AI For Business Transformation Imperial

AI For Business Transformation Imperial — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  AI literacy

  AI literacy or artificial intelligence literacy is "a set of competencies that enables individuals to critically evaluate AI technologies; communicate and collaborate effectively with AI; and use AI as a tool online, at home, and in the workplace." AI is employed in a variety of applications, including self-driving automobiles, virtual assistants and text generation by generative AI models. Users of these tools should be able to make informed decisions. AI literacy may have an impact on students' future employment prospects. With the rise of generative AI platforms, AI literacy has become a topic of conversation in the field of education. Some think AI literacy is essential for school and college students, while others restrict or prohibit the use of AI in assignments, viewing it as a form of academic dishonesty. However, many researchers and educational institutions promote a more nuanced approach, encouraging critical engagement with AI while developing policies that balance academic integrity with opportunities for learning. == Definitions == Other definitions of AI literacy include the ability to understand, use, monitor, and critically reflect on AI applications. That use of the term usually refers to teaching skills and knowledge to the general public, particularly those who are not adept in AI and the ability to understand, use, evaluate, and ethically navigate AI. As research into AI literacy is still emerging and focused on developing context-specific skills, there is not yet a single, broadly agreed-upon definition. AI literacy is linked to other forms of literacy. AI literacy requires digital literacy, whereas scientific and computational literacy may inform it. Data literacy also significantly overlaps with it. == Categories == AI literacy encompasses multiple categories, including a theoretical understanding of how artificial intelligence works, the usage of artificial intelligence technologies, and the critical appraisal of artificial intelligence, and its ethics. === Know and understand AI === Knowledge and understanding of AI refers to a basic understanding of what artificial intelligence is and how it works. This includes familiarity with machine learning algorithms and the limitations and biases present in AI systems. Users who know and understand AI should be familiar with various technologies that use artificial intelligence, including cognitive systems, robotics and machine learning. This includes recognizing that large language models (LLMs) are machine learning models trained on extensive datasets which generate new text rather than retrieving pre-written responses. === Use and apply AI === Using and applying AI refers to the ability to use AI tools to solve problems and perform tasks such as programming and analyzing big data. Some consider prompt engineering, the practice of designing effective prompts to guide generative AI platforms more effectively, as another competency within AI literacy. === Evaluate and create AI === Evaluation and creation refers to the ability to critically evaluate the quality and reliability of AI systems. It also refers to designing and building fair and ethical AI systems. To evaluate correctly, users should also learn in which areas AI is strong, and in which areas it is weak. === AI ethics === AI ethics refers to understanding the moral implications of AI, and the making informed decisions regarding the use of AI tools. This area includes considerations such as: Accountability: Hold AI actors accountable for the operation of AI systems and adherence to ethical ideals. Accuracy: Identify and report sources of error and uncertainty in algorithms and data. Auditability: Enable other parties to audit and assess algorithm behavior via transparent information sharing. Explainability: Make sure that algorithmic judgments and the underlying data can be presented in simple language. Fairness: Prevent biases and consider varied viewpoints. To do so, increase the diversity of researchers in the field. Human Centricity and Well-being: Prioritize human well-being in AI development and deployment. Human rights Alignment: Ensure that technology do not infringe internationally recognized human rights. Inclusivity: Make AI accessible to everyone. Progress: Choose high value initiatives. Responsibility, accountability, and transparency: Foster trust via responsibility, accountability, and fairness. Robustness and Security: Make AI systems safe, secure, and resistant to manipulation or data breach. Sustainability: Choose implementations that generate long-term, useful benefits. Environmental Implications: How this tool impacts the environment, any restrictions or laws, if this impact is worth the effects or not. === Enabling AI === Support AI by developing associated knowledge and skills such as programming and statistics. == Promoting AI literacy == Several governments have recognized the need to promote AI literacy, including among adults. Such programs have been published in the United States, China, Germany and Finland. Programs intended for the general public usually consist of short and easy to understand online study units. Programs intended for children are usually project-based. Programs for students at colleges and universities often address the specific professional needs of the student, depending on their field of study. Beyond the education system, AI literacy can also be developed in the community, for example in museums. === Schools === Schools use diverse pedagogies to promote AI literacy. These include: Performing a Turing test with an intelligent agent Creating chatbots Building apps using Blockly-based programming Project-based learning Building robots Data visualization Training AI models Artificial intelligence curricula can improve students' understanding of topics such as machine learning, neural networks, and deep learning. === Higher education === Before the second decade of the 21st century, artificial intelligence was studied mainly in STEM courses. Later, projects emerged to increase artificial intelligence education, specifically to promote AI literacy. Most courses start with one or more study units that deal with basic questions such as what artificial intelligence is, where it comes from, what it can do and what it can't do. Most courses also refer to machine learning and deep learning. Some of the courses deal with moral issues in artificial intelligence. In Ireland, the Higher Education Authority published Generative AI in Higher Education Teaching & Learning: Policy Framework in December 2025, which encouraged higher education institutions to embed AI literacy across programmes as a core graduate attribute. ==== Disciplinary policy ==== As a response to the increase of generative AI use in education, several disciplines formed committees or task forces to examine context-specific approaches toward AI literacy. In spring 2025, the Modern Language Association and Conference on College Composition and Communication Joint Task Force finished development of three working papers, a guide on AI literacy for students, and a collection of resources addressing AI use in writing. The task force emphasized the need for "a culture of critical AI literacy" and included guidelines not only for students but also educators and institutions, highlighting the need for modeling ethical AI use in planning processes. Similarly, a committee formed by the American Historical Association Council published "Guiding Principles for Artificial Intelligence in History Education" which encouraged "clear and transparent engagement with generative AI." The guidelines demonstrate the value of criticality when working with generative AI in thinking and research.

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  AI Code Generators Reviews: What Actually Works in 2026

  Trying to pick the best AI code generator? An AI code generator is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI code generator slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.

