Hubert Dreyfus was a critic of artificial intelligence research. In a series of papers and books, including Alchemy and AI (1965), What Computers Can't Do (1972; 1979; 1992) and Mind over Machine (1986), he presented a skeptical and cautious assessment of AI's progress and a critique of the philosophical foundations of the field. Dreyfus' objections are discussed in most introductions to the philosophy of artificial intelligence, including Russell & Norvig (2021), a standard AI textbook, and in Fearn (2007), a survey of contemporary philosophy. Dreyfus argued that human intelligence and expertise depend primarily on yet-to-be understood informal and unconscious processes rather than symbolic manipulation and that these essentially human skills cannot be fully captured in formal rules. His critique was based on the insights of modern continental philosophers such as Merleau-Ponty and Heidegger, and was directed at the first wave of AI research which tried to reduce intelligence to high level formal symbols. When Dreyfus' ideas were first introduced in the mid-1960s, they were met in the AI community with ridicule and outright hostility. By the 1980s, however, some of his perspectives were rediscovered by researchers working in robotics and the new field of connectionism—approaches that were called "sub-symbolic" at the time because they eschewed early AI research's emphasis on high level symbols. In the 21st century, "sub-symbolic" artificial neural networks and other statistics-based approaches to machine learning were highly successful. Historian and AI researcher Daniel Crevier wrote: "time has proven the accuracy and perceptiveness of some of Dreyfus's comments." Dreyfus said in 2007, "I figure I won and it's over—they've given up." == Dreyfus' critique == === The grandiose promises of artificial intelligence === In Alchemy and AI (1965) and What Computers Can't Do (1972), Dreyfus summarized the history of artificial intelligence and ridiculed the unbridled optimism that permeated the field. For example, Herbert A. Simon, following the success of his program General Problem Solver (1957), predicted that by 1967: A computer would be world champion in chess. A computer would discover and prove an important new mathematical theorem. Most theories in psychology will take the form of computer programs. The press dutifully reported these predictions of the imminent arrival of machine intelligence. Dreyfus felt that this optimism was unwarranted and, in 1965, argued forcefully that predictions like these would not come true. He would eventually be proven right. Pamela McCorduck explains Dreyfus' position: A great misunderstanding accounts for public confusion about thinking machines, a misunderstanding perpetrated by the unrealistic claims researchers in AI have been making, claims that thinking machines are already here, or at any rate, just around the corner. These predictions were based on the success of the cognitive revolution, which promoted an "information processing" model of the mind. It was articulated by Newell and Simon in their physical symbol systems hypothesis, and later expanded into a philosophical position known as computationalism by philosophers such as Jerry Fodor and Hilary Putnam. In AI, the approach is now called symbolic AI or "GOFAI". Dreyfus argued that "symbolic AI" was the latest version of the ancient program of rationalism in philosophy. Rationalism had come under heavy criticism in the 20th century from philosophers like Martin Heidegger and Edmund Husserl. The mind, according to modern continental philosophy, is not "rationalist" and is nothing like a digital computer. Cognitivism led early AI researchers to believe that they had successfully simulated the essential process of human thought, thus it seemed a short step to producing fully intelligent machines. Dreyfus' last paper detailed the ongoing history of the "first step fallacy", where AI researchers tend to wildly extrapolate initial success as promising, perhaps even guaranteeing, wild future successes. === Dreyfus' four assumptions of artificial intelligence research === In Alchemy and AI and What Computers Can't Do, Dreyfus identified four philosophical assumptions, at least one of which he deems necessary for AI to succeed. "In each case," Dreyfus writes, "the assumption is taken by workers in AI as an axiom, guaranteeing results, whereas it is, in fact, one hypothesis among others, to be tested by the success of such work." Dreyfus argues that AI would be impossible without accepting at least one of these four assumptions: The biological assumption The brain processes information in discrete operations by way of some biological equivalent of on/off switches. In the early days of research into neurology, scientists found that neurons fire in all-or-nothing pulses. Several researchers, such as Walter Pitts and Warren McCulloch, speculated with great confidence that neurons functioned similarly to the way Boolean logic gates operate, and so could be imitated by electronic circuitry at the level of the neuron. When digital computers became widely used in the early 50s, this argument was extended to suggest that the brain was a vast physical symbol system, manipulating the binary symbols of zero and one. Dreyfus was able to refute the biological assumption by citing research in neurology that suggested that the action and timing of neuron firing had analog components. But Daniel Crevier observes that "few still held that belief in the early 1970s, and nobody argued against Dreyfus" about the biological assumption. The psychological assumption The mind can be viewed as a device operating on bits of information according to formal rules. He refuted this assumption by showing that much of what we know about the world consists of complex attitudes or tendencies