AI Chatbot Interface

AI Chatbot Interface — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

  • Distribution management system

    Distribution management system

    A distribution management system (DMS) is a collection of applications designed to monitor and control the electric power distribution networks efficiently and reliably. It acts as a decision support system to assist the control room and field operating personnel with the monitoring and control of the electric distribution system. Improving the reliability and quality of service in terms of reducing power outages, minimizing outage time, maintaining acceptable frequency and voltage levels are the key deliverables of a DMS. Given the complexity of distribution grids, such systems may involve communication and coordination across multiple components. For example, the control of active loads may require a complex chain of communication through different components as described in US patent 11747849B2 In recent years, utilization of electrical energy increased exponentially and customer requirement and quality definitions of power were changed enormously. As electric energy became an essential part of daily life, its optimal usage and reliability became important. Real-time network view and dynamic decisions have become instrumental for optimizing resources and managing demands, leading to the need for distribution management systems in large-scale electrical networks. == Overview == Most distribution utilities have been comprehensively using IT solutions through their Outage Management System (OMS) that makes use of other systems like Customer Information System (CIS), Geographical Information System (GIS) and Interactive Voice Response System (IVRS). An outage management system has a network component/connectivity model of the distribution system. By combining the locations of outage calls from customers with knowledge of the locations of the protection devices (such as circuit breakers) on the network, a rule engine is used to predict the locations of outages. Based on this, restoration activities are charted out and the crew is dispatched for the same. In parallel with this, distribution utilities began to roll out Supervisory Control and Data Acquisition (SCADA) systems, initially only at their higher voltage substations. Over time, use of SCADA has progressively extended downwards to sites at lower voltage levels. DMSs access real-time data and provide all information on a single console at the control centre in an integrated manner. Their development varied across different geographic territories. In the US, for example, DMSs typically grew by taking Outage Management Systems to the next level, automating the complete sequences and providing an end to end, integrated view of the entire distribution spectrum. In the UK, by contrast, the much denser and more meshed network topologies, combined with stronger Health & Safety regulation, had led to early centralisation of high-voltage switching operations, initially using paper records and schematic diagrams printed onto large wallboards which were 'dressed' with magnetic symbols to show the current running states. There, DMSs grew initially from SCADA systems as these were expanded to allow these centralised control and safety management procedures to be managed electronically. These DMSs required even more detailed component/connectivity models and schematics than those needed by early OMSs as every possible isolation and earthing point on the networks had to be included. In territories such as the UK, therefore, the network component/connectivity models were usually developed in the DMS first, whereas in the USA these were generally built in the GIS. The typical data flow in a DMS has the SCADA system, the Information Storage & Retrieval (ISR) system, Communication (COM) Servers, Front-End Processors (FEPs) & Field Remote Terminal Units (FRTUs). == Why DMS? == Reduce the duration of outages Improve the speed and accuracy of outage predictions. Reduce crew patrol and drive times through improved outage locating. Improve the operational efficiency Determine the crew resources necessary to achieve restoration objectives. Effectively utilize resources between operating regions. Determine when best to schedule mutual aid crews. Increased customer satisfaction A DMS incorporates IVR and other mobile technologies, through which there is an improved outage communications for customer calls. Provide customers with more accurate estimated restoration times. Improve service reliability by tracking all customers affected by an outage, determining electrical configurations of every device on every feeder, and compiling details about each restoration process. == DMS Functions == In order to support proper decision making and O&M activities, DMS solutions should support the following functions: Network visualization & support tools Applications for Analytical & Remedial Action Utility Planning Tools System Protection Schemes The various sub functions of the same, carried out by the DMS are listed below:- === Network Connectivity Analysis (NCA) === Distribution network usually covers over a large area and catering power to different customers at different voltage levels. So locating required sources and loads on a larger GIS/Operator interface is often very difficult. Panning & zooming provided with normal SCADA system GUI does not cover the exact operational requirement. Network connectivity analysis is an operator specific functionality which helps the operator to identify or locate the preferred network or component very easily. NCA does the required analyses and provides display of the feed point of various network loads. Based on the status of all the switching devices such as circuit breaker (CB), Ring Main Unit (RMU) and/or isolators that affect the topology of the network modeled, the prevailing network topology is determined. The NCA further assists the operator to know operating state of the distribution network indicating radial mode, loops and parallels in the network. === Switching Schedule & Safety Management === In territories such as the UK a core function of a DMS has always been to support safe switching and work on the networks. Control engineers prepare switching schedules to isolate and make safe a section of network before work is carried out, and the DMS validates these schedules using its network model. Switching schedules can combine telecontrolled and manual (on-site) switching operations. When the required section has been made safe, the DMS allows a Permit To Work (PTW) document to be issued. After its cancellation when the work has been finished, the switching schedule then facilitates restoration of the normal running arrangements. Switching components can also be tagged to reflect any Operational Restrictions that are in force. The network component/connectivity model, and associated diagrams, must always be kept absolutely up to date. The switching schedule facility therefore also allows 'patches' to the network model to be applied to the live version at the appropriate stage(s) of the jobs. The term 'patch' is derived from the method previously used to maintain the wallboard diagrams. === State Estimation (SE) === The state estimator is an integral part of the overall monitoring and control systems for transmission networks. It is mainly aimed at providing a reliable estimate of the system voltages. This information from the state estimator flows to control centers and database servers across the network. The variables of interest are indicative of parameters like margins to operating limits, health of equipment and required operator action. State estimators allow the calculation of these variables of interest with high confidence despite the facts that the measurements may be corrupted by noise, or could be missing or inaccurate. Even though we may not be able to directly observe the state, it can be inferred from a scan of measurements which are assumed to be synchronized. The algorithms need to allow for the fact that presence of noise might skew the measurements. In a typical power system, the State is quasi-static. The time constants are sufficiently fast so that system dynamics decay away quickly (with respect to measurement frequency). The system appears to be progressing through a sequence of static states that are driven by various parameters like changes in load profile. The inputs of the state estimator can be given to various applications like Load Flow Analysis, Contingency Analysis, and other applications. === Load Flow Applications (LFA) === Load flow study is an important tool involving numerical analysis applied to a power system. The load flow study usually uses simplified notations like a single-line diagram and focuses on various forms of AC power rather than voltage and current. It analyzes the power systems in normal steady-state operation. The goal of a power flow study is to obtain complete voltage angle and magnitude information for each bus in a power system for specified load and generator real power and voltage conditions. Once this