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  Co-Büchi automaton

  In automata theory, a co-Büchi automaton is a variant of Büchi automaton. The only difference is the accepting condition: a Co-Büchi automaton accepts an infinite word w {\displaystyle w} if there exists a run, such that all the states occurring infinitely often in the run are in the final state set F {\displaystyle F} . In contrast, a Büchi automaton accepts a word w {\displaystyle w} if there exists a run, such that at least one state occurring infinitely often in the final state set F {\displaystyle F} . (Deterministic) Co-Büchi automata are strictly weaker than (nondeterministic) Büchi automata. == Formal definition == Formally, a deterministic co-Büchi automaton is a tuple A = ( Q , Σ , δ , q 0 , F ) {\displaystyle {\mathcal {A}}=(Q,\Sigma ,\delta ,q_{0},F)} that consists of the following components: Q {\displaystyle Q} is a finite set. The elements of Q {\displaystyle Q} are called the states of A {\displaystyle {\mathcal {A}}} . Σ {\displaystyle \Sigma } is a finite set called the alphabet of A {\displaystyle {\mathcal {A}}} . δ : Q × Σ → Q {\displaystyle \delta :Q\times \Sigma \rightarrow Q} is the transition function of A {\displaystyle {\mathcal {A}}} . q 0 {\displaystyle q_{0}} is an element of Q {\displaystyle Q} , called the initial state. F ⊆ Q {\displaystyle F\subseteq Q} is the final state set. A {\displaystyle {\mathcal {A}}} accepts exactly those words w {\displaystyle w} with the run ρ ( w ) {\displaystyle \rho (w)} , in which all of the infinitely often occurring states in ρ ( w ) {\displaystyle \rho (w)} are in F {\displaystyle F} . In a non-deterministic co-Büchi automaton, the transition function δ {\displaystyle \delta } is replaced with a transition relation Δ {\displaystyle \Delta } . The initial state q 0 {\displaystyle q_{0}} is replaced with an initial state set Q 0 {\displaystyle Q_{0}} . Generally, the term co-Büchi automaton refers to the non-deterministic co-Büchi automaton. For more comprehensive formalism see also ω-automaton. == Acceptance Condition == The acceptance condition of a co-Büchi automaton is formally ∃ i ∀ j : j ≥ i ρ ( w j ) ∈ F . {\displaystyle \exists i\forall j:\;j\geq i\quad \rho (w_{j})\in F.} The Büchi acceptance condition is the complement of the co-Büchi acceptance condition: ∀ i ∃ j : j ≥ i ρ ( w j ) ∈ F . {\displaystyle \forall i\exists j:\;j\geq i\quad \rho (w_{j})\in F.} == Properties == Co-Büchi automata are closed under union, intersection, projection and determinization.

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  Deep Learning Studio

  Deep Learning Studio is a software tool that aims to simplify the creation of deep learning models used in artificial intelligence. It is compatible with a number of open-source programming frameworks popularly used in artificial neural networks, including MXNet and Google's TensorFlow. Prior to the release of Deep Learning Studio in January 2017, proficiency in Python, among other programming languages, was essential in developing effective deep learning models. Deep Learning Studio sought to simplify the model creation process through a visual, drag-and-drop interface and the application of pre-trained learning models on available data. Irving, Texas–based Deep Cognition Inc. is the developer behind Deep Learning Studio. In 2017, the software allowed Deep Cognition to become a finalist for Best Innovation in Deep Learning in the Alconics Awards, which are given annually to the best artificial intelligence software. Deep Cognition launched version 2.0 of Deep Learning Studio at NVIDIA's GTC 2018 Conference in San Jose, California. Fremont, California–based computing products supplier Exxact Corp provides desktop computers specifically built to handle Deep Learning Studio workloads. == Features == Source: Deep Learning Studio is available in two versions: Desktop and Cloud, both of which are free software. The Desktop version is available on Windows and Ubuntu. The Cloud version is available in single-user and multi-user configurations. A Deep Cognition account is needed to access the Cloud version. Account registration is free. Deep Learning Studio can import existing Keras models; it also takes a data set as an input. Deep Learning Studio's AutoML feature allows automatic generation of deep learning models. More advanced users may choose to generate their own models using various types of layers and neural networks. Deep Learning Studio also has a library of loss functions and optimizers for use in hyperparameter tuning, a traditionally complicated area in neural network programming. Generated models can be trained using either CPUs or GPUs. Trained models can then be used for predictive analytics.

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  1 Second Everyday

  1 Second Everyday (1SE) is an application developed by Cesar Kuriyama. The application allows the user to record one second of video every day and then chronologically edits (mashes) them together into a single film. It is compatible with iOS and Android. The idea of the application was developed by Kuriyama's 1 Second Everyday — Age 30 video. The application was launched in January 2013. 1 Second Everyday played a part in the plot of Chef and also became the inspiration for the 2014 short animated clip Feast. == Background == === Kuriyama's video === In February 2011, when Cesar Kuriyama turned 30, after saving money, he quit his job in an advertising firm and took a year off to travel. During this time, he started working on a project he called 1 Second Everyday. As part of the project, every day he recorded one second of video – something that was supposed to help him remember that day. He started the project because he was frustrated with his memory. He planned to stockpile the 365 one-second clips into one film to serve as a memento of his year. While working on the project Kuriyama realized that recording one second every day impacted the decisions he made in a positive way. After a year he made a 365-second clip out of his recordings. The video called 1 Second Everyday – Age 30, went viral. According to Kuriyama, he was initially inspired to take a year off from work by a TED talk given by Stefan Sagmeister called "The Power of Time Off." Kuriyama also delivered a TED talk about 1 Second Everyday in 2012 at TED 2012 in Long Beach California. === Kickstarter campaign === After completing his own video, Kuriyama decided to develop an application that would allow the users to record one second every day and compile their own videos. He developed a prototype of the application and then in 2012, he launched a Kickstarter campaign to raise funds for completing the application. The campaign became one of the most backed app campaigns in the history of Kickstarter. It was backed by 11,281 backers who pledged a total of $56,959 on an initial goal of $20,000. Following the completion of the Kickstarter campaign, he partnered with an application design studio in Brooklyn to develop the application. 1 Second Everyday was released two weeks after the completion of its Kickstarter campaign. == Application == The application was released for iOS on 10 January 2013. An Android-compatible version of the application was developed later. Using it, the user can record the videos in the application or they can select one second portions from their libraries. 1 Second Everyday dates every snippet. The user can also set alarms to remember to record their daily video. In order to compile a video, the user selects the seconds they want and the application creates a compilation video. The user can keep multiple timelines. It also allows users to post directly on social networks. The main interface in 1 Second Everyday is a calendar, which shows the user which days have snippets and which they can still fill in. In the beginning, 1 Second Everyday restricted the recording to one second. However, the developers later released Super Seconds, which allowed users to record an additional half a second video. In 2014, 1 Second Everyday Crowds was launched, which is an area in the application featuring compilations of second clips from different users. == In the media == The Kickstarter campaign of 1 Second Everyday was featured in Entrepreneur's 3 Innovative Tech Startups on Kickstarter Right Now in 2012. The application was featured in The New York Times, The Washington Post, Gawker and other media outlets. By the end of the launch day, it was in Top 10 Free Apps on App Store. It was also selected as the App of the Week on GeekWire in 2013. Several other one-second compilation videos were also posted on the Internet after Kuriyama's video gained media attention. Sam Cornwell, an English photographer documented his son Indigo's growth using a montage of one-second iPhone clips. He shot these clips every single day from the moment of birth right up to the baby's first birthday. According to Cornwell, he was inspired by Kuriyama's project. The video of Cornwell's son gained considerable media attention after it was posted on YouTube. Save the Children also made a video commercial based on a similar format that showed a British girl oblivious of the Syrian war end up being a refugee. 1SE was a finalist for the Fast Company Innovation by Design Award in 2015, but lost to Google Maps. In 2015, Google Android created a gallery, Leap Second 2015, with the help of Droga5 and Kuriyama. The gallery showcased how people around the world enjoyed the one extra second of their lives. Through the 1 Second Everyday app available at Google Play, people were able to submit their extra second, which were then vetted and added to the gallery. The viewers were able to view other celebratory seconds from around the world as well as searching for them using different hashtags.

