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NASA AI Assisted-Air Quality Monitoring Project

The NASA Expert-System Ion Trap Mass Spectrometer (ES-ITMS) Project was a public-private partnership to develop an artificial intelligence assisted, air quality monitoring system and was qualified for use on the Space Shuttle. The partnership was also the first cost and intellectual property shared public-partnership implemented by NASA, which used the commercial Research and Development Limited Partnership (RDLP) model that had been adopted by the Reagan Administration for Department of Defense semiconductor development, and recommended for use by NASA for space commercialization. The project partners included NASA, the University of Florida and Finnigan MAT Corporation, was organized and administered by the NASA Joint Enterprise Institute (subsequently NASA Joint Sponsored Program) and ran from 1988 through 1990. The partnership concluded final testing in 1991, generating four patents, expert system software and application protocol reports. The system was space qualified for use on the Shuttle and elements of the ES-ITMS system were integrated into the product Improvements for Finnigan MAT corporation. The success of the partnership lead NASA to create a pilot program to develop partnership business models as an ongoing management practice. == Purpose and objectives == The need to monitor air quality in confined spaces represented an increasing challenge for NASA's planned space missions and private sector facility managers facing the increased scrutiny of possible air contaminants. Up to the early 1980's, air quality monitors generally required large spaces and human technicians to interpret readings. This created a need for miniaturized air quality monitors that could generate reliable and accurate analytic results without on-site technician presence. NASA initiated projects to develop..."mobile and/or portable mass spectrometers" that evaluated the "tradeoff between instrumentation capabilities and space, weight and power considerations." NASA selected a "commercial ITMS instrument capable of generating electron ionization, chemical ionization and mass spectrometry data", to develop a linked expert system to accomplish analysis without human intervention. The commercial instrumentation was from Finnigan MAT corporation while the scientific expertise to support expert system development was available at the University of Florida. The project managers at NASA Ames created a single, integrated project using the RDLP model with objectives to: Develop AI/expert system software for instrument control (NASA's role) Expand sensitivity, selectivity and speed of the spectrometer (Univ Florida role) Expand the spectrometer analytic capability and automate the screening (Finnigan role) == Membership == The partnership included seven specialists from five member organizations: Federal Government National Aeronautics and Space Administration (NASA) NASA Ames Research Center (ARC) NASA Kennedy Space Center (KSC) Commercial Finnigan MAT Corporation (Thermo-Fisher Scientific) TGS Technology, Inc. Research Management University of Florida == Organization, management and administration == The technical project was organized into two development teams, one located in at the NASA Ames Research Center covering expert systems and analytic capabilities and one in Florida covering improved sensitivity and testing. The partnership management and administration was provided by a non-profit, partnership support organization: the Joint Enterprise Institute operating through San Francisco State University Foundation (SFSUF) with a NASA employee liaison, Syed Shariq. == Public-private partnership == The partnership structure was as a prototype test of a pilot NASA program to develop public-private partnership business models. The pilot program was known as the NASA Joint Sponsored Research Program (JSRP), which operated as the NASA Joint Enterprise Institute between 1988 and 1991. The partnership was the first public-private, research and development partnership implemented by NASA in response to national policy shifts to increase technology transfer and space commercialization. The partnership structure included a two year technology development and testing plan that cost $610,000, of which NASA funded $310,000, Finnigan $175,000 and the University of Florida $95,000. == Results and commercialization == The project generated patents (4), software (2) and application protocol reports (8). NASA gained use of the patents and jointly development software while Finnigan received commercial utilization rights. The results were commercialized within eighteen months of project completion. == Recognition == NASA recognized the project as a space qualified instrument. Its achievements were reported to the NASA Administrator, directly leading to establishment of the agency-wide Joint Sponsored Research Program.
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How to Choose an AI Subtitle Generator

Shopping for the best AI subtitle generator? An AI subtitle generator is software that uses machine learning to help you get more done — it keeps getting smarter as the underlying models improve. Pricing, accuracy, and the size of the model behind the tool are the three factors that most affect daily usefulness. Whether you are a beginner or a pro, the right AI subtitle generator slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.
