maps knowledge from a source into a target. SME calls each description a dgroup. Dgroups contain a list of entities and predicates. Entities represent the objects or concepts in a description — such as an input gear or a switch. Predicates are one of three types and are a general way to express knowledge for SME. Relation predicates contain multiple arguments, which can be other predicates or entities. An example relation is: (transmit (what from to)). This relation has a functor transmit and takes three arguments: what, from, and to. Attribute predicates are the properties of an entity. An example of an attribute is (red gear) which means that gear has the attribute red. Function predicates map an entity into another entity or constant. An example of a function is (joules power source) which maps the entity power source onto the numerical quantity joules. Functions and attributes have different meanings, and consequently SME processes them differently. For example, in SME's true analogy rule set, attributes differ from functions because they cannot match unless there is a higher-order match between them. The difference between attributes and functions will be explained further in this section's examples. All predicates have four parameters. They have (1) a functor, which identifies it, and (2) a type, which is either relation, attribute, or function. The other two parameters (3 and 4) are for determining how to process the arguments in the SME algorithm. If the arguments have to be matched in order, commutative is false. If the predicate can take any number of arguments, N-ary is false. An example of a predicate definition is: (sme:defPredicate behavior-set (predicate) relation :n-ary? t :commutative? t) The predicate's functor is “behavior-set,” its type is “relation,” and its n-ary and commutative parameters are both set to true. The “(predicate)” part of the definition specifies that there will be one or more predicates inside an instantiation of behavior-set. == Algorithm details == The algorithm has several steps. The first step of the algorithm is to create a set of match hypotheses between source and target dgroups. A match hypothesis represents a possible mapping between any part of the source and the target. This mapping is controlled by a set of match rules. By changing the match rules, one can change the type of reasoning SME does. For example, one set of match rules may perform a kind of analogy called literal similarity, and another performs a kind of analogy called true-analogy. These rules are not the place where domain-dependent information is added, but rather where the analogy process is tweaked, depending on the type of cognitive function the user is trying to emulate. For a given match rule, there are two types of rules that further define how it will be applied: filter rules and intern rules. Intern rules use only the arguments of the expressions in the match hypotheses that the filter rules identify. This limitation makes the processing more efficient by constraining the number of match hypotheses that are generated. At the same time, it also helps to build the structural consistencies that are needed later on in the algorithm. An example of a filter rule from the true-analogy rule set creates match hypotheses between predicates that have the same functor. The true-analogy rule set has an intern rule that iterates over the arguments of any match hypothesis, creating more match hypotheses if the arguments are entities or functions, or if the arguments are attributes and have the same functor. In order to illustrate how the match rules produce match hypotheses consider these two predicates: transmit torque inputgear secondgear (p1) transmit signal switch div10 (p2) Here we use true analogy for the type of reasoning. The filter match rule generates a match between p1 and p2 because they share the same functor, transmit. The intern rules then produce three more match hypotheses: torque to signal, inputgear to switch, and secondgear to div10. The intern rules created these match hypotheses because all the arguments were entities. If the arguments were functions or attributes instead of entities, the predicates would be expressed as: transmit torque (inputgear gear) (secondgear gear) (p3) transmit signal (switch circuit) (div10 circuit) (p4) These additional predicates make inputgear, secondgear, switch, and div10 functions or attributes depending on the value defined in the language input file. The representation also contains additional entities for gear and circuit. Depending on what type inputgear, secondgear, switch, and div10 are, their meanings change. As attributes, each one is a property of the gear or circuit. For example, the gear has two attributes, inputgear and secondgear. The circuit has two attributes, switch and circuit. As functions inputgear, secondgear, switch, and div10 become quantities of the gear and circuit. In this example, the functions inputgear and secondgear now map to the numerical quantities “torque from inputgear” and “torque from secondgear,” For the circuit the quantities map to logical quantity “switch engaged” and the numerical quantity “current count on the divide by 10 counter.” SME processes these differently. It does not allow attributes to match unless they are part of a higher-order relation, but it does allow functions to match, even if they are not part of such a relation. It allows functions to match because they indirectly refer to entities and thus should be treated like relations that involve no entities. However, as next section shows, the intern rules assign lower weights to matches between functions than to matches between relations. The reason SME does not match attributes is because it is trying to create connected knowledge based on relationships and thus satisfy the systematicity principle. For example, if both a clock and a car have inputgear attributes, SME will not mark them as similar. If it did, it would be making a match between the clock and car based on their appearance — not on the relationships between them. When the additional predicates in p3 and p4 are functions, the results from matching p3 and p4 are similar to the results from p1 and p2 except there is an additional match between gear and circuit and the values for the match hypotheses between (inputgear gear) and (switch circuit), and (secondgear gear) and (div10 circuit), are lower. The next section describes the reason for this in more detail. If the inputgear, secondgear, switch, and div10 are attributes instead of entities, SME does not find matches between any of the attributes. It finds matches only between the transmit predicates and between torque and signal. Additionally, the structural-evaluation scores for the remaining two matches decrease. In order to get the two predicates to match, p3 would need to be replaced by p5, which is demonstrated below. transmit torque (inputgear gear) (div10 gear) (p5) Since the true-analogy rule set identifies that the div10 attributes are the same between p5 and p4 and because the div10 attributes are both part of the higher-relation match between torque and signal, SME makes a match between (div10 gear) and (div10 circuit) — which leads to a match between gear and circuit. Being part of a higher-order match is a requiremen

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.