that make us lean towards one interpretation over another. He argued that, even when we use explicit symbols, we are using them against an unconscious and informal background including commonsense knowledge and that without this background our symbols cease to mean anything. This background, in Dreyfus' view, was not implemented in individual brains as explicit individual symbols with explicit individual meanings. The epistemological assumption All knowledge can be formalized. This concerns the philosophical issue of epistemology, or the study of knowledge. Even if we agree that the psychological assumption is false, AI researchers could still argue (as AI founder John McCarthy has) that it is possible for a symbol processing machine to represent all knowledge, regardless of whether human beings represent knowledge the same way. Dreyfus argued that there is no justification for this assumption, since so much of human knowledge is not symbolic or even expressible using formal constructs. The ontological assumption The world consists of independent facts that can be represented by independent symbols AI researchers (and futurists and science fiction writers) often assume that there is no limit to formal, scientific knowledge, because they assume that any phenomenon in the universe can be described by symbols or scientific theories. This assumes that everything that exists can be understood as objects, properties of objects, classes of objects, relations of objects, and so on: precisely those things that can be described by logic, language and mathematics. The study of being or existence is called ontology, and so Dreyfus calls this the ontological assumption. If this is false, then it raises doubts about what we can ultimately know and what intelligent machines will ultimately be able to help us to do. === Knowing-how vs. knowing-that: the primacy of intuition === In Mind Over Machine (1986), written (with his brother) during the heyday of expert systems, Dreyfus analyzed the difference between human expertise and the programs that claimed to capture it. This expanded on ideas from What Computers Can't Do, where he had made a similar argument criticizing the "cognitive simulation" school of AI research practiced by Allen Newell and Herbert A. Simon in the 1960s. Dreyfus argued that human problem solving and expertise depend on our background sense of the context, of what is important and interesting given the situation, rather than on the process of searching through combinations of possibilities to find what we need. Dreyfus would describe it in 1986 as the difference between "knowing-that" and "knowing-how", based on Heidegger's distinction of present-at-hand and ready-to-hand. Knowing-that is our conscious, step-by-step problem solving abilities. We use these skills when we encounter a difficult problem that requires us to stop, step back and search through ideas one at time. At moments like this, the ideas become very precise and simple: they become context free symbols, which we manipulate using logic and language. These are the skills that Newell and Simon had demonstrated with both psy
Kolmogorov–Arnold Networks
Kolmogorov–Arnold Networks (KANs) are a type of artificial neural network architecture inspired by the Kolmogorov–Arnold representation theorem, also known as the superposition theorem. Unlike traditional multilayer perceptrons (MLPs), which rely on fixed activation functions and linear weights, KANs replace each weight with a learnable univariate function, often represented using splines. == History == KANs (Kolmogorov–Arnold Networks) were proposed by Liu et al. (2024) as a generalization of the Kolmogorov–Arnold representation theorem (KART), aiming to outperform MLPs in small-scale AI and scientific tasks. Before KANs, numerous studies explored KART's connections to neural networks or used it as a basis for designing new network architectures. In the 1980s and 1990s, early research applied KART to neural network design. Kůrková et al. (1992), Hecht-Nielsen (1987), and Nees (1994) established theoretical foundations for multilayer networks based on KART. Igelnik et al. (2003) introduced the Kolmogorov Spline Network using cubic splines to model complex functions. Sprecher (1996, 1997) introduced numerical methods for building network layers, while Nakamura et al. (1993) created activation functions with guaranteed approximation accuracy. These works linked KART's theoretical potential with practical neural network implementation. KART has also been used in other computational and theoretical fields. Coppejans (2004) developed nonparametric regression estimators using B-splines, Bryant (2008) applied it to high-dimensional image tasks, Liu (2015) investigated theoretical applications in optimal transport and image encryption, and more recently, Polar and Poluektov (2021) used Urysohn operators for efficient KART construction, while Fakhoury et al. (2022) introduced ExSpliNet, integrating KART with probabilistic trees and multivariate B-splines for improved function approximation. == Architecture == KANs are based on the Kolmogorov–Arnold representation theorem, which was linked to the 13th Hilbert problem. Given x = ( x 1 , x 2 , … , x n ) {\displaystyle x=(x_{1},x_{2},\dots ,x_{n})} consisting of n variables, a multivariate continuous function f ( x ) {\displaystyle f(x)} can be represented as: f ( x ) = f ( x 1 , … , x n ) = ∑ q = 1 2 n + 1 Φ q ( ∑ p = 1 n φ q , p ( x p ) ) {\displaystyle f(x)=f(x_{1},\dots ,x_{n})=\sum _{q=1}^{2n+1}\Phi _{q}\left(\sum _{p=1}^{n}\varphi _{q,p}(x_{p})\right)} (1) This formulation contains two nested summations: an outer and an inner sum. The outer sum ∑ q = 1 2 n + 1 {\displaystyle \sum _{q=1}^{2n+1}} aggregates 2 n + 1 {\displaystyle 2n+1} terms, each involving a function Φ q : R → R {\displaystyle \Phi _{q}:\mathbb {R} \to \mathbb {R} } . The inner sum ∑ p = 1 n {\displaystyle \sum _{p=1}^{n}} computes n terms for each q, where each term φ q , p : [ 0 , 1 ] → R {\displaystyle \varphi _{q,p}:[0,1]\to \mathbb {R} } is a continuous function of the single variable x p {\displaystyle x_{p}} . The inner continuous functions φ q , p {\displaystyle \varphi _{q,p}} are universal, independent of f {\displaystyle f} , while the outer functions Φ q {\displaystyle \Phi _{q}} depend on the specific function f {\displaystyle f} being represented. The representation (1) holds for all multivariate functions f {\displaystyle f} as proved in . If f {\displaystyle f} is continuous, then the outer functions Φ q {\displaystyle \Phi _{q}} are continuous; if f {\displaystyle f} is discontinuous, then the corresponding Φ q {\displaystyle \Phi _{q}} are generally discontinuous, while the inner functions φ q , p {\displaystyle \varphi _{q,p}} remain the same universal functions. Liu et al. proposed the name KAN. A general KAN network consisting of L layers takes x to generate the output as: K A N ( x ) = ( Φ L − 1 ∘ Φ L − 2 ∘ ⋯ ∘ Φ 1 ∘ Φ 0 ) x {\displaystyle \mathrm {KAN} (x)=(\Phi ^{L-1}\circ \Phi ^{L-2}\circ \cdots \circ \Phi ^{1}\circ \Phi ^{0})x} (3) Here, Φ l {\displaystyle \Phi ^{l}} is the function matrix of the l-th KAN layer or a set of pre-activations. Let i denote the neuron of the l-th layer and j the neuron of the (l+1)-th layer. The activation function φ j , i l {\displaystyle \varphi _{j,i}^{l}} connects (l, i) to (l+1, j): φ j , i l , l = 0 , … , L − 1 , i = 1 , … , n l , j = 1 , … , n l + 1 {\displaystyle \varphi _{j,i}^{l},\quad l=0,\dots ,L-1,\;i=1,\dots ,n_{l},\;j=1,\dots ,n_{l+1}} (4) where nl is the number of nodes of the l-th layer. Thus, the function matrix Φ l {\displaystyle \Phi ^{l}} can be represented as an n l + 1 × n l {\displaystyle n_{l+1}\times n_{l}} matrix of activations: x l + 1 = ( φ 1 , 1 l ( ⋅ ) φ 1 , 2 l ( ⋅ ) ⋯ φ 1 , n l l ( ⋅ ) φ 2 , 1 l ( ⋅ ) φ 2 , 2 l ( ⋅ ) ⋯ φ 2 , n l l ( ⋅ ) ⋮ ⋮ ⋱ ⋮ φ n l + 1 , 1 l ( ⋅ ) φ n l + 1 , 2 l ( ⋅ ) ⋯ φ n l + 1 , n l l ( ⋅ ) ) x l {\displaystyle x^{l+1}={\begin{pmatrix}\varphi _{1,1}^{l}(\cdot )&\varphi _{1,2}^{l}(\cdot )&\cdots &\varphi _{1,n_{l}}^{l}(\cdot )\\\varphi _{2,1}^{l}(\cdot )&\varphi _{2,2}^{l}(\cdot )&\cdots &\varphi _{2,n_{l}}^{l}(\cdot )\\\vdots &\vdots &\ddots &\vdots \\\varphi _{n_{l+1},1}^{l}(\cdot )&\varphi _{n_{l+1},2}^{l}(\cdot )&\cdots &\varphi _{n_{l+1},n_{l}}^{l}(\cdot )\end{pmatrix}}x^{l}} == Implementations == To make the KAN layers optimizable, the inner function is formed by the combination of spline and basic functions as the formula: φ ( x ) = w b b ( x ) + w s spline ( x ) {\displaystyle \varphi (x)=w_{b}\,b(x)+w_{s}\,{\text{spline}}(x)} where b ( x ) {\displaystyle b(x)} is the basic function, usually defined as s i l u ( x ) = x / ( 1 + e x ) {\displaystyle silu(x)=x/(1+e^{x})} and w b {\displaystyle w_{b}} is the base weight matrix. Also, w s {\displaystyle w_{s}} is the spline weight matrix and spline ( x ) {\displaystyle {\text{spline}}(x)} is the spline function. The spline function can be a sum of B-splines. spline ( x ) = ∑ i c i B i ( x ) {\displaystyle {\text{spline}}(x)=\sum _{i}c_{i}B_{i}(x)} Many studies suggested to use other polynomial and curve functions instead of B-spline to create new KAN variants. == Functions used == The choice of functional basis strongly influences the performance of KANs. Common function families include: B-splines: Provide locality, smoothness, and interpretability; they are the most widely used in current implementations. RBFs (include Gaussian RBFs): Capture localized features in data and are effective in approximating functions with non-linear or clustered structures. Chebyshev polynomials: Offer efficient approximation with minimized error in the maximum norm, making them useful for stable function representation. Rational function: Useful for approximating functions with singularities or sharp variations, as they can model asymptotic behavior better than polynomials. Fourier series: Capture periodic patterns effectively and are particularly useful in domains such as physics-informed machine learning. Wavelet functions (DoG, Mexican hat, Morlet, and Shannon): Used for feature extraction as they can capture both high-frequency and low-frequency data components. Piecewise linear functions: Provide efficient approximation for multivariate functions in KANs. == Usage == In some modern neural architectures like convolutional neural networks (CNNs), recurrent neural networks (RNNs), and Transformers, KANs are typically used as drop-in substitutes for MLP layers. Despite KANs' general-purpose design, researchers have created and used them for a number of tasks: Scientific machine learning (SciML): Function fitting, partial differential equations (PDEs) and physical/mathematical laws. Continual learning: KANs better preserve previously learned information during incremental updates, avoiding catastrophic forgetting due to the locality of spline adjustments. Graph neural networks: Extensions such as Kolmogorov–Arnold Graph Neural Networks (KA-GNNs) integrate KAN modules into message-passing architectures, showing improvements in molecular property prediction tasks. Sensor data processing: Kolmogorov–Arnold Networks (KANs) have recently been applied to sensor data processing due to their ability to model complex nonlinear relationships with relatively few parameters and improved interpretability compared to conventional multilayer perceptrons. Applications include industrial soft sensors, biomedical signal analysis, remote sensing, and environmental monitoring systems. == Drawbacks == KANs can be computationally intensive and require a large number of parameters due to their use of polynomial functions to capture data.
