The International Webmasters Association (IWA) is a non-profit association for education and certification of web professionals founded in 1996. It provides a Certified Web Professional certification. One of its objectives is to build a World Wide Web that is a true global community. According to the IWA, as of 2025 it has more than 100 official chapters with over 300,000 individual members in 106 countries. In 2001, the IWA merged with the HTML Writers Guild (HWG) and joined the World Wide Web Consortium (W3C). IWA's accomplishments include the publishing of the industry's first guidelines for ethical and professional standards, web certification and education programs, specialized employment resources, and technical assistance to individuals and businesses. IWA members participate to the activities of W3C WCAG Working Group, ATAG Working Group, and the XHTML Working Group. They have also participated in other initiatives such as the Multimodal Interaction Working Group which developed EMMA, the Extensible MultiModal Annotation markup language.
Graph cut optimization
Graph cut optimization is a combinatorial optimization method applicable to a family of functions of discrete variables, named after the concept of cut in the theory of flow networks. Thanks to the max-flow min-cut theorem, determining the minimum cut over a graph representing a flow network is equivalent to computing the maximum flow over the network. Given a pseudo-Boolean function f {\displaystyle f} , if it is possible to construct a flow network with positive weights such that each cut C {\displaystyle C} of the network can be mapped to an assignment of variables x {\displaystyle \mathbf {x} } to f {\displaystyle f} (and vice versa), and the cost of C {\displaystyle C} equals f ( x ) {\displaystyle f(\mathbf {x} )} (up to an additive constant) then it is possible to find the global optimum of f {\displaystyle f} in polynomial time by computing a minimum cut of the graph. The mapping between cuts and variable assignments is done by representing each variable with one node in the graph and, given a cut, each variable will have a value of 0 if the corresponding node belongs to the component connected to the source, or 1 if it belong to the component connected to the sink. Not all pseudo-Boolean functions can be represented by a flow network, and in the general case the global optimization problem is NP-hard. There exist sufficient conditions to characterise families of functions that can be optimised through graph cuts, such as submodular quadratic functions. Graph cut optimization can be extended to functions of discrete variables with a finite number of values, that can be approached with iterative algorithms with strong optimality properties, computing one graph cut at each iteration. Graph cut optimization is an important tool for inference over graphical models such as Markov random fields or conditional random fields, and it has applications in computer vision problems such as image segmentation, denoising, registration and stereo matching. == Representability == A pseudo-Boolean function f : { 0 , 1 } n → R {\displaystyle f:\{0,1\}^{n}\to \mathbb {R} } is said to be representable if there exists a graph G = ( V , E ) {\displaystyle G=(V,E)} with non-negative weights and with source and sink nodes s {\displaystyle s} and t {\displaystyle t} respectively, and there exists a set of nodes V 0 = { v 1 , … , v n } ⊂ V − { s , t } {\displaystyle V_{0}=\{v_{1},\dots ,v_{n}\}\subset V-\{s,t\}} such that, for each tuple of values ( x 1 , … , x n ) ∈ { 0 , 1 } n {\displaystyle (x_{1},\dots ,x_{n})\in \{0,1\}^{n}} assigned to the variables, f ( x 1 , … , x n ) {\displaystyle f(x_{1},\dots ,x_{n})} equals (up to a constant) the value of the flow determined by a minimum cut C = ( S , T ) {\displaystyle C=(S,T)} of the graph G {\displaystyle G} such that v i ∈ S {\displaystyle v_{i}\in S} if x i = 0 {\displaystyle x_{i}=0} and v i ∈ T {\displaystyle v_{i}\in T} if x i = 1 {\displaystyle x_{i}=1} . It is possible to classify pseudo-Boolean functions according to their order, determined by the maximum number of variables contributing to each single term. All first order functions, where each term depends upon at most one variable, are always representable. Quadratic functions f ( x ) = w 0 + ∑ i w i ( x i ) + ∑ i < j w i j ( x i , x j ) . {\displaystyle f(\mathbf {x} )=w_{0}+\sum _{i}w_{i}(x_{i})+\sum _{i