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  Corinna Cortes

  Corinna Cortes (born 31 March 1961) is a Danish computer scientist known for her contributions to machine learning. She is a Vice President at Google Research in New York City. Cortes is an ACM Fellow and a recipient of the Paris Kanellakis Award for her work on theoretical foundations of support vector machines. == Early life and education == Corinna Cortes was born in 1961 in Denmark. Cortes received her Master of Science degree in physics from University of Copenhagen in 1989. She received her PhD in computer science from the University of Rochester in 1993 for research supervised by Randal C. Nelson. == Career and research == Cortes joined AT&T Bell Labs as a researcher in 1993. Since 2003, she has served as Vice President of Google Research, New York City, and since 2011, as adjunct professor at the UCPH Department of Computer Science. She is serves as an editorial board member of the journal Machine Learning. Cortes' research covers a wide range of topics in machine learning, including support vector machines (SVM) and data mining. SVM is one of the most frequently used algorithms in machine learning, which is used in many practical applications, including medical diagnosis and weather forecasting. At AT&T, Cortes was a contributor to the design of Hancock programming language. === Awards and honours === In 2008, she jointly with Vladimir Vapnik received the Paris Kanellakis Award for the development of a highly effective algorithm for supervised learning known as support vector machines (SVM). She was named an ACM Fellow in 2023 for theoretical and practical contributions to machine learning, industrial leadership and service to the field. == Personal life == Corinna has two children and is also a competitive runner.

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  Raymond J. Mooney

  Raymond J. Mooney is an American computer scientist, professor of computer science, and director of the Artificial Intelligence laboratory at the University of Texas at Austin. His research focuses on machine learning and natural language processing. He was educated at O'Fallon Township High School in O'Fallon, Illinois and earned a BS, MS, and Ph.D. in computer science at the University of Illinois at Urbana-Champaign, where he was advised by Gerald DeJong. He is a fellow of the Association for Computing Machinery (ACM), Association for Computational Linguistics (ACL), and Association for the Advancement of Artificial Intelligence (AAAI).

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  Krohn–Rhodes theory

  In mathematics and computer science, the Krohn–Rhodes theory (or algebraic automata theory) is an approach to the study of finite semigroups and automata that seeks to decompose them in terms of elementary components. These components correspond to finite aperiodic semigroups and finite simple groups that are combined in a feedback-free manner (called a "wreath product" or "cascade"). Krohn and Rhodes found a general decomposition for finite automata. The authors discovered and proved an unexpected major result in finite semigroup theory, revealing a deep connection between finite automata and semigroups. Decidability of Krohn-Rhodes complexity long motivated much work in semigroup theory. In June 2024, Stuart Margolis, John Rhodes, and Anne Schilling announced a proof that the complexity is decidable. == Definitions and description of the Krohn–Rhodes theorem == Let T {\displaystyle T} be a semigroup. A semigroup S {\displaystyle S} that is a homomorphic image of a subsemigroup of T {\displaystyle T} is said to be a divisor of T {\displaystyle T} . The Krohn–Rhodes theorem for finite semigroups states that every finite semigroup S {\displaystyle S} is a divisor of a finite alternating wreath product of finite simple groups, each a divisor of S {\displaystyle S} , and finite aperiodic semigroups (which contain no nontrivial subgroups). In the automata formulation, the Krohn–Rhodes theorem for finite automata states that given a finite automaton A {\displaystyle A} with states Q {\displaystyle Q} and input alphabet I {\displaystyle I} , output alphabet U {\displaystyle U} , then one can expand the states to Q ′ {\displaystyle Q'} such that the new automaton A ′ {\displaystyle A'} embeds into a cascade of "simple", irreducible automata: In particular, A {\displaystyle A} is emulated by a feed-forward cascade of (1) automata whose transformation semigroups are finite simple groups and (2) automata that are banks of flip-flops running in parallel. The new automaton A ′ {\displaystyle A'} has the same input and output symbols as A {\displaystyle A} . Here, both the states and inputs of the cascaded automata have a very special hierarchical coordinate form. Moreover, each simple group (prime) or non-group irreducible semigroup (subsemigroup of the flip-flop monoid) that divides the transformation semigroup of A {\displaystyle A} must divide the transformation semigroup of some component of the cascade, and only the primes that must occur as divisors of the components are those that divide A {\displaystyle A} 's transformation semigroup. == Group complexity == The Krohn–Rhodes complexity (also called group complexity or just complexity) of a finite semigroup S is the least number of groups in a wreath product of finite groups and finite aperiodic semigroups of which S is a divisor. All finite aperiodic