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  • Dan Klein

    Dan Klein

    Daniel Klein (born c. 1976) is an American computer scientist and professor of computer science at the University of California, Berkeley. His research focuses on natural language processing and artificial intelligence. He was educated at Mt. Lebanon High School in Mt. Lebanon Township, Pennsylvania and earned a B.A. in mathematics, computer science, and linguistics from Cornell University (1998), a MSt in linguistics by Oxford University (1999) and a Ph.D. from Stanford University (2004), under Christopher D. Manning. He attended Oxford on a Marshall Scholarship. In addition to the Marshall scholarship, he has been awarded the ACM's Grace Murray Hopper Award, the Sloan Research Fellowship, the NSF CAREER Award, and the Microsoft New Faculty Fellowship.

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  • AI Resume Builders: Free vs Paid (2026)

    AI Resume Builders: Free vs Paid (2026)

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

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  • Bixby (software)

    Bixby (software)

    Bixby ( ) is a virtual assistant developed by Samsung Electronics that runs on various Samsung-branded appliances, primarily mobile devices but also some refrigerators televisions and PCs. It uses voice commands and a natural-language user interface to answer questions and perform tasks, while adapting to the users' preferences and behavior. Samsung first launched Bixby in 2017. Along with Bixby voice assistant, its other main component currently is Bixby Vision, a contextual and visual search augmented reality camera app. Formerly, the Bixby suite consisted of a number of other tools, but these have since been renamed, such as Bixby Routines (now Modes and Routines). == History == On 20 March 2017, Samsung announced the voice-powered digital assistant named "Bixby" as a replacement of the S Voice assistant. It was introduced alongside the Galaxy S8 and S8+ and the Galaxy Tab A (2017) during the Galaxy Unpacked 2017 event. Although released for these devices, it could also be sideloaded on older Galaxy devices running Android Nougat. Before the phone's release, the Bixby Button was reprogrammable and could be set to open other applications or assistants, such as Google Assistant. However, near the phone's release, this ability was removed with a firmware update. Remapping remained possible through third-party apps. Bixby was launched in Korean on 1 May 2017 (KST). Bixby Voice was intended to be made available in the US later that spring. However, Samsung postponed the release, as Bixby had issues understanding English. The English version was finally rolled out in July 2017, followed by a Chinese language version later that year. In October 2017, Samsung announced the release of Bixby 2.0 during its annual developer conference in San Francisco. The new version was rolled out across the company's line of connected products, including smartphones, TVs, and refrigerators. Third parties were allowed to develop applications for Bixby using the Samsung Developer Kit. In August 2018, Samsung announced the Bixby-integrated Galaxy Home smart speaker. In 2019, UX developers at Samsung stated that they intended to use AR Emoji avatars as a personified Bixby assistant. At SDC19, Samsung displayed the Galaxy Home Mini speaker, which also supported Bixby. Bixby 3.0 was released with One UI 3 at the start of 2021. With version 3.0, Home and Reminders features were separated from Bixby. In June 2021, screenshots surfaced for what some thought as a replacement for Bixby. The three-dimensional virtual assistant, Sam, was popular on social media, though it was not intended as a replacement for Bixby. Bixby launched for Microsoft Windows in October 2021, with distribution through the Microsoft Store. This version of Bixby was optimized for Samsung's Galaxy Book computers. Samsung launched an AI Bixby custom voice creator in 2023, allowing users to record their own voice commands. Most recently, in July 2024, Samsung confirmed that it plans to launch an upgraded version of Bixby later that year. This new Bixby would be powered by Samsung's proprietary large language model (LLM) technology, promising a significant boost to Bixby's capabilities with the help of generative AI. In January 2025, with the announcement of Galaxy S25 and the One UI 7 update, Bixby was no longer the default voice assistant, having been replaced by Google Gemini. Despite this, Bixby still continued to be developed and expanded by Samsung and was revamped at the same time with new AI capabilities. Samsung brought the "smarter" Bixby to Samsung televisions, allowing users to speak to their TV sets and control their homes with it. A visual refresh was planned for One UI 8.5. == Functionality == Bixby is a voice assistant developed by Samsung that provides device control, information retrieval, and task automation using voice input and artificial intelligence. It can answer contextual queries, adjust