Computing Machinery and Intelligence
"Computing Machinery and Intelligence" is a paper written by Alan Turing on the topic of artificial intelligence. The paper, published in 1950 in Mind, was the first to introduce his concept of what is now known as the Turing test to the general public. Turing's paper considers the question "Can machines think?" Turing says that since the words "think" and "machine" cannot clearly be defined, we should "replace the question by another, which is closely related to it and is expressed in relatively unambiguous words." To achieve this objective, Turing proposes a three-step approach. First, he identifies a simple and unambiguous concept to substitute for the term "think." Second, he delineates the specific "machines" under consideration. Third, armed with these tools, he poses a new question related to the first, which he believes he can answer in the affirmative. == Turing's test == Rather than trying to determine if a machine is thinking, Turing suggests we should ask if the machine can win a game, called the "Imitation Game". The original Imitation game, that Turing described, is a simple party game involving three players. Player A is a man, player B is a woman and player C (who plays the role of the interrogator) can be of either sex. In the Imitation Game, player C is unable to see either player A or player B (and knows them only as X and Y), and can communicate with them only through written notes or any other form that does not give away any details about their gender. By asking questions of player A and player B, player C tries to determine which of the two is the man and which is the woman. Player A's role is to trick the interrogator into making the wrong decision, while player B attempts to assist the interrogator in making the right one. Turing proposes a variation of this game that involves the computer: We now ask the question, "What will happen when a machine takes the part of A in this game?" Will the interrogator decide wrongly as often when the game is played like this as he does when the game is played between a man and a woman? These questions replace our original, "Can machines think?" So the modified game becomes one that involves three participants in isolated rooms: a computer (which is being tested), a human, and a (human) judge. The human judge can converse with both the human and the computer by typing into a terminal. Both the computer and the human try to convince the judge that they are the human. If the judge cannot consistently tell which is which, then the computer wins the game. Researchers in the United Kingdom had been exploring "machine intelligence" for up to ten years prior to the founding of the field of artificial intelligence (AI) research in 1956. It was a common topic among the members of the Ratio Club, an informal group of British cybernetics and electronics researchers that included Alan Turing. Turing, in particular, had been running the notion of machine intelligence since at least 1941 and one of the earliest-known mentions of "computer intelligence" was made by him in 1947. As Stevan Harnad notes, the question has become "Can machines do what we (as thinking entities) can do?" In other words, Turing is no longer asking whether a machine can "think"; he is asking whether a machine can act indistinguishably from the way a thinker acts. This question avoids the difficult philosophical problem of pre-defining the verb "to think" and focuses instead on the performance capacities that being able to think makes possible, and how a causal system can generate them. Since Turing introduced his test, it has been both highly influential and widely criticised, and has become an important concept in the philosophy of artificial intelligence. Some of its criticisms, such as John Searle's Chinese room, are themselves controversial. Some have taken Turing's question to have been "Can a computer, communicating over a teleprinter, fool a person into believing it is human?" but it seems clear that Turing was not talking about fooling people but about generating human cognitive capacity. == Digital machines == Turing