semigroups have complexity 0, while non-trivial finite groups have complexity 1. In fact, there are semigroups of every non-negative integer complexity. For example, for any n greater than 1, the multiplicative semigroup of all (n+1) × (n+1) upper-triangular matrices over any fixed finite field has complexity n (Kambites, 2007). A major open problem in finite semigroup theory is the decidability of complexity: is there an algorithm that will compute the Krohn–Rhodes complexity of a finite semigroup, given its multiplication table? Upper bounds and ever more precise lower bounds on complexity have been obtained (see, e.g. Rhodes & Steinberg, 2009). Rhodes has conjectured that the problem is decidable. In June 2024, Stuart Margolis, John Rhodes, and Anne Schilling announced a proof in the affirmative of the conjecture, though as of 2025 the result has yet to be confirmed. == History and applications == At a conference in 1962, Kenneth Krohn and John Rhodes announced a method for decomposing a (deterministic) finite automaton into "simple" components that are themselves finite automata. This joint work, which has implications for philosophy, comprised both Krohn's doctoral thesis at Harvard University and Rhodes' doctoral thesis at MIT. Simpler proofs, and generalizations of the theorem to infinite structures, have been published since then (see Chapter 4 of Rhodes and Steinberg's 2009 book The q-Theory of Finite Semigroups for an overview). In the 1965 paper by Krohn and Rhodes, the proof of the theorem on the decomposition of finite automata (or, equivalently sequential machines) made extensive use of the algebraic semigroup structure. Later proofs contained major simplifications using finite wreath products of finite transformation semigroups. The theorem generalizes the Jordan–Hölder decomposition for finite groups (in which the primes are the finite simple groups), to all finite transformation semigroups (for which the primes are again the finite simple groups plus all subsemigroups of the "flip-flop" (see above)). Both the group and more general finite automata decomposition require expanding the state-set of the general, but allow for the same number of input symbols. In the general case, these are embedded in a larger structure with a hierarchical "coordinate system". One must be careful in understanding the notion of "prime" as Krohn and Rhodes explicitly refer to their theorem as a "prime decomposition theorem" for automata. The components in the decomposition, however, are not prime automata (with prime defined in a naïve way); rather, the notion of prime is more sophisticated and algebraic: the semigroups and groups associated to the constituent automata of the decomposition are prime (or irreducible) in a strict and natural algebraic sense with respect to the wreath product (Eilenberg, 1976). Also, unlike earlier decomposition theorems, the Krohn–Rhodes decompositions usually require expansion of the state-set, so that the expanded automaton covers (emulates) the one being decomposed. These facts have made the theorem difficult to understand and challenging to apply in a practical way—until recently, when computational implementations became available (Egri-Nagy & Nehaniv 2005, 2008). H.P. Zeiger (1967) proved an important variant called the holonomy decomposition (Eilenberg 1976). The holonomy method appears to be relatively efficient and has been implemented computationally by A. Egri-Nagy (Egri-Nagy & Nehaniv 2005). Meyer and Thompson (1969) give a version of Krohn–Rhodes decomposition for finite automata that is equivalent to the decomposition previously developed by Hartmanis and Stearns, but for useful decompositions, the notion of expanding the state-set of the original automaton is essential (for the non-permutation automata case). Many proofs and constructions now exist of Krohn–Rhodes decompositions (e.g., [Krohn, Rhodes & Tilson 1968], [Ésik 2000], [Diekert et al. 2012]), with the holonomy method the most popular and efficient in general (although not in all cases). [Zimmermann 2010] gives an elementary proof of the theorem. Owing to the close relation between monoids and categories, a version of the Krohn–Rhodes theorem is applicable to category theory. This observation and a proof of an analogous result were offered by Wells (1980). The Krohn–Rhodes theorem for semigroups/monoids is an analogue of the Jordan–Hölder theorem for finite groups (for semigroups/monoids rather than groups). As such, the theorem is a deep and important result in semigroup/monoid theory. The theorem was also surprising to many mathematicians and computer scientists since it had previously been widely believed that the semigroup/monoid axioms were too weak to admit a structure theorem of any strength, and prior work (Hartmanis & Stearns) was only able to show much more rigid and less general decomposition results for finite automata. Work by Egri-Nagy and Nehaniv (2005, 2008–) continues to further automate the holonomy version of the Krohn–Rhodes decomposition extended with the related decomposition for finite groups (so-called Frobenius–Lagrange coordinates) using the computer algebra system GAP. Applications outside of the semigroup and monoid theories are now computationally feasible. They include computations in biology and biochemical systems (e.g. Egri-Nagy & Nehaniv 2008), artificial intelligence, finite-state physics, psychology, and game theory (see, for example, Rhodes 2009).

