AI Detector Just Done Free

AI Detector Just Done Free — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Comparison of video editing software

  This is a comparison of non-linear video editing software applications. See also a more complete list of video editing software. == General information == This table gives basic general information about the different editors: === Active === === Discontinued / Inactive === ==== Definition ==== professional: used for full length Hollywood movies; professional (small): mainly used for paid commercials, short films or podcasts/YouTube channels; prosumer: Mainly targeting private use, anything that can do more than just trimming a film; basic: trimming a film; == System requirements == This table lists the operating systems that different editors can run on without emulation, as well as other system requirements. Note that minimum system requirements are listed; some features (like High Definition support) may be unavailable with these specifications. "Unix" includes the similar Linux, BSD and Unix-like operating systems. == High definition/High resolution import == The table below indicates the ability of each program to import various High Definition video or High resolution video formats for editing. == Feature set == == Output options == Please note that recording to Blu-ray does not imply 1080@50p/60p . Most only support up to 1080i 25/30 frames per second recording. Also not all formats can be output.

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  Margin (machine learning)

  In machine learning, the margin of a single data point is defined to be the distance from the data point to a decision boundary. Note that there are many distances and decision boundaries that may be appropriate for certain datasets and goals. A margin classifier is a classification model that utilizes the margin of each example to learn such classification. There are theoretical justifications (based on the VC dimension) as to why maximizing the margin (under some suitable constraints) may be beneficial for machine learning and statistical inference algorithms. For a given dataset, there may be many hyperplanes that could classify it. One reasonable choice as the best hyperplane is the one that represents the largest separation, or margin, between the classes. Hence, one should choose the hyperplane such that the distance from it to the nearest data point on each side is maximized. If such a hyperplane exists, it is known as the maximum-margin hyperplane, and the linear classifier it defines is known as a maximum margin classifier (or, equivalently, the perceptron of optimal stability).

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  Noam Slonim

  Noam Slonim (Hebrew: נעם סלונים; born in Jerusalem) is an Israeli computer scientist, specializing in Natural Language Processing and the application of Large language models. He is a Research Scientist at Google Research Israel (since September 2025) and formerly an IBM Distinguished Engineer. He founded and served as Principal Investigator of Project Debater and led Language Model Utilization at IBM Research. Beyond his scientific achievements, Slonim had a writing and media career. He was a writer for Season 4 of The Cameric Five TV comedy show, published a weekly column in Haaretz on brain science, and co-created and wrote the Israeli sitcom Puzzle. He was also the head writer for Seasons 2 and 3 of the sitcom Ha-movilim and featured in the 2020 documentary The Debater. In October 2025, his debut novel, Questionable Memories, was published by Kinneret Publishing Group. == Education and research interests == Slonim graduated from the Hebrew University of Jerusalem in 1996 with a B.S. degree in Computer Science, Physics, and Mathematics. In 2002 he completed Ph.D. summa cum laude at the Interdisciplinary Center for Neural Computation at the Hebrew University, under the supervision of Professor Naftali Tishby. His thesis focused on the theory and applications of the Information Bottleneck method. From 2003 till 2006 he did post-doctoral studies at the Lewis-Sigler Institute for Integrative Genomics at Princeton University, working with Professor Bill Bialek and Professor Saeed Tavazoie. He joined IBM Research in 2007. Slonim holds over 30 patents (granted or pending) and has co-authored more than 100 scientific publications. In 2025, he joined Google Research Israel as a research scientist. == Research activities == From 1998 to 2003 he worked on the theory and applications of the Information Bottleneck method, suggesting various cluster analysis algorithms inspired by this method, and demonstrating the practical value of these algorithms on various domains. From 2003 to 2006 he worked on developing Machine Learning algorithms that rely on Information Theory concepts, and applied these algorithms to the analysis of various types of Genomics data. In 2011 he proposed to develop the first Artificial Intelligence system that can meaningfully participate in a full live debate with an expert human debater. This work gave rise to Project Debater, that debated expert human debaters in several live events during 2018 and 2019. In 2020, Slonim delivered the opening keynote at the EMNLP conference, describing the IBM Research work on developing Project Debater. From 2022 to 2025, he led IBM Research efforts applying large language models to practical use cases; in 2025 he moved to Google Research Israel as a Research Scientist. == Writing and video career == In 1996 Slonim was a writer for Season 4 of The Cameric Five TV comedy show. In 1997–1998 he published a weekly column in Haaretz newspaper, focused on brain science research. In 1997–1999 he co-created and co-wrote the Israeli sitcom, Puzzle. In 2008–2010 he was the head writer of Season 2 and Season 3 of the Israeli Sitcom, Ha-movilim. In 2020 he was featured in the documentary The Debater, an official selection of the 2020 Copenhagen International Documentary Film Festival. In 2025, his debut novel, Questionable Memories, was published by Kinneret Publishing Group.

