AI Content Youtube Monetization

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Imieliński–Lipski algebra

In database theory, Imieliński–Lipski algebra is an extension of relational algebra onto tables with different types of null values. It is used to operate on relations with incomplete information. Imieliński–Lipski algebras are defined to satisfy precise conditions for semantically meaningful extension of the usual relational operators, such as projection, selection, union, and join, from operators on relations to operators on relations with various kinds of "null values". These conditions require that the system be safe in the sense that no incorrect conclusion is derivable by using a specified subset F of the relational operators; and that it be complete in the sense that all valid conclusions expressible by relational expressions using operators in F are in fact derivable in this system. For example, it is well known that the three-valued logic approach to deal with null values, supported treatment of nulls values by SQL is not complete, see Ullman book. To show this, let T be: Take SQL query Q SQL query Q will return empty set (no results) under 3-valued semantics currently adopted by all variants of SQL. This is the case because in SQL, NULL is never equal to any constant – in this case, neither to “Spring” nor “Fall” nor “Winter” (if there is Winter semester in this school). NULL='Spring' will evaluate to MAYBE and so will NULL='Fall'. The disjunction MAYBE OR MAYBE evaluates to MAYBE (not TRUE). Thus Igor will not be part of the answer (and of course neither will Rohit). But Igor should be returned as the answer. Indeed, regardless what semester Igor took the Networks class (no matter what was the unknown value of NULL), the selection condition will be true. This “Igor” will be missed by SQL and the SQL answer would be incomplete according to completeness requirements specified in Tomasz Imieliński, Witold Lipski, 'Incomplete Information in Relational Databases'. It is also argued there that 3-valued logic (TRUE, FALSE, MAYBE) can never provide guarantee of complete answer for tables with incomplete information. Three algebras which satisfy conditions of safety and completeness are defined as Imielinski–Lipski algebras: the Codd-Tables algebra, the V-tables algebra and the Conditional tables (C-tables) algebra. == Codd-tables algebra == Codd-tables algebra is based on the usual Codd's single NULL values. The table T above is an example of Codd-table. Codd-table algebra supports projection and positive selections only. It is also demonstrated in [IL84 that it is not possible to correctly extend more relational operators over Codd-Tables. For example, such basic operation as join is not extendable over Codd-tables. It is not possible to define selections with Boolean conditions involving negation and preserve completeness. For example, queries like the above query Q cannot be supported. In order to be able to extend more relational operators, more expressive form of null value representation is needed in tables which are called V-table. == V-tables algebra == V-tables algebra is based on many different ("marked") null values or variables allowed to appear in a table. V-tables allow to show that a value may be unknown but the same for different tuples. For example, in the table below Gaurav and Igor order the same (but unknown) beer in two unknown bars (which may, or may not be different – but remain unknown). Gaurav and Jane frequent the same unknown bar (Y1). Thus, instead one NULL value, we use indexed variables, or Skolem constants . V-tables algebra is shown to correctly support projection, positive selection (with no negation occurring in the selection condition), union, and renaming of attributes, which allows for processing arbitrary conjunctive queries. A very desirable property enjoyed by the V-table algebra is that all relational operators on tables are performed in exactly the same way as in the case of the usual relations. === Conditional tables (c-tables) algebra === Example of conditional table (c-table) is shown below. It has additional column “con” which is a Boolean condition involving variables, null values – same as in V-tables. over the following table c-table Conditional tables algebra, mainly of theoretical interest, supports projection, selection, union, join, and renaming. Under closed-world assumption, it can also handle the operator of difference, thus it can support all relational operators. == History == Imieliński–Lipski algebras were introduced by Tomasz Imieliński and Witold Lipski Jr. in Incomplete Information in Relational Databases.
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The Best Free AI Essay Writer for Beginners

In search of the best AI essay writer? An AI essay writer is software that uses machine learning to help you get more done — it turns a rough idea into a polished result in seconds. When choosing one, weigh output quality, pricing, export formats, and how well it fits the tools you already use. Whether you are a beginner or a pro, the right AI essay writer 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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Best AI Analytics Tools in 2026

Curious about the best AI analytics tool? An AI analytics tool is software that uses machine learning to help you get more done — it combines speed, accuracy, and an interface that just works. Hands-on testing shows real-world results vary, so a short free trial is the smartest way to decide. Whether you are a beginner or a pro, the right AI analytics tool slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.
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BFR algorithm

