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
Pyramid (image processing)
Pyramid, or pyramid representation, is a type of multi-scale signal representation developed by the computer vision, image processing and signal processing communities, in which a signal or an image is subject to repeated smoothing and subsampling. Pyramid representation is a predecessor to scale-space representation and multiresolution analysis. == Pyramid generation == There are two main types of pyramids: lowpass and bandpass. A lowpass pyramid is made by smoothing the image with an appropriate smoothing filter and then subsampling the smoothed image, usually by a factor of 2 along each coordinate direction. The resulting image is then subjected to the same procedure, and the cycle is repeated multiple times. Each cycle of this process results in a smaller image with increased smoothing, but with decreased spatial sampling density (that is, decreased image resolution). If illustrated graphically, the entire multi-scale representation will look like a pyramid, with the original image on the bottom and each cycle's resulting smaller image stacked one atop the other. A bandpass pyramid is made by forming the difference between images at adjacent levels in the pyramid and performing image interpolation between adjacent levels of resolution, to enable computation of pixelwise differences. == Pyramid generation kernels == A variety of different smoothing kernels have been proposed for generating pyramids. Among the suggestions that have been given, the binomial kernels arising from the binomial coefficients stand out as a particularly useful and theoretically well-founded class. Thus, given a two-dimensional image, we may apply the (normalized) binomial filter (1/4, 1/2, 1/4) typically twice or more along each spatial dimension and then subsample the image by a factor of two. This operation may then proceed as many times as desired, leading to a compact and efficient multi-scale representation. If motivated by specific requirements, intermediate scale levels may also be generated where the subsampling stage is sometimes left out, leading to an oversampled or hybrid pyramid. With the increasing computational efficiency of CPUs available today, it is in some situations also feasible to use wider supported Gaussian filters as smoothing kernels in the pyramid generation steps. === Gaussian pyramid === In a Gaussian pyramid, subsequent images are weighted down using a Gaussian average (Gaussian blur) and scaled down. Each pixel containing a local average corresponds to a neighborhood pixel on a lower level of the pyramid. This technique is used especially in texture synthesis. === Laplacian pyramid === A Laplacian pyramid is very similar to a Gaussian pyramid but saves the difference image of the blurred versions between each levels. Only the smallest level is not a difference image to enable reconstruction of the high resolution image using the difference images on higher levels. This technique can be used in image compression. === Steerable pyramid === A steerable pyramid, developed by Simoncelli and others, is an implementation of a multi-scale, multi-orientation band-pass filter bank used for applications including image compression, texture synthesis, and object recognition. It can be thought of as an orientation selective version of a Laplacian pyramid, in which a bank of steerable filters are used at each level of the pyramid instead of a single Laplacian or Gaussian filter. == Applications of pyramids == === Alternative representation === In the early days of computer vision, pyramids were used as the main type of multi-scale representation for computing multi-scale image features from real-world image data. More recent techniques include scale-space representation, which has been popular among some researchers due to its theoretical foundation, the ability to decouple the subsampling stage from the multi-scale representation, the more powerful tools for theoretical analysis as well as the ability to compute a representation at any desired scale, thus avoiding the algorithmic problems of relating image representations at different resolution. Nevertheless, pyramids are still frequently used for expressing computationally efficient approximations to scale-space representation. === Detail manipulation === Levels of a Laplacian pyramid can be added to or removed from the original image to amplify or reduce detail at different scales. However, detail manipulation of this form is known to produce halo artifacts in many cases, leading to the development of alternatives such as the bilateral filter. Some image compression file formats use the Adam7 algorithm or some other interlacing technique. These can be seen as a kind of image pyramid. Because those file format store the "large-scale" features first, and fine-grain details later in the file, a particular viewer displaying a small "thumbnail" or on a small screen can quickly download just enough of the image to display it in the available pixels—so one file can support many viewer resolutions, rather than having to store or generate a different file for each resolution.
