AI Headshot Generator For Linkedin

AI Headshot Generator For Linkedin — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

Spatial embedding

Spatial embedding is one of feature learning techniques used in spatial analysis where points, lines, polygons or other spatial data types. representing geographic locations are mapped to vectors of real numbers. Conceptually it involves a mathematical embedding from a space with many dimensions per geographic object to a continuous vector space with a much lower dimension. Such embedding methods allow complex spatial data to be used in neural networks and have been shown to improve performance in spatial analysis tasks == Embedded data types == Geographic data can take many forms: text, images, graphs, trajectories, polygons. Depending on the task, there may be a need to combine multimodal data from different sources. The next section describes examples of different types of data and their uses. === Text === Geolocated posts on social media can be used to acquire a library of documents bound to a given place that can be later transformed to embedded vectors using word embedding techniques. === Image === Satellites and aircraft collect digital spatial data acquired from remotely sensed images which can be used in machine learning. They are sometimes hard to analyse using basic image analysis methods and convolutional neural networks can be used to acquire an embedding of images bound to a given geographical object or a region. === Point === A single point of interest (POI) can be assigned multiple features that can be used in machine learning. These could be demographic, transportation, meteorological, or economic data, for example. When embedding single points, it is common to consider the entire set of available points as nodes in a graph. === Line / multiline === Among other things, motion trajectories are represented as lines (multilines). Individual trajectories are embedded taking into account travel time, distances and also features of points visited along the way. Embedding of trajectories allows to improve performance of such tasks as clustering and also categorization. === Polygon === The geographic areas analyzed in machine learning are defined by both administrative boundaries and top-down division into grids of regular shapes such as rectangles, for example. Both types are represented as polygons and, like points, can be assigned different demographic, transportation, or economic features. A polygon can also have features related to the size of the area or shape it represents. === Graph === An example domain where graph representation is used is the street layout in a city, where vertices can be intersections and edges can be roads. The vertices can also be destination points like public transport stops or important points in the city, and the edges represent the flow between them. Embedding graphs or single vertices allows to improve accuracy of analysis methods in which the treated geographical domain can be represented as a network. == Usage == POI recommendation - generating personalized point of interest recommendations based on user preferences. Next/future location prediction - prediction of the next location a person will go to based on their historical trajectory. Zone functions classification - based on different mobility of people or POI distribution a function of a given area in a city can be predicted. Crime prediction - estimation of crime rate in different regions of a city. Local event detection - studying spatio-temporal changes in embeddings can provide valuable information in detection of local event occurring in specific location. Regional mobility popularity prediction - analysis of mobility can show patterns in popularity of different regions in a city. Shape matching - finding a similar shape of given polygon, for example finding building with the same shape as input building. Travel time estimation - predicting estimated travel time given current traffic conditions and special occurring events. Time estimation for on-demand food delivery - estimation of delivery time when placing an order through the website. == Temporal aspect == Some of the data analyzed has a timestamp associated with it. In some cases of data analysis this information is omitted and in others it is used to divide the set into groups. The most common division is the separation of weekdays from weekends or division into hours of the day. This is particularly important in the analysis of mobility data, because the characteristics of mobility during the week and at different times of the day are very different from each other. Another area in which time division into, for example, individual months can be used is in the analysis of tourism of a given region. In order to take such a split into account, embedding methods treat the time stamp specifically or separate versions of the model are developed for different subgroups of the analyzed set.
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Writer invariant

Writer invariant, also called authorial invariant or author's invariant, is a property of a text which is invariant of its author, that is, it will be similar in all texts of a given author and different in texts of different authors. It can be used to find plagiarism or discover who is real author of anonymously published text. Writer invariant is also an author's pattern of writing a letter in handwritten text recognition. While it is generally recognised that writer invariants exist, it is not agreed what properties of a text should be used. Among the first ones used was distribution of word lengths; other proposed invariants include average sentence length, average word length, noun, verb or adjective usage frequency, vocabulary richness, and frequency of function words, or specific function words. Of these, average sentence lengths can be very similar in works of different authors or vary significantly even within a single work; average word lengths likewise turn out to be very similar in works of different authors. Analysis of function words shows promise because they are used by authors unconsciously.
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Wang-Chiew Tan