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  Quantum finite automaton

  In quantum computing, quantum finite automata (QFA) or quantum state machines are a quantum analog of probabilistic automata or a Markov decision process. They provide a mathematical abstraction of real-world quantum computers. Several types of automata may be defined, including measure-once and measure-many automata. Quantum finite automata can also be understood as the quantization of subshifts of finite type, or as a quantization of Markov chains. QFAs are, in turn, special cases of geometric finite automata or topological finite automata. The automata work by receiving a finite-length string σ = ( σ 0 , σ 1 , … , σ k ) {\displaystyle \sigma =(\sigma _{0},\sigma _{1},\dots ,\sigma _{k})} of letters σ i {\displaystyle \sigma _{i}} from a finite alphabet Σ {\displaystyle \Sigma } , and assigning to each such string a probability Pr ⁡ ( σ ) {\displaystyle \operatorname {Pr} (\sigma )} indicating the probability of the automaton being in an accept state; that is, indicating whether the automaton accepted or rejected the string. The languages accepted by QFAs are not the regular languages of deterministic finite automata, nor are they the stochastic languages of probabilistic finite automata. Study of these quantum languages remains an active area of research. == Informal description == There is a simple, intuitive way of understanding quantum finite automata. One begins with a graph-theoretic interpretation of deterministic finite automata (DFA). A DFA can be represented as a labelled directed graph, with states as nodes in the graph, and arrows representing state transitions. Each arrow is labelled with a possible input symbol, so that, given a specific state and an input symbol, the arrow points at the next state. One way of representing such a graph is by means of a set of adjacency matrices, with one matrix for each input symbol. In this case, a list of possible DFA states is written as a column vector. For a given input symbol, the adjacency matrix indicates how any given state (row in the state vector) will transition to the next state; a state transition is given by matrix multiplication. One needs a distinct adjacency matrix for each possible input symbol, since each input symbol can result in a different transition. The entries in the adjacency matrix must be zero's and one's. For any given column in the matrix, only one entry can be non-zero: this is the entry that indicates the next (unique) state transition. Similarly, the state of the system is a column vector, in which only one entry is non-zero: this entry corresponds to the current state of the system. Let Σ {\displaystyle \Sigma } denote the set of input symbols. For a given input symbol α ∈ Σ {\displaystyle \alpha \in \Sigma } , write U α {\displaystyle U_{\alpha }} as the adjacency matrix that describes the evolution of the DFA to its next state. The set { U α | α ∈ Σ } {\displaystyle \{U_{\alpha }|\alpha \in \Sigma \}} then completely describes the state transition function of the DFA. Let Q represent the set of possible states of the DFA. If there are N states in Q, then each matrix U α {\displaystyle U_{\alpha }} is N by N-dimensional. The initial state q 0 ∈ Q {\displaystyle q_{0}\in Q} corresponds to a column vector with a one in the q0'th row. A general state q is then a column vector with a one in the q'th row. By abuse of notation, let q0 and q also denote these two vectors. Then, after reading input symbols α β γ ⋯ {\displaystyle \alpha \beta \gamma \cdots } from the input tape, the state of the DFA will be given by q = ⋯ U γ U β U α q 0 . {\displaystyle q=\cdots U_{\gamma }U_{\beta }U_{\alpha }q_{0}.} The state transitions are given by ordinary matrix multiplication (that is, multiply q0 by U α {\displaystyle U_{\alpha }} , etc.); the order of application is 'reversed' only because we follow the standard notation of linear algebra. The above description of a DFA, in terms of linear operators and vectors, almost begs for generalization, by replacing the state-vector q by some general vector, and the matrices { U α } {\displaystyle \{U_{\alpha }\}} by some general operators. This is essentially what a QFA does: it replaces q by a unit vector, and the { U α } {\displaystyle \{U_{\alpha }\}} by unitary matrices. Other, similar generalizations also become obvious: the vector q can be some distribution on a manifold; the set of transition matrices become automorphisms of the manifold; this defines a topological finite automaton. Similarly, the matrices could be taken as automorphisms of a homogeneous space; this defines a geometric finite automaton. Before moving on to the formal description of a QFA, there are two noteworthy generalizations that should be mentioned and understood. The first is the non-deterministic finite automaton (NFA). In this case, the vector q is replaced by a vector that can have more than one entry that is non-zero. Such a vector then represents an element of the power set of Q; it’s just an indicator function on Q. Likewise, the state transition matrices { U α } {\displaystyle \{U_{\alpha }\}} are defined in such a way that a given column can have several non-zero entries in it. Equivalently, the multiply-add operations performed during component-wise matrix multiplication should be replaced by Boolean and-or operations so that the semantics are kept intact. A well-known theorem states that, for each DFA, there is an equivalent NFA, and vice versa. This implies that the set of languages that can be recognized by DFA's and NFA's are the same; these are the regular languages. In the generalization to QFAs, the set of recognized languages will be different to the regular languages. Describing that set is one of the outstanding research problems in QFA theory. Another generalization that should be immediately apparent is to use a stochastic matrix for the transition matrices, and a probability vector for the state; this gives a probabilistic finite automaton. The entries in the state vector must be real numbers, positive, and sum to one, in order for the state vector to be interpreted as a probability. The transition matrices must preserve this property: this is why they must be stochastic. Each state vector should be imagined as specifying a point in a simplex; thus, this is a topological automaton, with the simplex being the manifold, and the stochastic matrices being linear automorphisms of the simplex onto itself. Since each transition is (essentially) independent of the previous (if we disregard the distinction between accepted and rejected languages), the PFA essentially becomes a kind of Markov chain. By contrast, in a QFA, the manifold is complex projective space C P N {\displaystyle \mathbb {C} P^{N}} , and the transition matrices are unitary matrices. Each point in C P N {\displaystyle \mathbb {C} P^{N}} corresponds to a (pure) quantum-mechanical state; the unitary matrices can be thought of as governing the time evolution of the system (viz in the Schrödinger picture). The generalization from pure states to mixed states should be straightforward: A mixed state is simply a measure-theoretic probability distribution on C P N {\displaystyle \mathbb {C} P^{N}} . A worthy point to contemplate is the distributions that result on the manifold during the input of a language. In order for an automaton to be 'efficient' in recognizing a language, that distribution should be 'as uniform as possible'. This need for uniformity is the underlying principle behind maximum entropy methods: these simply guarantee crisp, compact operation of the automaton. Put in other words, the machine learning methods used to train hidden Markov models generalize to QFAs as well: the Viterbi algorithm and the forward–backward algorithm generalize readily to the QFA. Although the study of QFA was popularized in the work of Kondacs and Watrous in 1997 and later by Moore and Crutchfeld, they were described as early as 1971, by Ion Baianu. == Measure-once automata == Measure-once automata were introduced by Cris Moore and James P. Crutchfield. They may be defined formally as follows. As with an ordinary finite automaton, the quantum automaton is considered to have N {\displaystyle N} possible internal states, represented in this case by an N {\displaystyle N} -level qudit | ψ ⟩ {\displaystyle |\psi \rangle } . More precisely, the N {\displaystyle N} -level qudit | ψ ⟩ ∈ P ( C N ) {\displaystyle |\psi \rangle \in P(\mathbb {C} ^{N})} is an element of ( N − 1 ) {\displaystyle (N-1)} -dimensional complex projective space, carrying an inner product ‖ ⋅ ‖ {\displaystyle \Vert \cdot \Vert } that is the Fubini–Study metric. The state transitions, transition matrices or de Bruijn graphs are represented by a collection of N × N {\displaystyle N\times N} unitary matrices U α {\displaystyle U_{\alpha }} , with one unitary matrix for each letter α ∈ Σ {\displaystyle \alpha \in \Sigma } . That is, given an input letter α {\displaystyle \alpha } , the unitary matrix describe

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  Database-as-IPC

  In computer programming, Database-as-IPC may be considered an anti-pattern where a disk persisted table in a database is used as the message queue store for routine inter-process communication (IPC) or subscribed data processing. If database performance is of concern, alternatives include sockets, network socket, or message queue. British computer scientist, Junade Ali, defined the Database-as-IPC Anti-Pattern as using a database to "schedule jobs or queue up tasks to be completed", noting that this anti-pattern centres around using a database for temporary messages instead of persistent data. == Controversy == The issue arises if there is a performance issue, and if additional systems (and servers) can be justified. In terms of performance, recent advancements in database systems provide more efficient mechanisms for signaling and messaging, and database systems also support memory (non-persisted) tables. There are databases with built-in notification mechanisms, such as PostgreSQL, SQL Server, and Oracle. These mechanisms and future improvements of database systems can make queuing much more efficient and avoid the need to set up a separate signaling or messaging queue system along with the server and management overhead. While MySQL doesn't have direct support for notifications, some workarounds are possible. However, they would be seen as non-standard and therefore more difficult to maintain.