Pushpak Bhattacharyya
Pushpak Bhattacharyya (3 July 1962 – 5 October 2025) was an Indian computer scientist and professor in the Department of Computer Science and Engineering at the IIT Bombay. He served as the Director of the IIT Patna from 2015 to 2021. He was a past President of the Association for Computational Linguistics (2016–17), and held the Vijay and Sita Vashee Chair Professorship at IIT Bombay. Bhattacharyya led the Natural Language Processing (NLP) research group at the Centre for Indian Language Technology (CFILT) at IIT Bombay until his death. At the inauguration of the Nilekani Centre at AI4Bharat, IIT Madras, Nandan Nilekani, Co-founder and Non-Executive Chairman of Infosys, referred to Bhattacharyya as the "Godfather of Indian NLP". == Early life and education == Bhattacharyya was born in Shillong in 1962. He completed his schooling at Jail Road Boys' High School, Shillong. He obtained a B.Tech. in Computer Science from the IIT Kharagpur, followed by an M.Tech. from the IIT Kanpur, and a Ph.D. in Computer Science from IIT Bombay in 1994. == Research == Bhattacharyya’s research areas includes Natural language processing, Artificial intelligence, Machine learning, Psycholinguistics, Eye tracking, and Information retrieval. He made contributions to the development of multilingual lexical databases such as IndoWordNet and other projects related to machine translation and computational linguistics. He authored and co-authored multiple academic works, including Investigations in Computational Sarcasm (with Aditya Joshi), Cognitively Inspired Natural Language Processing: An Investigation Based on Eye Tracking (with Abhijit Mishra), and Machine Translation and Transliteration of Low Resource Related Languages (with Anoop Kunchukuttan). Over his career, Bhattacharyya published more than 350 research papers in journals and conference proceedings and supervised over 300 undergraduate, master’s, and doctoral students. His projects often addressed computational challenges for Indian languages, such as developing wordnets, building translation systems for low-resource languages, and studying cognitive aspects of language processing. He also led government- and industry-funded research initiatives supported by organizations including IBM, Microsoft, Yahoo, and the United Nations. == Death == Bhattacharyya died on 5 October 2025, at the age of 63. == Awards == Patwardhan Award, IIT Bombay, for Technology Development VNMM Award, IIT Roorkee, for Technology Development Fellow, Indian National Academy of Engineering Eminent Engineer Award, Institution of Engineers (India)
Mona Diab
Mona Talat Diab (Arabic: منى طلعت دياب) is a computer science professor and director of Carnegie Mellon University's Language Technologies Institute. Previously, she was a professor at George Washington University and a research scientist with Facebook AI. Her research focuses on natural language processing, computational linguistics, cross lingual/multilingual processing, computational socio-pragmatics, Arabic language processing, and applied machine learning. == Education == Diab completed her M.Sc. in computer science with a major in machine learning and artificial intelligence at The George Washington University (1997) and her Ph.D. in computational linguistics at the University of Maryland, Linguistics Department and University of Maryland Institute for Advanced Computer Studies (UMIACS) in 2003, under the supervision of Philip Resnik. She was also a postdoctoral research scientist at Stanford University (2003–2005) under the mentorship of Dan Jurafsky, where she was a part of the Stanford NLP Group. == Career == After her postdoc at Stanford, Diab took a position as research scientist (principal investigator) at the Center for Computational Learning Systems (CCLS) in Columbia University, where she was also adjunct professor in the computer science department. In 2013 she joined the George Washington University as an associate professor, where she was promoted to full professor in 2017. Diab is the founder and director of the GW NLP lab CARE4Lang. Diab served as an elected faculty senator at Columbia University for 6 years (2007–2012) and an elected faculty senator at GW (2013–2014). She served the computational linguistics community as elected member, secretary and president of ACL SIGLEX (2005–2016) and elected president of ACL SIGSemitic. She currently serves as the elected VP-elect for ACL SIGDAT. In 2017 Diab joined Amazon AWS AI Deep Learning Group for Human Language Technologies, where she led the AWS Lex project for task oriented dialogue