The BFR algorithm, named after its inventors Bradley, Fayyad and Reina, is a variant of k-means algorithm that is designed to cluster data in a high-dimensional Euclidean space. It makes a very strong assumption about the shape of clusters: they must be normally distributed about a centroid. The mean and standard deviation for a cluster may differ for different dimensions, but the dimensions must be independent. In other words, the data must take the shape of axis-aligned ellipses.
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ViEWER

ViEWER, the Virtual Environment Workbench for Education and Research, is a proprietary, freeware computer program for Microsoft Windows written by researchers at the University of Idaho for the study of visual perception and complex immersive three-dimensional environments. It was created using C++ and OpenGL, and has been used by Dr. Brian Dyre, Dr. Steffen Werner, Dr. Ernesto Bustamante, Dr. Ben Barton, and their undergraduate and graduate researchers in visual perception, signal detection, and child-safety experiments.
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Synchronizing word

In computer science, more precisely, in the theory of deterministic finite automata (DFA), a synchronizing word or reset sequence is a word in the input alphabet of the DFA that sends any state of the DFA to one and the same state. That is, if an ensemble of copies of the DFA are each started in different states, and all of the copies process the synchronizing word, they will all end up in the same state. Not every DFA has a synchronizing word; for instance, a DFA with two states, one for words of even length and one for words of odd length, can never be synchronized. == Existence == Given a DFA, the problem of determining if it has a synchronizing word can be solved in polynomial time using a theorem due to Ján Černý. A simple approach considers the power set of states of the DFA, and builds a directed graph where nodes belong to the power set, and a directed edge describes the action of the transition function. A path from the node of all states to a singleton state shows the existence of a synchronizing word. This algorithm is exponential in the number of states. A polynomial algorithm results however, due to a theorem of Černý that exploits the substructure of the problem, and shows that a synchronizing word exists if and only if every pair of states has a synchronizing word. == Length == The problem of estimating the length of synchronizing words has a long history and was posed independently by several authors, but it is commonly known as the Černý conjecture. In 1969, Ján Černý conjectured that (n − 1)2 is the upper bound for the length of the shortest synchronizing word for any n-state complete DFA (a DFA with complete state transition graph). If this is true, it would be tight: in his 1964 paper, Černý exhibited a class of automata (indexed by the number n of states) for which the shortest reset words have this length. The best upper bound known is 0.1654n3, far from the lower bound. For n-state DFAs over a k-letter input alphabet, an algorithm by David Eppstein finds a synchronizing word of length at most 11n3/48 + O(n2), and runs in time complexity O(n3+kn2). This algorithm does not always find the shortest possible synchronizing word for a given automaton; as Eppstein also shows, the problem of finding the shortest synchronizing word is NP-complete. However, for a special class of automata in which all state transitions preserve the cyclic order of the states, he describes a different algorithm with time O(kn2) that always finds the shortest synchronizing word, proves that these automata always have a synchronizing word of length at most (n − 1)2 (the bound given in Černý's conjecture), and exhibits examples of automata with this special form whose shortest synchronizing word has length exactly (n − 1)2. == Road coloring == The road coloring problem is the problem of labeling the edges of a regular directed graph with the symbols of a k-letter input alphabet (where k is the outdegree of each vertex) in order to form a synchronizable DFA. It was conjectured in 1970 by Benjamin Weiss and Roy Adler that any strongly connected and aperiodic regular digraph can be labeled in this way; their conjecture was proven in 2007 by Avraham Trahtman. == Related: transformation semigroups == A transformation semigroup is synchronizing if it contains an element of rank 1, that is, an element whose image is of cardinality 1. A DFA corresponds to a transformation semigroup with a distinguished generator set.
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Vlado Keselj

Vlado Keselj (Vlado Kešelj) is a Serbian-Canadian computer scientist known for his research in natural language processing and authorship attribution. He is a professor at Dalhousie University. == Education == As a high school student in Yugoslavia, Keselj competed in the 1987 International Mathematical Olympiad, earning a bronze medal. He earned his Ph.D. in 2002 at the University of Waterloo, with the dissertation Modular Stochastic HPSGs for Question Answering supervised by Nick Cercone. == Awards == Vlado Keselj is a recipient of the 2019 CAIAC Distinguished Service Award, awarded by the Canadian Artificial Intelligence Association (CAIAC). == Selected publications == Kešelj, V., Peng, F., Cercone, N., & Thomas, C. (2003, August). N-gram-based author profiles for authorship attribution. In Proceedings of the Conference of the Pacific Association for Computational Linguistics, PACLING 2003 (Vol. 3, pp. 255–264).
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AI Writing Assistants Reviews: What Actually Works in 2026