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  Tomáš Mikolov

  Tomáš Mikolov is a Czech computer scientist working in the field of machine learning. In March 2020, Mikolov became a senior research scientist at the Czech Institute of Informatics, Robotics and Cybernetics. == Career == Mikolov obtained his PhD in Computer Science from Brno University of Technology for his work on recurrent neural network-based language models. He is the lead author of the 2013 paper that introduced the Word2vec technique in natural language processing and is an author on the FastText architecture. Mikolov came up with the idea to generate text from neural language models in 2007 and his RNNLM toolkit was the first to demonstrate the capability to train language models on large corpora, resulting in large improvements over the state of the art. Prior to joining Facebook in 2014, Mikolov worked as a visiting researcher at Johns Hopkins University, Université de Montréal, Microsoft and Google. He left Facebook at some time in 2019/2020 to join the Czech Institute of Informatics, Robotics and Cybernetics. Mikolov has argued that humanity might be at a greater existential risk if an artificial general intelligence is not developed.

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  Clement Farabet

  Clément Farabet is a computer scientist and AI expert known for his contributions to the field of deep learning. He served as a research scientist at the New York University. He serves as the Vice President of Research at Google DeepMind and previously served as the VP of AI Infrastructure at NVIDIA. His scholarly work received over 11,000 citations with an h-index of 21. == Education == In 2008, Farabet earned a master's degree in electrical engineering with honors from Institut national des sciences appliquées (INSA) de Lyon, France. In 2010, Farabet received his PhD at Université Paris-Est, co-advised by Professors Laurent Najman and Yann LeCun. His thesis focused on real-time image understanding and introduced multi-scale convolutional networks and graph-based techniques for efficient segmentations of class prediction maps. He successfully defended his thesis in 2013. == Career == In 2008, after completing his Master's degree, Farabet joined Professor Yann LeCun's laboratory at the Courant Institute of Mathematical Sciences at New York University. His Master's thesis work on reconfigurable hardware for deep neural networks resulted in a patent. He continued his collaboration with Yann LeCun, and in 2009, he began working with Yale University's e-Lab, led by Eugenio Culurciello. This collaboration eventually led to the creation of TeraDeep. He began his career as a researcher, contributing to the development of LuaTorch, one of the first AI frameworks, which later evolved into PyTorch, widely recognized and adopted globally. == Startups == Farabet co-founded MadBits, a startup with a focus on web-scale image understanding. The company was acquired by Twitter in 2014. Following this acquisition, Farabet co-founded Twitter Cortex, a team dedicated to building Twitter's deep learning platform for various applications, including recommendations, search, spam detection, and NSFW content and ads. == Publications == Farabet, Clement; Couprie, Camille; Najman, Laurent; LeCun, Yann (August 2013). "Learning Hierarchical Features for Scene Labeling". IEEE Transactions on Pattern Analysis and Machine Intelligence. 35 (8): 1915–1929. Bibcode:2013ITPAM..35.1915F. doi:10.1109/TPAMI.2012.231. PMID 23787344. S2CID 206765110. LeCun, Yann; Kavukcuoglu, Koray; Farabet, Clement (2010). "Convolutional networks and applications in vision". Proceedings of 2010 IEEE International Symposium on Circuits and Systems. pp. 253–256. doi:10.1109/ISCAS.2010.5537907. ISBN 978-1-4244-5308-5. S2CID 7625356. Collobert, Ronan; Kavukcuoglu, K.; Farabet, C. (2011). "Torch7: A Matlab-like Environment for Machine Learning". Neural Information Processing Systems. Couprie, Camille; Farabet, Clément; Najman, Laurent; LeCun, Yann (16 January 2013). "Indoor Semantic Segmentation using depth information". arXiv:1301.3572 [cs.CV]. Farabet, Clement (2011). "NeuFlow: A runtime reconfigurable dataflow processor for vision". CVPR 2011 Workshops. pp. 109–116. doi:10.1109/CVPRW.2011.5981829. ISBN 978-1-4577-0529-8. S2CID 851574. Farabet, Clement (2009). "CNP: An FPGA-based processor for Convolutional Networks". 2009 International Conference on Field Programmable Logic and Applications. pp. 32–37. doi:10.1109/FPL.2009.5272559. S2CID 5339694. Farabet, Clement (2010). "Hardware accelerated convolutional neural networks for synthetic vision systems". Proceedings of 2010 IEEE International Symposium on Circuits and Systems. pp. 257–260. doi:10.1109/ISCAS.2010.5537908. ISBN 978-1-4244-5308-5. S2CID 6542026.

