In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over the field F p {\displaystyle \mathbb {F} _{p}} with p {\displaystyle p} elements. The method was discovered by Elwyn Berlekamp in 1970 as an auxiliary to the algorithm for polynomial factorization over finite fields. The algorithm was later modified by Rabin for arbitrary finite fields in 1979. The method was also independently discovered before Berlekamp by other researchers. == History == The method was proposed by Elwyn Berlekamp in his 1970 work on polynomial factorization over finite fields. His original work lacked a formal correctness proof and was later refined and modified for arbitrary finite fields by Michael Rabin. In 1986 René Peralta proposed a similar algorithm for finding square roots in F p {\displaystyle \mathbb {F} _{p}} . In 2000 Peralta's method was generalized for cubic equations. == Statement of problem == Let p {\displaystyle p} be an odd prime number. Consider the polynomial f ( x ) = a 0 + a 1 x + ⋯ + a n x n {\textstyle f(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n}} over the field F p ≃ Z / p Z {\displaystyle \mathbb {F} _{p}\simeq \mathbb {Z} /p\mathbb {Z} } of remainders modulo p {\displaystyle p} . The algorithm should find all λ {\displaystyle \lambda } in F p {\displaystyle \mathbb {F} _{p}} such that f ( λ ) = 0 {\textstyle f(\lambda )=0} in F p {\displaystyle \mathbb {F} _{p}} . == Algorithm == === Randomization === Let f ( x ) = ( x − λ 1 ) ( x − λ 2 ) ⋯ ( x − λ n ) {\textstyle f(x)=(x-\lambda _{1})(x-\lambda _{2})\cdots (x-\lambda _{n})} . Finding all roots of this polynomial is equivalent to finding its factorization into linear factors. To find such factorization it is sufficient to split the polynomial into any two non-trivial divisors and factorize them recursively. To do this, consider the polynomial f z ( x ) = f ( x − z ) = ( x − λ 1 − z ) ( x − λ 2 − z ) ⋯ ( x − λ n − z ) {\textstyle f_{z}(x)=f(x-z)=(x-\lambda _{1}-z)(x-\lambda _{2}-z)\cdots (x-\lambda _{n}-z)} where z {\displaystyle z} is some element of F p {\displaystyle \mathbb {F} _{p}} . If one can represent this polynomial as the product f z ( x ) = p 0 ( x ) p 1 ( x ) {\displaystyle f_{z}(x)=p_{0}(x)p_{1}(x)} then in terms of the initial polynomial it means that f ( x ) = p 0 ( x + z ) p 1 ( x + z ) {\displaystyle f(x)=p_{0}(x+z)p_{1}(x+z)} , which provides needed factorization of f ( x ) {\displaystyle f(x)} . === Classification of === F p {\displaystyle \mathbb {F} _{p}} elements Due to Euler's criterion, for every monomial ( x − λ ) {\displaystyle (x-\lambda )} exactly one of following properties holds: The monomial is equal to x {\displaystyle x} if λ = 0 {\displaystyle \lambda =0} , The monomial divides g 0 ( x ) = ( x ( p − 1 ) / 2 − 1 ) {\textstyle g_{0}(x)=(x^{(p-1)/2}-1)} if λ {\displaystyle \lambda } is quadratic residue modulo p {\displaystyle p} , The monomial divides g 1 ( x ) = ( x ( p − 1 ) / 2 + 1 ) {\textstyle g_{1}(x)=(x^{(p-1)/2}+1)} if λ {\displaystyle \lambda } is quadratic non-residual modulo p {\displaystyle p} . Thus if f z ( x ) {\displaystyle f_{z}(x)} is not divisible by x {\displaystyle x} , which may be checked separately, then f z ( x ) {\displaystyle f_{z}(x)} is equal to the product of greatest common divisors gcd ( f z ( x ) ; g 0 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{0}(x))} and gcd ( f z ( x ) ; g 1 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{1}(x))} . === Berlekamp's method === The property above leads to the following algorithm: Explicitly calculate coefficients of f z ( x ) = f ( x − z ) {\displaystyle f_{z}(x)=f(x-z)} , Calculate remainders of x , x 2 , x 2 2 , x 2 3 , x 2 4 , … , x 2 ⌊ log 2 p ⌋ {\textstyle x,x^{2},x^{2^{2}},x^{2^{3}},x^{2^{4}},\ldots ,x^{2^{\lfloor \log _{2}p\rfloor }}} modulo f z ( x ) {\displaystyle f_{z}(x)} by squaring the current polynomial and taking remainder modulo f z ( x ) {\displaystyle f_{z}(x)} , Using exponentiation by squaring and polynomials calculated on the previous steps calculate the remainder of x ( p − 1 ) / 2 {\textstyle x^{(p-1)/2}} modulo f z ( x ) {\textstyle f_{z}(x)} , If x ( p − 1 ) / 2 ≢ ± 1 ( mod f z ( x ) ) {\textstyle x^{(p-1)/2}\not \equiv \pm 1{\pmod {f_{z}(x)}}} then gcd {\displaystyle \gcd } mentioned below provide a non-trivial factorization of f z ( x ) {\displaystyle f_{z}(x)} , Otherwise all roots of f z ( x ) {\displaystyle f_{z}(x)} are either residues or non-residues simultaneously and one has to choose another z {\displaystyle z} . If f ( x ) {\displaystyle f(x)} is divisible by some non-linear primitive polynomial g ( x ) {\displaystyle g(x)} over F p {\displaystyle \mathbb {F} _{p}} then when calculating gcd {\displaystyle \gcd } with g 0 ( x ) {\displaystyle