Looking for the best AI writing assistant? An AI writing assistant is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI writing assistant slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.
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Automated medical scribe

Automated medical scribes (also called artificial intelligence scribes, AI scribes, digital scribes, virtual scribes, ambient AI scribes, AI documentation assistants, and digital/virtual/smart clinical assistants) are tools for transcribing medical speech, such as patient consultations and dictated medical notes. Many also produce summaries of consultations. Automated medical scribes based on large language models (LLMs, commonly called "AI", short for "artificial intelligence") increased drastically in popularity in 2024. There are privacy and antitrust concerns. Accuracy concerns also exist, and intensify in situations in which tools try to go beyond transcribing and summarizing, and are asked to format information by its meaning, since LLMs do not deal well with meaning (see weak artificial intelligence). Medics using these scribes are generally expected to understand the ethical and legal considerations, and supervise the outputs. The privacy protections of automated medical scribes vary widely. While it is possible to do all the transcription and summarizing locally, with no connection to the internet, most closed-source providers require that data be sent to their own servers over the internet, processed there, and the results sent back (as with digital voice assistants). Some retailers say their tools use zero-knowledge encryption (meaning that the service provider can't access the data). Others explicitly say that they use patient data to train their AIs, or rent or resell it to third parties; the nature of privacy protections used in such situations is unclear, and they are likely not to be fully effective. Most providers have not published any safety or utility data in academic journals, and are not responsive to requests from medical researchers studying their products. == Privacy == Some providers unclear about what happens to user data. Some may sell data to third parties. Some explicitly send user data to for-profit tech companies for secondary purposes, which may not be specified. Some require users to sign consents to such reuse of their data. Some ingest user data to train the software, promising to anonymize it; however, deanonymization may be possible (that is, it may become obvious who the patient is). It is intrinsically impossible to prevent an LLM from correlating its inputs; they work by finding similar patterns across very large data sets. Some information on the patient will be known from other sources (for instance, information that they were injured in an incident on a certain day might be available from the news media; information that they attended specific appointment locations at specific times is probably available to their cellphone provider/apps/data brokers; information about when they had a baby is probably implied by their online shopping records; and they might mention lifestyle changes to their doctor and on a forum or blog). The software may correlate such information with the "anonymized" clinical consultation record, and, asked about the named patient, provide information which they only told their doctor privately. Because a patient's record is all about the same patient, it is all unavoidably linked; in very many cases, medical histories are intrinsically identifiable. Depending on how common a condition and what other data is available, K-anonymity may be useless. Differential privacy could theoretically preserve privacy. Data broker companies like Google, Amazon, Apple and Microsoft have produced or bought up medical scribes, some of which use user data for secondary purposes, which has led to antitrust concerns. Transfer of patient records for AI training has, in the past, prompted legal action. Open-source programs typically do all the transcription locally, on the doctor's own computer. Open-source software is widely used in healthcare, with some national public healthcare bodies holding hack days. === Data resale and commercialization === Several AI medical scribe providers include terms in their service agreements that allow the reuse, sale, or commercialization of de-identified or user-submitted data. Although such data are generally described as anonymized or aggregated, these practices have raised ethical concerns among clinicians and privacy advocates regarding secondary uses of medical information beyond clinical documentation. Freed, an AI transcription and scribe platform, states in its Terms of Use that it may "collect, use, publish, disseminate, sell, transfer, and otherwise exploit" de-identified and aggregated data derived from user inputs. OpenEvidence similarly states that it may "collect, use, transfer, sell, and disclose non-personal information and customer usage data for any purpose including commercial uses." Doximity, which offers an AI-enabled medical scribe as part of its physician platform, grants itself a "nonexclusive, irrevocable, worldwide, perpetual, unlimited, assignable, sublicensable, royalty-free" license to "copy, prepare derivative works from, improve, distribute, publish, ... analyze, index, tag, [and] commercialize" content submitted by users, subject to its privacy policy. Because these terms allow broad secondary use—including sale, licensing, model-training, derivative works, and commercial exploitation of de-identified or user-submitted data—some commentators have recommended that clinicians review data-handling provisions carefully when adopting AI-scribe tools, particularly in clinical environments where patient privacy and regulatory compliance are critical. === Encryption === Multifactor authentication for access to the data is expected practice. Typically, Diffie–Hellman key exchange is used for encryption; this is the standard method commonly used for things like online banking. This encryption is expensive but not impossible to break; it is not generally considered safe against eavesdroppers with the resources of a nation-state. If content is encrypted between the client and the service provider's remote server (transport cryptography), then the server has an unencrypted copy. This is necessary if the data is used by the service provider (for instance, to train the software). Zero-knowledge encryption implies that the only unencrypted copy is at the client, and the server cannot decrypt the data any more easily than a monster-in-the-middle attacker. == Platforms == Scribes may operate on desktops, laptop, or mobile computers, under a variety of operating systems. These vary in their risks; for instance, mobiles can be lost. The underlying mobile or desktop operating systems are also part of the trusted computing base, and if they are not secure, the software relying on them cannot be secure either. Some AI medical scribe platforms are designed to operate as cloud-based applications that generate structured clinical documentation from clinician–patient conversations. These systems may offer features such as real-time transcription, document generation, and integration with electronic health record (EHR) systems. == Confabulation, omissions, and other errors == Like other LLMs, medical-scribe LLMs are prone to hallucinations, where they make up content based on statistically associations between their training data and the transcription audio. LLMs do not distinguish between trying to transcribe the audio and guessing what words will come next, but perform both processes mixed together. They are especially likely to take short silences or non-speech noises and invent some sort of speech to transcribe them as. LLM medical scribes have been known to confabulate racist and otherwise prejudiced content; this is partly because the training datasets of many LLMs contain pseudoscientific texts about medical racism. They may misgender patients. A survey found that most doctors preferred, in principle, that scribes be trained on data reviewed by medical subject experts. Relevant, accurate training data increases the probability of an accurate transcription, but does not guarantee accuracy. Software trained on thousands of real clinical conversations generated transcripts with lower word error rates. Software trained on manually-transcribed training data did better than software trained with automatically transcribed training data such as YouTube captions. Autoscribes omit parts of the conversation classes as irrelevant. The may wrongly classify pertinent information as irrelevant and omit it. They may also confuse historic and current symptoms, or otherwise misclassify information. They may also simply wrongly transcribe the speech, writing something incorrect instead. If clinicians do not carefully check the recording, such mistakes could make their way into their medical records and cause patient harms. == Patient consent == Professional organizations generally require that scribes be used only with patient consent; some bodies may require written consent. Medics must also abide by local surveillance laws, which may criminalize recording pri
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Michael Kohlhase