system settings, perform searches, and manage reminders or schedules. The service also personalizes responses by recognizing individual user voices. Bixby itself was also formerly called Bixby Voice to differentiate from other Bixby tools in the suite. === Bixby Vision === Bixby Vision is a visual recognition feature that analyzes images captured through the device camera and provides context-specific information or actions. It combines on-device processing with cloud-based AI resources to identify objects, detect text, and interpret scenes within supported applications. It comes pre-installed on Samsung Galaxy phones. It is considered to be the imaging component of Bixby. ==== Translate ==== Detects foreign text in the camera view and provides real-time translation by overlaying translated text on the preview. ==== Text ==== Uses optical character recognition(OCR) to extract printed or handwritten text for copying, searching, or sharing. ==== Discover ==== Identifies consumer products, fashion items, or furniture and retrieves visually similar items or related online information. ==== Wine ==== Recognizes wine labels and provides information such as variety, region of origin, average price, and reviews. ==== Scene Describer ==== Generates written and spoken descriptions of captured scenes, supporting accessibility for users with visual impairments. ==== Object Identifier ==== Identifies plants, animals, food items, or landmarks and displays corresponding names or classification details. ==== Text Reader ==== Converts detected text into spoken audio using text-to-speech functionality. ==== Color Detector ==== Identifies and names colors within the frame, displaying or reading the recognized color aloud. === Former Bixby tools === Bixby Home was a vertically scrolling home screen displaying cards of information such as weather, fitness activity, and smart home controls. It was renamed Samsung Daily with the release of One UI 2.1 in 2020, then replaced by Samsung Free in One UI 3.0. Samsung Free was eventually discontinued in some markets. Its successor, Samsung News, now functions as a news aggregation service with optional home-screen integration similar to Bixby Home. Bixby Routines was an automation feature that allowed users to create custom rules based on triggers such as time, location, or device conditions. Beginning with One UI 5.0, it was renamed Modes and Routines. Bixby Text Call, introduced in One UI 5.0 (2022) in select regions, enabled users to handle incoming calls via speech-to-text conversion and vice versa. It is now named simply Text Call and can be found in the Phone app settings. Bixby Touch allowed users to trigger context-aware actions by touching on-screen content. It analyzed images, text, and other visual elements displayed on the device and provided related options such as translation, image search, product lookup, or other content-based information. Several of its capabilities overlapped with, or were later superseded by, features offered through Bixby Vision. Other legacy components including Bixby Touch, Bixby Global Action, Bixby Dictation, and Bixby Wakeup, formed part of the early Bixby suite and have since been phased out, though exact discontinuation details vary by region. == Regions and languages == As of April 2018, Bixby is available in over 195 countries, but only in Korean, English (American), and Chinese (Mandarin). The limitation is that the models not intended for the Japanese market, like S10e, are not allowed to login to Bixby services from Japan; therefore Bixby becomes blocked. The choice of languages has since expanded: Samsung has deployed Bixby's voice command function in French, and on 20 February 2019 Samsung announced the addition of further languages: English (British), German, Italian and Spanish (Spain). On 22 February 2020, Samsung announced the addition of Portuguese (Brazil), for Galaxy S10 & Note10, in Beta, and later for other models. == Compatible devices == === Flagship series === Galaxy S series: All models since Galaxy S7 Galaxy Tab S: All models since Galaxy Tab S4 Galaxy Note: All models since Galaxy Note FE and Galaxy Note 8 Galaxy Z series: All models === Other series === Galaxy A Galaxy A6/A6+ (Bixby Home, Reminder and Vision) Galaxy A7 (2017) (available to users in South Korea only; Bixby Home and Reminder only) Galaxy A7 (2018) (Bixby Home, Reminder and Vision only) Galaxy A8 (2018) (including A8 Star; Bixby Home, Reminder and Vision only; S Voice used instead) Galaxy A8s (Bixby Home, Reminder and Vision only) Galaxy A9 (2018)/A9s/A9 Star Pro (including A9 Star and A9 Star Lite; Bixby Home, Reminder and Vision only; S Voice used instead) Galaxy A9 Pro (2019) (Bixby Home, Reminder and Vision only) Galaxy A20 (Bixby Home and Service) Galaxy A21s Galaxy A30s (Bixby Home, Vision, Reminder and Routines) Galaxy A40 (Bixby Home and Reminder) Galaxy A41 (Bixby Home, Vision, Routines and Reminder) Galaxy A50 (Bixby Home, Voice, Vision, Reminder and Routines) Galaxy A50s (Bixby Home, Voice, Vision, Reminder and Routines) G