g_{0}(x)} and g 1 ( x ) {\displaystyle g_{1}(x)} one will obtain a non-trivial factorization of f z ( x ) / g z ( x ) {\displaystyle f_{z}(x)/g_{z}(x)} , thus algorithm allows to find all roots of arbitrary polynomials over F p {\displaystyle \mathbb {F} _{p}} . === Modular square root === Consider equation x 2 ≡ a ( mod p ) {\textstyle x^{2}\equiv a{\pmod {p}}} having elements β {\displaystyle \beta } and − β {\displaystyle -\beta } as its roots. Solution of this equation is equivalent to factorization of polynomial f ( x ) = x 2 − a = ( x − β ) ( x + β ) {\textstyle f(x)=x^{2}-a=(x-\beta )(x+\beta )} over F p {\displaystyle \mathbb {F} _{p}} . In this particular case problem it is sufficient to calculate only gcd ( f z ( x ) ; g 0 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{0}(x))} . For this polynomial exactly one of the following properties will hold: GCD is equal to 1 {\displaystyle 1} which means that z + β {\displaystyle z+\beta } and z − β {\displaystyle z-\beta } are both quadratic non-residues, GCD is equal to f z ( x ) {\displaystyle f_{z}(x)} which means that both numbers are quadratic residues, GCD is equal to ( x − t ) {\displaystyle (x-t)} which means that exactly one of these numbers is quadratic residue. In the third case GCD is equal to either ( x − z − β ) {\displaystyle (x-z-\beta )} or ( x − z + β ) {\displaystyle (x-z+\beta )} . It allows to write the solution as β = ( t − z ) ( mod p ) {\textstyle \beta =(t-z){\pmod {p}}} . === Example === Assume we need to solve the equation x 2 ≡ 5 ( mod 11 ) {\textstyle x^{2}\equiv 5{\pmod {11}}} . For this we need to factorize f ( x ) = x 2 − 5 = ( x − β ) ( x + β ) {\displaystyle f(x)=x^{2}-5=(x-\beta )(x+\beta )} . Consider some possible values of z {\displaystyle z} : Let z = 3 {\displaystyle z=3} . Then f z ( x ) = ( x − 3 ) 2 − 5 = x 2 − 6 x + 4 {\displaystyle f_{z}(x)=(x-3)^{2}-5=x^{2}-6x+4} , thus gcd ( x 2 − 6 x + 4 ; x 5 − 1 ) = 1 {\displaystyle \gcd(x^{2}-6x+4;x^{5}-1)=1} . Both numbers 3 ± β {\displaystyle 3\pm \beta } are quadratic non-residues, so we need to take some other z {\displaystyle z} . Let z = 2 {\displaystyle z=2} . Then f z ( x ) = ( x − 2 ) 2 − 5 = x 2 − 4 x − 1 {\displaystyle f_{z}(x)=(x-2)^{2}-5=x^{2}-4x-1} , thus gcd ( x 2 − 4 x − 1 ; x 5 − 1 ) ≡ x − 9 ( mod 11 ) {\textstyle \gcd(x^{2}-4x-1;x^{5}-1)\equiv x-9{\pmod {11}}} . From this follows x − 9 = x − 2 − β {\textstyle x-9=x-2-\beta } , so β ≡ 7 ( mod 11 ) {\displaystyle \beta \equiv 7{\pmod {11}}} and − β ≡ − 7 ≡ 4 ( mod 11 ) {\textstyle -\beta \equiv -7\equiv 4{\pmod {11}}} . A manual check shows that, indeed, 7 2 ≡ 49 ≡ 5 ( mod 11 ) {\textstyle 7^{2}\equiv 49\equiv 5{\pmod {11}}} and 4 2 ≡ 16 ≡ 5 ( mod 11 ) {\textstyle 4^{2}\equiv 16\equiv 5{\pmod {11}}} . == Correctness proof == The algorithm finds factorization of f z ( x ) {\displaystyle f_{z}(x)} in all cases except for ones when all numbers z + λ 1 , z + λ 2 , … , z + λ n {\displaystyle z+\lambda _{1},z+\lambda _{2},\ldots ,z+\lambda _{n}} are quadratic residues or non-residues simultaneously. According to theory of cyclotomy, the probability of such an event for the case when λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} are all residues or non-residues simultaneously (that is, when z = 0 {\displaystyle z=0} would fail) may be estimated as 2 − k {\displaystyle 2^{-k}} where k {\displaystyle k} is the number of distinct values in λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} . In this way even for the worst case of k = 1 {\displaystyle k=1} and f ( x ) = ( x − λ ) n {\displaystyle f(x)=(x-\lambda )^{n}} , the probability of error may be estimated as 1 / 2 {\displaystyle 1/2} and for modular square root case error probability is at most 1 / 4 {\displaystyle 1/4} . == Complexity == Let a polynomial have degree n {\displaystyle n} . We derive the algorithm's complexity as follows: Due to the binomial theorem ( x − z ) k = ∑ i = 0 k ( k i ) ( − z ) k − i x i {\textstyle (x-z)^{k}=\sum \limits _{i=0}^{k}{\binom {k}{i}}(-z)^{k-i}x^{i}} , we may transition from f ( x ) {\displaystyle f(x)} to f ( x − z ) {\displaystyle f(x-z)} in O ( n 2 ) {\displaystyle O(n^{2})} time. Polynomial multiplication a
Hierarchical Risk Parity