Michael Kohlhase (born 13 September 1964, in Erlangen) is a German computer scientist and professor at University of Erlangen–Nuremberg, where he is head of the KWARC research group (Knowledge Adaptation and Reasoning for Content). == Academic Positions == Michael Kohlhase is president of the OpenMath Society and a trustee of the Interest Group for Mathematical Knowledge Management (MKM). He was a trustee of the Conference on Automated Deduction and the CALCULEMUS Interest Group. He has been Conference Chair of CADE-21 and Program Chair of the KI-2006, MKM-2005, and CALCULEMUS-2000 conferences and has served on the Programme Committees of more than three dozen international conferences. Kohlhase holds an adjunct associate professorship at Carnegie Mellon University and was (2006–2008) vice director of the Department of Safe and Secure Cognitive Systems at German Research Centre for Artificial Intelligence (DFKI) Lab Bremen. In 2014, he became a member of the Global Digital Mathematics Library Working Group of the IMU. == Academic career == Michael Kohlhase obtained a degree in Mathematics (1989) from University of Bonn, a doctorate (1994) and habilitation (1999) in Computer Science at Saarland University. He has pursued his doctoral and post-doctoral research in extended research visits at Carnegie Mellon University, University of Amsterdam, the University of Edinburgh, and SRI International. From 2000–2003, he has conducted research and taught at the School of Computer Science at Carnegie Mellon University, where he was appointed to an adjunct associate professor. In September 2003 he was appointed as Professor of Computer Science at Jacobs University Bremen (International University Bremen until 2007), and 2006–2008 he was vice director of the Department of Safe and Secure Cognitive Systems of the German Research Centre for Artificial Intelligence (DFKI) Bremen. Since September 2016 he holds the Professorship for Knowledge Representation and Processing at University of Erlangen–Nuremberg. He has authored or edited four books and published almost 100 peer-reviewed papers. == Awards and Scholarships == 2000 3-year Heisenberg-Stipend of the Deutsche Forschungsgemeinschaft (DFG). 1996 AKI-prize, dissertation prize of the "Arbeitsgemeinschaft deutscher KI-Institute (AKI)" 1991 dissertation stipend of the Studienstiftung (German National Academic Foundation) 1986 masters stipend of Studienstiftung == Research interests == Michael Kohlhase's current research interests include Automated theorem proving and knowledge representation for mathematics, inference-based techniques for natural language processing and semantics, and computer-supported education. Much of his concrete work is based on web-based content markup formats like MathML, OpenMath, and OMDoc and systems for managing this data, e.g. semantic search engines for mathematical formulae, semantic extensions to LaTeX, or converting legacy LaTeX documents from the arXiv.
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How to Choose an AI Text-to-image Tool