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  Noémie Elhadad

  Noémie Elhadad is an American data scientist who is an associate professor of biomedical informatics at the Columbia University Vagelos College of Physicians and Surgeons. As of 2022, she serves as the chair of the Department of Biomedical Informatics. Her research considers machine learning in bioinformatics, natural language processing and medicine. == Early life and education == Elhadad studied computer software engineering at École nationale supérieure d'électronique, informatique, télécommunications, mathématique et mécanique de Bordeaux (ENSEIRB). She completed her doctoral research at Columbia University. She was based in the Department of Computer Science, where she developed patient-focused text summaries of clinical literature. == Research and career == Elhadad joined the faculty at the City College of New York. In 2007 she joined the Department of Biomedical Informatics at Columbia University. She was made Chair of the Health Analytics Center at the Columbia Data Science Institute in 2013. Her research considers how clinical data, electronic health records and patient-generated data can enhance access to information for researchers, patients and physicians. She developed an artificial intelligence tool that supported patients in the NewYork-Presbyterian Hospital. Elhadad is interested in using data to advance women's health. She led the Citizen Endo Project that looks to comprehensively describe how patients experience endometriosis. It was built using principles of citizen science, using patient testimonials from focus groups in New York City and data aggregation. She created the app, Phendo, which asks patients about their experience of the disease. The name Phendo is a portmanteau of phenotyping endometriosis. Elhadad was announced as chair of the Department of Biomedical Informatics in December 2022. == Selected publications == Caruana, Rich; Lou, Yin; Gehrke, Johannes; Koch, Paul; Sturm, Marc; Elhadad, Noemie (August 10, 2015). "Intelligible Models for HealthCare". Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. New York, NY, USA: ACM. pp. 1721–1730. doi:10.1145/2783258.2788613. ISBN 9781450336642. S2CID 14190268. Chaitanya Shivade; Preethi Raghavan; Eric Fosler-Lussier; Peter J Embi; Noemie Elhadad; Stephen B Johnson; Albert M Lai (November 7, 2013). "A review of approaches to identifying patient phenotype cohorts using electronic health records". Journal of the American Medical Informatics Association. 21 (2): 221–230. doi:10.1136/AMIAJNL-2013-001935. ISSN 1067-5027. PMC 3932460. PMID 24201027. Wikidata Q37598951. Shivade, Chaitanya; Raghavan, Preethi; Fosler-Lussier, Eric; Embi, Peter J; Elhadad, Noemie; Johnson, Stephen B; Lai, Albert M (March 2014). "A review of approaches to identifying patient phenotype cohorts using electronic health records". Journal of the American Medical Informatics Association. 21 (2): 221–230. doi:10.1136/amiajnl-2013-001935. ISSN 1067-5027. PMC 3932460. PMID 24201027. == Personal life == Elhadad suffers from endometriosis.

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  Hierarchical control system

  A hierarchical control system (HCS) is a form of control system in which a set of devices and governing software is arranged in a hierarchical tree. When the links in the tree are implemented by a computer network, then that hierarchical control system is also a form of networked control system. == Overview == A human-built system with complex behavior is often organized as a hierarchy. For example, a command hierarchy has among its notable features the organizational chart of superiors, subordinates, and lines of organizational communication. Hierarchical control systems are organized similarly to divide the decision making responsibility. Each element of the hierarchy is a linked node in the tree. Commands, tasks and goals to be achieved flow down the tree from superior nodes to subordinate nodes, whereas sensations and command results flow up the tree from subordinate to superior nodes. Nodes may also exchange messages with their siblings. The two distinguishing features of a hierarchical control system are related to its layers. Each higher layer of the tree operates with a longer interval of planning and execution time than its immediately lower layer. The lower layers have local tasks, goals, and sensations, and their activities are planned and coordinated by higher layers which do not generally override their decisions. The layers form a hybrid intelligent system in which the lowest, reactive layers are sub-symbolic. The higher layers, having relaxed time constraints, are capable of reasoning from an abstract world model and performing planning. A hierarchical task network is a good fit for planning in a hierarchical control system. Besides artificial systems, an animal's control systems are proposed to be organized as a hierarchy. In perceptual control theory, which postulates that an organism's behavior is a means of controlling its perceptions, the organism's control systems are suggested to be organized in a hierarchical pattern as their perceptions are constructed so. == Control system structure == The accompanying diagram is a general hierarchical model which shows functional manufacturing levels using computerised control of an industrial control system. Referring to the diagram; Level 0 contains the field devices such as flow and temperature sensors, and final control elements, such as control valves Level 1 contains the industrialised Input/Output (I/O) modules, and their associated distributed electronic processors. Level 2 contains the supervisory computers, which collate information from processor nodes on the system, and provide the operator control screens. Level 3 is the production control level, which does not directly control the process, but is concerned with monitoring production and monitoring targets Level 4 is the production scheduling level. == Applications == === Manufacturing, robotics and vehicles === Among the robotic paradigms is the hierarchical paradigm in which a robot operates in a top-down fashion, heavy on planning, especially motion planning. Computer-aided production engineering has been a research focus at NIST since the 1980s. Its Automated Manufacturing Research Facility was used to develop a five layer production control model. In the early 1990s DARPA sponsored research to develop distributed (i.e. networked) intelligent control systems for applications such as military command and control systems. NIST built on earlier research to develop its Real-Time Control System (RCS) and Real-time Control System Software which is a generic hierarchical control system that has been used to operate a manufacturing cell, a robot crane, and an automated vehicle. In November 2007, DARPA held the Urban Challenge. The winning entry, Tartan Racing employed a hierarchical control system, with layered mission planning, motion planning, behavior generation, perception, world modelling, and mechatronics. === Artificial intelligence === Subsumption architecture is a methodology for developing artificial intelligence that is heavily associated with behavior based robotics. This architecture is a way of decomposing complicated intelligent behavior into many "simple" behavior modules, which are in turn organized into layers. Each layer implements a particular goal of the software agent (i.e. system as a whole), and higher layers are increasingly more abstract. Each layer's goal subsumes that of the underlying layers, e.g. the decision to move forward by the eat-food layer takes into account the decision of the lowest obstacle-avoidance layer. Behavior need not be planned by a superior layer, rather behaviors may be triggered by sensory inputs and so are only active under circumstances where they might be appropriate. Reinforcement learning has been used to acquire behavior in a hierarchical control system in which each node can learn to improve its behavior with experience. James Albus, while at NIST, developed a theory for intelligent system design named the Reference Model Architecture (RMA), which is a hierarchical control system inspired by RCS. Albus defines each node to contain these components. Behavior generation is responsible for executing tasks received from the superior, parent node. It also plans for, and issues tasks to, the subordinate nodes. Sensory perception is responsible for receiving sensations from the subordinate nodes, then grouping, filtering, and otherwise processing them into higher level abstractions that update the local state and which form sensations that are sent to the superior node. Value judgment is responsible for evaluating the updated situation and evaluating alternative plans. World Model is the local state that provides a model for the controlled system, controlled process, or environment at the abstraction level of the subordinate nodes. At its lowest levels, the RMA can be implemented as a subsumption architecture, in which the world model is mapped directly to the controlled process or real world, avoiding the need for a mathematical abstraction, and in which time-constrained reactive planning can be implemented as a finite-state machine. Higher levels of the RMA however, may have sophisticated mathematical world models and behavior implemented by automated planning and scheduling. Planning is required when certain behaviors cannot be triggered by current sensations, but rather by predicted or anticipated sensations, especially those that come about as result of the node's actions.