Memetic algorithm
In computer science and operations research, a memetic algorithm (MA) is an extension of an evolutionary algorithm (EA) that aims to accelerate the evolutionary search for the optimum. An EA is a metaheuristic that reproduces the basic principles of biological evolution as a computer algorithm in order to solve challenging optimization or planning tasks, at least approximately. An MA uses one or more suitable heuristics or local search techniques to improve the quality of solutions generated by the EA and to speed up the search. The effects on the reliability of finding the global optimum depend on both the use case and the design of the MA. Memetic algorithms represent one of the recent growing areas of research in evolutionary computation. The term MA is now widely used as a synergy of evolutionary or any population-based approach with separate individual learning or local improvement procedures for problem search. Quite often, MAs are also referred to in the literature as Baldwinian evolutionary algorithms, Lamarckian EAs, cultural algorithms, or genetic local search. == Introduction == Inspired by both Darwinian principles of natural evolution and Dawkins' notion of a meme, the term memetic algorithm (MA) was introduced by Pablo Moscato in his technical report in 1989 where he viewed MA as being close to a form of population-based hybrid genetic algorithm (GA) coupled with an individual learning procedure capable of performing local refinements. The metaphorical parallels, on the one hand, to Darwinian evolution and, on the other hand, between memes and domain specific (local search) heuristics are captured within memetic algorithms thus rendering a methodology that balances well between generality and problem specificity. This two-stage nature makes them a special case of dual-phase evolution. The basic idea behind an MA is to combine the advantages of a global search performed by an EA (or another global search method) with the local refinement provided by one or more local search techniques, while avoiding their drawbacks. The main disadvantage of EAs is that, when searching in the vicinity of an optimum, they perform poorly in determining the exact position of that optimum. The downside of local search methods lies simply in the locality of their search relative to the chosen starting point. The combination of these two classes of methods aims to merge global and local search so that the advantages of both approaches can be leveraged. The idea of this approach can be illustrated by the search for the highest mountain in the Alps. A local search method would climb one of the mountains near the starting point, ignoring Mont Blanc as long as the starting point is not in its vicinity. An EA, on the other hand, will likely only find Mont Blanc after examining many other mountains, valleys, and hills, and then it will have difficulty identifying the summit cross. From the perspective of an MA’s global search procedure, however, only the summits of hills and mountains are seen, and its search is limited to finding the best summit. The open question is whether the additional effort required for the local search is worthwhile. This depends not only on the design of the MA but also on the specific application and the local search methods used. In the context of complex optimization, many different instantiations of memetic algorithms have been reported across a wide range of application domains, in general, converging to high-quality solutions more efficiently than their conventional evolutionary counterparts. In general, using the ideas of memetics within a computational framework is called memetic computing or memetic computation (MC). With MC, the traits of universal Darwinism are more appropriately captured. Viewed in this perspective, MA is a more constrained notion of MC. More specifically, MA covers one area of MC, in particular dealing with areas of evolutionary algorithms that marry other deterministic refinement techniques for solving optimization problems. MC extends the notion of memes to cover conceptual entities of knowledge-enhanced procedures or representations. == Theoretical Background == The no-free-lunch theorems of optimization and search state that all optimization strategies are equally effective with respect to the set of all optimization problems. Conversely, this means that one can expect the following: The more efficiently an algorithm solves a problem or class of problems, the less general it is and the more problem-specific knowledge it builds on. This insight leads directly to the recommendation to complement generally applicable metaheuristics with application-specific methods or heuristics, which fits well with the concept of MAs. == The development of MAs == === 1st generation === Pablo Moscato characterized an MA as follows: "Memetic algorithms are a marriage between a population-based global search and the heuristic local search made by each of the individuals. ... The mechanisms to do local search can be to reach a local optimum or to improve (regarding the objective cost function) up to a predetermined level." And he emphasizes "I am not constraining an MA to a genetic representation.". This original definition of MA although encompasses characteristics of cultural evolution (in the form of local refinement) in the search cycle, it may not qualify as a true evolving system according to universal Darwinism, since all the core principles of inheritance/memetic transmission, variation, and selection are missing. This suggests why the term MA stirred up criticisms and controversies among researchers when first introduced. The following pseudo code would correspond to this general definition of an MA: Pseudo code Procedure Memetic Algorithm Initialize: Generate an initial population, evaluate the individuals and assign a quality value to them; while Stopping conditions are not satisfied do Evolve a new population using stochastic search operators. Evaluate all individuals in the population and assign a quality value to them. Select the subset of individuals, Ω i l {\displaystyle \Omega _{il}} , that should undergo the individual