also notes that we need to determine which "machines" we wish to consider. He points out that a human clone, while man-made, would not provide a very interesting example. Turing suggested that we should focus on the capabilities of digital machinery—machines which manipulate the binary digits of 1 and 0, rewriting them into memory using simple rules. He gave two reasons. First, there is no reason to speculate whether or not they can exist. They already did in 1950. Second, digital machinery is "universal". Turing's research into the foundations of computation had proved that a digital computer can, in theory, simulate the behaviour of any other digital machine, given enough memory and time. (This is the essential insight of the Church–Turing thesis and the universal Turing machine.) Therefore, if any digital machine can "act like it is thinking", then every sufficiently powerful digital machine can. Turing writes, "all digital computers are in a sense equivalent." This allows the original question to be made even more specific. Turing now restates the original question as "Let us fix our attention on one particular digital computer C. Is it true that by modifying this computer to have an adequate storage, suitably increasing its speed of action, and providing it with an appropriate programme, C can be made to play satisfactorily the part of A in the imitation game, the part of B being taken by a man?" Hence, Turing states that the focus is not on "whether all digital computers would do well in the game nor whether the computers that are presently available would do well, but whether there are imaginable computers which would do well". What is more important is to consider the advancements possible in the state of our machines today regardless of whether we have the available resource to create one or not. == Nine common objections == Having clarified the question, Turing turned to answering it: he considered the following nine common objections, which include all the major arguments against artificial intelligence raised in the years since his paper was first published. Religious Objection: This states that thinking is a function of man's immortal soul; therefore, a machine cannot think. "In attempting to construct such machines," wrote Turing, "we should not be irreverently usurping His power of creating souls, any more than we are in the procreation of children: rather we are, in either case, instruments of His will providing mansions for the souls that He creates." 'Heads in the Sand' Objection: "The consequences of machines thinking would be too dreadful. Let us hope and believe that they cannot do so." This thinking is popular among intellectual people, as they believe superiority derives from higher intelligence and the possibility of being overtaken is a threat (as machines have efficient memory capacities and processing speed, machines exceeding the learning and knowledge capabilities are highly probable). This objection is a fallacious appeal to consequences, confusing what should not be with what can or cannot be (Wardrip-Fruin, 56). The Mathematical Objection: This objection uses mathematical theorems, such as Gödel's incompleteness theorem, to show that there are limits to what questions a computer system based on logic can answer. Turing suggests that humans are too often wrong themselves and pleased at the fallibility of a machine. (This argument would be made again by philosopher John Lucas in 1961 and physicist Roger Penrose in 1989, and later would be called Penrose–Lucas argument.) Argument From Consciousness: This argument, suggested by Professor Geoffrey Jefferson in his 1949 Lister Oration (acceptance speech for his 1948 award of Lister Medal) states that "not until a machine can write a sonnet or compose a concerto because of thoughts and emotions felt, and not by the chance fall of symbols, could we agree that machine equals brain." Turing replies by saying that we have no way of knowing that any individual other than ourselves experiences emotions, and that therefore we should accept the test. He adds, "I do not wish to give the impression that I think there is no mystery about consciousness ... [b]ut I do not think these mysteries necessarily need to be solved before we can answer the question [of whether machines can think]." (This argument, that a computer can't have conscious experiences or understanding, would be made in 1980 by philosopher John Searle in his Chinese room argument. Turing's reply is now known as the "other minds reply". See also Can a machine have a mind? in the philosophy of AI.) Arguments from various disabilities. These arguments all have the form "a computer will never do X". Turing offers a selection:Be kind, resourceful, beautiful, friendly, have initiative, have a sense of humour, tell right from wrong, make mistakes, fall in love, enjo