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  Chelsea Finn

  Chelsea Finn (born October 8, 1992) is an American computer scientist and assistant professor at Stanford University. Her research investigates intelligence through the interactions of robots, with the hope to create robotic systems that can learn how to learn. She previously worked for Google and currently is a co-founder of the startup Physical Intelligence. == Early life and education == Finn was an undergraduate student in electrical engineering and computer science at Massachusetts Institute of Technology. She then moved to the University of California, Berkeley, where she earned her Ph.D. in 2018 under Pieter Abbeel and Sergey Levine. Her work in the Berkeley Artificial Intelligence Lab (BAIR) focused on gradient based algorithms . Such algorithms allow machines to 'learn to learn', more akin to human learning than traditional machine learning systems. These “meta-learning” techniques train machines to quickly adapt, such that when they encounter new scenarios they can learn quickly. As a doctoral student she worked as an intern at Google Brain, where she worked on robot learning algorithms from deep predictive models. She delivered a massive open online course on deep reinforcement learning. She was the first woman to win the C.V. & Daulat Ramamoorthy Distinguished Research Award. == Research and career == Finn investigates the capabilities of robots to develop intelligence through learning and interaction. She has made use of deep learning algorithms to simultaneously learn visual perception and control robotic skills. She developed meta-learning approaches to train neural networks to take in student code and output useful feedback. She showed that the system could quickly adapt without too much input from the instructor. She trialled the programme on Code in Place, a 12,000 student course delivered by Stanford University every year. She found that 97.9% of the time the students agreed with the feedback being given. == Awards and honors == 2016 C.V. & Daulat Ramamoorthy Distinguished Research Award 2017 Electrical engineering and computer science rising star 2018 MIT Technology Review 35 Under 35 2018 ACM Doctoral Dissertation Award 2020 Samsung Advanced Institute of Technology AI Researcher of the Year 2020 Intel Rising Star Faculty Award 2021 Office of Naval Research Young Investigator Award 2022 IEEE Robotics and Automation Society Early Academic Career Award == Select publications == Finn, Chelsea; Abbeel, Pieter; Levine, Sergey (2017-07-17). "Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks". International Conference on Machine Learning. PMLR: 1126–1135. arXiv:1703.03400. Sergey Levine; Chelsea Finn; Trevor Darrell; Pieter Abbeel (2016). "End-to-End Training of Deep Visuomotor Policies". Journal of Machine Learning Research. 17 (39): 1–40. arXiv:1504.00702. ISSN 1533-7928. Wikidata Q90313375. Chelsea Finn; Ian Goodfellow; Sergey Levine (2016). "Unsupervised Learning for Physical Interaction through Video Prediction" (PDF). Advances in Neural Information Processing Systems 29. Advances in Neural Information Processing Systems. Wikidata Q46993574.

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  Best AI Website Builders in 2026

  Comparing the best AI website builder? An AI website builder is software that uses machine learning to help you get more done — it lowers the barrier so anyone can produce professional output. Privacy matters too: check whether your data trains the model and whether a no-log or enterprise tier is available. Whether you are a beginner or a pro, the right AI website builder slots into your workflow and pays for itself fast. Below we compare features, pricing, and real output so you can choose with confidence.

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  Sequential minimal optimization

  Sequential minimal optimization (SMO) is an algorithm for solving the quadratic programming (QP) problem that arises during the training of support-vector machines (SVM). It was invented by John Platt in 1998 at Microsoft Research. SMO is widely used for training support vector machines and is implemented by the popular LIBSVM tool. The publication of the SMO algorithm in 1998 has generated a lot of excitement in the SVM community, as previously available methods for SVM training were much more complex and required expensive third-party QP solvers. == Optimization problem == Consider a binary classification problem with a dataset (x1, y1), ..., (xn, yn), where xi is an input vector and yi ∈ {-1, +1} is a binary label corresponding to it. A soft-margin support vector machine is trained by solving a quadratic programming problem, which is expressed in the dual form as follows: max α ∑ i = 1 n α i − 1 2 ∑ i = 1 n ∑ j = 1 n y i y j K ( x i , x j ) α i α j , {\displaystyle \max _{\alpha }\sum _{i=1}^{n}\alpha _{i}-{\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}y_{i}y_{j}K(x_{i},x_{j})\alpha _{i}\alpha _{j},} subject to: 0 ≤ α i ≤ C , for i = 1 , 2 , … , n , {\displaystyle 0\leq \alpha _{i}\leq C,\quad {\mbox{ for }}i=1,2,\ldots ,n,} ∑ i = 1 n y i α i = 0 {\displaystyle \sum _{i=1}^{n}y_{i}\alpha _{i}=0} where C is an SVM hyperparameter and K(xi, xj) is the kernel function, both supplied by the user; and the variables α i {\displaystyle \alpha _{i}} are Lagrange multipliers. == Algorithm == SMO is an iterative algorithm for solving the optimization problem described above. SMO breaks this problem into a series of smallest possible sub-problems, which are then solved analytically. Because of the linear equality constraint involving the Lagrange multipliers α i {\displaystyle \alpha _{i}} , the smallest possible problem involves two such multipliers. Then, for any two multipliers α 1 {\displaystyle \alpha _{1}} and α 2 {\displaystyle \alpha _{2}} , the constraints are reduced to: 0 ≤ α 1 , α 2 ≤ C , {\displaystyle 0\leq \alpha _{1},\alpha _{2}\leq C,} y 1 α 1 + y 2 α 2 = k , {\displaystyle y_{1}\alpha _{1}+y_{2}\alpha _{2}=k,} and this reduced problem can be solved analytically: one needs to find a minimum of a one-dimensional quadratic function. k {\displaystyle k} is the negative of the sum over the rest of terms in the equality constraint, which is fixed in each iteration. The algorithm proceeds as follows: Find a Lagrange multiplier α 1 {\displaystyle \alpha _{1}} that violates the Karush–Kuhn–Tucker (KKT) conditions for the optimization problem. Pick a second multiplier α 2 {\displaystyle \alpha _{2}} and optimize the pair ( α 1 , α 2 ) {\displaystyle (\alpha _{1},\alpha _{2})} . Repeat steps 1 and 2 until convergence. When all the Lagrange multipliers satisfy the KKT conditions (within a user-defined tolerance), the problem has been solved. Although this algorithm is guaranteed to converge, heuristics are used to choose the pair of multipliers so as to accelerate the rate of convergence. This is critical for large data sets since there are n ( n − 1 ) / 2 {\displaystyle n(n-1)/2} possible choices for α i {\displaystyle \alpha _{i}} and α j {\displaystyle \alpha _{j}} . == Related work == The first approach to splitting large SVM learning problems into a series of smaller optimization tasks was proposed by Bernhard Boser, Isabelle Guyon, and Vladimir Vapnik. It is known as the "chunking algorithm". The algorithm starts with a random subset of the data, solves this problem, and iteratively adds examples which violate the optimality conditions. One disadvantage of this algorithm is that it is necessary to solve QP-problems scaling with the number of SVs. On real world sparse data sets, SMO can be more than 1000 times faster than the chunking algorithm. In 1997, E. Osuna, R. Freund, and F. Girosi proved a theorem which suggests a whole new set of QP algorithms for SVMs. By the virtue of this theorem a large QP problem can be broken down into a series of smaller QP sub-problems. A sequence of QP sub-problems that always add at least one violator of the Karush–Kuhn–Tucker (KKT) conditions is guaranteed to converge. The chunking algorithm obeys the conditions of the theorem, and hence will converge. The SMO algorithm can be considered a special case of the Osuna algorithm, where the size of the optimization is two and both Lagrange multipliers are replaced at every step with new multipliers that are chosen via good heuristics. The SMO algorithm is closely related to a family of optimization algorithms called Bregman methods or row-action methods. These methods solve convex programming problems with linear constraints. They are iterative methods where each step projects the current primal point onto each constraint.