systems for enterprises. A couple of years later, she moved to Facebook AI as a research scientist. In the fall of 2023, she became the director of CMU's Language Technologies Institute -- the first full time director since the passing of its founder Jaime Carbonell. == Research == Diab's research interests include several areas in computational linguistics/natural language processing, like conversational AI, computational lexical semantics, multilingual and cross lingual processing, social media processing with an emphasis on computational socio- pragmatics, information extraction & text analytics, machine translation. Besides this, she also has special interests in Arabic NLP and low resource scenarios. Diab co-established two research trends in the computational linguistics field, computational approaches to linguistic code switching in 2007 and semantic textual similarity in 2010. Diab together with Nizar Habash and Owen Rambow, co-founded CADIM in 2005, a global reference point in Arabic dialect processing. In 2012, Diab together with Eneko Agirre and Johan Bos, brought together two ACL communities SIGLEX and SIGSEM and established the 1st tier conference SEM. == Awards and recognition == Selected as one of top 150 leaders and visionaries in AI nationwide to participate in White House AI Summit in Government, Washington, D.C., US, September 2019 March 2017: 3 Muslim Women in STEM You Should Know About, Teen Vogue, March 2017 May 2017: Behind Every Strong Woman Is...Another Strong Woman: Ten women give thanks to the women who supported them on the way up. Elle, May 2017. Google Faculty Research Award – Tharwa++: Building a multidialectal Arabic Lexical Repository, (PI), 09.2015 –12.2016. Google Faculty Research Award – Nuanced Sentiment and Perspective Analysis for Arabic Social Media Text, (PI), 12.2014 –12.2015 QNRF Best Poster Award – Ossama Obeid, Houda Bouamor, Wajdi Zaghouani, Mahmoud Ghoneim, Abdelati Hawwari, Mona Diab, Kemal Oflazer. (2016) MANDIAC: A Web-based Annotation System For Manual Arabic Diacritization. Proceedings of the 2nd Workshop on Arabic Corpora and Processing Tools, LREC 2016. Best Paper Award – Aminian, Maryam, Mahmoud Ghoneim, Mona Diab. (2015) Unsupervised False Friend Disambiguation Using Contextual Word Clusters and Parallel Word Alignments. In Proceedings of Workshop 9th Semantics Syntax Statistical Translation, NAACL 2015, Denver CO, US. == Publications == Diab has over 250 publications, and she is an acting editor for several scientific journals. === Selected publications === Semeval-2012 task 6: A pilot on semantic textual similarity. E. Agirre, D. Cer, M. Diab, A. Gonzalez-Agirre. SEM 2012: The First Joint Conference on Lexical and Computational Semantics–Volume 1: Proceedings of the main conference and the shared task, and Volume 2: Proceedings of the Sixth International Workshop on Semantic Evaluation (SemEval 2012) Predictive linguistic features of schizophrenia. ES Kayi, M Diab, L Pauselli, M Compton, G Coppersmith. arXiv preprint arXiv:1810.09377 Ideological perspective detection using semantic features. H Elfardy, M Diab, C Callison-Burch – Proceedings of SEM 2015 DeSePtion: Dual sequence prediction and adversarial examples for improved fact-checking. Christopher Hidey, Tuhin Chakrabarty, Tariq Alhindi, Siddharth Varia, Kriste Krstovski, Mona Diab, Smaranda Muresan, 2020 Does Causal Coherence Predict Online Spread of Social Media? Pedram Hosseini, Mona Diab, David A Broniatowski. Proceedings of International Conference on Social Computing, Behavioral-Cultural Modeling and Prediction and Behavior Representation in Modeling and Simulation, 2019. Diversity, Density, and Homogeneity: Quantitative Characteristic Metrics for Text Collections. YA Lai, X Zhu, Y Zhang, M Diab, arXiv preprint arXiv:2003.08529, 2020 Readability of written medicine information materials in Arabic language: expert and consumer evaluation. S Al Aqeel, N Abanmy, A Aldayel, H Al-Khalifa, M Al-Yahya, M Diab. BMC health services research 18 (1), 1–7, 2019 Unsupervised word mapping using structural similarities in monolingual embeddings. H Aldarmaki, M Mohan, M Diab – Transactions of the Association for Computational Linguistics, 2018 An unsupervised method for word sense tagging using parallel corpora M Diab, P Resnik. Proceedings of ACL 2002 Overview for the first shared task on language identification in code-switched data. Thamar Solorio, Elizabeth Blair, Suraj Maharjan, Steven Bethard, Mona Diab, Mahmoud Ghoneim, Abdelati Hawwari, Fahad AlGhamdi, Julia Hirschberg, Alison Chang, Pascale Fung. Proceedings of the First Workshop on Computational Approaches to Code Switching, 2014 Modeling sentences in the latent space. W Guo, M Diab – ACL 20 12 Task-based evaluation of multiword expressions: a pilot study in statistical machine translation. M Carpuat, M Diab – NAACL-HLT 2010 Rumor detection and classification for twitter data. S Hamidian, MT Diab – arXiv preprint arXiv:1912.08926, 2019 Subgroup detection in ideological discussions. A Abu-Jbara, P Dasigi, M Diab, D Radev – ACL 2012 Madamira: A fast, comprehensive tool for morphological analysis and disambiguation of arabic. A. Pasha, M. Al-Badrashiny, M. Diab, A. El Kholy, R. Eskander, N. Habash, M. Pooleery, O. Rambow, R. Roth. LREC 14, 1094–1101. 