Cooperative coevolution
Cooperative Coevolution (CC) in the field of biological evolution is an evolutionary computation method. It divides a large problem into subcomponents, and solves them independently in order to solve the large problem. The subcomponents are also called species. The subcomponents are implemented as subpopulations and the only interaction between subpopulations is in the cooperative evaluation of each individual of the subpopulations. The general CC framework is nature inspired where the individuals of a particular group of species mate amongst themselves, however, mating in between different species is not feasible. The cooperative evaluation of each individual in a subpopulation is done by concatenating the current individual with the best individuals from the rest of the subpopulations as described by M. Potter. The cooperative coevolution framework has been applied to real world problems such as pedestrian detection systems, large-scale function optimization and neural network training. It has also be further extended into another method, called Constructive cooperative coevolution. == Pseudocode == i := 0 for each subproblem S do Initialise a subpopulation Pop0(S) calculate fitness of each member in Pop0(S) while termination criteria not satisfied do i := i + 1 for each subproblem S do select Popi(S) from Popi-1(S) apply genetic operators to Popi(S) calculate fitness of each member in Popi(S)
IJCAI Award for Research Excellence
The IJCAI Award for Research Excellence is a biannual award before given at the IJCAI conference to researcher in artificial intelligence as a recognition of excellence of their career. Beginning in 2016, the conference is held annually and so is the award. == Laureates == The recipients of this award have been: John McCarthy (1985) Allen Newell (1989) Marvin Minsky (1991) Raymond Reiter (1993) Herbert A. Simon (1995) Aravind Joshi (1997) Judea Pearl (1999) Donald Michie (2001) Nils Nilsson (2003) Geoffrey E. Hinton (2005) Alan Bundy (2007) Victor R. Lesser (2009) Robert Kowalski (2011) Hector Levesque (2013) Barbara Grosz (2015) for her pioneering research in Natural Language Processing and in theories and applications of Multiagent Collaboration. Michael I. Jordan (2016) for his groundbreaking and impactful research in both the theory and application of statistical machine learning. Andrew Barto (2017) for his pioneering work in the theory of reinforcement learning. Jitendra Malik (2018) Yoav Shoham (2019) Eugene Freuder (2020) Richard S. Sutton (2021) Stuart J. Russell (2022) Sarit Kraus (2023) for her pioneering work of the study of interactions among self-interested agents, creating the field of automated negotiation, and developing methods for coalition formation and teamwork, both as formal models and real-world implementations. == Winners of also Turing Award == John McCarthy (1971) Allen Newell (1975) Marvin Minsky (1969) Herbert A. Simon (1975) Judea Pearl (2011) Geoffrey Hinton (2018) Andrew Barto (2024) Richard S. Sutton (2024)
Proximal gradient methods for learning