Hierarchical Risk Parity (HRP) is an advanced investment portfolio optimization framework developed in 2016 by Marcos López de Prado at Guggenheim Partners and Cornell University. HRP is a probabilistic graph-based alternative to the prevailing mean-variance optimization (MVO) framework developed by Harry Markowitz in 1952, and for which he received the Nobel Prize in economic sciences. HRP algorithms apply discrete mathematics and machine learning techniques to create diversified and robust investment portfolios that outperform MVO methods out-of-sample. HRP aims to address the limitations of traditional portfolio construction methods, particularly when dealing with highly correlated assets. Following its publication, HRP has been implemented in numerous open-source libraries, and received multiple extensions. == Key features == HRP portfolios have been proposed as a robust alternative to traditional quadratic optimization methods, including the Critical Line Algorithm (CLA) of Markowitz. HRP addresses three central issues commonly associated with quadratic optimizers: numerical instability, excessive concentration in a small number of assets, and poor out-of-sample performance. HRP leverages techniques from graph theory and machine learning to construct diversified portfolios using only the information embedded in the covariance matrix. Unlike quadratic programming methods, HRP does not require the covariance matrix to be invertible. Consequently, HRP remains applicable even in cases where the covariance matrix is ill-conditioned or singular—conditions under which standard optimizers fail. Monte Carlo simulations indicate that HRP achieves lower out-of-sample variance than CLA, despite the fact that minimizing variance is the explicit optimization objective of CLA. Furthermore, HRP portfolios exhibit lower realized risk compared to those generated by traditional risk parity methodologies. Empirical backtests have demonstrated that HRP would have historically outperformed conventional portfolio construction techniques. Algorithms within the HRP framework are characterized by the following features: Machine Learning Approach: HRP employs hierarchical clustering, a machine learning technique, to group similar assets based on their correlations. This allows the algorithm to identify the underlying hierarchical structure of the portfolio, and avoid that errors spread through the entire network. Risk-Based Allocation: The algorithm allocates capital based on risk, ensuring that assets only compete with similar assets for representation in the portfolio. This approach leads to better diversification across different risk sources, while avoiding the instability associated with noisy returns estimates. Covariance Matrix Handling: Unlike traditional methods like Mean-Variance Optimization, HRP does not require inverting the covariance matrix. This makes it more stable and applicable to portfolios with a large number of assets, particularly when the covariance matrix's condition number is high. == The problem: Markowitz's Curse == Portfolio construction is perhaps the most recurrent financial problem. On a daily basis, investment managers must build portfolios that incorporate their views and forecasts on risks and returns. Despite the theoretical elegance of Markowitz's mean-variance framework, its practical implementation is hindered by several limitations that undermine the reliability of solutions derived from the Critical Line Algorithm (CLA). A principal concern is the high sensitivity of optimal portfolios to small perturbations in expected returns: even minor forecasting errors can result in significantly different allocations (Michaud, 1998). Given the inherent difficulty of producing accurate return forecasts, numerous researchers have advocated for approaches that forgo expected returns entirely and instead rely solely on the covariance structure of asset returns. This has given rise to risk-based allocation methods, among which risk parity is a widely cited example (Jurczenko, 2015). While eliminating return forecasts mitigates some instability, it does not eliminate it. Quadratic programming techniques employed in portfolio optimization require the inversion of a positive-definite covariance matrix, meaning all eigenvalues must be strictly positive. When the matrix is numerically ill-conditioned—that is, when the ratio of its largest to smallest eigenvalue (its condition number) is large—matrix inversion becomes unreliable and prone to significant numerical errors (Bailey and López de Prado, 2012). The condition number of a covariance, correlation, or any symmetric (and thus diagonalizable) matrix is defined as the absolute value of the ratio between its largest and smallest eigenvalues in modulus. The figure on the right presents the sorted eigenvalues of several correlation matrices; the condition number is represented by the ratio of the first to last eigenvalues in each sequence. A diagonal correlation matrix, which is equal to its own inverse, exhibits the minimum possible condition number. As the number of correlated (or multicollinear) assets in a portfolio increases, the condition number rises. At high levels, this leads to severe numerical instability, whereby slight modifications