In computer science, in particular in formal language theory, a quotient automaton can be obtained from a given nondeterministic finite automaton by joining some of its states. The quotient recognizes a superset of the given automaton; in some cases, handled by the Myhill–Nerode theorem, both languages are equal. == Formal definition == A (nondeterministic) finite automaton is a quintuple A = ⟨Σ, S, s0, δ, Sf⟩, where: Σ is the input alphabet (a finite, non-empty set of symbols), S is a finite, non-empty set of states, s0 is the initial state, an element of S, δ is the state-transition relation: δ ⊆ S × Σ × S, and Sf is the set of final states, a (possibly empty) subset of S. A string a1...an ∈ Σ is recognized by A if there exist states s1, ..., sn ∈ S such that ⟨si-1,ai,si⟩ ∈ δ for i=1,...,n, and sn ∈ Sf. The set of all strings recognized by A is called the language recognized by A; it is denoted as L(A). For an equivalence relation ≈ on the set S of A’s states, the quotient automaton A/≈ = ⟨Σ, S/≈, [s0], δ/≈, Sf/≈⟩ is defined by the input alphabet Σ being the same as that of A, the state set S/≈ being the set of all equivalence classes of states from S, the start state [s0] being the equivalence class of A’s start state, the state-transition relation δ/≈ being defined by δ/≈([s],a,[t]) if δ(s,a,t) for some s ∈ [s] and t ∈ [t], and the set of final states Sf/≈ being the set of all equivalence classes of final states from Sf. The process of computing A/≈ is also called factoring A by ≈. == Example == For example, the automaton A shown in the first row of the table is formally defined by ΣA = {0,1}, SA = {a,b,c,d}, sA0 = a, δA = { ⟨a,1,b⟩, ⟨b,0,c⟩, ⟨c,0,d⟩ }, and SAf = { b,c,d }. It recognizes the finite set of strings { 1, 10, 100 }; this set can also be denoted by the regular expression "1+10+100". The relation (≈) = { ⟨a,a⟩, ⟨a,b⟩, ⟨b,a⟩, ⟨b,b⟩, ⟨c,c⟩, ⟨c,d⟩, ⟨d,c⟩, ⟨d,d⟩ }, more briefly denoted as a≈b,c≈d, is an equivalence relation on the set {a,b,c,d} of automaton A’s states. Building the quotient of A by that relation results in automaton C in the third table row; it is formally defined by ΣC = {0,1}, SC = {a,c}, sC0 = a, δC = { ⟨a,1,a⟩, ⟨a,0,c⟩, ⟨c,0,c⟩ }, and SCf = { a,c }. It recognizes the finite set of all strings composed of arbitrarily many 1s, followed by arbitrarily many 0s, i.e. { ε, 1, 10, 100, 1000, ..., 11, 110, 1100, 11000, ..., 111, ... }; this set can also be denoted by the regular expression "10". Informally, C can be thought of resulting from A by glueing state a onto state b, and glueing state c onto state d. The table shows some more quotient relations, such as B = A/a≈b, and D = C/a≈c. == Properties == For every automaton A and every equivalence relation ≈ on its states set, L(A/≈) is a superset of (or equal to) L(A). Given a finite automaton A over some alphabet Σ, an equivalence relation ≈ can be defined on Σ by x ≈ y if ∀ z ∈ Σ: xz ∈ L(A) ↔ yz ∈ L(A). By the Myhill–Nerode theorem, A/≈ is a deterministic automaton that recognizes the same language as A. As a consequence, the quotient of A by every refinement of ≈ also recognizes the same language as A.