Wang-Chiew Tan is a Singaporean computer scientist specializing in data management and natural language processing. Her work in data management includes data provenance (or data lineage) and data integration. She is currently a Research Scientist at Facebook AI, and was previously the Director of Research at Megagon Labs in Mountain View, California. At Megagon Labs, Tan was the lead researcher on a study with the University of Tokyo that concluded that the company of other people is more effective than pets at making people happy. == Education and career == Tan earned her bachelor's degree in computer science (first-class) at the National University of Singapore, and completed her Ph.D. at the University of Pennsylvania. Her 2002 dissertation, Data Annotations, Provenance, and Archiving, was jointly supervised by Peter Buneman and Sanjeev Khanna. Before working at Megagon, she has been a professor of computer science at the University of California, Santa Cruz beginning in 2002, and, from 2010 to 2012, was on leave from Santa Cruz as a researcher at IBM Research - Almaden. == Recognition == Tan was named a Fellow of the Association for Computing Machinery in 2015 "for contributions to data provenance and to the foundations of information integration".
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Is an AI Paragraph Rewriter Worth It in 2026?

In search of the best AI paragraph rewriter? An AI paragraph rewriter 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 paragraph rewriter slots into your workflow and pays for itself fast. Below we compare features, pricing, and real output so you can choose with confidence.
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Video imprint (computer vision)

Proposed as an extension of image epitomes in the field of video content analysis, video imprint is obtained by recasting video contents into a fixed-sized tensor representation regardless of video resolution or duration. Specifically, statistical characteristics are retained to some degrees so that common video recognition tasks can be carried out directly on such imprints, e.g., event retrieval, temporal action localization. It is claimed that both spatio-temporal interdependences are accounted for and redundancies are mitigated during the computation of video imprints. The option of computing video imprints exploiting the epitome model has the advantage of more flexible input feature formats and more efficient training stage for video content analysis.
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How to Choose an AI Chatbot

Looking for the best AI chatbot? An AI chatbot 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 chatbot 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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Top 10 AI Virtual Assistants Compared (2026)

Looking for the best AI virtual assistant? An AI virtual 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 virtual assistant 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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Associative classifier

An associative classifier (AC) is a kind of supervised learning model that uses association rules to assign a target value. The term associative classification was coined by Bing Liu et al., in which the authors defined a model made of rules "whose right-hand side are restricted to the classification class attribute". == Model == The model generated by an AC and used to label new records consists of association rules, where the consequent corresponds to the class label. As such, they can also be seen as a list of "if-then" clauses: if the record matches some criteria (expressed in the left side of the rule, also called antecedent), it is then labeled accordingly to the class on the right side of the rule (or consequent). Most ACs read the list of rules in order, and apply the first matching rule to label the new record. == Metrics == The rules of an AC inherit some of the metrics of association rules, like the support or the confidence. Metrics can be used to order or filter the rules in the model and to evaluate their quality. == Implementations == The first proposal of a classification model made of association rules was FBM. The approach was popularized by CBA, although other authors had also previously proposed the mining of association rules for classification. Other authors have since then proposed multiple changes to the initial model, like the addition of a redundant rule pruning phase or the exploitation of Emerging Patterns. Notable implementations include: CMAR CPAR L3 CAEP GARC ADT.
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80 Million Tiny Images