in any matrix entry may result in drastically different inverses. This phenomenon, often referred to as Markowitz’s curse, encapsulates the paradox wherein increased correlation among assets heightens the theoretical need for diversification, yet simultaneously increases the likelihood of unstable optimization outcomes. Consequently, the potential benefits of diversification are frequently overshadowed by estimation errors. These problems are exacerbated as the dimensionality of the covariance matrix increases. The estimation of each covariance term consumes degrees of freedom, and in general, a minimum of 1 2 N ( N + 1 ) {\displaystyle {\frac {1}{2}}N(N+1)} independent and identically distributed (IID) observations is required to estimate a non-singular covariance matrix of dimension N {\displaystyle N} . For example, constructing an invertible covariance matrix of dimension 50 necessitates at least five years of daily IID observations. However, empirical evidence suggests that the correlation structure of financial assets is highly unstable over such extended periods. These difficulties are highlighted by the observation that even naïve allocation strategies—such as equally weighted portfolios—have frequently outperformed both mean-variance and risk-based optimizations in out-of-sample tests (De Miguel et al., 2009). == The solution: Hierarchical Risk Parity == The HRP algorithm addresses Markowitz's curse in three steps: Hierarchical Clustering: Assets are grouped into clusters based on their correlations, forming a hierarchical tree structure. Quasi-Diagonalization: The correlation matrix is reordered based on the clustering results, revealing a block diagonal structure. Recursive Bisection: Weights are assigned to assets through a top-down approach, splitting the portfolio into smaller sub-portfolios and allocating capital based on inverse variance. === Step 1: Hierarchical clustering === Given a T × N {\displaystyle T\times N} matrix of asset returns X {\displaystyle X} , where each column represents a time series of returns for one of N {\displaystyle N} assets over T {\displaystyle T} time periods, a hierarchical clustering process can be used to construct a tree-based representation of asset relationships. First, we compute the N × N {\displaystyle N\times N} correlation matrix ρ = ρ i , j i , j = 1 . . . N {\displaystyle \rho ={\rho _{i,j}}\;{i,j=1\;...\;N}} , where ρ i , j = c o r r ( X i , X j ) {\displaystyle \rho _{i,j}=\mathrm {corr} (X_{i},X_{j})} . From this, a pairwise distance matrix D = d i , j {\displaystyle D={d_{i,j}}} is defined using the transformation: d i , j = 1 2 ( 1 − ρ i , j ) {\displaystyle d_{i,j}={\sqrt {{\frac {1}{2}}(1-\rho _{i,j})}}} This distance function defines a proper metric space, satisfying non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. Next, a secondary distance matrix D ~ = d ~ i , j {\displaystyle {\tilde {D}}={{\tilde {d}}_{i,j}}} is computed, where each entry measures the Euclidean distance between the distance profiles of two assets: d ~ i , j = ∑ n = 1 N ( d n , i − d n , j ) 2 {\displaystyle {\tilde {d}}_{i,j}={\sqrt {\sum _{n=1}^{N}(d_{n,i}-d_{n,j})^{2}}}} While d i , j {\displaystyle d_{i,j}} reflects correlation-based proximity between two assets, d ~ i , j {\displaystyle {\tilde {d}}_{i,j}} quantifies dissimilarity across the entire system, as it depends on all pairwise distances. Hierarchical clustering proceeds by identifying the pair ( i , j ) {\displaystyle (i,j)} with the smallest value of d ~ i , j {\displaystyle {\tilde {d}}_{i,j}} (for i ≠ j {\displaystyle i\neq j} ), and forming a new cluster u [ 1 ] = ( i , j ) {\displaystyle u[1]=(i,j)} .
Compute (machine learning)
In machine learning and deep learning, compute is the amount of computing power or computational resources required to train machine learning models and large language models. More broadly, compute is the computational power or resources necessary for a computer or computer program to function. == Definition == Compute is commonly defined as the amount of computing power or computational resources required to train machine learning and large language models. The term "compute" has also been more broadly applied to cloud computing, referencing processing power, memory, networking, storage, and other resources required for the computation of any program. Compute is measured in petaflop/s-days and is used to document AI training. A petaflop/s-day (pfs-day) consists of performing 1015 neural net operations per second for one day, or a total of about 1020 operations. The compute-time product serves as a mental convenience, similar to kilowatt-hour for energy. An amount of compute is meant to give an idea of the number of actual