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  • Kubity

    Kubity

    Kubity is a cloud-based 3D communication tool that works on desktop computers, the web, smartphones, tablets, augmented reality gear, and virtual reality glasses. Kubity is powered by several proprietary 3D processing engines including "Paragone" and "Etna" that prepare the 3D file for transfer over mobile devices. Kubity has practical applications for architecture, interior design, engineering, product design, film, and video games among others. The majority of its users create 3D models using SketchUp or Autodesk Revit software. Kubity products include the Kubity web app and Kubity Go (a mobile application for iOS and Android). Kubity is compatible across many platforms, devices and operating systems including: iOS, Android, Firefox, Chrome, Windows, MacOS, and Linux. == History == Kubity was created by SPK Technology (ex Kubity S.A.S.), a Paris-based software company specializing in automatic 3D data optimization and visualization. Founded in 2012 by a group of software engineers and an urban projects developer, they united around a simple idea: create a way for anyone, anywhere to simply and intuitively explore 3D models on smartphones and computers. In order to bring architects, engineers and designers together with their clients around a 3D model, it was essential to develop an interactive platform that supported multiple desktop and mobile devices for instantaneous and fluid 3D navigation. With specifications in place, 15 engineers fused together several technologies: 3D design, data compression, decimation and rendering optimization, web and mobile transfer, and virtual reality headset integration. In January 2014, the first public Kubity prototype (1.0 Amethyst) was launched to a small group of beta testers with a plug-in that allowed users to import 3D models from SketchUp to their browser. A global release was announced in April 2014 at the SketchUp Basecamp in Vail, Colorado. In May 2015, Kubity launched a web application that worked using WebGL technology (2.0 Citrine). For the first time, users were able to drag and drop any SketchUp file in a web browser without having to install a plug-in. In December 2015, Kubity launched a mobile application on the App Store for iPhone, iPod, and iPad as well as on Google Play for Android devices (3.0 Druzy). In November 2016, Kubity launched support for Oculus Rift and HTC Vive (4.0 Emerald). Beginning in November 2017, Kubity launched a full suite rollout of mobile applications over six months that included Kubity AR for augmented reality, Kubity VR for virtual reality, and Kubity Mirror for remote presentations and screen mirroring (5.0 Feldspar). In September 2018, a one-click plugin for SketchUp and Revit (Kubity PRO), along with a mobile-first revamp of Kubity Go was launched, allowing PRO-to-Go device pairing for automatic mobile sync (6.0 Gypsum). In early 2019, the Kubity Go application was updated to include fully integrated AR, VR, and screen mirroring functionalities, killing off the dedicated companion apps Kubity AR, Kubity VR and Kubity Mirror in the process (7.0 Heliotrope). In January 2020, support for the Kubity PRO plugin for SketchUp and Revit was migrated to a SketchUp-only web app. == Technology == Kubity is powered by a proprietary 3D crystallization engine known as "Paragone"; a technology developed by SPK Technology. Paragone takes constrained information from a 3D file and runs it through the "BlockWave" algorithm (US Patent 10,482.629), also developed by SPK Technology. BlockWave is a multiphase optimization algorithm that combines 3D design, data compression, decimation and rendering optimization, web and mobile transfer, and mixed reality headset integration to create a crystallized universal format of the original file. One phase of the BlockWave algorithm is based on the quadric-based polygonal surface simplification algorithm, performed using predefined heuristics, and is associated with a plurality of simplified versions of the 3D model, each version being associated with a predefined level of detail adapted to the user specific end device. BlockWave extracts data content, geometry and textures, then sets quadrics for each top of the original 3D model, and identifies pairs of adjacent tops linked by vertices. The algorithm uses a local collapsing operator and a top-plan error metric to obtain a fixed number of faces or a maximum defined error; 3D meshing is simplified by replacing two points with one, then deleting the degrading faces and updating adjacent relations. Once decimation is completed, texture optimization is set using texture target parameters allowing maximized GPU memory to improve computing time. With texture encoding completed, the crystallized universal 3D file can now be easily opened on any user-specific end device and played across most digital devices with real-time rendering. == Features == === 3D Crystallization === A user converts (or crystallizes) a 3D file by exporting it with the Kubity web app. Crystallization adds features like AR/VR and cinematic fly-through tour as well as assigns the model a dedicated QR code. === Automatic Mobile Sync === When a 3D model is exported, it is automatically synced to Kubity Go on the user's mobile device. From there, it can be accessed, explored, and shared with others with or without an internet connection. === Security and Management === User models can be managed all in one place on Kubity Go or in a browser from their account. Models can be renamed, password-protected, shared, and played. === Augmented Reality === Developed using Apple ARKit and Google ARCore technology, Kubity Go's augmented reality feature maps the environment in a room detecting horizontal planes like tables and floors to track and place 3D objects. By blending digital objects and information with the environment, Kubity allows users to interact with 3D models in true augmented reality. Built-in communication features allows users to instantly share 3D models with anyone over text, email, social media, or direct device-to-device with a QR Code. Platform Support AR supports devices running iOS11 including: iPhone SE, iPhone 6s, iPhone 6s Plus, iPhone 7, iPhone 7 Plus, iPhone 8, iPhone X, all iPad Pro models, and iPad (2017). AR for Android requires Android 7.0 or later and access to the Google Play Store. === Virtual Reality === VR allows users to explore SketchUp models and Revit projects on-the-go right from a mobile device using Oculus Go, Google Cardboard, Samsung Gear VR, or the glasses-free Magic Window feature. Kubity's virtual reality feature is compatible with Oculus Go, Google Cardboard viewers and other cardboard compatible devices including clip-on style VR glasses like Homido Mini, as well as the mobile virtual reality headset, Samsung Gear VR. Samsung Gear VR supports: Galaxy S6, Galaxy S6 Edge, Galaxy S6 Edge+, Samsung Galaxy Note 5, Galaxy S7, Galaxy S7 Edge, Galaxy S8, Galaxy S8+, Samsung Galaxy Note Fan Edition, Samsung Galaxy Note 8, Samsung Galaxy A8/A8+ (2018), and Samsung Galaxy S9/Galaxy S9+. === Screen Mirroring === Screen mirroring allows a user to sync the sender device to a receiver on a webpage, then control from the sender device to give a remote presentation of the 3D model. Devices are easily synced by entering a six-digit number displayed on the receiving computer. == Platform support == On iOS, the Kubity application is compatible with devices running on the version 9.0 or higher. On Android, Kubity is compatible with devices running on the version 4.4 “Kit Kat” or higher. The web version of Kubity applications currently support web browsers compatible with WebGL2 : Mozilla Firefox and Google Chrome. AR is compatible with devices running iOS11 including: iPhone SE, iPhone 6s, iPhone 6s Plus, iPhone 7, iPhone 7 Plus, iPhone 8, iPhone X, all iPad Pro models, and iPad (2017), and Android devices. Requires Android 7.0 or later and access to the Google Play Store. VR is compatible with Google Cardboard viewers and other cardboard compatible devices including clip-on style VR glasses like Homido Mini, as well as the Samsung Gear VR and Oculus Go. Samsung Gear VR supports: Galaxy S6, Galaxy S6 Edge, Galaxy S6 Edge+, Samsung Galaxy Note 5, Galaxy S7, Galaxy S7 Edge, Galaxy S8, Galaxy S8+, Samsung Galaxy Note Fan Edition, Samsung Galaxy Note 8, Samsung Galaxy A8/A8+ (2018) and Samsung Galaxy S9/Galaxy S9+.

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  • Markov information source

    Markov information source

    In mathematics, a Markov information source, or simply, a Markov source, is an information source whose underlying dynamics are given by a stationary finite Markov chain. == Formal definition == An information source is a sequence of random variables ranging over a finite alphabet Γ {\displaystyle \Gamma } , having a stationary distribution. A Markov information source is then a (stationary) Markov chain M {\displaystyle M} , together with a function f : S → Γ {\displaystyle f:S\to \Gamma } that maps states S {\displaystyle S} in the Markov chain to letters in the alphabet Γ {\displaystyle \Gamma } . A unifilar Markov source is a Markov source for which the values f ( s k ) {\displaystyle f(s_{k})} are distinct whenever each of the states s k {\displaystyle s_{k}} are reachable, in one step, from a common prior state. Unifilar sources are notable in that many of their properties are far more easily analyzed, as compared to the general case. == Applications == Markov sources are commonly used in communication theory, as a model of a transmitter. Markov sources also occur in natural language processing, where they are used to represent hidden meaning in a text. Given the output of a Markov source, whose underlying Markov chain is unknown, the task of solving for the underlying chain is undertaken by the techniques of hidden Markov models, such as the Viterbi algorithm.