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METAL MT

A machine translation system developed at the University of Texas and at Siemens which ran on Lisp Machines. == Background == Originally titled the Linguistics Research System (LRS), it was later renamed METAL (Mechanical Translation and Analysis of Languages). It started life as a German-English system funded by the USAF. == 1980 == A copy of the Weidner Multi-Lingual Word Processing software was requested by the German Government for the Siemens Corporation of Germany in September 1980 and was nicknamed the Siemens-Weidner Engine (originally English-German). This revolutionary multilingual word processing engine became foundational in the development of the Metal MT project, according to John White of the Siemens Corporation. After the Metal MT, development Rights to the Siemens-Weidner Engine were sold to a Belgium company, Lernout & Hauspie. The Siemens copy of the Weidner Multilingual Word Processing software has since been acquired through the purchase of assets of Lernout & Hauspie by Bowne Global Solutions, Inc., which was later acquired by Lionbridge Technologies, Inc. and is demonstrated in their itranslator software.
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Top 10 AI Video Generators Compared (2026)

Shopping for the best AI video generator? An AI video generator is software that uses machine learning to help you get more done — it keeps getting smarter as the underlying models improve. Pricing, accuracy, and the size of the model behind the tool are the three factors that most affect daily usefulness. Whether you are a beginner or a pro, the right AI video generator slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.
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Ideonomy

Ideonomy is a combinatorial "science of ideas" developed by American independent scholar Patrick M. Gunkel (1947–2017). Specifically, Ideonomy is concerned with the systematic organization of ideas and the discovery of the rules behind how ideas combine, diverge, and transform. Gunkel defined ideonomy as "the science of the laws of ideas and of the application of such laws to the generation of all possible ideas in connection with any subject, idea, or thing." In his 1992 book A History of Knowledge, Charles Van Doren compared ideonomy to a "mining operation" that excavates meanings and thought to discover treasures hidden deep within language. Sources from the 1980s and 1990s demonstrate that ideonomy was useful to academic researchers in fields including biology, toxicology, and nursing/patient care. Beginning in the 2010s, academics in a wide range of fields including machine learning, marketing, computational modeling, and cybersecurity have relied on materials generated for ideonomy to provide methodological support for their research. == Etymology and definition == The word "ideonomy" combines the Greek roots ideo- (from idea, meaning pattern or form) and -nomy (from nomos, meaning law or custom). The suffix -nomy suggests the laws concerning or the totality of knowledge about a given subject, as in astronomy or taxonomy. In a note posted on the MIT ideonomy website, Gunkel states that the word was supposedly first coined by the French Encyclopedists to refer to a science of ideas. No evidence is provided for this statement, however. The concept bears some relationship to Antoine Destutt de Tracy's "ideology" (1796), which originally meant a systematic science of ideas before acquiring its modern political connotations. Gunkel provided several metaphorical descriptions of ideonomy: An "idea bank": a computer network enabling systematic exploration of infinite possible ideas A "kaleidoscope" that can exhibit all possible combinations and transformations of ideas A "prism" capable of diffracting any idea into its cognitive components A "gigantic microscope for magnifying the ideocosm" == History and development == In 1984, Gunkel received a five-year unsolicited grant from the Richard Lounsbery Foundation of New York to develop ideonomy. A June 1, 1987 article on the front page of The Wall Street Journal brought Gunkel and ideonomy to wider public attention. Some academics were interested in using ideonomy's techniques, including biologist Betsey Dyer, who published several contemporaneous peer-reviewed studies citing ideonomy. Academic researchers in the field of toxicology and nursing/patient care also used ideonomy. However, ideonomy's broadest contribution to date came beginning in the 2010s, as a list of personality traits generated for combinatorial matching was used by researchers in artificial intelligence to code human emotions for machine-learning tasks, develop computational models related to personality, develop a measurement framework for influencer-brand recommender systems, and aid information awareness/cybersecurity assessment. == Methodology == The foundational empirical method of ideonomy involves the systematic creation of extensive lists. Gunkel's apartment reportedly contained thousands of lists on every conceivable topic. Gunkel termed each list an "organon," which he described as expanding through "combination, permutation, transformation, generalization, specialization, intersection, interaction, reapplication, recursive use, etc. of existing organons." The ideonomic process follows a progressive structure. The ideonomist begins with a simple list of examples of a particular