improvement procedure. for each individual in Ω i l {\displaystyle \Omega _{il}} do Perform individual learning using meme(s) with frequency or probability of f i l {\displaystyle f_{il}} , with an intensity of t i l {\displaystyle t_{il}} . Proceed with Lamarckian or Baldwinian learning. end for end while Lamarckian learning in this context means to update the chromosome according to the improved solution found by the individual learning step, while Baldwinian learning leaves the chromosome unchanged and uses only the improved fitness. This pseudo code leaves open which steps are based on the fitness of the individuals and which are not. In question are the evolving of the new population and the selection of Ω i l {\displaystyle \Omega _{il}} . Since most MA implementations are based on EAs, the pseudo code of a corresponding representative of the first generation is also given here, following Krasnogor: Pseudo code Procedure Memetic Algorithm Based on an EA Initialization: t = 0 {\displaystyle t=0} ; // Initialization of the generation counter Randomly generate an initial population P ( t ) {\displaystyle P(t)} ; Compute the fitness f ( p ) ∀ p ∈ P ( t ) {\displaystyle f(p)\ \ \forall p\in P(t)} ; while Stopping conditions are not satisfied do Selection: Accordingly to f ( p ) {\displaystyle f(p)} choose a subset of P ( t ) {\displaystyle P(t)} and store it in M ( t ) {\displaystyle M(t)} ; Offspring: Recombine and mutate individuals p ∈ M ( t ) {\displaystyle p\in M(t)} and store them in M ′ ( t ) {\displaystyle M'(t)} ; Learning: Improve p ′ {\displaystyle p'} by local search or heuristic ∀ p ′ ∈ M ′ ( t ) {\displaystyle \forall p'\in M'(t)} ; Evaluation: Compute the fitness f ( p ′ ) ∀ p ′ ∈ M ′ ( t ) {\displaystyle f(p')\ \ \forall p'\in M'(t)} ; if Lamarckian learning then Update chromosome of p ′ {\displaystyle p'} according to improvement ∀ p ′ ∈ M ′ ( t ) {\displaystyle \forall p'\in M'(t)} ; fi New generation: Generate P ( t + 1 ) {\displaystyle P(t+1)} by selecting some individuals from P ( t ) {\displaystyle P(t)} and M ′ ( t ) {\displaystyle M'(t)} ; t = t + 1 {\displaystyle t=t+1} ; // Increment the generation counter end while Return best individual p ∈ P ( t − 1 ) {\displaystyle p\in P(t-1)} as result; There are some alternatives for this MA scheme. For example: All or some of the initial individuals may be improved by the meme(s). The parents may be locally improved instead of the offspring. Instead of all offspring, only a randomly selected or fitness-dependent fraction may undergo local improvement. The latter requires the evaluation of the offspring in M ′ ( t ) {\displaystyle M'(t)} prior to the Learning step. === 2nd generation === Multi-meme, hyper-heuristic and meta-Lamarckian MA are referred to as second generation MA exhibiting the principles of me
Gaussian adaptation
Gaussian adaptation (GA), also called normal or natural adaptation (NA) is an evolutionary algorithm designed for the maximization of manufacturing yield due to statistical deviation of component values of signal processing systems. In short, GA is a stochastic adaptive process where a number of samples of an n-dimensional vector x[xT = (x1, x2, ..., xn)] are taken from a multivariate Gaussian distribution, N(m, M), having mean m and moment matrix M. The samples are tested for fail or pass. The first- and second-order moments of the Gaussian restricted to the pass samples are m and M. The outcome of x as a pass sample is determined by a function s(x), 0 < s(x) < q ≤ 1, such that s(x) is the probability that x will be selected as a pass sample. The average probability of finding pass samples (yield) is P ( m ) = ∫ s ( x ) N ( x − m ) d x {\displaystyle P(m)=\int s(x)N(x-m)\,dx} Then the theorem of GA states: For any s(x) and for any value of P < q, there always exist a Gaussian p. d. f. [ probability density function ] that is adapted for maximum dispersion. The necessary conditions for a local optimum are m = m and M proportional to M. The dual problem is also solved: P is maximized while keeping the dispersion constant (Kjellström, 1991). Proofs of the theorem may be found in the papers by Kjellström, 1970, and Kjellström & Taxén, 1981. Since dispersion is defined as the exponential of entropy/disorder/average information it immediately follows that the theorem is valid also for those concepts. Altogether, this means that Gaussian adaptation may carry out a simultaneous maximisation of yield and average information (without any need for the yield or the average information to be defined as criterion functions). The theorem is valid for all regions of acceptability and all Gaussian distributions. It may be used by cyclic repetition of random variation and selection (like the natural evolution). In every cycle a sufficiently large number of Gaussian distributed points are sampled and tested for membership in the region of acceptability. The centre of gravity of the Gaussian, m, is then moved to the centre of gravity of the approved (selected) points, m. Thus, the process converges to a state of equilibrium fulfilling the theorem. A solution is always approximate because the centre of gravity is always determined for a limited number of points. It was used for the first time in 1969 as a pure optimization algorithm making the regions of acceptability smaller and smaller (in analogy to simulated annealing, Kirkpatrick 1983). Since 1970 it has been used for both ordinary optimization and yield maximization. == Natural evolution and Gaussian adaptation == It has also been compared to the natural evolution of populations of living organisms. In this case s(x) is the probability that the individual having an array x of phenotypes will survive by giving offspring to the next generation; a definition of individual fitness given by Hartl 1981. The yield, P, is replaced by the mean fitness determined as a mean over the set of individuals in a large population. Phenotypes are often Gaussian distributed in a large population and a necessary condition for the natural evolution to be able to fulfill the theorem of Gaussian adaptation, with respect to all Gaussian quantitative characters, is that it