80 Million Tiny Images is a dataset intended for training machine-learning systems constructed by Antonio Torralba, Rob Fergus, and William T. Freeman in a collaboration between MIT and New York University. It was published in 2008. The dataset has size 760 GB. It contains 79,302,017 32×32-pixel color images, scaled down from images scraped from the World Wide Web over 8 months. The images are classified into 75,062 classes. Each class is a non-abstract noun in WordNet. Images may appear in more than one class. The dataset was motivated by non-parametric models of neural activations in the visual cortex upon seeing images. The CIFAR-10 dataset uses a subset of the images in this dataset, but with independently generated labels, as the original labels were not reliable. The CIFAR-10 set has 6000 examples of each of 10 classes, and the CIFAR-100 set has 600 examples of each of 100 non-overlapping classes. == Construction == It was first reported in a technical report in April 2007, during the middle of the construction process, when there were only 73 million images. The full dataset was published in 2008. They began with all 75,846 non-abstract nouns in WordNet, and then for each of these nouns, they scraped 7 image search engines: Altavista, Ask.com, Flickr, Cydral, Google, Picsearch, and Webshots. After 8 months of scraping, they obtained 97,245,098 images. Since they did not have enough storage, they downsized the images to 32×32 as they were scraped. After gathering, they removed images with zero variance and intra-word duplicate images, resulting in the final dataset. Out of the 75,846 nouns, only 75,062 classes had any results, so the other nouns did not appear in the final dataset. The number of images per noun follows a Zipf-like distribution, with 1056 images per noun on average. To prevent a few nouns taking up too many images, they put an upper bound of at most 3000 images per noun. == Retirement == The 80 Million Tiny Images dataset was retired from use by its creators in 2020, after a paper by researchers Abeba Birhane and Vinay Prabhu found that some of the labeling of several publicly available image datasets, including 80 Million Tiny Images, contained racist and misogynistic slurs which were causing models trained on them to exhibit racial and sexual bias. The dataset also contained offensive images. Following the release of the paper, the dataset's creators removed the dataset from distribution, and requested that other researchers not use it for further research and to delete their copies of the dataset.
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Monica S. Lam

Monica Sin-Ling Lam is an American computer scientist. She is a professor in the Computer Science Department at Stanford University. == Education == Monica Lam received a B.Sc. from University of British Columbia in 1980 and a Ph.D. in computer science from Carnegie Mellon University in 1987. == Career == Lam joined the faculty of Computer Science at Stanford University in 1988. She has contributed to the research of a wide range of computer systems topics including compilers, program analysis, operating systems, security, computer architecture, and high-performance computing. More recently, she is working in natural language processing, and virtual assistants with an emphasis on privacy protection. She is the faculty director of the Open Virtual Assistant Lab, which organized the first workshop for the World Wide Voice Web. The lab developed the open-source Almond voice assistant, which is sponsored by the National Science Foundation. Almond received Popular Science's Best of What's New award in 2019. Previously, Lam led the SUIF (Stanford University Intermediate Format) Compiler project, which produced a widely used compiler infrastructure known for its locality optimizations and interprocedural parallelization. Many of the compiler techniques she developed have been adopted by industry. Her other research projects included the architecture and compiler for the CMU Warp machine, a systolic array of VLIW processors, and the Stanford DASH distributed shared memory machine. In 1998, she took a sabbatical leave from Stanford to help start Tensilica Inc., a company that specializes in configurable processor cores. In another research project, her program analysis group developed a collection of tools for improving software security and reliability. They developed the first scalable context-sensitive inclusion-based pointer analysis and a freely available tool called BDDBDDB, that allows programmers to express context-sensitive analyses simply by writing Datalog queries. Other tools developed include Griffin, static and dynamic analysis for finding security vulnerabilities in Web applications such as SQL injection, a static and dynamic program query language called QL, a static memory leak detector called Clouseau, a dynamic buffer overrun detector called CRED, and a dynamic error diagnosis tool called DIDUCE. In the Collective project, her research group and she developed the concept of a livePC: subscribers of the livePC will automatically run the latest of the published PC virtual images with each reboot. This approach allows computers to be managed scalably and securely. In 2005, the group started a company called MokaFive to transfer the technology to industry. She also directed the MobiSocial laboratory at Stanford, as part of the Programmable Open Mobile Internet 2020 initiative. Lam is also the cofounder of Omlet, which launched in 2014. Omlet is the first product from MobiSocial. Omlet is an open, decentralized social networking tool, based on an extensible chat platform. Lam chaired the ACM SIGPLAN Programming Languages Design and Implementation Conference in 2000, served on the Editorial Board of ACM Transactions on Computer Systems and numerous program committees for conferences on languages and compilers (PLDI, POPL), operating systems (SOSP), and computer architecture (ASPLOS, ISCA). == Awards and honors == National Academy of Engineering member, 2019 University of British Columbia Computer Science 50th Anniversary Research Award, 2018 Fellow of the ACM, 2007 ACM Programming Language Design and Implementation Best Paper Award in 2004 ACM SIGSOFT Distinguished Paper Award in 2002 ACM Most Influential Programming Language Design and Implementation Paper Award in 2001 NSF Young Investigator award in 1992 Two of her papers were recognized in "20 Years of PLDI--a Selection (1979-1999)" One of her papers was recognized in the "25 Years of the International Symposia on Computer Architecture", 1988. == Selected works == Compilers: Principles, Techniques and Tools (2d Ed) (2006) (the "Dragon Book") by Alfred V. Aho, Monica S. Lam, Ravi Sethi, and Jeffrey D. Ullman (ISBN 0-321-48681-1) A Systolic Array Optimizing Compiler (1989) (ISBN 0-89838-300-5) Monica Lam, Dissertation
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Collocation