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  Harris corner detector

  The Harris corner detector is a corner detection operator that is commonly used in computer vision algorithms to extract corners and infer features of an image. It was first introduced by Chris Harris and Mike Stephens in 1988 upon the improvement of Moravec's corner detector. Compared to its predecessor, Harris' corner detector takes the differential of the corner score into account with reference to direction directly, instead of using shifting patches for every 45 degree angles, and has been proved to be more accurate in distinguishing between edges and corners. Since then, it has been improved and adopted in many algorithms to preprocess images for subsequent applications. == Introduction == A corner is a point whose local neighborhood stands in two dominant and different edge directions. In other words, a corner can be interpreted as the junction of two edges, where an edge is a sudden change in image brightness. Corners are the important features in the image, and they are generally termed as interest points which are invariant to translation, rotation and illumination. Although corners are only a small percentage of the image, they contain the most important features in restoring image information, and they can be used to minimize the amount of processed data for motion tracking, image stitching, building 2D mosaics, stereo vision, image representation and other related computer vision areas. In order to capture the corners from the image, researchers have proposed many different corner detectors including the Kanade-Lucas-Tomasi (KLT) operator and the Harris operator which are most simple, efficient and reliable for use in corner detection. These two popular methodologies are both closely associated with and based on the local structure matrix. Compared to the Kanade-Lucas-Tomasi corner detector, the Harris corner detector provides good repeatability under changing illumination and rotation, and therefore, it is more often used in stereo matching and image database retrieval. Although there still exist drawbacks and limitations, the Harris corner detector is still an important and fundamental technique for many computer vision applications. == Development of Harris corner detection algorithm == Source: Without loss of generality, we will assume a grayscale 2-dimensional image is used. Let this image be given by I {\displaystyle I} . Consider taking an image patch ( x , y ) ∈ W {\displaystyle (x,y)\in W} (window) and shifting it by ( Δ x , Δ y ) {\displaystyle (\Delta x,\Delta y)} . The sum of squared differences (SSD) between these two patches, denoted f {\displaystyle f} , is given by: f ( Δ x , Δ y ) = ∑ ( x k , y k ) ∈ W ( I ( x k , y k ) − I ( x k + Δ x , y k + Δ y ) ) 2 {\displaystyle f(\Delta x,\Delta y)={\underset {(x_{k},y_{k})\in W}{\sum }}\left(I(x_{k},y_{k})-I(x_{k}+\Delta x,y_{k}+\Delta y)\right)^{2}} I ( x + Δ x , y + Δ y ) {\displaystyle I(x+\Delta x,y+\Delta y)} can be approximated by a Taylor expansion. Let I x {\displaystyle I_{x}} and I y {\displaystyle I_{y}} be the partial derivatives of I {\displaystyle I} , such that I ( x + Δ x , y + Δ y ) ≈ I ( x , y ) + I x ( x , y ) Δ x + I y ( x , y ) Δ y {\displaystyle I(x+\Delta x,y+\Delta y)\approx I(x,y)+I_{x}(x,y)\Delta x+I_{y}(x,y)\Delta y} This produces the approximation f ( Δ x , Δ y ) ≈ ∑ ( x , y ) ∈ W ( I x ( x , y ) Δ x + I y ( x , y ) Δ y ) 2 , {\displaystyle f(\Delta x,\Delta y)\approx {\underset {(x,y)\in W}{\sum }}\left(I_{x}(x,y)\Delta x+I_{y}(x,y)\Delta y\right)^{2},} which can be written in matrix form: f ( Δ x , Δ y ) ≈ ( Δ x Δ y ) M ( Δ x Δ y ) , {\displaystyle f(\Delta x,\Delta y)\approx {\begin{pmatrix}\Delta x&\Delta y\end{pmatrix}}M{\begin{pmatrix}\Delta x\\\Delta y\end{pmatrix}},} where M is the structure tensor, M = ∑ ( x , y ) ∈ W [ I x 2 I x I y I x I y I y 2 ] = [ ∑ ( x , y ) ∈ W I x 2 ∑ ( x , y ) ∈ W I x I y ∑ ( x , y ) ∈ W I x I y ∑ ( x , y ) ∈ W I y 2 ] {\displaystyle M={\underset {(x,y)\in W}{\sum }}{\begin{bmatrix}I_{x}^{2}&I_{x}I_{y}\\I_{x}I_{y}&I_{y}^{2}\end{bmatrix}}={\begin{bmatrix}{\underset {(x,y)\in W}{\sum }}I_{x}^{2}&{\underset {(x,y)\in W}{\sum }}I_{x}I_{y}\\{\underset {(x,y)\in W}{\sum }}I_{x}I_{y}&{\underset {(x,y)\in W}{\sum }}I_{y}^{2}\end{bmatrix}}} == Process of Harris corner detection algorithm == Commonly, Harris corner detector algorithm can be divided into five steps. Color to grayscale Spatial derivative calculation Structure tensor setup Harris response calculation Non-maximum suppression === Color to grayscale === If we use Harris corner detector in a color image, the first step is to convert it into a grayscale image, which will enhance the processing speed. The value of the gray scale pixel can be computed as a weighted sums of the values R, B and G of the color image, ∑ C ∈ { R , G , B } w C ⋅ C {\displaystyle \sum _{C\,\in \,\{R,G,B\}}w_{C}\cdot C} , where, e.g., w R = 0.299 , w G = 0.587 , w B = 1 − ( w R + w G ) = 0.114. {\displaystyle w_{R}=0.299,\ w_{G}=0.587,\ w_{B}=1-(w_{R}+w_{G})=0.114.} === Spatial derivative calculation === Next, we are going to find the derivative with respect to x and the derivative with respect to y, I x ( x , y ) {\displaystyle I_{x}(x,y)} and I y ( x , y ) {\displaystyle I_{y}(x,y)} . This can be approximated by applying Sobel operators. === Structure tensor setup === With I x ( x , y ) {\displaystyle I_{x}(x,y)} , I y ( x , y ) {\displaystyle I_{y}(x,y)} , we can construct the structure tensor M {\displaystyle M} . === Harris response calculation === For x ≪ y {\displaystyle x\ll y} , one has x ⋅ y x + y = x 1 1 + x / y ≈ x . {\displaystyle {\tfrac {x\cdot y}{x+y}}=x{\tfrac {1}{1+x/y}}\approx x.} In this step, we compute the smallest eigenvalue of the structure tensor using that approximation: λ min ≈ λ 1 λ 2 ( λ 1 + λ 2 ) = det ( M ) tr ⁡ ( M ) {\displaystyle \lambda _{\min }\approx {\frac {\lambda _{1}\lambda _{2}}{(\lambda _{1}+\lambda _{2})}}={\frac {\det(M)}{\operatorname {tr} (M)}}} with the trace t r ( M ) = m 11 + m 22 {\displaystyle \mathrm {tr} (M)=m_{11}+m_{22}} . Another commonly used Harris response calculation is shown as below, R = λ 1 λ 2 − k ( λ 1 + λ 2 ) 2 = det ( M ) − k tr ⁡ ( M ) 2 {\displaystyle R=\lambda _{1}\lambda _{2}-k(\lambda _{1}+\lambda _{2})^{2}=\det(M)-k\operatorname {tr} (M)^{2}} where k {\displaystyle k} is an empirically determined constant; k ∈ [ 0.04 , 0.06 ] {\displaystyle k\in [0.04,0.06]} . === Non-maximum suppression === In order to pick up the optimal values to indicate corners, we find the local maxima as corners within the window which is a 3 by 3 filter. == Improvement == Sources: Harris-Laplace Corner Detector Differential Morphological Decomposition Based Corner Detector Multi-scale Bilateral Structure Tensor Based Corner Detector == Applications == Image Alignment, Stitching and Registration 2D Mosaics Creation 3D Scene Modeling and Reconstruction Motion Detection Object Recognition Image Indexing and Content-based Retrieval Video Tracking

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  TAUM system

  TAUM (Traduction Automatique à l'Université de Montréal) is the name of a research group which was set up at the Université de Montréal in 1965. Most of its research was done between 1968 and 1980. It gave birth to the TAUM-73 and TAUM-METEO machine translation prototypes, using the Q-Systems programming language created by Alain Colmerauer, which were among the first attempts to perform automatic translation through linguistic analysis. The prototypes were never used in actual production. The TAUM-METEO name has been erroneously used for many years to designate the METEO System subsequently developed by John Chandioux.