2014 Context-Aware Self-Attentive Natural Language Understanding for Task-Oriented Chatbots. A. Gupta, P. Zhang, G. Lalwani, M. Diab. EMNLP 2019 A multitask learning approach for diacritic restoration. S. Alqahtani, A. Mishra, M. Diab. ACL 2020
Tom M. Mitchell
Tom Michael Mitchell (born August 9, 1951) is an American computer scientist and the Founders University Professor at Carnegie Mellon University (CMU). He is a founder and former chair of the Machine Learning Department at CMU. Mitchell is known for his contributions to the advancement of machine learning, artificial intelligence, and cognitive neuroscience and is the author of the textbook Machine Learning. He is a member of the United States National Academy of Engineering since 2010. He is also a Fellow of the American Academy of Arts and Sciences, the American Association for the Advancement of Science and a Fellow and past president of the Association for the Advancement of Artificial Intelligence. In October 2018, Mitchell was appointed as the Interim Dean of the School of Computer Science at Carnegie Mellon. == Early life and education == Mitchell was born in Blossburg, Pennsylvania and grew up in Upstate New York, in the town of Vestal. He received his bachelor of Science degree in electrical engineering from the Massachusetts Institute of Technology in 1973 and a Ph.D. from Stanford University under the direction of Bruce G. Buchanan in 1979. == Career == Mitchell began his teaching career at Rutgers University in 1978. During his tenure at Rutgers, he held the positions of assistant and associate professor in the Department of Computer Science. In 1986, he left Rutgers and joined Carnegie Mellon University, Pittsburgh as a professor. In 1999, he became the E. Fredkin Professor in the School of Computer Science. In 2006 Mitchell was appointed as the first chair of the Machine Learning Department within the School of Computer Science. He became university professor in 2009, and served as Interim Dean of the Carnegie Mellon School of Computer Science during 2018–2019. Mitchell currently serves on the Scientific Advisory Board of the Allen Institute for AI and on the Science Board of the Santa Fe Institute. == Honors and awards == He was elected into the United States National Academy of Engineering in 2010 "for pioneering contributions and leadership in the methods and applications of machine learning." He is also a Fellow of the American Association for the Advancement of Science (AAAS) since 2008 and a Fellow the Association for the Advancement of Artificial Intelligence (AAAI) since 1990. In 2016 he became a Fellow of the American Academy of Arts and Sciences. Mitchell was awarded an Honorary Doctor of Laws degree from Dalhousie University in 2015 for his contributions to machine learning and to cognitive neuroscience, and the President's Medal from Stevens Institute of Technology in 2018. He is a recipient of the NSF Presidential Young Investigator Award in 1984. == Publications == Mitchell is a prolific author of scientific works on various topics in computer science, including machine learning, artificial intelligence, robotics, and cognitive neuroscience. He has authored hundreds of scientific articles. Mitchell published one of the first textbooks in machine learning, entitled Machine Learning, in 1997 (publisher: McGraw Hill Education). He is also a coauthor of the following books: J. Franklin, T. Mitchell, and S. Thrun (eds.), Recent Advances in Robot Learning, Kluwer Academic Publishers, 1996. T. Mitchell, J. Carbonell, and R. Michalski (eds.), Machine Learning: A Guide to Current Research, Kluwer Academic Publishers, 1986. R. Michalski, J. Carbonell, and T. Mitchell (eds.), Machine Learning: An Artificial Intelligence Approach, Volume 2, Morgan Kaufmann, 1986. R. Michalski, J. Carbonell, and T. Mitchell (eds.), Machine Learning: An Artificial Intelligence Approach, Tioga Press, 1983.