Simulation noise
Simulation noise is a function that creates a divergence-free vector field. This signal can be used in artistic simulations for the purpose of increasing the perception of extra detail. The function can be calculated in three dimensions by dividing the space into a regular lattice grid. With each edge is associated a random value, indicating a rotational component of material revolving around the edge. By following rotating material into and out of faces, one can quickly sum the flux passing through each face of the lattice. Flux values at lattice faces are then interpolated to create a field value for all positions. Perlin noise is the earliest form of lattice noise, which has become very popular in computer graphics. Perlin Noise is not suited for simulation because it is not divergence-free. Noises based on lattices, such as simulation noise and Perlin noise, are often calculated at different frequencies and summed together to form band-limited fractal signals. Other approaches developed later that use vector calculus identities to produce divergence free fields, such as "Curl-Noise" as suggested by Rook Bridson, and "Divergence-Free Noise" due to Ivan DeWolf. These often require calculation of lattice noise gradients, which sometimes are not readily available. A naive implementation would call a lattice noise function several times to calculate its gradient, resulting in more computation than is strictly necessary. Unlike these noises, simulation noise has a geometric rationale in addition to its mathematical properties. It simulates vortices scattered in space, to produce its pleasing aesthetic. == Curl noise == The vector field is created as follows, for every point (x,y,z) in the space a vector field G is created, every component x, y and z of the vector field (Gx, Gy, Gz) is defined by a 3D perlin or simplex noise function with x, y and z as parameters. The partial derivative of Gx, Gy, and Gz respect to x, y and z is obtained with the gradient of the perlin or simplex noise by finite differences of implicit calculation inside the simplex noise. The partial derivatives are used to calculate F as the curl of G given by F = ( ∂ G z ∂ y − ∂ G y ∂ z , ∂ G x ∂ z − ∂ G z ∂ x , ∂ G y ∂ x − ∂ G x ∂ y ) {\displaystyle F=({\frac {\partial Gz}{\partial y}}-{\frac {\partial Gy}{\partial z}},{\frac {\partial Gx}{\partial z}}-{\frac {\partial Gz}{\partial x}},{\frac {\partial Gy}{\partial x}}-{\frac {\partial Gx}{\partial y}})} == Bitangent noise == This method is based in the fact that the curl of the gradient of scalar field is zero and the identity that expand the divergence of a cross product of two vectors A and B as the difference of the dot products of each vector with the curl of the other: ∇ × ( ∇ φ ) = 0 . {\displaystyle \nabla \times (\nabla \varphi )=\mathbf {0} .} ∇ ⋅ ( A × B ) = ( ∇ × A ) ⋅ B − A ⋅ ( ∇ × B ) {\displaystyle \nabla \cdot (\mathbf {A} \times \mathbf {B} )=\ (\nabla {\times }\mathbf {A} )\cdot \mathbf {B} \,-\,\mathbf {A} \cdot (\nabla {\times }\mathbf {B} )} which means that if the curl of both vector fields is zero then the divergence of the product of two vectors that are the gradients of scalar fields is zero too. This result in a divergence free vector field by construction only calling two noise functions to create the scalar fields. The vector field es created as follows, two scalar fields are calculated ϕ {\displaystyle \phi } and ψ {\displaystyle \psi } using 3D perlin or simplex noise functions, then the gradients A and B of each of this fields is calculated, the cross product of A and B gives a divergence free vector field. == Signed distance noise == The vector field is created based on a closed and differentiable implicit surface S = F(x,y,z) = 0. For every point in the space, frequently outside or near the surface, we get a vector g that is normal to the surface, this is the gradient of S or the partial derivatives respect to x, y and z, this vector is not unitary, but we can get a unitary normal n by dividing each component of the point by the magnitude of the gradient g. Outside of the surface all these normals point away from the surface. g = ∇ F ( x , y , z ) = ( ∂ F ∂ x , ∂ F ∂ y , ∂ F ∂ z ) {\displaystyle g=\nabla F(x,y,z)=\left({\frac {\partial F}{\partial x}},{\frac {\partial F}{\partial y}},{\frac {\partial F}{\partial z}}\right)} n = g ( x , y , z ) ‖ ∇ F ( x , y , z ) ‖ {\displaystyle \mathbf {n} ={\frac {g(x,y,z)}{\|\nabla F(x,y,z)\|}}} ‖ ∇ F ( x , y , z ) ‖ = ( ∂ F ∂ x ) 2 + ( ∂ F ∂ y ) 2 + ( ∂ F ∂ z ) 2 {\displaystyle \|\nabla F(x,y,z)\|={\sqrt {\left({\frac {\partial F}{\partial x}}\right)^{2}+\left({\frac {\partial F}{\partial y}}\right)^{2}+\left({\frac {\partial F}{\partial z}}\right)^{2}}}} Afterwards we calculate a scalar value p for that point in the space using a 3D perlin or simplex noise function. Now we create a vector field V = pn pointing outside of the surface. The curl of this vector field gives the direction in every point in the space where the particles should move. S D N = ( ∂ V z ∂ y − ∂ V y ∂ z , ∂ V x ∂ z − ∂ V z ∂ x , ∂ V y ∂ x − ∂ V x ∂ y ) {\displaystyle SDN=({\frac {\partial Vz}{\partial y}}-{\frac {\partial Vy}{\partial z}},{\frac {\partial Vx}{\partial z}}-{\frac {\partial Vz}{\partial x}},{\frac {\partial Vy}{\partial x}}-{\frac {\partial Vx}{\partial y}})} By construction this vector SDN will point in a tangent direction to an isosurface at the level of the signed distance to the original surface and can be used to confine the movements of the particles to stay in that surface.