In corpus linguistics, a collocation is a series of words or terms that co-occur more often than would be expected by chance. In phraseology, a collocation is a type of compositional phraseme, meaning that it can be understood from the words that make it up. This contrasts with an idiom, where the meaning of the whole cannot be inferred from its parts, and may be completely unrelated. There are about seven main types of collocations: adjective + noun, noun + noun (such as collective nouns), noun + verb, verb + noun, adverb + adjective, verbs + prepositional phrase (phrasal verbs), and verb + adverb. Collocation extraction is a computational technique that finds collocations in a document or corpus, using various computational linguistics elements resembling data mining. == Expanded definition == Collocations are partly or fully fixed expressions that become established through repeated context-dependent use. Such terms as crystal clear, middle management, nuclear family, and cosmetic surgery are examples of collocated pairs of words. Collocations can be in a syntactic relation (such as verb–object: make and decision), lexical relation (such as antonymy), or they can be in no linguistically defined relation. Knowledge of collocations is vital for the competent use of a language: a grammatically correct sentence will stand out as awkward if collocational preferences are violated. This makes collocation a common focus for language teaching. Corpus linguists specify a key word in context (KWIC) and identify the words immediately surrounding them, to illustrate the way words are used in practice. The processing of collocations involves a number of parameters, the most important of which is the measure of association, which evaluates whether the co-occurrence is purely by chance or statistically significant. Due to the non-random nature of language, most collocations are classed as significant, and the association scores are simply used to rank the results. Commonly used measures of association include mutual information, t scores, and log-likelihood. Rather than select a single definition, Gledhill proposes that collocation involves at least three different perspectives: co-occurrence, a statistical view, which sees collocation as the recurrent appearance in a text of a node and its collocates; construction, which sees collocation either as a correlation between a lexeme and a lexical-grammatical pattern, or as a relation between a base and its collocative partners; and expression, a pragmatic view of collocation as a conventional unit of expression, regardless of form. These different perspectives contrast with the usual way of presenting collocation in phraseological studies. Traditionally speaking, collocation is explained in terms of all three perspectives at once, in a continuum: == In dictionaries == In 1933, Harold Palmer's Second Interim Report on English Collocations highlighted the importance of collocation as a key to producing natural-sounding language, for anyone learning a foreign language. Thus from the 1940s onwards, information about recurrent word combinations became a standard feature of monolingual learner's dictionaries. As these dictionaries became "less word-centred and more phrase-centred", more attention was paid to collocation. This trend was supported, from the beginning of the 21st century, by the availability of large text corpora and intelligent corpus-querying software, making it possible to provide a more systematic account of collocation in dictionaries. Using these tools, dictionaries such as the Macmillan English Dictionary and the Longman Dictionary of Contemporary English included boxes or panels with lists of frequent collocations. There are also a number of specialized dictionaries devoted to describing the frequent collocations in a language. These include (for Spanish) Redes: Diccionario combinatorio del español contemporaneo (2004), (for French) Le Robert: Dictionnaire des combinaisons de mots (2007), and (for English) the LTP Dictionary of Selected Collocations (1997) and the Macmillan Collocations Dictionary (2010). == Statistically significant collocation == Student's t-test can be used to determine whether the occurrence of a collocation in a corpus is statistically significant. For a bigram w 1 w 2 {\displaystyle w_{1}w_{2}} , let P ( w 1 ) = # w 1 N {\displaystyle P(w_{1})={\frac {\#w_{1}}{N}}} be the unconditional probability of occurrence of w 1 {\displaystyle w_{1}} in a corpus with size N {\displaystyle N} , and let P ( w 2 ) = # w 2 N {\displaystyle P(w_{2})={\frac {\#w_{2}}{N}}} be the unconditional probability of occurrence of w 2 {\displaystyle w_{2}} in the corpus. The t-score for the bigram w 1 w 2 {\displaystyle w_{1}w_{2}} is calculated as: where x ¯ = # w i w j N {\displaystyle {\bar {x}}={\frac {\#w_{i}w_{j}}{N}}} is the sample mean of the occurrence of w 1 w 2 {\displaystyle w_{1}w_{2}} , # w 1 w 2 {\displaystyle \#w_{1}w_{2}} is the number of occurrences of w 1 w 2 {\displaystyle w_{1}w_{2}} , μ = P ( w i ) P ( w j ) {\displaystyle \mu =P(w_{i})P(w_{j})} is the probability of w 1 w 2 {\displaystyle w_{1}w_{2}} under the null-hypothesis that w 1 {\displaystyle w_{1}} and w 2 {\displaystyle w_{2}} appear independently in the text, and s 2 = x ¯ ( 1 − x ¯ ) ≈ x ¯ {\displaystyle s^{2}={\bar {x}}(1-{\bar {x}})\approx {\bar {x}}} is the sample variance. With a large N {\displaystyle N} , the t-test is equivalent to a Z-test.
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Top 10 AI Presentation Makers Compared (2026)