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  Yorick Wilks

  Yorick Alexander Wilks FBCS (27 October 1939 – 14 April 2023) was a British computer scientist. He was an emeritus professor of artificial intelligence at the University of Sheffield, visiting professor of artificial intelligence at Gresham College (a post created especially for him), senior research fellow at the Oxford Internet Institute, senior scientist at the Florida Institute for Human and Machine Cognition, and a member of the Epiphany Philosophers. In February 2023, Wilks joined WiredVibe as Director of AI and a Board Member, with the goal of commercializing his previous research and ideas. He remained in this role until his death, which occurred shortly before WiredVibe was acquired by AKY X, a company that continues to build on his legacy and contributions. == Biography == Wilks was born in Gerrards Cross, Buckinghamshire in England. He was educated at Torquay Boys' Grammar School, followed by Pembroke College, Cambridge, where he read Philosophy, joined the Epiphany Philosophers and obtained his Doctor of Philosophy degree (1968) under Professor R. B. Braithwaite for the thesis 'Argument and Proof'; he was an early pioneer in meaning-based approaches to the understanding of natural language content by computers. His main early contribution in the 1970s was called "Preference Semantics" (Wilks, 1973; Wilks and Fass, 1992), an algorithmic method for assigning the "most coherent" interpretation to a sentence in terms of having the maximum number of internal preferences of its parts (normally verbs or adjectives) satisfied. That early work was hand-coded with semantic entries (of the order of some hundreds) as was normal at the time, but since then has led to the empirical determinations of preferences (chiefly of English verbs) in the 1980s and 1990s. A key component of the notion of preference in semantics was that the interpretation of an utterance is not a well- or ill-formed notion, as was argued in Chomskyan approaches, such as those of Jerry Fodor and Jerrold Katz. It was rather that a semantic interpretation was the best available, even though some preferences might not be satisfied. So, in "The machine answered the question with a low whine" the agent of "answer" does not satisfy that verb's preference for a human answerer—which would cause it to be deemed ill-formed by Fodor and Katz—but is accepted as sub-optimal or metaphorical, and, now, conventional. The function of the algorithm is not to determine well-formedness at all but to make the optimal selection of word-senses to participate in the overall interpretation. Thus, in "The Pole answered..." the system will always select the human sense of the agent and not the inanimate one if it gives a more coherent interpretation overall. Preference Semantics is thus some of the earliest computational work—with programs run at Systems Development Corporation in Santa Monica in 1967 in LISP on an IBM360—in the now established field of word sense disambiguation. This approach was used in the first operational machine translation system based principally on meaning structures and built by Wilks at Stanford Artificial Intelligence Laboratory in the early 1970s (Wilks, 1973) at the same time and place as Roger Schank was applying his "Conceptual Dependency" approach to machine translation. The LISP code of Wilks' system was in The Computer Museum, Boston. Wilks was elected a fellow of the American and European Associations for Artificial Intelligence, of the British Computer Society, a member of the UK Computing Research Committee, and a permanent member of ICCL, the International Committee on Computational Linguistics. He was professor of artificial intelligence at the University of Sheffield and a senior research fellow at the Oxford Internet Institute. In 1991 he received a Defense Advanced Projects Agency grant on interlingual pragmatics-based machine translation and in 1994 he received a grant by the Engineering and Physical Sciences Research Council to investigate in the field of large-scale information extraction (LaSIE); in the following years he would obtain more grants to carry on exploring the field of information extraction (AVENTINUS, ECRAN, PASTA...). In the 1990s Wilks also became interested in modelling human-computer dialogue and the team led by David Levy and him as chief researcher won the Loebner Prize in 1997. He was the founding director of the EU funded Companions Project on creating long-term computer companions for people. At his Festschrift in 2007 at the British Computer Society in London a volume of his own papers was presented along with a volume of essays in his honour. He was awarded the Antonio Zampolli prize in honour of his lifetime work at the LREC 2008 conference on 28 May 2008, and the Lifetime Achievement Award at the ACL 2008 conference on 18 June 2008. In 2009, he was awarded the British Computer Society's Lovelace Medal, its annual award for research achievement, and was awarded the Fellowship of the Association for Computing Machinery. In 1998, Wilks became head of the Department of Computer Science of the University of Sheffield, where he had started working in the year 1993 as professor of artificial intelligence, a post he still held. In 1993 he became the founding director of the Institute of Language, Speech and Hearing (ILASH). Wilks also set up the Natural Language Processing Group of the University of Sheffield. In 1994 he (along with Rob Gaizauskas and Hamish Cunningham) designed GATE, an advanced NLP architecture that has been widely distributed. National Life Stories conducted an oral history interview (C1672/24) with Yorick Wilks in 2016 for its Science and Religion collection held by the British Library. Wilks died on 14 April 2023, at the age of 83. == Awards == Wilks received many awards: (2009) Elected Fellow of the Association for Computing Machinery (2009) Lovelace Medal by the British Computer Society (2008) Zampolli Prize (ELRA, awarded at LREC in Marrakech, Morocco) (2008) Lifetime Achievement Award (Association for Computational Linguistics, in Columbus) (2006) Visiting Professor, University of Oxford (2004) Elected to UK Computing Research Committee (2004) Elected Fellow, British Computer Society (2003) Visiting Fellow, Oxford Internet Institute (1998) Elected Fellow of European Association for Artificial Intelligence (1997) Elected Fellow, EPSRC College of Computing (1991) Visiting Fellow, Trinity Hall, Cambridge (1991) Elected Fellow of the American Association for Artificial Intelligence (1983) Royal Society Travel Fellowship (1983) Commonwealth of Australia Visiting Professor (1981) Visiting Sloan Fellow, University of California, Berkeley (1980) Invited Participant in the Nobel Symposium on Language, Stockholm (1979) NATO Senior Scientist Fellowship (1979) Visiting Sloan Fellow, Yale University (1975) SRC Senior Visiting Fellowship, University of Edinburgh == Membership == Wilks was an active member of the following associations: Association for Computational Linguistics Society for the Study of AI and Simulation of Behaviour Association for Computing Machinery Cognitive Science Society British Society for the Philosophy of Science American Association for Artificial Intelligence Aristotelian Society == Selected works == === Books === Wilks, Y. (2019) Artificial Intelligence: Modern Magic or Dangerous Future?.Icon Books. New illustrated edition, 2023, MIT Press. Wilks, Y. (2015) Machine Translation: its scope and limits. Springer Wilks, Y (ed.) (2010) Close Engagements with Artificial Companions: Key Social, Psychological and Design issues. John Benjamins; Amsterdam Wilks, Y., Brewster, C. (2009) Natural Language Processing as a Foundation of the Semantic Web. Now Press: London. Wilks, Y. (2007) Words and Intelligence I, Selected papers by Yorick Wilks. In K. Ahmad, C. Brewster & M. Stevenson (eds.), Springer: Dordrecht. Wilks, Y. (ed. and with introduction and commentaries). (2006) Language, cohesion and form: selected papers of Margaret Masterman. Cambridge: Cambridge University Press. Wilks, Y., Nirenburg, S., Somers, H. (eds.) (2003) Readings in Machine Translation. Cambridge, MA: MIT Press. Wilks, Y.(ed.). (1999) Machine Conversations. Kluwer: New York. Wilks, Y., Slator, B., Guthrie, L. (1996) Electric Words: dictionaries, computers and meanings. Cambridge, MA: MIT Press. Ballim, A., Wilks, Y. (1991) Artificial Believers. Norwood, NJ: Erlbaum. Wilks, Y.(ed.). (1990) Theoretical Issues in Natural Language Processing. Norwood, NJ: Erlbaum. Wilks, Y., Partridge, D. (eds. plus three YW chapters and an introduction). (1990) The Foundations of Artificial Intelligence: a sourcebook. Cambridge: Cambridge University Press. Wilks, Y., Sparck-Jones, K.(eds.). (1984) Automatic Natural Language Processing, paperback edition. New York: Wiley. Originally published by Ellis Horwood. Wilks, Y., Charniak, E. (eds and principal authors). (1976) Computational Semantics—an Introduction to Artificial Intelligence and