Fuzzy cognitive map
A fuzzy cognitive map (FCM) is a cognitive map within which the relations between the elements (e.g. concepts, events, project resources) of a "mental landscape" can be used to compute the "strength of impact" of these elements. Fuzzy cognitive maps were introduced by Bart Kosko. Robert Axelrod introduced cognitive maps as a formal way of representing social scientific knowledge and modeling decision making in social and political systems, then brought in the computation. == Details == Fuzzy cognitive maps are signed fuzzy directed graphs. Spreadsheets or tables are used to map FCMs into matrices for further computation. FCM is a technique used for causal knowledge acquisition and representation, it supports causal knowledge reasoning process and belong to the neuro-fuzzy system that aim at solving decision making problems, modeling and simulate complex systems. Learning algorithms have been proposed for training and updating FCMs weights mostly based on ideas coming from the field of Artificial Neural Networks. Adaptation and learning methodologies used to adapt the FCM model and adjust its weights. Kosko and Dickerson (Dickerson & Kosko, 1994) suggested the Differential Hebbian Learning (DHL) to train FCM. There have been proposed algorithms based on the initial Hebbian algorithm; others algorithms come from the field of genetic algorithms, swarm intelligence and evolutionary computation. Learning algorithms are used to overcome the shortcomings that the traditional FCM present i.e. decreasing the human intervention by suggested automated FCM candidates; or by activating only the most relevant concepts every execution time; or by making models more transparent and dynamic. Fuzzy cognitive maps (FCMs) have gained considerable research interest due to their ability in representing structured knowledge and model complex systems in various fields. This growing interest led to the need for enhancement and making more reliable models that can better represent real situations. A first simple application of FCMs is described in a book of William R. Taylor, where the war in Afghanistan and Iraq is analyzed. In Bart Kosko's book Fuzzy Thinking, several Hasse diagrams illustrate the use of FCMs. As an example, one FCM quoted from Rod Taber describes 11 factors of the American cocaine market and the relations between these factors. For computations, Taylor uses pentavalent logic (scalar values out of {-1,-0.5,0,+0.5,+1}). That particular map of Taber uses trivalent logic (scalar values out of {-1,0,+1}). Taber et al. also illustrate the dynamics of map fusion and give a theorem on the convergence of combination in a related article. While applications in social sciences introduced FCMs to the public, they are used in a much wider range of applications, which all have to deal with creating and using models of uncertainty and complex processes and systems. Examples: In business FCMs can be used for product planning and decision support. In economics, FCMs support the use of game theory in more complex settings. In education for modeling Critical Success Factors of Learning Management Systems. In medical applications to model systems, provide diagnosis, develop decision support systems and medical assessment. In engineering for modeling and control mainly of complex systems and reliability engineering In project planning FCMs help to analyze the mutual dependencies between project resources. In robotics FCMs support machines to develop fuzzy models of their environments and to use these models to make crisp decisions. In computer assisted learning FCMs enable computers to check whether students understand their lessons. In expert systems a few or many FCMs can be aggregated into one FCM in order to process estimates of knowledgeable persons. In IT project management, a FCM-based methodology helps to success modelling, risk analysis and assessment, IT scenarios FCMappers is an international online community for the analysis and the visualization of fuzzy cognitive maps. FCMappers offer support for starting with FCM and also provide a Microsoft Excel-based tool that is able to check and analyse FCMs. The output is saved as Pajek file and can be visualized within third party software like Pajek, Visone, etc. They also offer to adapt the software to specific research needs. Additional FCM software tools, such as Mental Modeler, have recently been developed as a decision-support tool for use in social science research, collaborative decision-making, and natural resource planning.