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  Artificial intelligence

  Artificial intelligence (AI) is the capability of computational systems to perform tasks typically associated with human intelligence, such as learning, reasoning, problem-solving, perception, and decision-making. It is a field of research in engineering, mathematics and computer science that develops and studies methods and software that enable machines to perceive their environment and use learning and intelligence to take actions that maximize their chances of achieving defined goals. High-profile applications of AI include advanced web search engines, chatbots, virtual assistants, autonomous vehicles, and play and analysis in strategy games (e.g., chess and Go). Since the 2020s, generative AI has become widely available to generate images, audio, and videos from text prompts. The traditional goals of AI research include learning, reasoning, knowledge representation, planning, natural language processing, and perception, as well as support for robotics. To reach these goals, AI researchers have used techniques including state space search and mathematical optimization, formal logic, artificial neural networks, and methods based on statistics, operations research, and economics. AI also draws upon psychology, linguistics, philosophy, neuroscience, and other fields. Some companies, such as OpenAI, Google DeepMind and Meta, aim to create artificial general intelligence (AGI) – AI that can complete virtually any cognitive task at least as well as a human. Artificial intelligence was founded as an academic discipline in 1956, and the field went through multiple cycles of optimism throughout its history, followed by periods of disappointment and loss of funding, known as AI winters. Funding and interest increased substantially after 2012, when graphics processing units began being used to accelerate neural networks, and deep learning outperformed previous AI techniques. This growth accelerated further after 2017 with the transformer architecture. In the 2020s, an AI boom has coincided with advances in generative AI, which allowed for the creation and modification of media. In addition to AI safety and unintended consequences and harms from the use of AI, ethical concerns, AI's long-term effects, and potential existential risks have prompted discussions of AI regulation. == Goals == The general problem of simulating (or creating) intelligence has been broken into subproblems. These consist of particular traits or capabilities that researchers expect an intelligent system to display. The traits described below have received the most attention and cover the scope of AI research. === Reasoning and problem-solving === Early researchers developed algorithms that imitated step-by-step reasoning that humans use when they solve puzzles or make logical deductions. By the late 1980s and 1990s, methods were developed for dealing with uncertain or incomplete information, employing concepts from probability and economics. Many of these algorithms are insufficient for solving large reasoning problems because they experience a "combinatorial explosion": They become exponentially slower as the problems grow. Even humans rarely use the step-by-step deduction that early AI research could model. They solve most of their problems using fast, intuitive judgments. Accurate and efficient reasoning is an unsolved problem. === Knowledge representation === Knowledge representation and knowledge engineering allow AI programs to answer questions intelligently and make deductions about real-world facts. Formal knowledge representations are used in content-based indexing and retrieval, scene interpretation, clinical decision support, knowledge discovery (mining "interesting" and actionable inferences from large databases), and other areas. A knowledge base is a body of knowledge represented in a form that can be used by a program. An ontology is the set of objects, relations, concepts, and properties used by a particular domain of knowledge. Knowledge bases need to represent things such as objects, properties, categories, and relations between objects; situations, events, states, and time; causes and effects; knowledge about knowledge (what we know about what other people know); default reasoning (things that humans assume are true until they are told differently and will remain true even when other facts are changing); and many other aspects and domains of knowledge. Among the most difficult problems in knowledge representation are the breadth of commonsense knowledge (the set of atomic facts that the average person knows is enormous); and the sub-symbolic form of most commonsense knowledge (much of what people know is not represented as "facts" or "statements" that they could express verbally). There is also the difficulty of knowledge acquisition, the problem of obtaining knowledge for AI applications. === Planning and decision-making === An "agent" is any entity (artificial or not) that perceives and takes actions in the world. A rational agent has goals or preferences and takes actions to make them happen. In automated planning, the agent has a specific goal. In automated decision-making, the agent has preferences—there are some situations it would prefer to be in, and some situations it is trying to avoid. The decision-making agent assigns a number to each situation (called the "utility") that measures how much the agent prefers it. For each possible action, it can calculate the "expected utility": the utility of all possible outcomes of the action, weighted by the probability that the outcome will occur. It can then choose the action with the maximum expected utility. In classical planning, the agent knows exactly what the effect of any action will be. In most real-world problems, however, the agent may not be certain about the situation they are in (it is "unknown" or "unobservable") and it may not know for certain what will happen after each possible action (it is not "deterministic"). It must choose an action by making a probabilistic guess and then reassess the situation to see if the action worked. Alongside thorough testing and improvement based on previous decisions, having an explanation for why the agent took certain decisions is a way to build trust, especially when the decisions have to be relied upon. In some problems, the agent's preferences may be uncertain, especially if there are other agents or humans involved. These can be learned (e.g., with inverse reinforcement learning), or the agent can seek information to improve its preferences. Information value theory can be used to weigh the value of exploratory or experimental actions. The space of possible future actions and situations is typically intractably large, so the agents must take actions and evaluate situations while being uncertain of what the outcome will be. A Markov decision process has a transition model that describes the probability that a particular action will change the state in a particular way and a reward function that supplies the utility of each state and the cost of each action. A policy associates a decision with each possible state. The policy could be calculated (e.g., by iteration), be heuristic, or it can be learned. Game theory describes the rational behavior of multiple interacting agents and is used in AI programs that make decisions that involve other agents. === Learning === Machine learning is the study of programs that can improve their performance on a given task automatically. It has been a part of AI from the beginning. There are several kinds of machine learning. Unsupervised learning analyzes a stream of data and finds patterns and makes predictions without any other guidance. Supervised learning requires labeling the training data with the expected answers, and comes in two main varieties: classification (where the program must learn to predict what category the input belongs in) and regression (where the program must deduce a numeric function based on numeric input). In reinforcement learning, the agent is rewarded for good responses and punished for bad ones. The agent learns to choose responses that are classified as "good". Transfer learning is when the knowledge gained from one problem is applied to a new problem. Deep learning is a type of machine learning that runs inputs through biologically inspired artificial neural networks for all of these types of learning. Computational learning theory can assess learners by computational complexity, by sample complexity (how much data is required), or by other notions of optimization. === Natural language processing === Natural language processing (NLP) allows programs to read, write and communicate in human languages. Specific problems include speech recognition, speech synthesis, machine translation, information extraction, information retrieval and question answering. Early work, based on Noam Chomsky's generative grammar and semantic networks, had difficulty with word-sense disambiguation unless