operations performed. == History == In a 2018 analysis titled "AI and compute", artificial intelligence company OpenAI introduced the concept of compute. OpenAI identified two eras of training AI systems in terms of compute-usage. From 1959 to 2012, compute roughly followed Moore’s law. Between 2012 and 2018, the amount of compute used in the largest AI training runs increased exponentially, growing by more than 300,000 times — roughly doubling every 3.4 months. By comparison, Moore’s Law doubled every two years over the same period. One of the largest models, released in 2020, used 600,000 times more computing power than the 2012 model. After 2020, compute growth began to slow down, with the compute needed for the largest AI models continuing to slow down in 2023. The notion of compute has become increasingly used from the mid-2020s onwards. == Compute growth and AI progress == Larger AI models trained on more data and using more computational resources, tend to perform better. This happens even if the algorithms themselves remain unchanged. As early as 2018, OpenAI noted the exponential increase in compute to be have a key role in AI progress. OpenAI considers three factors drive the advance of AI: algorithmic innovation, data, and the amount of compute available for training. AI models with more compute not only improve in the tasks they were trained on but can develop emergent abilities. Incremental improvements can lead to more abrupt leaps in capabilities. AI provider SpaceXAI said in 2026 that their AI progress is driven by compute and used it a key metric in the AI training of its supercomputer Colossus, the which contains 1 million GPUs. Anthropic has a contract of $1.25 billion per month with SpaceXAI to buy all the compute capacity at Colossus 1 data center. === Criticism and policy === Increasing, promoting or constraining progress in artificial intelligence has often be done via controlling the amount of compute. Policymarkers have enacted policies and provided support to make compute resources more accessible to domestic AI researchers. In a January 2022 report, the Center for Security and Emerging Technology (CSET) suggested to institutions that increasingly powerful and generalizable AI (AGI) will likely require other strategies than maximizing compute. Some AI researchers are also concerned that government might exclusively focus on scaling compute instead of other strategies. The CSET has reported on the various bottlenecks which could explain why deep learning needs for compute have slow down: training is expensive and training extremely large models generates traffic jams across many processors that are difficult to manage. there is a limited supply of AI chips (see AI chip memory shortage). CSET advances that the main resource is human capital, specifically talented researchers — according to a 2023 published survey of more than 400 AI researchers, academic and private sector workers. The survey found that AI researchers are not primarily or exclusively constrained by compute access. However, both academic and industry AI researchers equally report concerns that insufficient compute could prevent them from contributing meaningfully to AI research in the future. High compute users are more concerned about compute access. When asked about which resource provided by the government would be the most useful to them, some AI researchers select compute, other prefer grant funding. For this goal, CSET advised policymakers to ensure that even researchers with smaller budgets could effectively contribute to AI research. Other proposed strategies include using contemporary AI algorithms, managing modern AI infrastructure or focusing on interdisciplinary work between the AI field and other fields of computer science. A 2024 study on compute access found that academic-only AI research teams often have less compute intensive research topics, especially foundation models, compared to industry AI labs. As a consequence, academia is likely to play a smaller role in advancing such techniques. The researchers suggest nationally-sponsored computing infrastructure as well as open science initiatives to boost academic compute access. === Data === A 2022 study found that current large language models are significantly under-trained, a consequence of focusing on scaling language models whilst keeping the amount of training data constant. By training over 400 language models of various parameter and token size, they found that "for compute-optimal training", the model size and the number of training tokens should ideally be scaled equally: for every doubling of model size the number of training tokens should also be doubled.
Cognitive robotics