Proximal gradient (forward backward splitting) methods for learning is an area of research in optimization and statistical learning theory which studies algorithms for a general class of convex regularization problems where the regularization penalty may not be differentiable. One such example is ℓ 1 {\displaystyle \ell _{1}} regularization (also known as Lasso) of the form min w ∈ R d 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 + λ ‖ w ‖ 1 , where x i ∈ R d and y i ∈ R . {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}+\lambda \|w\|_{1},\quad {\text{ where }}x_{i}\in \mathbb {R} ^{d}{\text{ and }}y_{i}\in \mathbb {R} .} Proximal gradient methods offer a general framework for solving regularization problems from statistical learning theory with penalties that are tailored to a specific problem application. Such customized penalties can help to induce certain structure in problem solutions, such as sparsity (in the case of lasso) or group structure (in the case of group lasso). == Relevant background == Proximal gradient methods are applicable in a wide variety of scenarios for solving convex optimization problems of the form min x ∈ H F ( x ) + R ( x ) , {\displaystyle \min _{x\in {\mathcal {H}}}F(x)+R(x),} where F {\displaystyle F} is convex and differentiable with Lipschitz continuous gradient, R {\displaystyle R} is a convex, lower semicontinuous function which is possibly nondifferentiable, and H {\displaystyle {\mathcal {H}}} is some set, typically a Hilbert space. The usual criterion of x {\displaystyle x} minimizes F ( x ) + R ( x ) {\displaystyle F(x)+R(x)} if and only if ∇ ( F + R ) ( x ) = 0 {\displaystyle \nabla (F+R)(x)=0} in the convex, differentiable setting is now replaced by 0 ∈ ∂ ( F + R ) ( x ) , {\displaystyle 0\in \partial (F+R)(x),} where ∂ φ {\displaystyle \partial \varphi } denotes the subdifferential of a real-valued, convex function φ {\displaystyle \varphi } . Given a convex function φ : H → R {\displaystyle \varphi :{\mathcal {H}}\to \mathbb {R} } an important operator to consider is its proximal operator prox φ : H → H {\displaystyle \operatorname {prox} _{\varphi }:{\mathcal {H}}\to {\mathcal {H}}} defined by prox φ ( u ) = arg min x ∈ H φ ( x ) + 1 2 ‖ u − x ‖ 2 2 , {\displaystyle \operatorname {prox} _{\varphi }(u)=\operatorname {arg} \min _{x\in {\mathcal {H}}}\varphi (x)+{\frac {1}{2}}\|u-x\|_{2}^{2},} which is well-defined because of the strict convexity of the ℓ 2 {\displaystyle \ell _{2}} norm. The proximal operator can be seen as a generalization of a projection. We see that the proximity operator is important because x ∗ {\displaystyle x^{}} is a minimizer to the problem min x ∈ H F ( x ) + R ( x ) {\displaystyle \min _{x\in {\mathcal {H}}}F(x)+R(x)} if and only if x ∗ = prox γ R ( x ∗ − γ ∇ F ( x ∗ ) ) , {\displaystyle x^{}=\operatorname {prox} _{\gamma R}\left(x^{}-\gamma \nabla F(x^{})\right),} where γ > 0 {\displaystyle \gamma >0} is any positive real number. === Moreau decomposition === One important technique related to proximal gradient methods is the Moreau decomposition, which decomposes the identity operator as the sum of two proximity operators. Namely, let φ : X → R {\displaystyle \varphi :{\mathcal {X}}\to \mathbb {R} } be a lower semicontinuous, convex function on a vector space X {\displaystyle {\mathcal {X}}} . We define its Fenchel conjugate φ ∗ : X → R {\displaystyle \varphi ^{}:{\mathcal {X}}\to \mathbb {R} } to be the function φ ∗ ( u ) := sup x ∈ X ⟨ x , u ⟩ − φ ( x ) . {\displaystyle \varphi ^{}(u):=\sup _{x\in {\mathcal {X}}}\langle x,u\rangle -\varphi (x).} The general form of Moreau's decomposition states that for any x ∈ X {\displaystyle x\in {\mathcal {X}}} and any γ > 0 {\displaystyle \gamma >0} that x = prox γ φ ( x ) + γ prox φ ∗ / γ ( x / γ ) , {\displaystyle x=\operatorname {prox} _{\gamma \varphi }(x)+\gamma \operatorname {prox} _{\varphi ^{}/\gamma }(x/\gamma ),} which for γ = 1 {\displaystyle \gamma =1} implies that x = prox φ ( x ) + prox φ ∗ ( x ) {\displaystyle x=\operatorname {prox} _{\varphi }(x)+\operatorname {prox} _{\varphi ^{}}(x)} . The Moreau decomposition can be seen to be a generalization of the usual orthogonal decomposition of a vector space, analogous with the fact that proximity operators are generalizations of projections. In certain situations it may be easier to compute the proximity operator for the conjugate φ ∗ {\displaystyle \varphi ^{}} instead of the function φ {\displaystyle \varphi } , and therefore the Moreau decomposition can be applied. This is the case for group lasso. == Lasso regularization == Consider the regularized empirical risk minimization problem with square loss and with the ℓ 1 {\displaystyle \ell _{1}} norm as the regularization penalty: min w ∈ R d 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 + λ ‖ w ‖ 1 , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}+\lambda \|w\|_{1},} where x i ∈ R d and y i ∈ R . {\displaystyle x_{i}\in \mathbb {R} ^{d}{\text{ and }}y_{i}\in \mathbb {R} .