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  • Statistical machine translation

    Statistical machine translation

    Statistical machine translation (SMT) is a machine translation approach where translations are generated on the basis of statistical models whose parameters are derived from the analysis of bilingual text corpora. The statistical approach contrasts with the rule-based approaches to machine translation as well as with example-based machine translation, that superseded the previous rule-based approach that required explicit description of each and every linguistic rule, which was costly, and which often did not generalize to other languages. The first ideas of statistical machine translation were introduced by Warren Weaver in 1949, including the ideas of applying Claude Shannon's information theory. Statistical machine translation was re-introduced in the late 1980s and early 1990s by researchers at IBM's Thomas J. Watson Research Center. Before the introduction of neural machine translation, it was by far the most widely studied machine translation method. == Basis == The idea behind statistical machine translation comes from information theory. A document is translated according to the probability distribution p ( e | f ) {\displaystyle p(e|f)} that a string e {\displaystyle e} in the target language (for example, English) is the translation of a string f {\displaystyle f} in the source language (for example, French). The problem of modeling the probability distribution p ( e | f ) {\displaystyle p(e|f)} has been approached in a number of ways. One approach which lends itself well to computer implementation is to apply Bayes' theorem, that is p ( e | f ) ∝ p ( f | e ) p ( e ) {\displaystyle p(e|f)\propto p(f|e)p(e)} , where the translation model p ( f | e ) {\displaystyle p(f|e)} is the probability that the source string is the translation of the target string, and the language model p ( e ) {\displaystyle p(e)} is the probability of seeing that target language string. This decomposition is attractive as it splits the problem into two subproblems. Finding the best translation e ~ {\displaystyle {\tilde {e}}} is done by picking up the one that gives the highest probability: e ~ = a r g max e ∈ e ∗ p ( e | f ) = a r g max e ∈ e ∗ p ( f | e ) p ( e ) {\displaystyle {\tilde {e}}=arg\max _{e\in e^{}}p(e|f)=arg\max _{e\in e^{}}p(f|e)p(e)} . For a rigorous implementation of this one would have to perform an exhaustive search by going through all strings e ∗ {\displaystyle e^{}} in the native language. Performing the search efficiently is the work of a machine translation decoder that uses the foreign string, heuristics and other methods to limit the search space and at the same time keeping acceptable quality. This trade-off between quality and time usage can also be found in speech recognition. As the translation systems are not able to store all native strings and their translations, a document is typically translated sentence by sentence. Language models are typically approximated by smoothed n-gram models, and similar approaches have been applied to translation models, but this introduces additional complexity due to different sentence lengths and word orders in the languages. Statistical translation models were initially word based (Models 1-5 from IBM Hidden Markov model from Stephan Vogel and Model 6 from Franz-Joseph Och), but significant advances were made with the introduction of phrase based models. Later work incorporated syntax or quasi-syntactic structures. == Benefits == The most frequently cited benefits of statistical machine translation (SMT) over rule-based approach are: More efficient use of human and data resources There are many parallel corpora in machine-readable format and even more monolingual data. Generally, SMT systems are not tailored to any specific pair of languages. More fluent translations owing to use of a language model == Shortcomings == Corpus creation can be costly. Specific errors are hard to predict and fix. Results may have superficial fluency that masks translation problems. Statistical machine translation usually works less well for language pairs with significantly different word order. The benefits obtained for translation between Western European languages are not representative of results for other language pairs, owing to smaller training corpora and greater grammatical differences. == Word-based translation == In word-based translation, the fundamental unit of translation is a word in some natural language. Typically, the number of words in translated sentences are different, because of compound words, morphology and idioms. The ratio of the lengths of sequences of translated words is called fertility, which tells how many foreign words each native word produces. Necessarily it is assumed by information theory that each covers the same concept. In practice this is not really true. For example, the English word corner can be translated in Spanish by either rincón or esquina, depending on whether it is to mean its internal or external angle. Simple word-based translation cannot translate between languages with different fertility. Word-based translation systems can relatively simply be made to cope with high fertility, such that they could map a single word to multiple words, but not the other way about. For example, if we were translating from English to French, each word in English could produce any number of French words— sometimes none at all. But there is no way to group two English words producing a single French word. An example of a word-based translation system is the freely available GIZA++ package (GPLed), which includes the training program for IBM models and HMM model and Model 6. The word-based translation is not widely used today; phrase-based systems are more common. Most phrase-based systems are still using GIZA++ to align the corpus. The alignments are used to extract phrases or deduce syntax rules. And matching words in bi-text is still a problem actively discussed in the community. Because of the predominance of GIZA++, there are now several distributed implementations of it online. == Phrase-based translation == In phrase-based translation, the aim is to reduce the restrictions of word-based translation by translating whole sequences of words, where the lengths may differ. The sequences of words are called blocks or phrases. These are typically not linguistic phrases, but phrasemes that were found using statistical methods from corpora. It has been shown that restricting the phrases to linguistic phrases (syntactically motivated groups of words, see syntactic categories) decreased the quality of translation. The chosen phrases are further mapped one-to-one based on a phrase translation table, and may be reordered. This table could be learnt based on word-alignment, or directly from a parallel corpus. The second model is trained using the expectation maximization algorithm, similarly to the word-based IBM model. == Syntax-based translation == Syntax-based translation is based on the idea of translating syntactic units, rather than single words or strings of words (as in phrase-based MT), i.e. (partial) parse trees of sentences/utterances. Until the 1990s, with advent of strong stochastic parsers, the statistical counterpart of the old idea of syntax-based translation did not take off. Examples of this approach include DOP-based MT and later synchronous context-free grammars. == Hierarchical phrase-based translation == Hierarchical phrase-based translation combines the phrase-based and syntax-based approaches to translation. It uses synchronous context-free grammar rules, but the grammars can be constructed by an extension of methods for phrase-based translation without reference to linguistically motivated syntactic constituents. This idea was first introduced in Chiang's Hiero system (2005). == Language models == A language model is an essential component of any statistical machine translation system, which aids in making the translation as fluent as possible. It is a function that takes a translated sentence and returns the probability of it being said by a native speaker. A good language model will for example assign a higher probability to the sentence "the house is small" than to "small the is house". Other than word order, language models may also help with word choice: if a foreign word has multiple possible translations, these functions may give better probabilities for certain translations in specific contexts in the target language. == Systems implementing statistical machine translation == Google Translate (started transition to neural machine translation in 2016) Microsoft Translator (started transition to neural machine translation in 2016) Yandex.Translate (switched to hybrid approach incorporating neural machine translation in 2017) == Challenges with statistical machine translation == Problems with statistical machine translation include: === Sentence alignment === Single sentences in one language can be found translated into several sentences in the o

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  • AI Paragraph Rewriters Reviews: What Actually Works in 2026