Halloween Problem
In computing, the Halloween Problem refers to a phenomenon in databases in which an update operation causes a change in the physical location of a row, potentially allowing the row to be visited again later in the same update operation. This could even cause an infinite loop in some cases where updates continually place the updated record ahead of the scan performing the update operation. The potential for this database error was first discovered by Don Chamberlin, Pat Selinger, and Morton Astrahan in the mid-1970s, on Halloween day, while working on query optimization. They wrote a SQL query supposed to give a ten percent raise to every employee who earned less than $25,000. This query would run successfully, with no errors, but when finished all the employees in the database earned at least $25,000, because it kept giving them a raise until they reached that level. The expectation was that the query would iterate over each of the employee records with a salary less than $25,000 precisely once. In fact, because even updated records were visible to the query execution engine and so continued to match the query's criteria, salary records were matching multiple times and each time being given a 10% raise until they were all greater than $25,000. Contrary to what some believe, the name is not descriptive of the nature of the problem but rather was given due to the day it was discovered on. As recounted by Don Chamberlin: Pat and Morton discovered this problem on Halloween... I remember they came into my office and said, "Chamberlin, look at this. We have to make sure that when the optimizer is making a plan for processing an update, it doesn't use an index that is based on the field that is being updated. How are we going to do that?" It happened to be on a Friday, and we said, "Listen, we are not going to be able to solve this problem this afternoon. Let's just give it a name. We'll call it the Halloween Problem and we'll work on it next week." And it turns out it has been called that ever since.
Weighted automaton
In theoretical computer science and formal language theory, a weighted automaton or weighted finite-state machine is a generalization of a finite-state machine in which the edges have weights, for example real numbers or integers. Finite-state machines are only capable of answering decision problems; they take as input a string and produce a Boolean output, i.e. either "accept" or "reject". In contrast, weighted automata produce a quantitative output, for example a count of how many answers are possible on a given input string, or a probability of how likely the input string is according to a probability distribution. They are one of the simplest studied models of quantitative automata. The definition of a weighted automaton is generally given over an arbitrary semiring R {\displaystyle R} , an abstract set with an addition operation + {\displaystyle +} and a multiplication operation × {\displaystyle \times } . The automaton consists of a finite set of states, a finite input alphabet of characters Σ {\displaystyle \Sigma } and edges which are labeled with both a character in Σ {\displaystyle \Sigma } and a weight in R {\displaystyle R} . The weight of any path in the automaton is defined to be the product of weights along the path, and the weight of a string is the sum of the weights of all paths which are labeled with that string. The weighted automaton thus defines a function from Σ ∗ {\displaystyle \Sigma ^{}} to R {\displaystyle R} . Weighted automata generalize deterministic finite automata (DFAs) and nondeterministic finite automata (NFAs), which correspond to weighted automata over the Boolean semiring, where addition is logical disjunction and multiplication is logical conjunction. In the DFA case, there is only one accepting path for any input string, so disjunction is not applied. When the weights are real numbers and the outgoing weights for each state add to one, weighted automata can be considered a probabilistic model and are also known as probabilistic automata. These machines define a probability distribution over all strings, and are related to other probabilistic models such as Markov decision processes and Markov chains. Weighted automata have applications in natural language processing where they are used to assign weights to words and sentences, as well as in image compression. They were first introduced by Marcel-Paul Schützenberger in his 1961 paper On the definition of a family of automata. Since their introduction, many extensions have been proposed, for example nested weighted automata, cost register automata, and weighted finite-state transducers. Researchers have studied weighted automata from the perspective of learning a machine from its input-output behavior (see computational learning theory) and studying decidability questions. == Definition == A commutative semiring (or rig) is a set R equipped with two distinguished elements 0 ≠ 1 {\displaystyle 