} The ℓ 1 {\displaystyle \ell _{1}} regularization problem is sometimes referred to as lasso (least absolute shrinkage and selection operator). Such ℓ 1 {\displaystyle \ell _{1}} regularization problems are interesting because they induce sparse solutions, that is, solutions w {\displaystyle w} to the minimization problem have relatively few nonzero components. Lasso can be seen to be a convex relaxation of the non-convex problem min w ∈ R d 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 + λ ‖ w ‖ 0 , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}+\lambda \|w\|_{0},} where ‖ w ‖ 0 {\displaystyle \|w\|_{0}} denotes the ℓ 0 {\displaystyle \ell _{0}} "norm", which is the number of nonzero entries of the vector w {\displaystyle w} . Sparse solutions are of particular interest in learning theory for interpretability of results: a sparse solution can identify a small number of important factors. === Solving for L1 proximity operator === For simplicity we restrict our attention to the problem where λ = 1 {\displaystyle \lambda =1} . To solve the problem min w ∈ R d 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 + ‖ w ‖ 1 , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}+\|w\|_{1},} we consider our objective function in two parts: a convex, differentiable term F ( w ) = 1 n ∑ i = 1 n ( y i − ⟨ w , x i ⟩ ) 2 {\displaystyle F(w)={\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}} and a convex function R ( w ) = ‖ w ‖ 1 {\displaystyle R(w)=\|w\|_{1}} . Note that R {\displaystyle R} is not strictly convex. Let us compute the proximity operator for R ( w ) {\displaystyle R(w)} . First we find an alternative characterization of the proximity operator prox R ( x ) {\displaystyle \operatorname {prox} _{R}(x)} as follows: u = prox R ( x ) ⟺ 0 ∈ ∂ ( R ( u ) + 1 2 ‖ u − x ‖ 2 2 ) ⟺ 0 ∈ ∂ R ( u ) + u − x ⟺ x − u ∈ ∂ R ( u ) . {\displaystyle {\begin{aligned}u=\operatorname {prox} _{R}(x)\iff &0\in \partial \left(R(u)+{\frac {1}{2}}\|u-x\|_{2}^{2}\right)\\\iff &0\in \partial R(u)+u-x\\\iff &x-u\in \partial R(u).\end{aligned}}} For R ( w ) = ‖ w ‖ 1 {\displaystyle R(w)=\|w\|_{1}} it is easy to compute ∂ R ( w ) {\displaystyle \partial R(w)} : the i {\displaystyle i} th entry of ∂ R ( w ) {\displaystyle \partial R(w)} is precisely ∂ | w i | = { 1 , w i > 0 − 1 , w i < 0 [ − 1 , 1 ] , w i = 0. {\displaystyle \partial |w_{i}|={\begin{cases}1,&w_{i}>0\\-1,&w_{i}<0\\\left[-1,1\right],&w_{i}=0.\end{cases}}} Using the recharacterization of the proximity operator given above, for the choice of R ( w ) = ‖ w ‖ 1 {\displaystyle R(w)=\|w\|_{1}} and γ > 0 {\displaystyle \gamma >0} we have that prox γ R ( x ) {\displaystyle \operatorname {prox} _{\gamma R}(x)} is defined entrywise by ( prox γ R ( x ) ) i = { x i − γ , x i > γ 0 , | x i | ≤ γ x i + γ , x i < − γ , {\displaystyle \left(\operatorname {prox} _{\gamma R}(x)\right)_{i}={\begin{cases}x_{i}-\gamma ,&x_{i}>\gamma \\0,&|x_{i}|\leq \gamma \\x_{i}+\gamma ,&x_{i}<-\gamma ,\end{cases}}} which is known as the soft thresholding operator S γ ( x ) = prox γ ‖ ⋅ ‖ 1 ( x ) {\displaystyle S_{\gamma }(x)=\operatorname {prox} _{\gamma \|\cdot \|_{1}}(x)} . === Fixed point iterative schemes === To finally solve the lasso problem we consider the fixed point equation shown earlier: x ∗ = prox γ R ( x ∗ − γ ∇ F ( x ∗ ) ) . {\displaystyle x^{}=\operatorname {prox} _{\gamma R}\left(x^{}-\gamma \nabla F(x^{})\right).} Given that we have computed the form of the proximity operator explicitly, then we can define a standard fixed point iteration procedure. Namely, fix some initial w 0 ∈ R d {\displaystyle w^{0}\in \mathbb {R} ^{d}} , and for k = 1 , 2 , … {\displaystyle k=1,2,\ldots } define w k + 1 = S γ ( w k − γ ∇ F ( w k ) ) . {\displaystyle w^{k+1}=S_{\gamma }\left(w^{k}-\gamma \nabla F\l
Fuzzy concept