Trying to pick the best AI presentation maker? An AI presentation maker is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI presentation maker 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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SurveyLab

SurveyLab is an online system designed for creating and deploying surveys, questionnaires, web forms, tests, and quizzes. The platform functions as a web application, without the need for additional software installation. Founded in 2006, by the Polish company 7 Points, SurveyLab is used by businesses and professional users for market research, human resources assessments, customer feedback, and academic research. == History == SurveyLab was launched in 2006 under the name MySurveyLab, developed by the Warsaw-based company 7 Points. Early media coverage described the system as supporting online survey creation, real-time reporting, group collaboration and question logic, and noted that the platform was opened to custom feature development. MySurveyLab featured multi-user accounts, SSL-secured surveys, and support for right-to-left languages. Further 2010s updates improved reporting capabilities, expanded question types, and integration options. In 2020, the platform was rebranded to SurveyLab. By the early 2020s, the software supported integrations with external tools including Zapier, and offered additional analytics features. In 2025, 7 Points reported that SurveyLab had over 85,000 registered users and had processed over 7 million surveys. == Functionalities == SurveyLab is a web-based platform used for creating online surveys, questionnaires, and forms. Independent reviewers and software directories describe it as a tool used for market research, customer feedback management, and human resources-related assessments, including employee feedback surveys. According to the creators at 7 Points, SurveyLab supports customer satisfaction measurement, survey analysis, and 360-degree feedback evaluations. The platform allows users to create surveys with no limits on the number of questions or responses. Independent reviews describe SurveyLab as offering multiple-choice, matrix, rating-scale, and open-ended questions. According to 7 Points, the platform manages market-research workflows, including Net Promoter Score, Customer Satisfaction, and Customer Effort Score questions. The tool can also re-use previous answers in later questions, and create A/B survey variants. SurveyLab can integrate with external services and applications through APIs and third-party connectors. According to its developers, the platform can connect with customer service tools, as well as CRM, marketing automation, e-commerce, and data-storage tools An industry review cited workflow integrations with CINT, Slack, Salesforce, and Zendesk Other integrations included Aquera (SSO), Sona Systems (internet research), and Synerise (customer data management). == Data collection and aggregation == Independent descriptions note that SurveyLab can combine results from emails, SMS, website widgets and pop-ups, QR codes, and social media. Its surveys are also accessible through mobile apps on iOS and Android, used for online and offline data collection in the field. Developers state that the tool supports exporting data as CSV, Excel, and SPSS, with independent reviews also mentioning PDF and PowerPoint. SurveyLab can automate response collection through a multi-channel survey distribution and reporting. It includes data trends, offline responses, and reminders to non-respondents. According to its documentation, newer versions include AI-based tools that detect and analyze sentiment, and a survey builder generating questionnaires based on user prompts. === Data security and compliance === According to 7 Points, SurveyLab provides password-protected surveys, token-based access, IP-address filtering, and two-factor authentication for user accounts, and it complies with the General Data Protection Regulation. == Awards and accolades == In 2017, SurveyLab was listed in Capterra’s Top 20 Survey Software ranking, among 20 highest-scoring survey tools based on market presence and user base. In 2018, a software review platform FinancesOnline awarded SurveyLab the Rising Star Award and the Great User Experience Award, distinctions given to products that demonstrate positive user satisfaction and strong usability characteristics.
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Human-readable medium and data