Cognitive robotics or cognitive technology is a subfield of robotics concerned with endowing a robot with intelligent behavior by providing it with a processing architecture that will allow it to learn and reason about how to behave in response to complex goals in a complex world. Cognitive robotics may be considered the engineering branch of embodied cognitive science and embodied embedded cognition, consisting of robotic process automation, artificial intelligence, machine learning, deep learning, optical character recognition, image processing, process mining, analytics, software development and system integration. == Core issues == While traditional cognitive modeling approaches have assumed symbolic coding schemes as a means for depicting the world, translating the world into these kinds of symbolic representations has proven to be problematic if not untenable. Perception and action and the notion of symbolic representation are therefore core issues to be addressed in cognitive robotics. == Starting point == Cognitive robotics views human or animal cognition as a starting point for the development of robotic information processing, as opposed to more traditional artificial intelligence techniques. Target robotic cognitive capabilities include perception processing, attention allocation, anticipation, planning, complex motor coordination, reasoning about other agents and perhaps even about their own mental states. Robotic cognition embodies the behavior of intelligent agents in the physical world (or a virtual world, in the case of simulated cognitive robotics). Ultimately, the robot must be able to act in the real world. == Learning techniques == === Motor Babble === A preliminary robot learning technique called motor babbling involves correlating pseudo-random complex motor movements by the robot with resulting visual and/or auditory feedback such that the robot may begin to expect a pattern of sensory feedback given a pattern of motor output. Desired sensory feedback may then be used to inform a motor control signal. This is thought to be analogous to how a baby learns to reach for objects or learns to produce speech sounds. For simpler robot systems, where, for instance, inverse kinematics may feasibly be used to transform anticipated feedback (desired motor result) into motor output, this step may be skipped. === Imitation === Once a robot can coordinate its motors to produce a desired result, the technique of learning by imitation may be used. The robot monitors the performance of another agent and then the robot tries to imitate that agent. It is often a challenge to transform imitation information from a complex scene into a desired motor result for the robot. Note that imitation is a high-level form of cognitive behavior and imitation is not necessarily required in a basic model of embodied animal cognition. === Knowledge acquisition === A more complex learning approach is "autonomous knowledge acquisition": the robot is left to explore the environment on its own. A system of goals and beliefs is typically assumed. A somewhat more directed mode of exploration can be achieved by "curiosity" algorithms, such as Intelligent Adaptive Curiosity or Category-Based Intrinsic Motivation. These algorithms generally involve breaking sensory input into a finite number of categories and assigning some sort of prediction system (such as an artificial neural network) to each. The prediction system keeps track of the error in its predictions over time. Reduction in prediction error is considered learning. The robot then preferentially explores categories in which it is learning (or reducing prediction error) the fastest. == Other architectures == Some researchers in cognitive robotics have tried using architectures such as (ACT-R and Soar (cognitive architecture)) as a basis of their cognitive robotics programs. These highly modular symbol-processing architectures have been used to simulate operator performance and human performance when modeling simplistic and symbolized laboratory data. The idea is to extend these architectures to handle real-world sensory input as that input continuously unfolds through time. What is needed is a way to somehow translate the world into a set of symbols and their relationships. == Questions == Some of the fundamental questions to be answered in cognitive robotics are: How much human programming should or can be involved to support the learning processes? How can one quantify progress? Some of the adopted ways are reward and punishment. But what kind of reward and what kind of punishment? In humans, when teaching a child, for example, the reward would be candy or some encouragement, and the punishment can take many forms. But what is an effective way with robots?