may push the centre of gravity of the Gaussian to the centre of gravity of the selected individuals. This may be accomplished by the Hardy–Weinberg law. This is possible because the theorem of Gaussian adaptation is valid for any region of acceptability independent of the structure (Kjellström, 1996). In this case the rules of genetic variation such as crossover, inversion, transposition etcetera may be seen as random number generators for the phenotypes. So, in this sense Gaussian adaptation may be seen as a genetic algorithm. == How to climb a mountain == Mean fitness may be calculated provided that the distribution of parameters and the structure of the landscape is known. The real landscape is not known, but figure below shows a fictitious profile (blue) of a landscape along a line (x) in a room spanned by such parameters. The red curve is the mean based on the red bell curve at the bottom of figure. It is obtained by letting the bell curve slide along the x-axis, calculating the mean at every location. As can be seen, small peaks and pits are smoothed out. Thus, if evolution is started at A with a relatively small variance (the red bell curve), then climbing will take place on the red curve. The process may get stuck for millions of years at B or C, as long as the hollows to the right of these points remain, and the mutation rate is too small. If the mutation rate is sufficiently high, the disorder or variance may increase and the parameter(s) may become distributed like the green bell curve. Then the climbing will take place on the green curve, which is even more smoothed out. Because the hollows to the right of B and C have now disappeared, the process may continue up to the peaks at D. But of course the landscape puts a limit on the disorder or variability. Besides — dependent on the landscape — the process may become very jerky, and if the ratio between the time spent by the process at a local peak and the time of transition to the next peak is very high, it may as well look like a punctuated equilibrium as suggested by Gould (see Ridley). == Computer simulation of Gaussian adaptation == Thus far the theory only considers mean values of continuous distributions corresponding to an infinite number of individuals. In reality however, the number of individuals is always limited, which gives rise to an uncertainty in the estimation of m and M (the moment matrix of the Gaussian). And this may also affect the efficiency of the process. Unfortunately very little is known about this, at least theoretically. The implementation of normal adaptation on a computer is a fairly simple task. The adaptation of m may be done by one sample (individual) at a time, for example m(i + 1) = (1 – a) m(i) + ax where x is a pass sample, and a < 1 a suitable constant so that the inverse of a represents the number of individuals in the population. M may in principle be updated after every step y leading to a feasible point x = m + y according to: M(i + 1) = (1 – 2b) M(i) + 2byyT, where yT is the transpose of y and b << 1 is another suitable constant. In order to guarantee a suitable increase of average information, y should be normally distributed with moment matrix μ2M, where the scalar μ > 1 is used to increase average information (information entropy, disorder, diversity) at a suitable rate. But M will never be used in the calculations. Instead we use the matrix W defined by WWT = M. Thus, we have y = Wg, where g is normally distributed with the moment matrix μU, and U is the unit matrix. W and WT may be updated by the formulas W = (1 – b)W + bygT and WT = (1 – b)WT + bgyT because multiplication gives M = (1 – 2b)M + 2byyT, where terms including b2 have been neglected. Thus, M will be indirectly adapted with good approximation. In practice it will suffice to update W only W(i + 1) = (1 – b)W(i) + bygT. This is the formula used in a simple 2-dimensional model of a brain satisfying the Hebbian rule of associative learning; see the next section (Kjellström, 1996 and 1999). The figure below illustrates the effect of increased average information in a Gaussian p.d.f. used to climb a mountain Crest (the two lines represent the contour line). Both the red and green cluster have equal mean fitness, about 65%, but the green cluster has a much higher average information making the green process much more efficient. The effect of this adaptation is not very salient in a 2-dimensional case, but in a high-dimensional case, the efficiency of the search process may be increased by many orders of magnitude. == The evolution in the brain == In the brain the evolution of DNA-messages is supposed to be replaced by an evolution of signal patterns and the phenotypic landscape is replaced by a mental landscape, the complexity of which will hardly be second to the former. The metaphor with the mental landscape is based on the assumption that certain signal patterns give rise to a better well-being or performance. For instance, the control of a group of muscles leads to a better pronunciation of a word or performance of a piece of music. In this simple model it is assumed that the brain consists of interconnected components that may add, multiply and delay signal values. A nerve cell kernel may add signal values, a synapse may multiply with a constant and An axon may delay values. This is a basis of the theory of digital filters and neural networks consisting of components that may add, multiply and delay signalvalues and also of many brain models, Levine 1991. In the figure below the brain stem is supposed to deliver Gaussian distributed signal patterns. This may be possible since certai
Santa Fe Trail problem
The Santa Fe Trail problem is a genetic programming exercise in which artificial ants search for food pellets according to a programmed set of instructions. The layout of food pellets in the Santa Fe Trail problem has become a standard for comparing different genetic programming algorithms and solutions. One method for programming and testing algorithms on the Santa Fe Trail problem is by using the NetLogo application. There is at least one case of a student creating a Lego robotic ant to solve the problem.