In computing, a human-readable medium or human-readable format is any encoding of data or information that can be naturally read by humans, resulting in human-readable data. It is often encoded as ASCII or Unicode text, rather than as binary data. In most contexts, the alternative to a human-readable representation is a machine-readable format or medium of data primarily designed for reading by electronic, mechanical or optical devices, or computers. For example, Universal Product Code (UPC) barcodes are very difficult to read for humans, but very effective and reliable with the proper equipment, whereas the strings of numerals that commonly accompany the label are the human-readable form of the barcode information. Since any type of data encoding can be parsed by a suitably programmed computer, the decision to use binary encoding rather than text encoding is usually made to conserve storage space. Encoding data in a binary format typically requires fewer bytes of storage and increases efficiency of access (input and output) by eliminating format parsing or conversion. With the advent of standardized, highly structured markup languages, such as Extensible Markup Language (XML), the decreasing costs of data storage, and faster and cheaper data communication networks, compromises between human-readability and machine-readability are now more common-place than they were in the past. This has led to humane markup languages and modern configuration file formats that are far easier for humans to read. In addition, these structured representations can be compressed very effectively for transmission or storage. Human-readable protocols greatly reduce the cost of debugging. Various organizations have standardized the definition of human-readable and machine-readable data and how they are applied in their respective fields of application, e.g., the Universal Postal Union. Often the term human-readable is also used to describe shorter names or strings, that are easier to comprehend or to remember than long, complex syntax notations, such as some Uniform Resource Locator strings. Occasionally "human-readable" is used to describe ways of encoding an arbitrary integer into a long series of English words. Compared to decimal or other compact binary-to-text encoding systems, English words are easier for humans to read, remember, and type in.
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Semiautomaton