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  Suffix automaton

  In computer science, a suffix automaton is an efficient data structure for representing the substring index of a given string which allows the storage, processing, and retrieval of compressed information about all its substrings. The suffix automaton of a string S {\displaystyle S} is the smallest directed acyclic graph with a dedicated initial vertex and a set of "final" vertices, such that paths from the initial vertex to final vertices represent the suffixes of the string. In terms of automata theory, a suffix automaton is the minimal partial deterministic finite automaton that recognizes the set of suffixes of a given string S = s 1 s 2 … s n {\displaystyle S=s_{1}s_{2}\dots s_{n}} . The state graph of a suffix automaton is called a directed acyclic word graph (DAWG), a term that is also sometimes used for any deterministic acyclic finite state automaton. Suffix automata were introduced in 1983 by a group of scientists from the University of Denver and the University of Colorado Boulder. They suggested a linear time online algorithm for its construction and showed that the suffix automaton of a string S {\displaystyle S} having length at least two characters has at most 2 | S | − 1 {\textstyle 2|S|-1} states and at most 3 | S | − 4 {\textstyle 3|S|-4} transitions. Further works have shown a close connection between suffix automata and suffix trees, and have outlined several generalizations of suffix automata, such as compacted suffix automaton obtained by compression of nodes with a single outgoing arc. Suffix automata provide efficient solutions to problems such as substring search and computation of the largest common substring of two and more strings. == History == The concept of suffix automaton was introduced in 1983 by a group of scientists from University of Denver and University of Colorado Boulder consisting of Anselm Blumer, Janet Blumer, Andrzej Ehrenfeucht, David Haussler and Ross McConnell, although similar concepts had earlier been studied alongside suffix trees in the works of Peter Weiner, Vaughan Pratt and Anatol Slissenko. In their initial work, Blumer et al. showed a suffix automaton built for the string S {\displaystyle S} of length greater than 1 {\displaystyle 1} has at most 2 | S | − 1 {\displaystyle 2|S|-1} states and at most 3 | S | − 4 {\displaystyle 3|S|-4} transitions, and suggested a linear algorithm for automaton construction. In 1983, Mu-Tian Chen and Joel Seiferas independently showed that Weiner's 1973 suffix-tree construction algorithm while building a suffix tree of the string S {\displaystyle S} constructs a suffix automaton of the reversed string S R {\textstyle S^{R}} as an auxiliary structure. In 1987, Blumer et al. applied the compressing technique used in suffix trees to a suffix automaton and invented the compacted suffix automaton, which is also called the compacted directed acyclic word graph (CDAWG). In 1997, Maxime Crochemore and Renaud Vérin developed a linear algorithm for direct CDAWG construction. In 2001, Shunsuke Inenaga et al. developed an algorithm for construction of CDAWG for a set of words given by a trie. == Definitions == Usually when speaking about suffix automata and related concepts, some notions from formal language theory and automata theory are used, in particular: "Alphabet" is a finite set Σ {\displaystyle \Sigma } that is used to construct words. Its elements are called "characters"; "Word" is a finite sequence of characters ω = ω 1 ω 2 … ω n {\displaystyle \omega =\omega _{1}\omega _{2}\dots \omega _{n}} . "Length" of the word ω {\displaystyle \omega } is denoted as | ω | = n {\displaystyle |\omega |=n} ; "Formal language" is a set of words over given alphabet; "Language of all words" is denoted as Σ ∗ {\displaystyle \Sigma ^{}} (where the "" character stands for Kleene star), "empty word" (the word of zero length) is denoted by the character ε {\displaystyle \varepsilon } ; "Concatenation of words" α = α 1 α 2 … α n {\displaystyle \alpha =\alpha _{1}\alpha _{2}\dots \alpha _{n}} and β = β 1 β 2 … β m {\displaystyle \beta =\beta _{1}\beta _{2}\dots \beta _{m}} is denoted as α ⋅ β {\displaystyle \alpha \cdot \beta } or α β {\displaystyle \alpha \beta } and corresponds to the word obtained by writing β {\displaystyle \beta } to the right of α {\displaystyle \alpha } , that is, α β = α 1 α 2 … α n β 1 β 2 … β m {\displaystyle \alpha \beta =\alpha _{1}\alpha _{2}\dots \alpha _{n}\beta _{1}\beta _{2}\dots \beta _{m}} ; "Concatenation of languages" A {\displaystyle A} and B {\displaystyle B} is denoted as A ⋅ B {\displaystyle A\cdot B} or A B {\displaystyle AB} and corresponds to the set of pairwise concatenations A B = { α β : α ∈ A , β ∈ B } {\displaystyle AB=\{\alpha \beta :\alpha \in A,\beta \in B\}} ; If the word ω ∈ Σ ∗ {\displaystyle \omega \in \Sigma ^{}} may be represented as ω = α γ β {\displaystyle \omega =\alpha \gamma \beta } , where α , β , γ ∈ Σ ∗ {\displaystyle \alpha ,\beta ,\gamma \in \Sigma ^{}} , then words α {\displaystyle \alpha } , β {\displaystyle \beta } and γ {\displaystyle \gamma } are called "prefix", "suffix" and "subword" (substring) of the word ω {\displaystyle \omega } correspondingly; If T = T 1 … T n {\displaystyle T=T_{1}\dots T_{n}} and T l T l + 1 … T r = S {\displaystyle T_{l}T_{l+1}\dots T_{r}=S} (with 1 ≤ l ≤ r ≤ n {\displaystyle 1\leq l\leq r\leq n} ) then S {\displaystyle S} is said to "occur" in T {\displaystyle T} as a subword. Here l {\displaystyle l} and r {\displaystyle r} are called left and right positions of occurrence of S {\displaystyle S} in T {\displaystyle T} correspondingly. == Automaton structure == Formally, deterministic finite automaton is determined by 5-tuple A = ( Σ , Q , q 0 , F , δ ) {\displaystyle {\mathcal {A}}=(\Sigma ,Q,q_{0},F,\delta )} , where: Σ {\displaystyle \Sigma } is an "alphabet" that is used to construct words, Q {\displaystyle Q} is a set of automaton "states", q 0 ∈ Q {\displaystyle q_{0}\in Q} is an "initial" state of automaton, F ⊂ Q {\displaystyle F\subset Q} is a set of "final" states of automaton, δ : Q × Σ ↦ Q {\displaystyle \delta :Q\times \Sigma \mapsto Q} is a partial "transition" function of automaton, such that δ ( q , σ ) {\displaystyle \delta (q,\sigma )} for q ∈ Q {\displaystyle q\in Q} and σ ∈ Σ {\displaystyle \sigma \in \Sigma } is either undefined or defines a transition from q {\displaystyle q} over character σ {\displaystyle \sigma } . Most commonly, deterministic finite automaton is represented as a directed graph ("diagram") such that: Set of graph vertices corresponds to the state of states Q {\displaystyle Q} , Graph has a specific marked vertex corresponding to initial state q 0 {\displaystyle q_{0}} , Graph has several marked vertices corresponding to the set of final states F {\displaystyle F} , Set of graph arcs corresponds to the set of transitions δ {\displaystyle \delta } , Specifically, every transition δ ( q 1 , σ ) = q 2 {\textstyle \delta (q_{1},\sigma )=q_{2}} is represented by an arc from q 1 {\displaystyle q_{1}} to q 2 {\displaystyle q_{2}} marked with the character σ {\displaystyle \sigma } . This transition also may be denoted as q 1 σ ⟶ q 2 {\textstyle q_{1}{\begin{smallmatrix}{\sigma }\\[-5pt]{\longrightarrow }\end{smallmatrix}}q_{2}} . In terms of its diagram, the automaton recognizes the word ω = ω 1 ω 2 … ω m {\displaystyle \omega =\omega _{1}\omega _{2}\dots \omega _{m}} only if there is a path from the initial vertex q 0 {\displaystyle q_{0}} to some final vertex q ∈ F {\displaystyle q\in F} such that concatenation of characters on this path forms ω {\displaystyle \omega } . The set of words recognized by an automaton forms a language that is set to be recognized by the automaton. In these terms, the language recognized by a suffix automaton of S {\displaystyle S} is the language of its (possibly empty) suffixes. === Automaton states === "Right context" of the word ω {\displaystyle \omega } with respect to language L {\displaystyle L} is a set [ ω ] R = { α : ω α ∈ L } {\displaystyle [\omega ]_{R}=\{\alpha :\omega \alpha \in L\}} that is a set of words α {\displaystyle \alpha } such that their concatenation with ω {\displaystyle \omega } forms a word from L {\displaystyle L} . Right contexts induce a natural equivalence relation [ α ] R = [ β ] R {\displaystyle [\alpha ]_{R}=[\beta ]_{R}} on the set of all words. If language L {\displaystyle L} is recognized by some deterministic finite automaton, there exists unique up to isomorphism automaton that recognizes the same language and has the minimum possible number of states. Such an automaton is called a minimal automaton for the given language L {\displaystyle L} . Myhill–Nerode theorem allows it to define it explicitly in terms of right contexts: In these terms, a "suffix automaton" is the minimal deterministic finite automaton recognizing the language of suffixes of the word S = s 1 s 2 … s n {\displaystyle S=s_{1}s_{2}\dots s_{n}} . The right context of the word ω {\displaystyle \omeg