Vanish (computer science)
Vanish was a project to "give users control over the lifetime of personal data stored on the web." It was led by Roxana Geambasu at the University of Washington. The project proposed to allow a user to enter information to send across the internet, thereby relinquishing control of it. However, the user can include an "expiration date," after which the information is no longer usable by anyone who may have a copy of it, even the creator. The Vanish approach was found to be vulnerable to a Sybil attack and thus insecure by a team called Unvanish from the University of Texas, University of Michigan, and Princeton. == Theory == Vanish acts by automating the encryption of information entered by the user with an encryption key that is unknown to the user. Along with the information the user enters, the user also enters metadata concerning how long the information should remain available. The system then encrypts the information but does not store either the encryption key or the original information. Instead, it breaks up the decryption key into smaller components that are disseminated across distributed hash tables, or DHTs, via the Internet. The DHTs refresh information within their nodes on a set schedule unless configured to make the information persistent. The time delay entered by the user in the metadata controls how long the DHTs should allow the information to persist, but once that time period is over, the DHTs will reuse those nodes, making the information about the decryption stored irretrievable. As long as the decryption key may be reassembled from the DHTs, the information is retrievable. However, once the period entered by the user has lapsed, the information is no longer recoverable, as the user never possessed the decryption key. == Implementation == Vanish currently exists as a Firefox plug-in which allows a user to enter text into either a standard Gmail email or Facebook message and choose to send the message via Vanish. The message is then encrypted and sent via the normal networking pathways through the cloud to the recipient. The recipient must have the same Firefox plug-in to decrypt the message. The plugin accesses BitTorrent DHTs, which have 8-hour lifespans. This means the user may select an expiration date for the message in increments of 8 hours. After the expiration of the user-defined time span, the information in the DHT is overwritten, thereby eliminating the key. While both the user and recipient may have copies of the original encrypted message, the key used to turn it back into plain text is now gone. Although this particular instance of the data has become inaccessible, it's important to note that the information can always be saved by other means before expiration (copied or even via screen shots) and published again.
Inverted pendulum
An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is unstable and falls over without additional help. It can be suspended stably in this inverted position by using a control system to monitor the angle of the pole and move the pivot point horizontally back under the center of mass when it starts to fall over, keeping it balanced. The inverted pendulum is a classic problem in dynamics and control theory and is used as a benchmark for testing control strategies. It is often implemented with the pivot point mounted on a cart that can move horizontally under control of an electronic servo system as shown in the photo; this is called a cart and pole apparatus. Most applications limit the pendulum to 1 degree of freedom by affixing the pole to an axis of rotation. Whereas a normal pendulum is stable when hanging downward, an inverted pendulum is inherently unstable, and must be actively balanced in order to remain upright; this can be done either by applying a torque at the pivot point, by moving the pivot point horizontally as part of a feedback system, changing the rate of rotation of a mass mounted on the pendulum on an axis parallel to the pivot axis and thereby generating a net torque on the pendulum, or by oscillating the pivot point vertically. A simple demonstration of moving the pivot point in a feedback system is achieved by balancing an upturned broomstick on the end of one's finger. A second type of inverted pendulum is a tiltmeter for tall structures, which consists of a wire anchored to the bottom of the foundation and attached to a float in a pool of oil at the top of the structure that has devices for measuring movement of the neutral position of the float away from its original position. == Overview == A pendulum with its bob hanging directly below the support pivot is at a stable equilibrium point, where it remains motionless because there is no torque on the pendulum. If displaced from this position, it experiences a restoring torque that returns it toward the equilibrium position. A pendulum with its bob in an inverted position, supported on a rigid rod directly above the pivot, 180° from its stable equilibrium position, is at an unstable equilibrium point. At this point again there is no torque on the pendulum, but the slightest displacement away from this position causes a gravitation torque on the pendulum that accelerates it away from equilibrium, causing it to fall over. In order to stabilize a pendulum in this inverted position, a feedback control system can be used, which monitors the pendulum's angle and moves the position of the pivot point sideways when the pendulum starts to fall over, to keep it balanced. The inverted pendulum is a classic problem in dynamics and control theory and is widely used as a benchmark for testing control algorithms (PID controllers, state-space representation, neural networks, fuzzy control, genetic algorithms, etc.). Variations on this problem include multiple links, allowing the motion of the cart to be commanded while maintaining the