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  Lübke English

  The term Lübke English (or, in German, Lübke-Englisch) refers to nonsensical English created by literal word-by-word translation of German phrases, disregarding differences between the languages in syntax and meaning. Lübke English is named after Heinrich Lübke, a president of Germany in the 1960s, whose limited English made him a target of German humorists. In 2006, the German magazine konkret revealed that most of the statements ascribed to Lübke were in fact invented by the editorship of Der Spiegel, mainly by staff writer Ernst Goyke and subsequent letters to the editor. In the 1980s, comedian Otto Waalkes had a routine called "English for Runaways", which is a nonsensical literal translation of Englisch für Fortgeschrittene (actually an idiom for 'English for advanced speakers' in German – note that fortschreiten divides into fort, meaning "away" or "forward", and schreiten, meaning "to walk in steps"). In this mock "course", he translates every sentence back or forth between English and German at least once (usually from German literally into English). Though there are also other, more complex language puns, the title of this routine has gradually replaced the term Lübke English when a German speaker wants to point out naive literal translations.

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  Autonomic networking

  Autonomic networking follows the concept of Autonomic Computing, an initiative started by IBM in 2001. Its ultimate aim is to create self-managing networks to overcome the rapidly growing complexity of the Internet and other networks and to enable their further growth, far beyond the size of today. == Increasing size and complexity == The ever-growing management complexity of the Internet caused by its rapid growth is seen by some experts as a major problem that limits its usability in the future. What's more, increasingly popular smartphones, PDAs, networked audio and video equipment, and game consoles need to be interconnected. Pervasive Computing not only adds features, but also burdens existing networking infrastructure with more and more tasks that sooner or later will not be manageable by human intervention alone. Another important aspect is the price of manually controlling huge numbers of vitally important devices of current network infrastructures. == Autonomic nervous system == The autonomic nervous system (ANS) is the part of complex biological nervous systems that is not consciously controlled. It regulates bodily functions and the activity of specific organs. As proposed by IBM, future communication systems might be designed in a similar way to the ANS. == Components of autonomic networking == As autonomics conceptually derives from biological entities such as the human autonomic nervous system, each of the areas can be metaphorically related to functional and structural aspects of a living being. In the human body, the autonomic system facilitates and regulates a variety of functions including respiration, blood pressure and circulation, and emotive response. The autonomic nervous system is the interconnecting fabric that supports feedback loops between internal states and various sources by which internal and external conditions are monitored. === Autognostics === Autognostics includes a range of self-discovery, awareness, and analysis capabilities that provide the autonomic system with a view on high-level state. In metaphor, this represents the perceptual sub-systems that gather, analyze, and report on internal and external states and conditions – for example, this might be viewed as the eyes, visual cortex and perceptual organs of the system. Autognostics, or literally "self-knowledge", provides the autonomic system with a basis for response and validation. A rich autognostic capability may include many different "perceptual senses". For example, the human body gathers information via the usual five senses, the so-called sixth sense of proprioception (sense of body position and orientation), and through emotive states that represent the gross wellness of the body. As conditions and states change, they are detected by the sensory monitors and provide the basis for adaptation of related systems. Implicit in such a system are imbedded models of both internal and external environments such that relative value can be assigned to any perceived state - perceived physical threat (e.g. a snake) can result in rapid shallow breathing related to fight-flight response, a phylogenetically effective model of interaction with recognizable threats. In the case of autonomic networking, the state of the network may be defined by inputs from: individual network elements such as switches and network interfaces including specification and configuration historical records and current state traffic flows end-hosts application performance data logical diagrams and design specifications Most of these sources represent relatively raw and unprocessed views that have limited relevance. Post-processing and various forms of analysis must be applied to generate meaningful measurements and assessments against which current state can be derived. The autognostic system interoperates with: configuration management - to control network elements and interfaces policy management - to define performance objectives and constraints autodefense - to identify attacks and accommodate the impact of defensive responses === Configuration management === Configuration management is responsible for the interaction with network elements and interfaces. It includes an accounting capability with historical perspective that provides for the tracking of configurations over time, with respect to various circumstances. In the biological metaphor, these are the hands and, to some degree, the memory of the autonomic system. On a network, remediation and provisioning are applied via configuration setting of specific devices. Implementation affecting access and selective performance with respect to role and relationship are also applied. Almost all the "actions" that are currently taken by human engineers fall under this area. With only a few exceptions, interfaces are set by hand, or by extension of the hand, through automated scripts. Implicit in the configuration process is the maintenance of a dynamic population of devices under management, a historical record of changes and the directives which invoked change. Typical to many accounting functions, configuration management should be capable of operating on devices and then rolling back changes to recover previous configurations. Where change may lead to unrecoverable states, the sub-system should be able to qualify the consequences of changes prior to issuing them. As directives for change must originate from other sub-systems, the shared language for such directives must be abstracted from the details of the devices involved. The configuration management sub-system must be able to translate unambiguously between directives and hard actions or to be able to signal the need for further detail on a directive. An inferential capacity may be appropriate to support sufficient flexibility (i.e. configuration never takes place because there is no unique one-to-one mapping between directive and configuration settings). Where standards are not sufficient, a learning capacity may also be required to acquire new knowledge of devices and their configuration. Configuration management interoperates with all of the other sub-systems including: autognostics - receives direction for and validation of changes policy management - implements policy models through mapping to underlying resources security - applies access and authorization constraints for particular policy targets autodefense - receives direction for changes === Policy management === Policy management includes policy specification, deployment, reasoning over policies, updating and maintaining policies, and enforcement. Policy-based management is required for: constraining different kinds of behavior including security, privacy, resource access, and collaboration configuration management describing business processes and defining performance defining role and relationship, and establishing trust and reputation It provides the models of environment and behavior that represent effective interaction according to specific goals. In the human nervous system metaphor, these models are implicit in the evolutionary "design" of biological entities and specific to the goals of survival and procreation. Definition of what constitutes a policy is necessary to consider what is involved in managing it. A relatively flexible and abstract framework of values, relationships, roles, interactions, resources, and other components of the network environment is required. This sub-system extends far beyond the physical network to the applications in use and the processes and end-users that employ the network to achieve specific goals. It must express the relative values of various resources, outcomes, and processes and include a basis for assessing states and conditions. Unless embodied in some system outside the autonomic network or implicit to the specific policy implementation, the framework must also accommodate the definition of process, objectives and goals. Business process definitions and descriptions are then an integral part of the policy implementation. Further, as policy management represents the ultimate basis for the operation of the autonomic system, it must be able to report on its operation with respect to the details of its implementation. The policy management sub-system interoperates (at least) indirectly with all other sub-systems but primarily interacts with: autognostics - providing the definition of performance and accepting reports on conditions configuration management - providing constraints on device configuration security - providing definitions of roles, access and permissions === Autodefense === Autodefense represents a dynamic and adaptive mechanism that responds to malicious and intentional attacks on the network infrastructure, or use of the network infrastructure to attack IT resources. As defensive measures tend to impede the operation of IT, it is optimally capable of balancing performance objectives with typically over-riding threat management actions. In the