    AI Paragraph Rewriters Reviews: What Actually Works in 2026

    Looking for the best AI paragraph rewriter? An AI paragraph rewriter 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 paragraph rewriter 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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  • Human–AI interaction

    Human–AI interaction

    Human–AI interaction is a developing field of research and a sub-field of human–computer interaction (HCI). HCI is a field of research that explores the interactions between humans and computer-based technology, focusing on design implementation, user experience, and psychological factors. With the proliferation of artificial intelligence (AI), there has developed a sub-section of HCI research dedicated specifically to artificial intelligence and how people interact with and are impacted by it. This is human–AI interaction, abbreviated either as HAX or HAII. == Introduction == Artificial intelligence (AI), in general, has fluid definitions and varied research applications, but in brief can be applied to mechanizing tasks that would require human intelligence to complete. AI are tools designed to replicate the human abilities of navigating uncertainty, active learning, and processing information in different contexts. Within the context of HCI and HAX research, artificial intelligence can be broken into two sub-fields, natural language processing (NLP) and computer vision (CV). AI technologies notably include machine-learning, deep-learning and neural networks, and large-language models (LLMs). As a new and rapidly developing technology, AI is changing how computers work and therefore changing how humans interact with computers. Unlike the traditional human-computer interaction, where a human directs a machine, human-AI interaction is characterized by a more collaborative relationship between the computer program (the AI) and the human user, as AI is perceived as an active agent rather than a tool. This changing dynamic creates new questions and necessitates new research methods that are not present in traditional HCI research. According to a scoping review on the state of the discipline, the HAX field comprises research on the "design, development, and evaluation of AI systems" and encompasses the themes of human-AI collaboration, human-AI competition, human-AI conflict, and human-AI symbiosis. == Design == Machine learning and artificial intelligence have been used for decades in targeted advertising and to recommend content in social media. Ethical Guidelines (Framework for ethical AI development) == User Experience (UX) == This section should handle research on how users interact with tools. What techniques do they use, do they develop habits, what types of programs and devices are they using to access these tools, what do they use these tools to do exactly. === Cognitive Frameworks in AI Tool Users === AI has been viewed with various expectations, attributions, and often misconceptions. Many people exclusively understand AI as the LLM chatbots they interact with, like ChatGPT or Claude, or other generative AI programs. [Insert section: discuss how people interact with these specific AI tools as a connection to the following paragraphs] Most fundamentally, humans have a mental model of understanding AI's reasoning and motivation for its decision recommendations, and building a holistic and precise mental model of AI helps people create prompts to receive more valuable responses from AI. However, these mental models are not whole because people can only gain more information about AI through their limited interaction with it; more interaction with AI builds a better mental model that a person may build to produce better prompt outcomes. Research on human-AI interaction has emphasized that users develop mental models of AI systems and revise those models through repeated use, feedback, and explanation, while design research has stressed the importance of communicating capabilities and limitations early and supporting trust calibration through explanation and correction. In a 2025 SSRN working paper, John DeVadoss proposed "Hypothetico-Deductive Interaction" (HDI), a framework that describes human-AI interaction as a mutual process of conjecture and refutation in which users test assumptions about an AI system's capabilities while the system infers and updates assumptions about user goals through its responses and clarifying questions. DeVadoss argued that this framing helps explain prompt iteration, weak capability awareness, and trust miscalibration, and suggested design responses such as clearer communication of uncertainty, easier correction, actionable explanations, and safer failure modes. == Research themes == === Human-AI collaboration === Human-AI collaboration occurs when the human and AI supervise the task on the same level and extent to achieve the same goal. Some collaboration occurs in the form of augmenting human capability. AI may help human ability in analysis and decision-making through providing and weighing a volume of information, and learning to defer to the human decision when it recognizes its unreliability. It is especially beneficial when the human can detect a task that AI can be trusted to make few errors so that there is not a lot of excessive checking process required on the human's end. Some findings show signs of human-AI augmentation, or human–AI symbiosis, in which AI enhances human ability in a way that co-working on a task with AI produces better outcomes than a human working alone. For example: the quality and speed of customer service tasks increase when a human agent collaborates with AI, training on specific models allows AI to improve diagnoses in clinical settings, and AI with human-intervention can improve creativity of artwork while fully AI-generated haikus were rated negatively. Human-AI synergy, a concept in which human-AI collaboration would produce more optimal outcomes than either human or AI working alone could explain why AI does not always help with performance. Some AI features and development may accelerate human-AI synergy, while others may stagnate it. For example, when AI updates for better performance, it sometimes worsens the team performance with human and AI by reducing the compatibility with the new model and the mental model a user has developed on the previous version. Research has found that AI often supports human capabilities in the form of human-AI augmentation and not human-AI synergy, potentially because people rely too much on AI and stop thinking on their own. Prompting people to actively engage in analysis and think when to follow AI recommendations reduces their over-reliance, especially for individuals with higher need for cognition. === Human-AI competition === Robots and computers have substituted routine tasks historically completed by humans, but agentic AI has made it possible to also replace cognitive tasks including taking phone calls for appointments and driving a car. At the point of 2016, research has estimated that 45% of paid activities could be replaced by AI by 2030. Perceived autonomy of robots is known to increase people's negative attitude toward them, and worry about the technology taking over leads people to reject it. There has been a consistent tendency of algorithm aversion in which people prefer human advice over AI advice. However, people are not always able to tell apart tasks completed by AI or other humans. See AI takeover for more information. It is also notable that this sentiment is more prominent in the Western cultures as Westerners tend to show less positive views about AI compared to East Asians. == Research on the psychological impacts of AI == === Perception on others who use AI === As much as people perceive and make judgment about AI itself, they also form impressions of themselves and others who use AI. In the workplace, employees who disclose the use of AI in their tasks are more likely to receive feedback that they are not as hardworking as those who are in the same job who receive non-AI help to complete the same tasks. AI use disclosure diminishes the perceived legitimacy in the employee's task and decision making which ultimately leads observers to distrust people who use AI. Although these negative effects of AI use disclosure are weakened by the observers who use AI frequently themselves, the effect is still not attenuated by the observers' positive attitude towards AI. === Bias, AI, and human === Although AI provides a wide range of information and suggestions to its users, AI itself is not free of biases and stereotypes, and it does not always help people reduce their cognitive errors and biases. People are prone to such errors by failing to see other potential ideas and cases that are not listed by AI responses and committing to a decision suggested by AI that directly contradicts the correct information and directions that they are already aware of. Gender bias is also reflected as the female gendering of AI technologies which conceptualizes females as a helpful assistant. == Emotional connection with AI == Human-AI interaction has been theorized in the context of interpersonal relationships mainly in social psychology, communications and media studies, and as a technology interface through the lens of hu