pendulum, and balancing the cart-pendulum system on a see-saw. The inverted pendulum is related to rocket or missile guidance, where the center of gravity is located behind the center of drag causing aerodynamic instability. The understanding of a similar problem can be shown by simple robotics in the form of a balancing cart. Balancing an upturned broomstick on the end of one's finger is a simple demonstration, and the problem is solved by self-balancing personal transporters such as the Segway PT, the self-balancing hoverboard and the self-balancing unicycle. Another way that an inverted pendulum may be stabilized, without any feedback or control mechanism, is by oscillating the pivot rapidly up and down. This is called Kapitza's pendulum. If the oscillation is sufficiently strong (in terms of its acceleration and amplitude) then the inverted pendulum can recover from perturbations in a strikingly counterintuitive manner. If the driving point moves in simple harmonic motion, the pendulum's motion is described by the Mathieu equation. == Equations of motion == The equations of motion of inverted pendulums are dependent on what constraints are placed on the motion of the pendulum. Inverted pendulums can be created in various configurations resulting in a number of Equations of Motion describing the behavior of the pendulum. === Stationary pivot point === In a configuration where the pivot point of the pendulum is fixed in space, the equation of motion is similar to that for an uninverted pendulum. The equation of motion below assumes no friction or any other resistance to movement, a rigid massless rod, and the restriction to 2-dimensional movement. θ ¨ − g ℓ sin θ = 0 {\displaystyle {\ddot {\theta }}-{g \over \ell }\sin \theta =0} Where θ ¨ {\displaystyle {\ddot {\theta }}} is the angular acceleration of the pendulum, g {\displaystyle g} is the standard gravity on the surface of the Earth, ℓ {\displaystyle \ell } is the length of the pendulum, and θ {\displaystyle \theta } is the angular displacement measured from the equilibrium position. When θ ¨ {\displaystyle {\ddot {\theta }}} added to both sides, it has the same sign as the angular acceleration term: θ ¨ = g ℓ sin θ {\displaystyle {\ddot {\theta }}={g \over \ell }\sin \theta } Thus, the inverted pendulum accelerates away from the vertical unstable equilibrium in the direction initially displaced, and the acceleration is inversely proportional to the length. Tall pendulums fall more slowly than short ones. Derivation using torque and moment of inertia: The pendulum is assumed to consist of a point mass, of mass m {\displaystyle m} , affixed to the end of a massless rigid rod, of length ℓ {\displaystyle \ell } , attached to a pivot point at the end opposite the point mass. The net torque of the system must equal the moment of inertia times the angular acceleration: τ n e t = I θ ¨ {\displaystyle {\boldsymbol {\tau }}_{\mathrm {net} }=I{\ddot {\theta }}} The torque due to gravity providing the net torque: τ n e t = m g ℓ sin θ {\displaystyle {\boldsymbol {\tau }}_{\mathrm {net} }=mg\ell \sin \theta \,\!} Where θ {\displaystyle \theta \ } is the angle measured from the inverted equilibrium position. The resulting equation: I θ ¨ = m g ℓ sin θ {\displaystyle I{\ddot {\theta }}=mg\ell \sin \theta \,\!} The moment of inertia for a point mass: I = m R 2 {\displaystyle I=mR^{2}} In the case of the inverted pendulum the radius is the length of the rod, ℓ {\displaystyle \ell } . Substituting in I = m ℓ 2 {\displaystyle I=m\ell ^{2}} m ℓ 2 θ ¨ = m g ℓ sin θ {\displaystyle m\ell ^{2}{\ddot {\theta }}=mg\ell \sin \theta \,\!} Mass and ℓ 2 {\displaystyle \ell ^{2}} is divided from each side resulting in: θ ¨ = g ℓ sin θ {\displaystyle {\ddot {\theta }}={g \over \ell }\sin \theta } === Inverted pendulum on a cart === An inverted pendulum on a cart consists of a mass m {\displaystyle m} at the top of a pole of length ℓ {\displaystyle \ell } pivoted on a horizontally moving base as shown in the adjacent image. The cart is restricted to linear motion and is subject to forces resulting in or hindering motion. === Essentials of stabilization === The essentials of stabilizing the inverted pendulum can be summarized qualitatively in three steps. 1. If the tilt angle θ {\displaystyle \theta } is to the right, the cart must accelerate to the right and vice versa. 2. The position of the cart x {\displaystyle x} relative to track center is stabilized by slightly modulating the null angle (the angle error that the control system tries to null) by the position of the cart, that is, null angle = θ + k x {\displaystyle =\theta +kx} where k {\displaystyle k} is small. This makes the pole want to lean slightly toward track center and stabilize at track center where the tilt angle is exactly vertical. Any offset in the tilt sensor or track slope that would otherwise cause instability translates into a stable position offset. A further added offset gives position control. 3. A normal pendulum subject to a moving pivot point such as a load lifted by a crane, has a peaked response at the pendulum radian frequency of ω p = g / ℓ {\displaystyle \omega _{p}={\sqrt {g/\ell }}} . To prevent uncontrolled swinging, the frequency spectrum of the pivot motion should be suppressed near ω p {\displaystyle \omega _{p}} . The inverted pendulum requires the same suppression filter to achieve stability. As a consequence of the null angle modulation strategy, the position feedback is positive, that is, a sudden command to move right produces an initial cart motion to the left followed by a move right to rebalance the pendulum. The interaction of the pendulum instability and the positive position feedback instability to produce a stable system is a feature that makes the mathematical analysis an interesting and challenging problem. === From Lagrange's equations === The equations of motion c