idea, concept, or thing. The list need not be exhaustive. By studying this list, the ideonomist isolates and identifies types. This categorical analysis then reveals missing items, allowing the primary list to be improved and refined. Gunkel emphasized that list items must not only cover genuine categories of nature but also be formulated in ways that yield the largest possible number of syntactically coherent possibilities when combined. The core technique of ideonomy is "ideocombinatorics"—the systematic intersection and combination of items from different lists to generate novel composite concepts. Gunkel developed computer programs to automate this process. For example, combining a list of 230 Universal Elementary Shapes (pits, pyramids, trenches, hemispheres, needles) with a list of 74 Types of Order (recurrence, identity, likeness of parts) yields 17,020 possible "shapes of order." These combinations, when phrased as questions ("Can there be pits of recurrence?"), could suggest new categories of phenomena worthy of investigation. The computer-generated output is typically repetitive and often meaningless. However, with sufficient frequency, the combinations yield results that are unexpectedly interesting and fruitful. In one documented case, Gunkel's programs generated 45,540 questions about toxins for microbiologist David Bermudes. One question—"Can hierarchies of cell process be used as a basis for classifying toxic action?"—prompted Bermudes to develop a novel approach to classifying biological toxins by the type of molecule they attack, rather than by chemical structure or physiological system affected. According to one contemporaneous account of ideonomy, "Gunkel takes for his field all fields and all ideas about anything. He uses a computer to generate lists of words and phrases and by juxtaposition reviews the resultant patterns for novel ideas. The computer is ideal for this task because the mind would rebel at the formidable processing task ideonomy involves. What we have here is computer generated originality." == Applications == Gunkel and his supporters identified several practical applications for ideonomic methods: Scientific research: Biologist Betsey Dyer of Wheaton College published research crediting ideonomy for helping to generate ideas. Medical science: When Austin pathologist Michael T. O'Brien was presented with the ideonomically-generated question "Can arteries have rashes?", he initially dismissed it as nonsense. Upon reflection, he realized that large arteries are supplied with blood by tiny vessels that might become inflamed and dilated, analogous to skin vessels in a rash—a phenomenon potentially worth researching. Analogical thinking: Harvard law professor Robert Clark used ideonomic analogies to write a research paper comparing plant structure with human hierarchies. Artificial intelligence: Douglas Lenat, a researcher at Microelectronics and Computer Technology Corporation (MCC) in Austin, suggested that Gunkel's lists enumerating types of human mistakes could help design AI systems capable of recognizing and correcting their own errors. == Reception and criticism == Ideonomy received mixed reactions from the academic and scientific communities. Prominent supporters included: Edward Fredkin, former director of MIT's computer science laboratory, who praised Gunkel's "provocative ideas on artificial intelligence." Marvin Minsky, AI scientist and MIT professor, who described ideonomy as "perhaps the most extensive study of ways to generate ideas." Frederick Seitz, president emeritus of Rockefeller University, who noted Gunkel's "encyclopedic scope" Robert C. Clark, Harvard law professor, who called Gunkel "the most intelligent person I ever met" However, skeptics questioned whether ideonomy constituted a genuine science. Fredkin himself noted that Gunkel "pours out about 60 ideas a minute, and 59 of them are bad," though he added that "even with one good idea out of 60, it's still an amazing accomplishment." Douglas Lenat observed that brainstorming with Gunkel was "a bit like being hit over the head by the muse with a sledgehammer" and that "he puts people off." Gunkel himself acknowledged that ideonomy was in its infancy and might seem "absurdly utopian." His planned magnum opus on ideonomy remained incomplete, and was posted on an MIT website thanks to faculty advisor Whitman Richards. Gunkel wrote: "Pioneering in a completely new field, yes in a new science, is almost unreal. It is heartbreaking, it is pitiable, it is almost inhuman. Honestly, it is a hell. There is nothing heroic about it." == Related concepts == Gunkel identified several historical precedents for ideonomic thinking: Gottfried Wilhelm Leibniz (1646–1716): The philosopher's work on a universal characteristic (characteristica universalis) and calculus of reasoning Peter Mark Roget (1779–1869): Creator of Roget's Thesaurus, which organized concepts into a systematic taxonomy Dmitri Mendeleev (1834–1907): Developer of the periodic table, demonstrating how combining lists of element families could reveal previously unseen connections Fritz Zwicky (1898–1974): The Caltech astrophysicist whom Gunkel called the "grandfather of ideonomy" for his development of "morphological research"—systematic exploration of all possible solutions t
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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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Constrained conditional model