Artificial intelligence in spirituality
Some users of artificial intelligence (AI) technologies, especially chatbots, may develop beliefs that AI has or can attain supernatural or spiritual powers. AI models such as ChatGPT are turned to for fortune telling, mysticism and remote viewing. Recent and sudden advances in large language models have led to folk myths about their origin or capabilities, as well as their deification or worship by some users. Tucker Carlson has made similar claims, including directly to Sam Altman. Pope Leo XIV advised priests against using LLM models when it came to the creation of sermons.
Lucy–Hook coaddition method
The Lucy–Hook coaddition method is an image processing technique for combining sub-stepped astronomical image data onto a finer grid. The method allows the option of resolution and contrast enhancement or the choice of a conservative, re-convolved, output. Tests with very deep Hubble Space Telescope Wide Field and Planetary Camera 2 (WFPC2) imaging data of excellent quality show that these methods can be very effective and allow fine-scale features to be studied better than on the unprocessed images. The Lucy–Hook coaddition method is an extension of the standard Richardson–Lucy deconvolution iterative restoration method. For many purposes it may be more convenient to combine dithered datasets using the Drizzle method.
Grammar systems theory
Grammar systems theory is a field of theoretical computer science that studies systems of finite collections of formal grammars generating a formal language. Each grammar works on a string, a so-called sequential form that represents an environment. Grammar systems can thus be used as a formalization of decentralized or distributed systems of agents in artificial intelligence. Let A {\displaystyle \mathbb {A} } be a simple reactive agent moving on the table and trying not to fall down from the table with two reactions, t for turning and ƒ for moving forward. The set of possible behaviors of A {\displaystyle \mathbb {A} } can then be described as formal language L A = { ( f m t n f r ) + : 1 ≤ m ≤ k ; 1 ≤ n ≤ ℓ ; 1 ≤ r ≤ k } , {\displaystyle \mathbb {L_{A}} =\{(f^{m}t^{n}f^{r})^{+}:1\leq m\leq k;1\leq n\leq \ell ;1\leq r\leq k\},} where ƒ can be done maximally k times and t can be done maximally ℓ times considering the dimensions of the table. Let G A {\displaystyle \mathbb {G_{A}} } be a formal grammar which generates language L A {\displaystyle \mathbb {L_{A}} } . The behavior of A {\displaystyle \mathbb {A} } is then described by this grammar. Suppose the A {\displaystyle \mathbb {A} } has a subsumption architecture; each component of this architecture can be then represented as a formal grammar, too, and the final behavior of the agent is then described by this system of grammars. The schema on the right describes such a system of grammars which shares a common string representing an environment. The shared sequential form is sequentially rewritten by each grammar, which can represent either a component or generally an agent. If grammars communicate together and work on a shared sequential form, it is called a Cooperating Distributed (DC) grammar system. Shared sequential form is a similar concept to the blackboard approach in AI, which is inspired by an idea of experts solving some problem together while they share their proposals and ideas on a shared blackboard. Each grammar in a grammar system can also work on its own string and communicate with other grammars in a system by sending their sequential forms on request. Such a grammar system is then called a Parallel Communicating (PC) grammar system. PC and DC are inspired by distributed AI. If there is no communication between grammars, the system is close to the decentralized approaches in AI. These kinds of grammar systems are sometimes called colonies or Eco-Grammar systems, depending (besides others) on whether the environment is changing on its own (Eco-Grammar system) or not (colonies).