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  • Myhill–Nerode theorem

    Myhill–Nerode theorem

    In the theory of formal languages, the Myhill–Nerode theorem provides a necessary and sufficient condition for a language to be regular. The theorem is named for John Myhill and Anil Nerode, who proved it at the University of Chicago in 1957 (Nerode & Sauer 1957, p. ii). == Statement == Given a language L {\displaystyle L} , and a pair of strings x {\displaystyle x} and y {\displaystyle y} , define a distinguishing extension to be a string z {\displaystyle z} such that exactly one of the two strings x z {\displaystyle xz} and y z {\displaystyle yz} belongs to L {\displaystyle L} . Define a relation ∼ L {\displaystyle \sim _{L}} on strings as x ∼ L y {\displaystyle x\;\sim _{L}\ y} if there is no distinguishing extension for x {\displaystyle x} and y {\displaystyle y} . It is easy to show that ∼ L {\displaystyle \sim _{L}} is an equivalence relation on strings, and thus it divides the set of all strings into equivalence classes. The Myhill–Nerode theorem states that a language L {\displaystyle L} is regular if and only if ∼ L {\displaystyle \sim _{L}} has a finite number of equivalence classes, and moreover, that this number is equal to the number of states in the minimal deterministic finite automaton (DFA) accepting L {\displaystyle L} . Furthermore, every minimal DFA for the language is isomorphic to the canonical one (Hopcroft & Ullman 1979). Generally, for any language, the constructed automaton is a state automaton acceptor. However, it does not necessarily have finitely many states. The Myhill–Nerode theorem shows that finiteness is necessary and sufficient for language regularity. Some authors refer to the ∼ L {\displaystyle \sim _{L}} relation as Nerode congruence, in honor of Anil Nerode. == Use and consequences == The Myhill–Nerode theorem may be used to show that a language L {\displaystyle L} is regular by proving that the number of equivalence classes of ∼ L {\displaystyle \sim _{L}} is finite. This may be done by an exhaustive case analysis in which, beginning from the empty string, distinguishing extensions are used to find additional equivalence classes until no more can be found. For example, the language consisting of binary representations of numbers that can be divided by 3 is regular. Given two binary strings x , y {\displaystyle x,y} , extending them by one digit gives 2 x + b , 2 y + b {\displaystyle 2x+b,2y+b} , so 2 x + b ≡ 2 y + b mod 3 {\displaystyle 2x+b\equiv 2y+b\mod 3} iff x ≡ y mod 3 {\displaystyle x\equiv y\mod 3} . Thus, 00 {\displaystyle 00} (or 11 {\displaystyle 11} ), 01 {\displaystyle 01} , and 10 {\displaystyle 10} are the only distinguishing extensions, resulting in the 3 classes. The minimal automaton accepting our language would have three states corresponding to these three equivalence classes. Another immediate corollary of the theorem is that if for a language L {\displaystyle L} the relation ∼ L {\displaystyle \sim _{L}} has infinitely many equivalence classes, it is not regular. It is this corollary that is frequently used to prove that a language is not regular. == Generalizations == The Myhill–Nerode theorem can be generalized to tree automata.

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  • AI Photo Editors Reviews: What Actually Works in 2026

    AI Photo Editors Reviews: What Actually Works in 2026

    Curious about the best AI photo editor? An AI photo editor 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 photo editor 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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  • Best AI Avatar Generators in 2026

    Best AI Avatar Generators in 2026

    Looking for the best AI avatar generator? An AI avatar generator 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 avatar generator 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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  • VGACAD

    VGACAD

    VGACAD was the parent of a suite of shareware graphic utilities made for the MS-DOS operating system used in the IBM PC and clones. It was popular for editing and capturing images using BSAVE (graphics image format) and provided an early graphic editing suite compatible with multiple graphic cards and resolutions, used on the IBM PC. == Usage == Written by Lawrence Gozum in 1987, it was the genesis of multiple versions and improvements over 10 years. Ran with his brother, Marvin initially helped with design ideas, strategic focus, technical support calls, and managing the early shareware business. The growth of the VGACAD suite grew quickly to preoccupy most of their time. Lawrence then focused more of his efforts on software and formed Applied Insights, to manage VGACAD and its offspring, VidFun, and Ai Picture Explorer. At its peak, its users ranged from individuals, Federal government offices, museums and major newspapers. == Features == VGACAD was a misnomer, and meant VGA-Computer Assisted Drawing, rather than computer-aided design, as CAD is commonly referred to today. Its longevity was due to its color accuracy, speed, small size, and that its suite of small utilities often worked stand-alone. One called VGACAP, for 'capture', dumped video memory into a file that could later be converted to popular graphic image formats, later made commonplace when Microsoft Windows programmed the print screen key to dump graphics into the clipboard. However, VGACAP ran insulated apart from early versions of Windows, and thus could capture screens were applications prohibited such function.