A constrained conditional model (CCM) is a machine learning and inference framework that augments the learning of conditional (probabilistic or discriminative) models with declarative constraints. The constraint can be used as a way to incorporate expressive prior knowledge into the model and bias the assignments made by the learned model to satisfy these constraints. The framework can be used to support decisions in an expressive output space while maintaining modularity and tractability of training and inference. Models of this kind have recently attracted much attention within the natural language processing (NLP) community. Formulating problems as constrained optimization problems over the output of learned models has several advantages. It allows one to focus on the modeling of problems by providing the opportunity to incorporate domain-specific knowledge as global constraints using a first order language. Using this declarative framework frees the developer from low level feature engineering while capturing the problem's domain-specific properties and guarantying exact inference. From a machine learning perspective it allows decoupling the stage of model generation (learning) from that of the constrained inference stage, thus helping to simplify the learning stage while improving the quality of the solutions. For example, in the case of generating compressed sentences, rather than simply relying on a language model to retain the most commonly used n-grams in the sentence, constraints can be used to ensure that if a modifier is kept in the compressed sentence, its subject will also be kept. == Motivation == Making decisions in many domains (such as natural language processing and computer vision problems) often involves assigning values to sets of interdependent variables where the expressive dependency structure can influence, or even dictate, what assignments are possible. These settings are applicable not only to Structured Learning problems such as semantic role labeling, but also for cases that require making use of multiple pre-learned components, such as summarization, textual entailment and question answering. In all these cases, it is natural to formulate the decision problem as a constrained optimization problem, with an objective function that is composed of learned models, subject to domain- or problem-specific constraints. Constrained conditional models form a learning and inference framework that augments the learning of conditional (probabilistic or discriminative) models with declarative constraints (written, for example, using a first-order representation) as a way to support decisions in an expressive output space while maintaining modularity and tractability of training and inference. These constraints can express either hard restrictions, completely prohibiting some assignments, or soft restrictions, penalizing unlikely assignments. In most applications of this framework in NLP, following, Integer Linear Programming (ILP) was used as the inference framework, although other algorithms can be used for that purpose. == Formal Definition == Given a set of feature functions { ϕ i ( x , y ) } {\displaystyle \{\phi _{i}(x,y)\}} and a set of constraints { C i ( x , y ) } {\displaystyle \{C_{i}(x,y)\}} , defined over an input structure x ∈ X {\displaystyle x\in X} and an output structure y ∈ Y {\displaystyle y\in Y} , a constraint conditional model is characterized by two weight vectors, w and ρ {\displaystyle \rho } , and is defined as the solution to the following optimization problem: a r g m a x y ∑ i w i ϕ i ( x , y ) − ∑ ρ i C i ( x , y ) {\displaystyle argmax_{y}\sum _{i}w_{i}\phi _{i}(x,y)-\sum \rho _{i}C_{i}(x,y)} . Each constraint C i ∈ C {\displaystyle C_{i}\in C} is a boolean mapping indicating if the joint assignment ( x , y ) {\displaystyle (x,y)} violates a constraint, and ρ {\displaystyle \rho } is the penalty incurred for violating the constraints. Constraints assigned an infinite penalty are known as hard constraints, and represent unfeasible assignments to the optimization problem. == Training paradigms == === Learning local vs. global models === The objective function used by CCMs can be decomposed and learned in several ways, ranging from a complete joint training of the model along with the constraints to completely decoupling the learning and the inference stage. In the latter case, several local models are learned independently and the dependency between these models is considered only at decision time via a global decision process. The advantages of each approach are discussed in which studies the two training paradigms: (1) local models: L+I (learning + inference) and (2) global model: IBT (Inference based training), and shows both theoretically and experimentally that while IBT (joint training) is best in the limit, under some conditions (basically, ”good” components) L+I can generalize better. The ability of CCM to combine local models is especially beneficial in cases where joint learning is computationally intractable or when training data are not available for joint learning. This flexibility distinguishes CCM from the other learning frameworks that also combine statistical information with declarative constraints, such as Markov logic network, that emphasize joint training. === Minimally supervised CCM === CCM can help reduce supervision by using domain knowledge (expressed as constraints) to