Computational law
Computational law is the branch of legal informatics concerned with the automation of legal reasoning. What distinguishes Computational Law systems from other instances of legal technology is their autonomy, i.e. the ability to answer legal questions without additional input from human legal experts. While there are many possible applications of Computational Law, the primary focus of work in the field today is compliance management, i.e. the development and deployment of computer systems capable of assessing, facilitating, or enforcing compliance with rules and regulations. Some systems of this sort already exist. TurboTax is a good example. And the potential is particularly significant now due to recent technological advances – including the prevalence of the Internet in human interaction and the proliferation of embedded computer systems (such as smart phones, self-driving cars, and robots). There are also applications that do not involve governmental laws. The regulations can just as well be the terms of contracts (e.g. delivery schedules, insurance covenants, real estate transactions, financial agreements). They can be the policies of corporations (e.g. constraints on travel, expenditure reporting, pricing rules). They can even be the rules of games (embodied in computer game playing systems). == History == Speculation about potential benefits to legal practice through applying methods from computational science and AI research to automate parts of the law date back at least to the middle 1940s. Further, AI and law and computational law do not seem easily separable, as perhaps most of AI research focusing on the law and its automation appears to utilize computational methods. The forms that speculation took are multiple and not all related in ways to readily show closeness to one another. This history will sketch them as they were, attempting to show relationships where they can be found to have existed. By 1949, a minor academic field aiming to incorporate electronic and computational methods to legal problems had been founded by American legal scholars, called jurimetrics. Though broadly said to be concerned with the application of the "methods of science" to the law, these methods were actually of a quite specifically defined scope. Jurimetrics was to be "concerned with such matters as the quantitative analysis of judicial behavior, the application of communication and information theory to legal expression, the use of mathematical logic in law, the retrieval of legal data by electronic and mechanical means, and the formulation of a calculus of legal predictability". These interests led in 1959 to the founding a journal, Modern Uses of Logic in Law, as a forum wherein articles would be published about the applications of techniques such as mathematical logic, engineering, statistics, etc. to the legal study and development. In 1966, this Journal was renamed as Jurimetrics. Today, however, the journal and meaning of jurimetrics seems to have broadened far beyond what would fit under the areas of applications of computers and computational methods to law. Today the journal not only publishes articles on such practices as found in computational law, but has broadened jurimetrical concerns to mean also things like the use of social science in law or the "policy implications [of] and legislative and administrative control of science". Independently in 1958, at the Conference for the Mechanization of Thought held at the National Physical Laboratory in Teddington, Middlesex, UK, the French jurist Lucien Mehl presented a paper both on the benefits of using computational methods for law and on the potential means to use such methods to automate law for a discussion that included AI luminaries like Marvin Minsky. Mehl believed that the law could by automated by two basic distinct, though not wholly separable, types of machine. These were the "documentary or information machine", which would provide the legal researcher quick access to relevant case precedents and legal scholarship, and the "consultation machine", which would be "capable of answering any question put to it over a vast field of law". The latter type of machine would be able to basically do much of a lawyer's job by simply giving the "exact answer to a [legal] problem put to it". By 1970, Mehl's first type of machine, one that would be able to retrieve information, had been accomplished but there seems to have been little consideration of further fruitful intersections between AI and legal research. There were, however, still hopes that computers could model the lawyer's thought processes through computational methods and then apply that capacity to solve legal problems, thus automating and improving legal services via increased efficiency as well as shedding light on the nature of legal reasoning. By the late 1970s, computer science and the affordability of computer technology had progressed enough that the retrieval of "legal data by electronic and mechanical means" had been achieved by machines fitting Mehl's first type and were in common use in American law firms. During this time, research focused on improving the goals of the early 1970s occurred, with programs like Taxman being worked on in order to both bring useful computer technology into the law as practical aids and to help specify the exact nature of legal concepts. Nonetheless, progress on the second type of machine, one that would more fully automate the law, remained relatively inert. Research into machines that could answer questions in the way that Mehl's consultation machine would picked up somewhat in the late 1970s and 1980s. A 1979 convention in Swansea, Wales marked the first international effort solely to focus upon applying artificial intelligence research to legal problems in order to "consider how computers can be used to discover and apply the legal norms embedded within the written sources of the law". Considerable progress on the development of the second type of machine was made in the following decade, with the development of a variety of expert systems. According to Thorne McCarty, "these systems all have the following characteristics: They do backward chaining inference from a specified goal; they ask questions to elicit information from the user; and they produce a suggested answer along with a trace of the supporting legal rules." According to Prakken and Sartor the representation of the British