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  Markov property

  In probability theory and statistics, the Markov property is the memoryless property of a stochastic process, which means that its future evolution is independent of its history. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time. The term Markov assumption is used to describe a model where the Markov property is assumed to hold, such as a hidden Markov model. A Markov random field extends this property to two or more dimensions or to random variables defined for an interconnected network of items. An example of a model for such a field is the Ising model. A discrete-time stochastic process satisfying the Markov property is known as a Markov chain. == Introduction == A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present values) depends only upon the present state; that is, given the present, the future does not depend on the past. A process with this property is said to be Markov or Markovian and known as a Markov process. Two famous classes of Markov process are the Markov chain and Brownian motion. Note that there is a subtle, often overlooked and very important point that is often missed in the plain English statement of the definition: the statespace of the process is constant through time. The conditional description involves a fixed "bandwidth". For example, without this restriction we could augment any process to one which includes the complete history from a given initial condition and it would be made to be Markovian. But the state space would be of increasing dimensionality over time and does not meet the definition. == History == == Definition == Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} be a probability space with a filtration ( F s , s ∈ I ) {\displaystyle ({\mathcal {F}}_{s},\ s\in I)} , for some (totally ordered) index set I {\displaystyle I} ; and let ( S , Σ ) {\displaystyle (S,\Sigma )} be a measurable space. An ( S , Σ ) {\displaystyle (S,\Sigma )} -valued stochastic process X = { X t : Ω → S } t ∈ I {\displaystyle X=\{X_{t}:\Omega \to S\}_{t\in I}} adapted to the filtration is said to possess the Markov property if, for each A ∈ Σ {\displaystyle A\in \Sigma } and each s , t ∈ I {\displaystyle s,t\in I} with s < t {\displaystyle s Read more →