drive learning. These settings were studied in and. These works introduce semi-supervised Constraints Driven Learning (CODL) and show that by incorporating domain knowledge the performance of the learned model improves significantly. === Learning over latent representations === CCMs have also been applied to latent learning frameworks, where the learning problem is defined over a latent representation layer. Since the notion of a correct representation is inherently ill-defined, no gold-standard labeled data regarding the representation decision is available to the learner. Identifying the correct (or optimal) learning representation is viewed as a structured prediction process and therefore modeled as a CCM. This problem was covered in several papers, in both supervised and unsupervised settings. In all cases research showed that explicitly modeling the interdependencies between representation decisions via constraints results in an improved performance. == Integer linear programming for natural language processing applications == The advantages of the CCM declarative formulation and the availability of off-the-shelf solvers have led to a large variety of natural language processing tasks being formulated within the framework, including semantic role labeling, syntactic parsing, coreference resolution, summarization, transliteration, natural language generation and joint information extraction. Most of these works use an integer linear programming (ILP) solver to solve the decision problem. Although theoretically solving an Integer Linear Program is exponential in the size of the decision problem, in practice using state-of-the-art solvers and approximate inference techniques large scale problems can be solved efficiently. The key advantage of using an ILP solver for solving the optimization problem defined by a constrained conditional model is the declarative formulation used as input for the ILP solver, consisting of a linear objective function and a set of linear constraints. == Resources == CCM Tutorial Predicting Structures in NLP: Constrained Conditional Models and Integer Linear Programming in NLP
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Bidyut Baran Chaudhuri

Bidyut Baran Chaudhuri (B. B. Chauduri) is a senior computer scientist and an emeritus professor of Techno India University in West Bengal, India. He is also adjuncted to Indian Statistical Institute, where he was a professor for about three decades. He was the founding Head of Computer Vision and Pattern Recognition Unit (which was established in 1994) of ISI. Moreover, he was a J.C. Bose Fellow and Indian National Academy of Engineering Distinguished Professor at ISI. He was the vice-president of the Society for Natural Language Technology Research (SNLTR). His primary research contributes to the fields of computer vision, image processing and pattern recognition. He is a pioneer of "Indian language script OCR". == Education == Chaudhuri received his BSc (Hons.), BTech and MTech degrees from University of Calcutta, India in 1969, 1972 and 1974, respectively and PhD Degree from Indian Institute of Technology Kanpur in 1980. He did his post-doc work during 1981-1982 from Queen's University, U.K, through Leverhulme Overseas Fellowship. He also worked as a visiting faculty at Tech University, Hannover during 1986-87 as well as at GSF Institute of Radiation Protection (now Leibnitz Institute), Munich in 1990 and 1992. == Awards and recognition == Chaudhuri has been elected as a Life Fellow of IEEE "for contributions to pattern recognition, especially Indian language script OCR, document processing and natural language processing". He has become a Fellow of International Association for Pattern Recognition (IAPR) "for contributions to character recognition and speech synthesis in Indian language". He is also Fellow of The World Academy of Sciences (TWAS), Indian National Science Academy (INSA), Indian National Academy of Engineering (INAE), National Academy of Sciences (NASI), and Institute of Electronics and Telecommunication Engineering (IETE). In 2011, Chaudhuri received the Om Prakash Bhasin Award for his contribution in the field of electronics and information technology. Chaudhuri's interview on some of his works has been reported in Indian newspaper as well. He is within world's top 2% scientists and top-10 Indian AI scientists according to a study conducted by Stanford University. He has also been featured as top-10 machine learning researcher from India.
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Cooliris (plugin)

Cooliris (for Desktop), formerly known as PicLens, was a web browser extension developed by Cooliris, Inc, and later acquired by Yahoo. The plugin provides an interactive 3D-like experience for viewing digital images and videos from the web and from desktop applications. The software places a small icon atop image thumbnails that appear on a webpage. Clicking on the icon loads the Cooliris 3D Wall, a browsing environment that gives the user the effect of flying through a three-dimensional space. Released to the public in January 2008, The New York Times described Cooliris as the "new immersive approach to Web navigation". Cooliris went out to win the 2008 Crunchies Award for Best Design. The plugin has received over 50 million downloads. As of May 2014 browser plugins are unavailable from the official website. There are only links to tablet apps - for iOS and Android.