Trazzler

Trazzler is a travel destination app that specializes in unique and local destinations. The initial concept was developed by Adam Rugel and Biz Stone in 2006 at Twitter's original offices under the name "71 miles". More than 10,000 writers and photographers have contributed and more than $350,000 in freelance contracts have been issued as a result of Trazzeler's weekly writing and photography contests. Investors in the company include SV Angel, AOL Founder Steve Case, and the Twitter founders, Evan Williams, Jack Dorsey, and Biz Stone. The company's partners are the City of Chicago, Hawaii Tourism Authority, Fairmont Hotels & Resorts, Salon.com, and Air New Zealand. Trazzler is designed for use on the iOS, Android, and Facebook.

SpreeAI

SpreeAI (stylized as SPREEAI) is an American fashion technology company headquartered in Incline Village, Nevada that develops artificial intelligence software for the apparel and retail industries, including photorealistic virtual try-on, AI-powered sizing recommendations, and digital model generation. Founded in 2022 by John Imah and Bob Davidson, the company achieved unicorn status in 2025 following a Series B round led by Davidson Group that valued the company at approximately US$1.5 billion. TechCrunch identified SpreeAI as one of the more than 100 new tech unicorns minted in 2025. Its board of directors includes supermodel Naomi Campbell and hospitality executive Larry Ruvo. == History == SpreeAI was founded in 2022 by John Imah and Bob Davidson with a focus on artificial intelligence applications in fashion retail. By 2024, the company had raised approximately US$60 million in venture funding. In May 2025, SpreeAI announced a Series B round led by Davidson Group; reporting at the time placed the company's valuation at approximately US$1.5 billion, making it one of a small number of fashion-technology companies to reach unicorn status. In January 2026, TechCrunch listed SpreeAI among the more than 100 new tech unicorns minted in 2025. == Technology == SpreeAI develops a suite of artificial intelligence tools for the apparel industry. Its consumer-facing platform allows shoppers to upload a single photograph or select a digital model and then visualize clothing items on that figure with photorealistic rendering, while a complementary sizing engine generates fit recommendations intended to reduce returns. The platform is designed for integration with online retailers so that shoppers can preview garments before purchase. The company has stated that its models were developed in part through research collaborations with the Massachusetts Institute of Technology and Carnegie Mellon University. == Leadership and board == John Imah, a Nigerian-American technology executive who previously held roles at Samsung, Twitch, Meta Platforms, and Snap Inc., is co-founder and chief executive officer. Co-founder Bob Davidson, through Davidson Group, led the company's Series B financing. The company's board of directors includes supermodel Naomi Campbell, who joined in 2024, and Las Vegas hospitality executive Larry Ruvo. == Partnerships == SpreeAI has formed partnerships across both academia and the fashion industry. Council of Fashion Designers of America (CFDA). In 2025, SpreeAI entered a partnership with the CFDA to support American designers and brands with AI-driven tools; the CFDA described SpreeAI as "a fashion technology leader delivering innovative solutions to help designers and brands thrive." Massachusetts Institute of Technology and Carnegie Mellon University. The company has cited ongoing research and talent collaborations with both institutions. Sergio Hudson and Kai Collective. In 2025, SpreeAI made what WWD described as its Met Gala debut through a custom collaboration with designer Sergio Hudson and Nigerian-British label Kai Collective; the collaboration paired Hudson's couture with SpreeAI's virtual try-on platform. == Recognition == In 2025, TechCrunch named SpreeAI among the new tech unicorns of the year. In 2025, SpreeAI was named an honoree in Inc.'s Best in Business awards, and CEO John Imah was included on Inc.'s list of 40 business leaders who "propelled their organizations to success." In 2025, Imah was named to the Observer's AI Power Index, a list of 100 leaders shaping the future of artificial intelligence. In 2025, Imah was included in AfroTech's Future 50, recognizing Black innovators in technology. SpreeAI and Imah have been the subject of profile coverage in The Washington Post, Rolling Stone UK, WWD, Vogue UA, L'Officiel Arabia, GQ South Africa, and Inc..