A fuzzy concept is an idea of which the boundaries of application can vary considerably according to context or conditions, instead of being fixed once and for all. That means the idea is somewhat vague or imprecise. Yet it is not unclear or meaningless. It has a definite meaning, which can often be made more exact with further elaboration and specification — including a closer definition of the context in which the concept is used. The inverse of a "fuzzy concept" is a "crisp concept" (i.e. a precise concept). Fuzzy concepts are often used to navigate imprecision in the real world, when precise information is not available and an approximate indication is sufficient to be helpful. Although the linguist George Philip Lakoff already defined the semantics of a fuzzy concept in 1973 (inspired by an unpublished 1971 paper by Eleanor Rosch,) the term "fuzzy concept" rarely received a standalone entry in dictionaries, handbooks and encyclopedias. Sometimes it was defined in encyclopedia articles on fuzzy logic, or it was simply equated with a mathematical “fuzzy set”. A fuzzy concept can be "fuzzy" for many different reasons in different contexts. This makes it harder to provide a precise definition that covers all cases. Paradoxically, the definition of fuzzy concepts may itself be somewhat "fuzzy". Lotfi A. Zadeh, known as "the father of fuzzy logic", claimed that "vagueness connotes insufficient specificity, whereas fuzziness connotes unsharpness of class boundaries". Not all scholars agree. With increasing academic literature on the subject, the term "fuzzy concept" is now more widely recognized as a philosophical, linguistic or scientific category, and the study of the characteristics of fuzzy concepts and fuzzy language is known as fuzzy semantics. “Fuzzy logic” has become a generic term for many different kinds of many-valued logics, and is applied in many different areas of research, computer programming and industrial design. For engineers, "Fuzziness is imprecision or vagueness of definition." For computer scientists, a fuzzy concept is an idea which is "to an extent applicable" in a situation. It means that the concept can have gradations of significance or unsharp (variable) boundaries of application — a "fuzzy statement" is a statement which is true "to some extent", and that extent can often be represented by a scaled value (a score). For mathematicians, a "fuzzy concept" is usually a fuzzy set or a combination of such sets (see fuzzy mathematics and fuzzy set theory). In cognitive linguistics, the things that belong to a "fuzzy category" exhibit gradations of family resemblance, and the borders of the category are not clearly defined. Through most of the 20th century, the idea of reasoning with fuzzy concepts faced considerable resistance from Western academic elites. They did not want to endorse the use of imprecise concepts in research or argumentation, and they often regarded fuzzy logic with suspicion, derision or even hostility. That may partly explain why the idea of a "fuzzy concept" did not get a separate entry in encyclopedias, handbooks and dictionaries. Yet although people might not be aware of it, the use of fuzzy concepts has risen gigantically in all walks of life from the 1970s onward. That is mainly due to advances in electronic engineering, fuzzy mathematics and digital computer programming. The new technology allows very complex inferences about "variations on a theme" to be anticipated and fixed in a program. The Perseverance Mars rover, a driverless NASA vehicle used to explore the Jezero crater on the planet Mars, features fuzzy logic programming that steers it through rough terrain. Similarly, to the North, the Chinese Mars rover Zhurong used fuzzy logic algorithms to calculate its travel route in Utopia Planitia from sensor data. New neuro-fuzzy computational methods make it possible for machines to identify, measure, adjust and respond to fine gradations of significance with great precision. It means that practically useful concepts can be coded, sharply defined, and applied to all kinds of tasks, even if ordinarily these concepts are never exactly defined. Nowadays engineers, statisticians and programmers often represent fuzzy concepts mathematically, using fuzzy logic, fuzzy values, fuzzy variables and fuzzy sets (see also fuzzy set theory). Fuzzy logic is not "woolly thinking", but a "precise logic of imprecision" which reasons with graded concepts and gradations of truth. Fuzzy concepts and fuzzy logic often play a significant role in artificial intelligence programming, for example because they can model human cognitive processes more easily than other methods. == Origins == Vagueness and fuzziness have probably always been a part of human experience. In the West, ancient texts show that philosophers and scientists were already thinking critically about this in classical antiquity. Most often, they regarded vagueness as a problem: as an obstacle to clear thinking, as a source of confusion, or as an evasive tactic. It got in the way of providing clear orientation, guidance, direction and leadership. Therefore, vagueness became associated with a hermeneutic of suspicion — it was considered as something to avoid, as something undesirable. By contrast, in the ancient Chinese tradition of Daoist thought of Laozi and Zhuang Zhou, "vagueness is not regarded with suspicion, but is simply an acknowledged characteristic of the world around us" — a subject for meditation and a source of insight. === Sorites paradox === The ancient Sorites paradox raised the logical problem, of how we could exactly define the threshold at which a change in quantitative gradation turns into a qualitative or categorical difference. With some physical processes, this threshold seems relatively easy to identify. For example, water turns into steam at 100 °C or 212 °F. Of course, the boiling point depends partly on atmospheric pressure, which decreases at higher altitudes; it is also affected by the level of humidity — in that sense, the boiling point is "somewhat fuzzy", because it can vary under different conditions. Nevertheless, for every altitude, level of air pressure and degree of humidity, we can predict accurately what the boiling point will be, if we know the relevant conditions. With many other processes and gradations, however, the point of change is much more difficult to locate, and remains somewhat vague. Thus, the boundaries between qualitatively different things may be unsharp: we know that there are boundaries, but we cannot define them exactly. For example, to identify "the oldest city in the world", we have to define what counts as a city, and at what point a growing human settlement becomes a city. === The continuum fallacy and Loki's wager === According to the modern idea of the continuum fallacy, the fact that a statement is to an extent vague, does not automatically mean that it has no validity. The question then arises, of how (by what method or approach) we could ascertain and define the validity that the fuzzy statement does have. The Nordic myth of Loki's wager suggested that concepts that lack precise meanings or lack precise boundaries of application cannot be operated with, because they evade any clear definition. However, the 20th-century idea of "fuzzy concepts" proposes that "somewhat vague terms" can be operated with, because we can explicate and define the variability of their application — by assigning numbers to gradations of applicability. This idea sounds simple enough, but it had large implications. === Precursors and pioneers === In Western civilization, the intellectual recognition of fuzzy concepts has been traced back to a diversity of famous and less well-known thinkers, including (among many others) Eubulides, Epicurus, Plato, Cicero, William Ockham and John Buridan, Georg Wilhelm Friedrich Hegel, Karl Marx and Friedrich Engels, Friedrich Nietzsche, William James, Hugh MacColl, Charles S. Peirce, Hans Reichenbach, Carl Gustav Hempel, Max Black, Arto Salomaa, Ludwig Wittgenstein, Jan Łukasiewicz, Emil Leon Post, Alfred Tarski, Georg Cantor, Nicolai A. Vasiliev, Kurt Gödel, Stanisław Jaśkowski, Willard Van Orman Quine, George J. Klir, Petr Hájek, Joseph Goguen, Ronald R. Yager, Enrique Héctor Ruspini, Jan Pavelka, Didier Dubois, Bernadette Bouchon-Meunier, and Donald Knuth. Across at least two and a half millennia, all of them had something to say about graded concepts with unsharp boundaries. This suggests at least that the awareness of the existence of concepts with "fuzzy" characteristics, in one form or another, has a very long history in human thought. Quite a few 20th century logicians, mathematicians and philosophers also tried to analyze the characteristics of fuzzy concepts as a recognized species, sometimes with the aid of some kind of many-valued logic or substructural logic. An early attempt in the post-WW2 era to create a mathematical theory of sets with gradations of