Curious about the best AI text-to-image tool? An AI text-to-image tool is software that uses machine learning to help you get more done — it combines speed, accuracy, and an interface that just works. Hands-on testing shows real-world results vary, so a short free trial is the smartest way to decide. Whether you are a beginner or a pro, the right AI text-to-image tool slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.
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Tagged Deterministic Finite Automaton

In the automata theory, a tagged deterministic finite automaton (TDFA) is an extension of deterministic finite automaton (DFA). In addition to solving the recognition problem for regular languages, TDFA is also capable of submatch extraction and parsing. While canonical DFA can find out if a string belongs to the language defined by a regular expression, TDFA can also extract substrings that match specific subexpressions. More generally, TDFA can identify positions in the input string that match tagged positions in a regular expression (tags are meta-symbols similar to capturing parentheses, but without the pairing requirement). == History == TDFA were first described by Ville Laurikari in 2000. Prior to that it was unknown whether it is possible to perform submatch extraction in one pass on a deterministic finite-state automaton, so this paper was an important advancement. Laurikari described TDFA construction and gave a proof that the determinization process terminates, however the algorithm did not handle disambiguation correctly. In 2007 Chris Kuklewicz implemented TDFA in a Haskell library Regex-TDFA with POSIX longest-match semantics. Kuklewicz gave an informal description of the algorithm and answered the principal question whether TDFA are capable of POSIX longest-match disambiguation, which was doubted by other researchers. In 2017 Ulya Trafimovich described TDFA with one-symbol lookahead. The use of a lookahead symbol reduces the number of registers and register operations in a TDFA, which makes it faster and often smaller than Laurikari TDFA. Trafimovich called TDFA variants with and without lookahead TDFA(1) and TDFA(0) by analogy with LR parsers LR(1) and LR(0). The algorithm was implemented in the open-source lexer generator RE2C. Trafimovich formalized Kuklewicz disambiguation algorithm. In 2018 Angelo Borsotti worked on an experimental Java implementation of TDFA; it was published later in 2021. In 2019 Borsotti and Trafimovich adapted POSIX disambiguation algorithm by Okui and Suzuki to TDFA. They gave a formal proof of correctness of the new algorithm and showed that it is faster than Kuklewicz algorithm in practice. In 2020 Trafimovich published an article about TDFA implementation in RE2C. In 2022 Borsotti and Trafimovich published a paper with a detailed description of TDFA construction. The paper incorporated their past research and presented multi-pass TDFA that are better suited to just-in-time determinization. They also compared TDFA against other algorithms and provided benchmarks. == Formal definition == TDFA have the same basic structure as ordinary DFA: a finite set of states linked by transitions. In addition to that, TDFA have a fixed set of registers that hold tag values, and register operations on transitions that set or copy register values. The values may be scalar offsets, or offset lists for tags that match repeatedly (the latter can be represented efficiently using a trie structure). There is no one-to-one mapping between tags in a regular expression and registers in a TDFA: a single tag may need many registers, and the same register may hold values of different tags. The following definition is according to Trafimovich and Borsotti. The original definition by Laurikari is slightly different. A tagged deterministic finite automaton F {\displaystyle F} is a tuple ( Σ , T , S , S f , s 0 , R , R f , δ , φ ) {\displaystyle (\Sigma ,T,S,S_{f},s_{0},R,R_{f},\delta ,\varphi )} , where: Σ {\displaystyle \Sigma } is a finite set of symbols (alphabet) T {\displaystyle T} is a finite set of tags S {\displaystyle S} is a finite set of states with initial state s 0 {\displaystyle s_{0}} and a subset of final states S f ⊆ S {\displaystyle S_{f}\subseteq S} R {\displaystyle R} is a finite set of registers with a subset of final registers R f {\displaystyle R_{f}} (one per tag) δ : S × Σ → S × O ∗ {\displaystyle \delta :S\times \Sigma \rightarrow S\times O^{}} is a transition function φ : S f → O ∗ {\displaystyle \varphi :S_{f}\rightarrow O^{}} is a final function, where O {\displaystyle O} is a set of register operations of the following types: set register i {\displaystyle i} to nil or to the current position: i ← v {\displaystyle i\leftarrow v} , where v ∈ { n , p } {\displaystyle v\in \{\mathbf {n} ,\mathbf {p} \}} copy register j {\displaystyle j} to register i {\displaystyle i} : i ← j {\displaystyle i\leftarrow j} copy register j {\displaystyle j} to register i {\displaystyle i} and append history: i ← j ⋅ h {\displaystyle i\leftarrow j\cdot h} , where h {\displaystyle h} is a string over { n , p } {\displaystyle \{\mathbf {n} ,\mathbf {p} \}} === Example === Figure 0 shows an example TDFA for regular expression ( 1 a 2 ) ∗ 3 ( a | 4 b ) 5 b ∗ {\displaystyle (1a2)^{}3(a|4b)5b^{}} with alphabet Σ = { a , b } {\displaystyle \Sigma =\{a,b\}} and a set of tags T = { 1 , 2 , 3 , 4 , 5 } {\displaystyle T=\{1,2,3,4,5\}} that matches strings of the form a … a b … b {\displaystyle a\dots ab\dots b} with at least one symbol. TDFA has four states S = { 0 , 1 , 2 , 3 } {\displaystyle S=\{0,1,2,3\}} three of which are final S f = { 1 , 2 , 3 } {\displaystyle S_{f}=\{1,2,3\}} . The set of registers is R = { r 1 , r 2 , r 3 , r 4 , r 5 } {\displaystyle R=\{r_{1},r_{2},r_{3},r_{4},r_{5}\}} with a subset of final registers R f = { r 1 , r 2 , r 3 , r 4 , r 5 } {\displaystyle R_{f}=\{r_{1},r_{2},r_{3},r_{4},r_{5}\}} where register r i {\displaystyle r_{i}} corresponds to i {\displaystyle i} -th tag. Transitions have operations defined by the δ {\displaystyle \delta } function, and final states have operations defined by the φ {\displaystyle \varphi } function (marked with wide-tipped arrow). For example, to match string a a b {\displaystyle aab} , one starts in state 0, matches the first a {\displaystyle a} and moves to state 1 (setting registers r 1 , r 2 {\displaystyle r_{1},r_{2}} to undefined and r 3 {\displaystyle r_{3}} to the current position 0), matches the second a {\displaystyle a} and loops to state 1 (register values are now r 1 = 0 , r 2 = r 3 = 1 {\displaystyle r_{1}=0,r_{2}=r_{3}=1} ), matches b {\displaystyle b} and moves to state 2 (register values are now r 1 = 1 , r 2 = r 3 = r 4 = 2 {\displaystyle r_{1}=1,r_{2}=r_{3}=r_{4}=2} ), executes the final operations in state 2 (register values are now r 1 = 1 , r 2 = r 3 = r 4 = 2 , r 5 = 3 {\displaystyle r_{1}=1,r_{2}=r_{3}=r_{4}=2,r_{5}=3} ) and finally exits TDFA. == Complexity == Canonical DFA solve the recognition problem in linear time. The same holds for TDFA, since the number of registers and register operations is fixed and depends only on the regular expression, but not on the length of input. The overhead on submatch extraction depends on tag density in a regular expression and nondeterminism degree of each tag (the maximum number of registers needed to track all possible values of the tag in a single TDFA state). On one extreme, if there are no tags, a TDFA is identical to a canonical DFA. On the other extreme, if every subexpression is tagged, a TDFA effectively performs full parsing and has many operations on every transition. In practice for real-world regular expressions with a few submatch groups the overhead is negligible compared to matching with canonical DFA. == TDFA construction == TDFA construction is performed in a few steps. First, a regular expression is converted to a tagged nondeterministic finite automaton (TNFA). Second, a TNFA is converted to a TDFA using a determinization procedure; this step also includes disambiguation that resolves conflicts between ambiguous TNFA paths. After that, a TDFA can optionally go through a number of optimizations that reduce the number of registers and operations, including minimization that reduces the number of states. Algorithms for all steps of TDFA construction with pseudocode are given in the paper by Borsotti and Trafimovich. This section explains TDFA construction on the example of a regular expression a ∗ t b ∗ | a b {\displaystyle a^{}tb^{}|ab} , where t {\displaystyle t} is a tag and { a , b } {\displaystyle \{a,b\}} are alphabet symbols. === Tagged NFA === TNFA is a nondeterministic finite automaton with tagged ε-transitions. It was first described by Laurikari, although similar constructions were known much earlier as Mealy machines and nondeterministic finite-state transducers. TNFA construction is very similar to Thompson's construction: it mirrors the structure of a regular expression. Importantly, TNFA preserves ambiguity in a regular expression: if it is possible to match a string in two different ways, then TNFA for this regular expression has two different accepting paths for this string. TNFA definition by Borsotti and Trafimovich differs from the original one by Laurikari in that TNFA can have negative tags on transitions: they are needed to make the absence of match explicit in cases when there is a bypass for a tagged transition. Figure 1 shows TNFA for the example regu
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Scale-space axioms