Nationality Act as a logic program, which introduced this approach, was "hugely influential for the development of computational representations of legislation, showing how logic programming enables intuitively appealing representations that can be directly deployed to generate automatic inferences". In 2021, this work received the Inaugural CodeX Prize as "one of the first and best-known works in computational law, and one of the most widely cited papers in the field." In a 1988 review of Anne Gardner's book An Artificial Intelligence Approach to Legal Reasoning (1987), the Harvard academic legal scholar and computer scientist Edwina Rissland wrote that "She plays, in part, the role of pioneer; artificial intelligence ("AI") techniques have not yet been widely applied to perform legal tasks. Therefore, Gardner, and this review, first describe and define the field, then demonstrate a working model in the domain of contract offer and acceptance." Eight years after the Swansea conference had passed, and still AI and law researchers merely trying to delineate the field could be described by their own kind as "pioneer[s]". In the 1990s and early 2000s more progress occurred. Computational research generated insights for law. The First International Conference on AI and the Law occurred in 1987, but it is in the 1990s and 2000s that the biannual conference began to build up steam and to delve more deeply into the issues involved with work intersecting computational methods, AI, and law. Classes began to be taught to undergraduates on the uses of computational methods to automating, understanding, and obeying the law. Further, by 2005, a team largely composed of Stanford computer scientists from the Stanford Logic group had devoted themselves to studying the uses of computational techniques to the law. Computational methods in fact advanced enough that members of the legal profession began in the 2000s to both analyze, predict and worry about the potential future of computational law and a new academic field of computational legal studies seems to be now well established. As insight into what such scholars see in the law's future due in part to computational law, here is quote from a recent conference about the "New Normal" for the legal profession: "Over the last 5 years, in the fallout of the Great Recession, the legal profession has entered the era of the New Normal. Notably, a series of forces related to technological change, globalization, and the pressure to do more with less (in both corpo
Ensemble averaging (machine learning)
In machine learning, ensemble averaging is the process of creating multiple models (typically artificial neural networks) and combining them to produce a desired output, as opposed to creating just one model. Ensembles of models often outperform individual models, as the various errors of the ensemble constituents "average out". == Overview == Ensemble averaging is one of the simplest types of committee machines. Along with boosting, it is one of the two major types of static committee machines. In contrast to standard neural network design, in which many networks are generated but only one is kept, ensemble averaging keeps the less satisfactory networks, but with less weight assigned to their outputs. The theory of ensemble averaging relies on two properties of artificial neural networks: In any network, the bias can be reduced at the cost of increased variance In a group of networks, the variance can be reduced at no cost to the bias. This is known as the bias–variance tradeoff. Ensemble averaging creates a group of networks, each with low bias and high variance, and combines them to form a new network which should theoretically exhibit low bias and low variance. Hence, this can be thought of as a resolution of the bias–variance tradeoff. The idea of combining experts can be traced back to Pierre-Simon Laplace. == Method == The theory mentioned above gives an obvious strategy: create a set of experts with low bias and high variance, and average them. Generally, what this means is to create a set of experts with varying parameters; frequently, these are the initial synaptic weights of a neural network, although other factors (such as learning rate, momentum, etc.) may also be varied. Some authors recommend against varying weight decay and early stopping. The steps are therefore: Generate N experts, each with their own initial parameters (these values are usually sampled randomly from a distribution) Train each expert separately Combine the experts and average their values. Alternatively, domain knowledge may be used to generate several classes of experts. An expert from each class is trained, and then combined. A more complex version of ensemble average views the final result not as a mere average of all the experts, but rather as a weighted sum. If each expert is y i {\displaystyle y_{i}} , then the overall result y ~ {\displaystyle {\tilde {y}}} can be defined as: y ~ ( x ; α ) = ∑ j = 1 p α j y j ( x ) {\displaystyle {\tilde {y}}(\mathbf {x} ;\mathbf {\alpha } )=\sum _{j=1}^{p}\alpha _{j}y_{j}(\mathbf {x} )} where α {\displaystyle \mathbf {\alpha } } is a set of weights. The optimization problem of finding alpha is readily solved through neural networks, hence a "meta-network" where each "neuron" is in fact an entire neural network can be trained, and the synaptic weights of the final network is the weight applied to each expert. This is known as a linear combination of experts. It can be seen that most forms of neural network are some subset of a linear combination: the standard neural net (where only one expert is used) is simply a linear combination with all α j = 0 {\displaystyle \alpha _{j}=0} and one α k = 1 {\displaystyle \alpha _{k}=1} . A raw average is where all α j {\displaystyle \alpha _{j}} are equal to some constant value, namely one over the total number of experts. A more recent ensemble averaging method is negative correlation learning, proposed by Y. Liu and X. Yao. This method has been widely used in evolutionary computing. == Benefits == The resulting committee is almost always less complex than a single network that would achieve the same level of performance The resulting committee can be trained more easily on smaller datasets The resulting committee often has improved performance over any single model The risk of overfitting is lessened, as there are fewer parameters (e.g. neural network weights) which need to be set.