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  Dan Roth

  Dan Roth (Hebrew: דן רוט) is the Eduardo D. Glandt Distinguished Professor of Computer and Information Science at the University of Pennsylvania and the Chief AI Scientist at Oracle. Until June 2024 Roth was a VP and distinguished scientist at AWS AI. In his role at AWS, Roth led over the last three years the scientific effort behind the first-generation Generative AI products from AWS, including Titan Models, Amazon Q efforts, and Bedrock, from inception until they became generally available. Roth got his B.A. summa cum laude in mathematics from the Technion, Israel, and his Ph.D. in computer science from Harvard University in 1995. He taught at the University of Illinois at Urbana-Champaign from 1998 to 2017 before moving to the University of Pennsylvania. == Professional career == Roth is a Fellow of the American Association for the Advancement of Science (AAAS), the Association for Computing Machinery (ACM), the Association for the Advancement of Artificial Intelligence (AAAI), and the Association of Computational Linguistics (ACL). Roth’s research focuses on the computational foundations of intelligent behavior. He develops theories and systems pertaining to intelligent behavior using a unified methodology, at the heart of which is the idea that learning has a central role in intelligence. His work centers around the study of machine learning and inference methods to facilitate natural language understanding. In doing that he has pursued several interrelated lines of work that span multiple aspects of this problem - from fundamental questions in learning and inference and how they interact, to the study of a range of natural language processing (NLP) problems and developing advanced machine learning based tools for natural language applications. Roth has made seminal contribution to the fusion of Learning and Reasoning, Machine Learning with weak, incidental supervision, and to machine learning and inference approaches to natural language understanding. He has written the first paper on zero-shot learning in natural language processing, a 2008 paper by Chang, Ratinov, Roth, and Srikumar that was published at AAAI’08, but the name given to the learning paradigm there was dataless classification. Roth has worked on probabilistic reasoning (including its complexity and probabilistic lifted inference ), Constrained Conditional Models (ILP formulations of NLP problems) and constraints-driven learning, part-based (constellation) methods in object recognition, response based Learning, He has developed NLP and Information extraction tools that are being used broadly by researchers and commercially, including NER, coreference resolution, wikification, SRL, and ESL text correction. Roth is a co-founder of NexLP, Inc., a startup that applies natural language processing and machine learning in the legal and compliance domains. In 2020, NexLP was acquired by Reveal, Inc., an e-discovery software company. He is currently on the scientific advisory board of the Allen Institute for AI.

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  Cobham's theorem

  Cobham's theorem is a theorem in combinatorics on words that has important connections with number theory, notably transcendental numbers, and automata theory. Informally, the theorem gives the condition for the members of a set S of natural numbers written in bases b1 and base b2 to be recognised by finite automata. Specifically, consider bases b1 and b2 such that they are not powers of the same integer. Cobham's theorem states that S written in bases b1 and b2 is recognised by finite automata if and only if S differs by a finite set from a finite union of arithmetic progressions. The theorem was proved by Alan Cobham in 1969 and has since given rise to many extensions and generalisations. == Definitions == Let n > 0 {\displaystyle n>0} be an integer. The representation of a natural number n {\textstyle n} in base b {\textstyle b} is the sequence of digits n 0 n 1 ⋯ n h {\displaystyle n_{0}n_{1}\cdots n_{h}} such that n = n 0 + n 1 b + ⋯ + n h b h {\displaystyle n=n_{0}+n_{1}b+\cdots +n_{h}b^{h}} where 0 ≤ n 0 , n 1 , … , n h < b {\displaystyle 0\leq n_{0},n_{1},\ldots ,n_{h} 0 {\displaystyle n_{h}>0} . The word n 0 n 1 ⋯ n h {\displaystyle n_{0}n_{1}\cdots n_{h}} is often denoted ⟨ n ⟩ b {\displaystyle \langle n\rangle _{b}} , or more simply, n b {\displaystyle n_{b}} . A set of natural numbers S is recognisable in base b {\textstyle b} or more simply b {\textstyle b} -recognisable or b {\textstyle b} -automatic if the set { n b ∣ n ∈ S } {\displaystyle \{n_{b}\mid n\in S\}} of the representations of its elements in base b {\displaystyle b} is a language recognisable by a finite automaton on the alphabet { 0 , 1 , … , b − 1 } {\displaystyle \{0,1,\ldots ,b-1\}} . Two positive integers k {\displaystyle k} and ℓ {\displaystyle \ell } are multiplicatively independent if there are no non-negative integers p {\displaystyle p} and q {\displaystyle q} such that k p = ℓ q {\displaystyle k^{p}=\ell ^{q}} . For example, 2 and 3 are multiplicatively independent, but 8 and 16 are not since 8 4 = 16 3 {\displaystyle 8^{4}=16^{3}} . Two integers are multiplicatively dependent if and only if they are powers of a same third integer. == Problem statements == === Original problem statement === More equivalent statements of the theorem have been given. The original version by Cobham is the following: Another way to state the theorem is by using automatic sequences. Cobham himself calls them "uniform tag sequences." The following form is found in Allouche and Shallit's book:We can show that the characteristic sequence of a set of natural numbers S recognisable by finite automata in base k is a k-automatic sequence and that conversely, for all k-automatic sequences u {\displaystyle u} and all integers 0 ≤ i < k {\displaystyle 0\leq i 1 {\displaystyle \alpha >1} is the dominant eigenvalue of the matrix of morphism f {\displaystyle f} , namely, the matrix M ( f ) = ( m x , y ) x ∈ B , y ∈ A {\displaystyle M(f)=(m_{x,y})_{x\in B,y\in A}} , where m x , y {\displaystyle m_{x,y}} is the number of occurrences of the letter x {\displaystyle x} in the word f ( y ) {\displaystyle f(y)} . A set S of natural numbers is α {\displaystyle \alpha } -recognisable if its characteristic sequence s {\displaystyle s} is α {\displaystyle \alpha } -substitutive. A last definition: a Perron number is an algebraic number z > 1 {\displaystyle z>1} such that all its conjugates belong to the disc { z ′ ∈ C , | z ′ | < z } {\displaystyle \{z'\in \mathbb {C} ,|z'| Read more →