In image processing and computer vision, a scale space framework can be used to represent an image as a family of gradually smoothed images. This framework is very general and a variety of scale space representations exist. A typical approach for choosing a particular type of scale space representation is to establish a set of scale-space axioms, describing basic properties of the desired scale-space representation and often chosen so as to make the representation useful in practical applications. Once established, the axioms narrow the possible scale-space representations to a smaller class, typically with only a few free parameters. A set of standard scale space axioms, discussed below, leads to the linear Gaussian scale-space, which is the most common type of scale space used in image processing and computer vision. == Scale space axioms for the linear scale-space representation == The linear scale space representation L ( x , y , t ) = ( T t f ) ( x , y ) = g ( x , y , t ) ∗ f ( x , y ) {\displaystyle L(x,y,t)=(T_{t}f)(x,y)=g(x,y,t)f(x,y)} of signal f ( x , y ) {\displaystyle f(x,y)} obtained by smoothing with the Gaussian kernel g ( x , y , t ) {\displaystyle g(x,y,t)} satisfies a number of properties 'scale-space axioms' that make it a special form of multi-scale representation: linearity T t ( a f + b h ) = a T t f + b T t h {\displaystyle T_{t}(af+bh)=aT_{t}f+bT_{t}h} where f {\displaystyle f} and h {\displaystyle h} are signals while a {\displaystyle a} and b {\displaystyle b} are constants, shift invariance T t S ( Δ x , Δ y ) f = S ( Δ x , Δ y ) T t f {\displaystyle T_{t}S_{(\Delta x,\Delta _{y})}f=S_{(\Delta x,\Delta _{y})}T_{t}f} where S ( Δ x , Δ y ) {\displaystyle S_{(\Delta x,\Delta _{y})}} denotes the shift (translation) operator ( S ( Δ x , Δ y ) f ) ( x , y ) = f ( x − Δ x , y − Δ y ) {\displaystyle (S_{(\Delta x,\Delta _{y})}f)(x,y)=f(x-\Delta x,y-\Delta y)} semi-group structure g ( x , y , t 1 ) ∗ g ( x , y , t 2 ) = g ( x , y , t 1 + t 2 ) {\displaystyle g(x,y,t_{1})g(x,y,t_{2})=g(x,y,t_{1}+t_{2})} with the associated cascade smoothing property L ( x , y , t 2 ) = g ( x , y , t 2 − t 1 ) ∗ L ( x , y , t 1 ) {\displaystyle L(x,y,t_{2})=g(x,y,t_{2}-t_{1})L(x,y,t_{1})} existence of an infinitesimal generator A {\displaystyle A} ∂ t L ( x , y , t ) = ( A L ) ( x , y , t ) {\displaystyle \partial _{t}L(x,y,t)=(AL)(x,y,t)} non-creation of local extrema (zero-crossings) in one dimension, non-enhancement of local extrema in any number of dimensions ∂ t L ( x , y , t ) ≤ 0 {\displaystyle \partial _{t}L(x,y,t)\leq 0} at spatial maxima and ∂ t L ( x , y , t ) ≥ 0 {\displaystyle \partial _{t}L(x,y,t)\geq 0} at spatial minima, rotational symmetry g ( x , y , t ) = h ( x 2 + y 2 , t ) {\displaystyle g(x,y,t)=h(x^{2}+y^{2},t)} for some function h {\displaystyle h} , scale invariance g ^ ( ω x , ω y , t ) = h ^ ( ω x φ ( t ) , ω x φ ( t ) ) {\displaystyle {\hat {g}}(\omega _{x},\omega _{y},t)={\hat {h}}({\frac {\omega _{x}}{\varphi (t)}},{\frac {\omega _{x}}{\varphi (t)}})} for some functions φ {\displaystyle \varphi } and h ^ {\displaystyle {\hat {h}}} where g ^ {\displaystyle {\hat {g}}} denotes the Fourier transform of g {\displaystyle g} , positivity g ( x , y , t ) ≥ 0 {\displaystyle g(x,y,t)\geq 0} , normalization ∫ x = − ∞ ∞ ∫ y = − ∞ ∞ g ( x , y , t ) d x d y = 1 {\displaystyle \int _{x=-\infty }^{\infty }\int _{y=-\infty }^{\infty }g(x,y,t)\,dx\,dy=1} . In fact, it can be shown that the Gaussian kernel is a unique choice given several different combinations of subsets of these scale-space axioms: most of the axioms (linearity, shift-invariance, semigroup) correspond to scaling being a semigroup of shift-invariant linear operator, which is satisfied by a number of families integral transforms, while "non-creation of local extrema" for one-dimensional signals or "non-enhancement of local extrema" for higher-dimensional signals are the crucial axioms which relate scale-spaces to smoothing (formally, parabolic partial differential equations), and hence select for the Gaussian. The Gaussian kernel is also separable in Cartesian coordinates, i.e. g ( x , y , t ) = g ( x , t ) g ( y , t ) {\displaystyle g(x,y,t)=g(x,t)\,g(y,t)} . Separability is, however, not counted as a scale-space axiom, since it is a coordinate dependent property related to issues of implementation. In addition, the requirement of separability in combination with rotational symmetry per se fixates the smoothing kernel to be a Gaussian. There exists a generalization of the Gaussian scale-space theory to more general affine and spatio-temporal scale-spaces. In addition to variabilities over scale, which original scale-space theory was designed to handle, this generalized scale-space theory also comprises other types of variabilities, including image deformations caused by viewing variations, approximated by local affine transformations, and relative motions between objects in the world and the observer, approximated by local Galilean transformations. In this theory, rotational symmetry is not imposed as a necessary scale-space axiom and is instead replaced by requirements of affine and/or Galilean covariance. The generalized scale-space theory leads to predictions about receptive field profiles in good qualitative agreement with receptive field profiles measured by cell recordings in biological vision. In the computer vision, image processing and signal processing literature there are many other multi-scale approaches, using wavelets and a variety of other kernels, that do not exploit or require the same requirements as scale space descriptions do; please see the article on related multi-scale approaches. There has also been work on discrete scale-space concepts that carry the scale-space properties over to the discrete domain; see the article on scale space implementation for examples and references.
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Diane Litman