Content Threat Removal
Content Threat Removal (CTR) is a cybersecurity technology intended to defeat the threat posed by handling digital content in the cyberspace. Unlike other defenses, including antivirus software and sandboxed execution, CTR does not rely on being able to detect threats. Similar to Content Disarm and Reconstruction, CTR is designed to remove the threat without knowing whether it has done so and acts without knowing if data contains a threat or not. Detection strategies work by detecting unsafe content, and then blocking or removing that content. Content that is deemed safe is delivered to its destination. In contrast, Content Threat Removal assumes all data is hostile and delivers none of it to the destination, regardless of whether it is actually hostile. Although no data is delivered, the business information carried by the data is delivered using new data created for the purpose. == Threat == Advanced attacks continuously defeat defenses that are based on detection. These are often referred to as zero-day attacks, because as soon as they are discovered attack detection mechanisms must be updated to identify and neutralize the attack, and until they are, all systems are unprotected. These attacks succeed because attackers find new ways of evading detection. Polymorphic code can be used to evade the detection of known unsafe data and sandbox detection allows attacks to evade dynamic analysis. == Method == A Content Threat Removal defence works by intercepting data on its way to its destination. The business information carried by the data is extracted and the data is discarded. Then entirely new, clean and safe data is built to carry the information to its destination. The effect of building new data to carry the business information is that any unsafe elements of the original data are left behind and discarded. This includes executable data, macros, scripts and malformed data that trigger vulnerabilities in applications. While CTR is a form of content transformation, not all transformations provide a complete defence against the content threat. == Applicability == CTR is applicable to user-to-user traffic, such as email and chat, and machine-to-machine traffic, such as web services. Data transfers can be intercepted by in-line application layer proxies and these can transform the way information content is delivered to remove any threat. CTR works by extracting business information from data and it is not possible to extract information from executable code. This means CTR is not directly applicable to web browsing, since most web pages are code. It can, however, be applied to content that is downloaded from, and uploaded to, websites. Although most web pages cannot be transformed to render them safe, web browsing can be isolated and the remote access protocols used to reach the isolated environment can be subjected to CTR. CTR provides a solution to the problem of stegware. It naturally removes detectable steganography and eliminates symbiotic and permutation steganography through normalisation.
Degree of truth
In classical logic, propositions are typically unambiguously considered as being true or false. For instance, the proposition one is both equal and not equal to itself is regarded as simply false, being contrary to the Law of Noncontradiction; while the proposition one is equal to one is regarded as simply true, by the Law of Identity. However, some mathematicians, computer scientists, and philosophers have been attracted to the idea that a proposition might be more or less true, rather than wholly true or wholly false. Consider this pizza is hot. In mathematics, this idea can be developed in terms of fuzzy logic. In computer science, it has found application in artificial intelligence. In philosophy, the idea has proved particularly appealing in the case of vagueness. Degrees of truth is an important concept in law. The term is an older concept than conditional probability. Instead of determining the objective probability, only a subjective assessment is defined. In adjudicative processes, 'substantive truth' is distinct from 'formal legal truth' which comes in four degrees: hearsay, balance of probabilities, proven beyond reasonable doubt and absolute truth (knowledge reserved unto God).