In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set Q of states, a set Σ called the input alphabet, and a function T: Q × Σ → Q called the transition function. Associated with any semiautomaton is a monoid called the characteristic monoid, input monoid, transition monoid or transition system of the semiautomaton, which acts on the set of states Q. This may be viewed either as an action of the free monoid of strings in the input alphabet Σ, or as the induced transformation semigroup of Q. In older books like Clifford and Preston (1967) semigroup actions are called "operands". In category theory, semiautomata essentially are functors. == Transformation semigroups and monoid acts == A transformation semigroup or transformation monoid is a pair ( M , Q ) {\displaystyle (M,Q)} consisting of a set Q (often called the "set of states") and a semigroup or monoid M of functions, or "transformations", mapping Q to itself. They are functions in the sense that every element m of M is a map m : Q → Q {\displaystyle m\colon Q\to Q} . If s and t are two functions of the transformation semigroup, their semigroup product is defined as their function composition ( s t ) ( q ) = ( s ∘ t ) ( q ) = s ( t ( q ) ) {\displaystyle (st)(q)=(s\circ t)(q)=s(t(q))} . Some authors regard "semigroup" and "monoid" as synonyms. Here a semigroup need not have an identity element; a monoid is a semigroup with an identity element (also called "unit"). Since the notion of functions acting on a set always includes the notion of an identity function, which when applied to the set does nothing, a transformation semigroup can be made into a monoid by adding the identity function. === M-acts === Let M be a monoid and Q be a non-empty set. If there exists a multiplicative operation μ : Q × M → Q {\displaystyle \mu \colon Q\times M\to Q} ( q , m ) ↦ q m = μ ( q , m ) {\displaystyle (q,m)\mapsto qm=\mu (q,m)} which satisfies the properties q 1 = q {\displaystyle q1=q} for 1 the unit of the monoid, and q ( s t ) = ( q s ) t {\displaystyle q(st)=(qs)t} for all q ∈ Q {\displaystyle q\in Q} and s , t ∈ M {\displaystyle s,t\in M} , then the triple ( Q , M , μ ) {\displaystyle (Q,M,\mu )} is called a right M-act or simply a right act. In long-hand, μ {\displaystyle \mu } is the right multiplication of elements of Q by elements of M. The right act is often written as Q M {\displaystyle Q_{M}} . A left act is defined similarly, with μ : M × Q → Q {\displaystyle \mu \colon M\times Q\to Q} ( m , q ) ↦ m q = μ ( m , q ) {\displaystyle (m,q)\mapsto mq=\mu (m,q)} and is often denoted as M Q {\displaystyle \,_{M}Q} . An M-act is closely related to a transformation monoid. However the elements of M need not be functions per se, they are just elements of some monoid. Therefore, one must demand that the action of μ {\displaystyle \mu } be consistent with multiplication in the monoid (i.e. μ ( q , s t ) = μ ( μ ( q , s ) , t ) {\displaystyle \mu (q,st)=\mu (\mu (q,s),t)} ), as, in general, this might not hold for some arbitrary μ {\displaystyle \mu } , in the way that it does for function composition. Once one makes this demand, it is completely safe to drop all parenthesis, as the monoid product and the action of the monoid on the set are completely associative. In particular, this allows elements of the monoid to be represented as strings of letters, in the computer-science sense of the word "string". This abstraction then allows one to talk about string operations in general, and eventually leads to the concept of formal languages as being composed of strings of letters. Another difference between an M-act and a transformation monoid is that for an M-act Q, two distinct elements of the monoid may determine the same transformation of Q. If we demand that this does not happen, then an M-act is essentially the same as a transformation monoid. === M-homomorphism === For two M-acts Q M {\displaystyle Q_{M}} and B M {\displaystyle B_{M}} sharing the same monoid M {\displaystyle M} , an M-homomorphism f : Q M → B M {\displaystyle f\colon Q_{M}\to B_{M}} is a map f : Q → B {\displaystyle f\colon Q\to B} such that f ( q m ) = f ( q ) m {\displaystyle f(qm)=f(q)m} for all q ∈ Q M {\displaystyle q\in Q_{M}} and m ∈ M {\displaystyle m\in M} . The set of all M-homomorphisms is commonly written as H o m ( Q M , B M ) {\displaystyle \mathrm {Hom} (Q_{M},B_{M})} or H o m M ( Q , B ) {\displaystyle \mathrm {Hom} _{M}(Q,B)} . The M-acts and M-homomorphisms together form a category called M-Act. == Semiautomata == A semiautomaton is a triple ( Q , Σ , T ) {\displaystyle (Q,\Sigma ,T)} where Σ {\displaystyle \Sigma } is a non-empty set, called the input alphabet, Q is a non-empty set, called the set of states, and T is the transition function T : Q × Σ → Q . {\displaystyle T\colon Q\times \Sigma \to Q.