Diane Litman is an American professor of computer science at the University of Pittsburgh. She also jointly holds the positions of senior scientist with the Learning Research and Development Center and faculty with the Intelligent Systems department. Litman is noted for her work in the areas of artificial intelligence, computational linguistics, knowledge representation and reasoning, natural language processing, and user modeling. == Education == Litman did her undergraduate studies at the College of William and Mary and her master's and PhD degrees at the University of Rochester. == Career == Before joining the University of Pittsburgh, she was an assistant professor at Columbia University. She additionally held the position of a research scientist in the Artificial Intelligence Principles Research Department Laboratory at AT&T Labs. Litman has held the position of Chair of the North American Chapter of the Association for Computational Linguistics two times, elected twice for the position, whose tenure lasts four years. She is also a distinguished member of the executive committee of the Association for Computational Linguistics, and a member of the editorial boards of Computational Linguistics and User Modeling and User-Adapted Interaction. She has also held the position of Leverhulme Professor at the University of Edinburgh. Litman was the keynote speaker at the Speech and Language Technology in Education 2013 symposium, the 2006 SIGdial Meeting on Discourse and Dialogue, and at the 2008 Symposium of the Annual Meeting of the Society for the Study of Artificial Intelligence and Simulation of Behaviour. She also sits on the board of the several interest groups, including the International Speech Communication Association's Special Interest Group on Speech and Language Technology in Education. Litman has served as chair, organizer, and a senior member of numerous committees of peer-reviewed scientific journals. == Awards and recognition == She has also co-authored numerous award-winning papers and was awarded senior member status by the Association for the Advancement of Artificial Intelligence in 2011, an award designed to honor those who have "achieved significant accomplishments within the field of artificial intelligence."
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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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