0\neq 1} and addition and multiplication operations ⊕ {\displaystyle \oplus } and ⊗ {\displaystyle \otimes } such that ⊕ {\displaystyle \oplus } is commutative and associative with identity 0 {\displaystyle 0} , ⊗ {\displaystyle \otimes } is commutative and associative with identity 1 {\displaystyle 1} , ⊗ {\displaystyle \otimes } distributes over ⊕ {\displaystyle \oplus } , and 0 is an absorbing element for ⊗ {\displaystyle \otimes } . A weighted automaton over R {\displaystyle R} is a tuple A = ( Q , Σ , Δ , I , F ) {\displaystyle {\mathcal {A}}=(Q,\Sigma ,\Delta ,I,F)} where: Q {\displaystyle Q} is a finite set of states. Σ {\displaystyle \Sigma } is a finite alphabet. Δ ⊆ Q × Σ × R × Q {\displaystyle \Delta \subseteq Q\times \Sigma \times R\times Q} is a finite set of transitions ( q , σ , w , q ′ ) {\displaystyle (q,\sigma ,w,q')} , where σ {\displaystyle \sigma } is called a character and w {\displaystyle w} is called a weight. I : Q → R {\displaystyle I:Q\to R} is an initial weight function. F : Q → R {\displaystyle F:Q\to R} is a final weight function. A path on input w ∈ Σ ∗ {\displaystyle w\in \Sigma ^{}} is a finite path in the graph, where the concatenation of the character labels equals w {\displaystyle w} . The weight of the path q 0 , q 1 , … , q n {\displaystyle q_{0},q_{1},\ldots ,q_{n}} is the product ( ⊗ {\displaystyle \otimes } ) of the weights along the path, additionally multiplied by the initial and final weights I ( q 0 ) ⊗ F ( q n ) {\displaystyle I(q_{0})\otimes F(q_{n})} . The weight of the word w {\displaystyle w} is the sum ( ⊕ {\displaystyle \oplus } ) of the weights of all paths on input w {\displaystyle w} (or 0 if there are no accepting paths). In this way the machine defines a function [ [ A ] ] : Σ ∗ → R {\displaystyle [\![{\mathcal {A}}]\!]:\Sigma ^{}\to R} . == Ambiguity and determinism == Since Δ {\displaystyle \Delta } is a set of transitions, weighted automata allow multiple transitions (or paths) on a single input string. Therefore a weighted automaton can be considered analogous to a nondeterministic finite automaton (NFA). As is the case with NFAs, restrictions of weighted automata are considered that correspond to the concepts of deterministic finite automaton and unambiguous finite automaton (deterministic weighted automata and unambiguous weighted automata, respectively). First, a preliminary definition: the underlying NFA of A {\displaystyle {\mathcal {A}}} is an NFA formed by removing all transitions with weight 0 {\displaystyle 0} and then erasing all of the weights on the transitions Δ {\displaystyle \Delta } , so that the new transition set lies in Q × Σ × Q {\displaystyle Q\times \Sigma \times Q} . The initial states and final states are the set of states q {\displaystyle q} such that I ( q ) ≠ 0 {\displaystyle I(q)\neq 0} and F ( q ) ≠ 0 {\displaystyle F(q)\neq 0} , respectively. A weighted automaton is deterministic if the underlying NFA is deterministic and unambiguous if the underlying NFA is unambiguous. Every deterministic weighted automaton is unambiguous. In both the deterministic and unambiguous cases, there is always at most one accepting path, so the ⊕ {\displaystyle \oplus } operation is never applied and can be omitted from the definition. == Variations == The requirement that there is a zero element for ⊕ {\displaystyle \oplus } is sometimes omitted; in this case the machine defines a partial function from Σ ∗ {\displaystyle \Sigma ^{}} to R {\displaystyle R} rather than a total function. It is possible to extend the definition to allow epsilon transitions ( q , ϵ , w , q ′ ) {\displaystyle (q,\epsilon ,w,q')} , where ϵ {\displaystyle \epsilon } is the empty string. In this case, one must then require that there are no cycles of epsilon transitions. This does not increase the expressiveness of weighted automata. If epsilon transitions are allowed, the initial weights and final weights can be replaced by initial and final sets of states without loss of expressiveness. Some authors omit the initial and final weight functions I {\displaystyle I} and F {\displaystyle F} . Instead, I {\displaystyle I} and F {\displaystyle F} are replaced by a set of initial and final states. If epsilon transitions are not present, this technically decreases expressiveness as it forces [ [ A ] ] ( ε ) {\displaystyle [\![{\mathcal {A}}]\!](\varepsilon )} to depend only on the number of states that are both initial and final. The transition function can be given as a matrix Δ σ ∈ R Q × Q {\displaystyle \Delta _{\sigma }\in R^{Q\times Q}} with entries in R {\displaystyle R} for each σ {\displaystyle \sigma } , rather than a set of transitions. The entry of the matrix at ( q , q ′ ) {\displaystyle (q,q')} is the sum of all transitions labeled ( q , σ , q ′ ) {\displaystyle (q,\sigma ,q')} . Some authors restrict to specific semirings, such as N {\displaystyle \mathbb {N} } or Z {\displaystyle \mathbb {Z} } , particularly when studying decidability results.