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Levenshtein automaton

In computer science, a Levenshtein automaton for a string w and a number n is a finite-state automaton that can recognize the set of all strings whose Levenshtein distance from w is at most n. That is, a string x is in the formal language recognized by the Levenshtein automaton if and only if x can be transformed into w by at most n single-character insertions, deletions, and substitutions. == Applications == Levenshtein automata may be used for spelling correction, by finding words in a given dictionary that are close to a misspelled word. In this application, once a word is identified as being misspelled, its Levenshtein automaton may be constructed, and then applied to all of the words in the dictionary to determine which ones are close to the misspelled word. If the dictionary is stored in compressed form as a trie, the time for this algorithm (after the automaton has been constructed) is proportional to the number of nodes in the trie, significantly faster than using dynamic programming to compute the Levenshtein distance separately for each dictionary word. It is also possible to find words in a regular language, rather than a finite dictionary, that are close to a given target word, by computing the Levenshtein automaton for the word, and then using a Cartesian product construction to combine it with an automaton for the regular language, giving an automaton for the intersection language. Alternatively, rather than using the product construction, both the Levenshtein automaton and the automaton for the given regular language may be traversed simultaneously using a backtracking algorithm. Levenshtein automata are used in Lucene for full-text searches that can return relevant documents even if the query is misspelled. == Construction == For any fixed constant n, the Levenshtein automaton for w and n may be constructed in time O(|w|). Mitankin studies a variant of this construction called the universal Levenshtein automaton, determined only by a numeric parameter n, that can recognize pairs of words (encoded in a certain way by bitvectors) that are within Levenshtein distance n of each other. Touzet proposed an effective algorithm to build this automaton. Yet a third finite automaton construction of Levenshtein (or Damerau–Levenshtein) distance are the Levenshtein transducers of Hassan et al., who show finite state transducers implementing edit distance one, then compose these to implement edit distances up to some constant.
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Generalized filtering

Generalized filtering is a generic Bayesian filtering scheme for nonlinear state-space models. It is based on a variational principle of least action, formulated in generalized coordinates of motion. Note that "generalized coordinates of motion" are related to—but distinct from—generalized coordinates as used in (multibody) dynamical systems analysis. Generalized filtering furnishes posterior densities over hidden states (and parameters) generating observed data using a generalized gradient descent on variational free energy, under the Laplace assumption. Unlike classical (e.g. Kalman-Bucy or particle) filtering, generalized filtering eschews Markovian assumptions about random fluctuations. Furthermore, it operates online, assimilating data to approximate the posterior density over unknown quantities, without the need for a backward pass. Special cases include variational filtering, dynamic expectation maximization and generalized predictive coding. == Definition == Definition: Generalized filtering rests on the tuple ( Ω , U , X , S , p , q ) {\displaystyle (\Omega ,U,X,S,p,q)} : A sample space Ω {\displaystyle \Omega } from which random fluctuations ω ∈ Ω {\displaystyle \omega \in \Omega } are drawn Control states U ∈ R {\displaystyle U\in \mathbb {R} } – that act as external causes, input or forcing terms Hidden states X : X × U × Ω → R {\displaystyle X:X\times U\times \Omega \to \mathbb {R} } – that cause sensory states and depend on control states Sensor states S : X × U × Ω → R {\displaystyle S:X\times U\times \Omega \to \mathbb {R} } – a probabilistic mapping from hidden and control states Generative density p ( s ~ , x ~ , u ~ ∣ m ) {\displaystyle p({\tilde {s}},{\tilde {x}},{\tilde {u}}\mid m)} – over sensory, hidden and control states under a generative model m {\displaystyle m} Variational density q ( x ~ , u ~ ∣ μ ~ ) {\displaystyle q({\tilde {x}},{\tilde {u}}\mid {\tilde {\mu }})} – over hidden and control states with mean μ ~ ∈ R {\displaystyle {\tilde {\mu }}\in \mathbb {R} } Here ~ denotes a variable in generalized coordinates of motion: u ~ = [ u , u ′ , u ″ , … ] T {\displaystyle {\tilde {u}}=[u,u',u'',\ldots ]^{T}} === Generalized filtering === The objective is to approximate the posterior density over hidden and control states, given sensor states and a generative model – and estimate the (path integral of) model evidence p ( s ~ ( t ) | m ) {\displaystyle p({\tilde {s}}(t)\vert m)} to compare different models. This generally involves an intractable marginalization over hidden states, so model evidence (or marginal likelihood) is replaced with a variational free energy bound. Given the following definitions: μ ~ ( t ) = a r g m i n μ ~ { F ( s ~ ( t ) , μ ~ ) } {\displaystyle {\tilde {\mu }}(t)={\underset {\tilde {\mu }}{\operatorname {arg\,min} }}\{F({\tilde {s}}(t),{\tilde {\mu }})\}} G ( s ~ , x ~ , u ~ ) = − ln ⁡ p ( s ~ , x ~ , u ~ | m ) {\displaystyle G({\tilde {s}},{\tilde {x}},{\tilde {u}})=-\ln p({\tilde {s}},{\tilde {x}},{\tilde {u}}\vert m)} Denote the Shannon entropy of the density q {\displaystyle q} by H [ q ] = E q [ − log ⁡ ( q ) ] {\displaystyle H[q]=E_{q}[-\log(q)]} . We can then write the variational free energy in two ways: F ( s ~ , μ ~ ) = E q [ G ( s ~ , x ~ , u ~ ) ] − H [ q ( x ~ , u ~ | μ ~ ) ] = − ln ⁡ p ( s ~ | m ) + D K L [ q ( x ~ , u ~ | μ ~ ) | | p ( x ~ , u ~ | s ~ , m ) ] {\displaystyle F({\tilde {s}},{\tilde {\mu }})=E_{q}[G({\tilde {s}},{\tilde {x}},{\tilde {u}})]-H[q({\tilde {x}},{\tilde {u}}\vert {\tilde {\mu }})]=-\ln p({\tilde {s}}\vert m)+D_{KL}[q({\tilde {x}},{\tilde {u}}\vert {\tilde {\mu }})\vert \vert p({\tilde {x}},{\tilde {u}}\vert {\tilde {s}},m)]} The second equality shows that minimizing variational free energy (i) minimizes the Kullback-Leibler divergence between the variational and true posterior density and (ii) renders the variational free energy (a bound approximation to) the negative log evidence (because the divergence can never be less than zero). Under the Laplace assumption q ( x ~ , u ~ ∣ μ ~ ) = N ( μ ~ , C ) {\displaystyle q({\tilde {x}},{\tilde {u}}\mid {\tilde {\mu }})={\mathcal {N}}({\tilde {\mu }},C)} the variational density is Gaussian and the precision that minimizes free energy is C − 1 = Π = ∂ μ ~ μ ~ G ( μ ~ ) {\displaystyle C^{-1}=\Pi =\partial _{{\tilde {\mu }}{\tilde {\mu }}}G({\tilde {\mu }})} . This means that free-energy can be expressed in terms of the variational mean (omitting constants): F = G ( μ ~ ) + 1 2 ln ⁡ | ∂ μ ~ μ ~ G ( μ ~ ) | {\displaystyle F=G({\tilde {\mu }})+\textstyle {1 \over 2}\ln \vert \partial _{{\tilde {\mu }}{\tilde {\mu }}}G({\tilde {\mu }})\vert } The variational means that minimize the (path integral) of free energy can now be recovered by solving the generalized filter: μ ~ ˙ = D μ ~ − ∂ μ ~ F ( s ~ , μ ~ ) {\displaystyle {\dot {\tilde {\mu }}}=D{\tilde {\mu }}-\partial _{\tilde {\mu }}F({\tilde {s}},{\tilde {\mu }})} where D {\displaystyle D} is a block matrix derivative operator of identify matrices such that D u ~ = [ u ′ , u ″ , … ] T {\displaystyle D{\tilde {u}}=[u',u'',\ldots ]^{T}} === Variational basis === Generalized filtering is