} When the set of states Q is a finite set—it need not be—, a semiautomaton may be thought of as a deterministic finite automaton ( Q , Σ , T , q 0 , A ) {\displaystyle (Q,\Sigma ,T,q_{0},A)} , but without the initial state q 0 {\displaystyle q_{0}} or set of accept states A. Alternately, it is a finite-state machine that has no output, and only an input. Any semiautomaton induces an act of a monoid in the following way. Let Σ ∗ {\displaystyle \Sigma ^{}} be the free monoid generated by the alphabet Σ {\displaystyle \Sigma } (so that the superscript is understood to be the Kleene star); it is the set of all finite-length strings composed of the letters in Σ {\displaystyle \Sigma } . For every word w in Σ ∗ {\displaystyle \Sigma ^{}} , let T w : Q → Q {\displaystyle T_{w}\colon Q\to Q} be the function, defined recursively, as follows, for all q in Q: If w = ε {\displaystyle w=\varepsilon } , then T ε ( q ) = q {\displaystyle T_{\varepsilon }(q)=q} , so that the empty word ε {\displaystyle \varepsilon } does not change the state. If w = σ {\displaystyle w=\sigma } is a letter in Σ {\displaystyle \Sigma } , then T σ ( q ) = T ( q , σ ) {\displaystyle T_{\sigma }(q)=T(q,\sigma )} . If w = σ v {\displaystyle w=\sigma v} for σ ∈ Σ {\displaystyle \sigma \in \Sigma } and v ∈ Σ ∗ {\displaystyle v\in \Sigma ^{}} , then T w ( q ) = T v ( T σ ( q ) ) {\displaystyle T_{w}(q)=T_{v}(T_{\sigma }(q))} . Let M ( Q , Σ , T ) {\displaystyle M(Q,\Sigma ,T)} be the set M ( Q , Σ , T ) = { T w | w ∈ Σ ∗ } . {\displaystyle M(Q,\Sigma ,T)=\{T_{w}\vert w\in \Sigma ^{}\}.} The set M ( Q , Σ , T ) {\displaystyle M(Q,\Sigma ,T)} is closed under function composition; that is, for all v , w ∈ Σ ∗ {\displaystyle v,w\in \Sigma ^{}} , one has T w ∘ T v = T v w {\displaystyle T_{w}\circ T_{v}=T_{vw}} . It also contains T ε {\displaystyle T_{\varepsilon }} , which is the identity function on Q. Since function composition is associative, the set M ( Q , Σ , T ) {\displaystyle M(Q,\Sigma ,T)} is a monoid: it is called the input monoid, characteristic monoid, characteristic semigroup or transition monoid of the semiautomaton ( Q , Σ , T ) {\displaystyle (Q,\Sigma ,T)} . == Properties == If the set of states Q is finite, then the transition functions are commonly represented as state transition tables. The structure of all possible transitions driven by strings in the free monoid has a graphical depiction as a de Bruijn graph. The set of states Q need not be finite, or even countable. As an example, semiautomata underpin the concept of quantum finite automata. There, the set of states Q are given by the complex projective space C P n {\displaystyle \mathbb {C} P^{n}} , and individual states are referred to as n-state qubits. State transitions are given by unitary n×n matrices. The input alphabet Σ {\displaystyle \Sigma } remains finite, and other typical concerns of automata theory remain in play. Thus, the quantum semiautomaton may be simply defined as the triple ( C P n , Σ , { U σ 1 , U σ 2 , … , U σ p } ) {\displaystyle (\mathbb {C} P^{n},\Sigma ,\{U_{\sigma _{1}},U_{\sigma _{2}},\dotsc ,U_{\sigma _{p}}\})} when the alphabet Σ {\displaystyle \Sigma } has p letters, so that there is one unitary matrix U σ {\displaystyle U_{\sigma }} for each letter σ ∈ Σ {\displaystyle \sigma \in \Sigma } . Stated in this way, the quantum semiautomaton has many geometrical generalizations. Thus, for example, one may take a Riemannian symmetric space in place of C P n {\displaystyle \mathbb {C} P^{n}} , and selections from its group of isometries as transition functions. The syntactic monoid of a regular language is isomorphic to the transition monoid of the minimal automaton accepting the language. == Literature == A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups. American Mathematical Society, volume 2 (1967), ISBN 978-0-8218-0272-4. F. Gecseg and I. Peak, Algebraic Theory of Automata (1972), Akademiai Kiado, Budapest. W. M. L. Holcombe, Algebraic Automata Theory (1982), Cambridge University Press J. M. Howie, Automata and Languages, (1991), Cla
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