based on the following lemma: The self-consistent solution to μ ~ ˙ = D μ ~ − ∂ μ ~ F ( s , μ ~ ) {\displaystyle {\dot {\tilde {\mu }}}=D{\tilde {\mu }}-\partial _{\tilde {\mu }}F(s,{\tilde {\mu }})} satisfies the variational principle of stationary action, where action is the path integral of variational free energy S = ∫ d t F ( s ~ ( t ) , μ ~ ( t ) ) {\displaystyle S=\int dt\,F({\tilde {s}}(t),{\tilde {\mu }}(t))} Proof: self-consistency requires the motion of the mean to be the mean of the motion and (by the fundamental lemma of variational calculus) μ ~ ˙ = D μ ~ ⇔ ∂ μ ~ F ( s ~ , μ ~ ) = 0 ⇔ δ μ ~ S = 0 {\displaystyle {\dot {\tilde {\mu }}}=D{\tilde {\mu }}\Leftrightarrow \partial _{\tilde {\mu }}F({\tilde {s}},{\tilde {\mu }})=0\Leftrightarrow \delta _{\tilde {\mu }}S=0} Put simply, small perturbations to the path of the mean do not change variational free energy and it has the least action of all possible (local) paths. Remarks: Heuristically, generalized filtering performs a gradient descent on variational free energy in a moving frame of reference: μ ~ ˙ − D μ ~ = − ∂ μ ~ F ( s , μ ~ ) {\displaystyle {\dot {\tilde {\mu }}}-D{\tilde {\mu }}=-\partial _{\tilde {\mu }}F(s,{\tilde {\mu }})} , where the frame itself minimizes variational free energy. For a related example in statistical physics, see Kerr and Graham who use ensemble dynamics in generalized coordinates to provide a generalized phase-space version of Langevin and associated Fokker-Planck equations. In practice, generalized filtering uses local linearization over intervals Δ t {\displaystyle \Delta t} to recover discrete updates Δ μ ~ = ( exp ⁡ ( Δ t ⋅ J ) − I ) J − 1 μ ~ ˙ J = ∂ μ ~ μ ~ ˙ = D − ∂ μ ~ μ ~ F ( s ~ , μ ~ ) {\displaystyle {\begin{aligned}\Delta {\tilde {\mu }}&=(\exp(\Delta t\cdot J)-I)J^{-1}{\dot {\tilde {\mu }}}\\J&=\partial _{\tilde {\mu }}{\dot {\tilde {\mu }}}=D-\partial _{{\tilde {\mu }}{\tilde {\mu }}}F({\tilde {s}},{\tilde {\mu }})\end{aligned}}} This updates the means of hidden variables at each interval (usually the interval between observations). == Generative (state-space) models in generalized coordinates == Usually, the generative density or model is specified in terms of a nonlinear input-state-output model with continuous nonlinear functions: s = g ( x , u ) + ω s x ˙ = f ( x , u ) + ω x {\displaystyle {\begin{aligned}s&=g(x,u)+\omega _{s}\\{\dot {x}}&=f(x,u)+\omega _{x}\end{aligned}}} The corresponding generalized model (under local linearity assumptions) obtains the from the chain rule s ~ = g ~ ( x ~ , u ~ ) + ω ~ s s = g ( x , u ) + ω s s ′ = ∂ x g ⋅ x ′ + ∂ u g ⋅ u ′ + ω s ′ s ″ = ∂ x g ⋅ x ″ + ∂ u g ⋅ u ″ + ω s ″ ⋮ x ~ ˙ = f ~ ( x ~ , u ~ ) + ω ~ x x ˙ = f ( x , u ) + ω x x ˙ ′ = ∂ x f ⋅ x ′ + ∂ u f ⋅ u ′ + ω x ′ x ˙ ″ = ∂ x f ⋅ x ″ + ∂ u f ⋅ u ″ + ω x ″ ⋮ {\displaystyle {\begin{aligned}{\tilde {s}}&={\tilde {g}}({\tilde {x}},{\tilde {u}})+{\tilde {\omega }}_{s}\\\\s&=g(x,u)+\omega _{s}\\s'&=\partial _{x}g\cdot x'+\partial _{u}g\cdot u'+\omega '_{s}\\s''&=\partial _{x}g\cdot x''+\partial _{u}g\cdot u''+\omega ''_{s}\\&\vdots \\\end{aligned}}\qquad {\begin{aligned}{\dot {\tilde {x}}}&={\tilde {f}}({\tilde {x}},{\tilde {u}})+{\tilde {\omega }}_{x}\\\\{\dot {x}}&=f(x,u)+\omega _{x}\\{\dot {x}}'&=\partial _{x}f\cdot x'+\partial _{u}f\cdot u'+\omega '_{x}\\{\dot {x}}''&=\partial _{x}f\cdot x''+\partial _{u}f\cdot u''+\omega ''_{x}\\&\vdots \end{aligned}}} Gaussian assumptions about the random fluctuations ω {\displaystyle \omega } then prescribe the likelihood and empirical priors on the motion of hidden states p ( s ~ , x ~ , u ~ | m ) = p ( s ~ | x ~ , u ~ , m ) p ( D x ~ | x , u ~ , m ) p ( x | m ) p ( u ~ | m ) p ( s ~ | x ~ , u ~ , m ) = N ( g ~ ( x ~ , u ~ ) , Σ ~ ( x ~ , u ~ ) s ) p ( D x ~ | x , u ~ , m ) = N ( f ~ ( x ~ , u ~ ) , Σ ~ ( x ~ , u ~ ) x ) {\displayst
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