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  • Büchi automaton

    Büchi automaton

    In computer science and automata theory, a deterministic Büchi automaton is a theoretical machine which either accepts or rejects infinite inputs. Such a machine has a set of states and a transition function, which determines which state the machine should move to from its current state when it reads the next input character. Some states are accepting states and one state is the start state. The machine accepts an input if and only if it will pass through an accepting state infinitely many times as it reads the input. A non-deterministic Büchi automaton, later referred to just as a Büchi automaton, has a transition function which may have multiple outputs, leading to many possible paths for the same input; it accepts an infinite input if and only if some possible path is accepting. Deterministic and non-deterministic Büchi automata generalize deterministic finite automata and nondeterministic finite automata to infinite inputs. Each are types of ω-automata. Büchi automata recognize the ω-regular languages, the infinite word version of regular languages. They are named after the Swiss mathematician Julius Richard Büchi, who invented them in 1962. Büchi automata are often used in model checking as an automata-theoretic version of a formula in linear temporal logic. == Formal definition == Formally, a deterministic Büchi automaton is a tuple A = ( Q , Σ , δ , q 0 , F ) {\textstyle A=(Q,\Sigma ,\delta ,q_{0},\mathbf {F} )} that consists of the following components: Q {\textstyle Q} is a finite set. The elements of Q {\textstyle Q} are called the states of A {\textstyle A} . Σ {\textstyle \Sigma } is a finite set called the alphabet of A {\textstyle A} . δ : Q × Σ → Q {\textstyle \delta \colon Q\times \Sigma \to Q} is a function, called the transition function of A {\textstyle A} . q 0 {\textstyle q_{0}} is an element of Q {\textstyle Q} , called the initial state of A {\textstyle A} . F ⊆ Q {\textstyle \mathbf {F} \subseteq Q} is the acceptance condition. A run i _ = i 0 i 1 i 2 ⋯ ∈ Σ ω {\displaystyle {\underline {i}}=i_{0}i_{1}i_{2}\cdots \in \Sigma ^{\omega }} is an infinite string of inputs of A {\displaystyle A} . By calling δ {\displaystyle \delta } recursively, we can extend it to a function δ ω : Σ ω → Q ω {\displaystyle \delta ^{\omega }:\Sigma ^{\omega }\to Q^{\omega }} . A state q ∈ Q {\displaystyle q\in Q} is said to occur infinitely often for a run i _ {\displaystyle {\underline {i}}} when the set { n ∈ N ∣ δ ω ( i _ ) n = q } {\displaystyle \{n\in \mathbb {N} \mid \delta ^{\omega }({\underline {i}})_{n}=q\}} is infinite. Let I n f ( i _ ) {\displaystyle \mathrm {Inf} ({\underline {i}})} be the set of states occurring infinitely often for i _ {\displaystyle {\underline {i}}} . The language of A {\displaystyle A} is then the set of runs of A {\displaystyle A} in which at least one of the infinitely-often occurring states is in F {\textstyle \mathbf {F} } ; in symbols: L ( A ) = { i _ ∈ Σ ω ∣ I n f ( i _ ) ∩ F ≠ ∅ } . {\displaystyle L(A)=\{{\underline {i}}\in \Sigma ^{\omega }\mid \mathrm {Inf} ({\underline {i}})\cap \mathbf {F} \neq \varnothing \}.} In a (non-deterministic) Büchi automaton, the transition function δ {\textstyle \delta } is replaced with a transition relation Δ {\textstyle \Delta } that returns a set of states, and the single initial state q 0 {\textstyle q_{0}} is replaced by a set I {\textstyle I} of initial states. Generally, the term Büchi automaton without qualifier refers to non-deterministic Büchi automata. For more comprehensive formalism see also ω-automaton. == Closure properties == The set of Büchi automata is closed under the following operations. Let A = ( Q A , Σ , Δ A , I A , F A ) {\displaystyle A=(Q_{A},\Sigma ,\Delta _{A},I_{A},{F}_{A})} and B = ( Q B , Σ , Δ B , I B , F B ) {\displaystyle B=(Q_{B},\Sigma ,\Delta _{B},I_{B},{F}_{B})} be Büchi automata and C = ( Q C , Σ , Δ C , I C , F C ) {\displaystyle C=(Q_{C},\Sigma ,\Delta _{C},I_{C},{F}_{C})} be a finite automaton. Union: There is a Büchi automaton that recognizes the language L ( A ) ∪ L ( B ) . {\displaystyle L(A)\cup L(B).} Proof: If we assume, w.l.o.g., Q A ∩ Q B {\displaystyle Q_{A}\cap Q_{B}} is empty then L ( A ) ∪ L ( B ) {\displaystyle L(A)\cup L(B)} is recognized by the Büchi automaton ( Q A ∪ Q B , Σ ∪ Σ , Δ A ∪ Δ B , I A ∪ I B , F A ∪ F B ) . {\displaystyle (Q_{A}\cup Q_{B},\Sigma \cup \Sigma ,\Delta _{A}\cup \Delta _{B},I_{A}\cup I_{B},{F}_{A}\cup {F}_{B}).} Intersection: There is a Büchi automaton that recognizes the language L ( A ) ∩ L ( B ) . {\displaystyle L(A)\cap L(B).} Proof: The Büchi automaton A ′ = ( Q ′ , Σ , Δ ′ , I ′ , F ′ ) {\displaystyle A'=(Q',\Sigma ,\Delta ',I',F')} recognizes L ( A ) ∩ L ( B ) , {\displaystyle L(A)\cap L(B),} where Q ′ = Q A × Q B × { 1 , 2 } {\displaystyle Q'=Q_{A}\times Q_{B}\times \{1,2\}} Δ ′ = Δ 1 ∪ Δ 2 {\displaystyle \Delta '=\Delta _{1}\cup \Delta _{2}} Δ 1 = { ( ( q A , q B , 1 ) , a , ( q A ′ , q B ′ , i ) ) | ( q A , a , q A ′ ) ∈ Δ A and ( q B , a , q B ′ ) ∈ Δ B and if q A ∈ F A then i = 2 else i = 1 } {\displaystyle \Delta _{1}=\{((q_{A},q_{B},1),a,(q'_{A},q'_{B},i))|(q_{A},a,q'_{A})\in \Delta _{A}{\text{ and }}(q_{B},a,q'_{B})\in \Delta _{B}{\text{ and if }}q_{A}\in F_{A}{\text{ then }}i=2{\text{ else }}i=1\}} Δ 2 = { ( ( q A , q B , 2 ) , a , ( q A ′ , q B ′ , i ) ) | ( q A , a , q A ′ ) ∈ Δ A and ( q B , a , q B ′ ) ∈ Δ B and if q B ∈ F B then i = 1 else i = 2 } {\displaystyle \Delta _{2}=\{((q_{A},q_{B},2),a,(q'_{A},q'_{B},i))|(q_{A},a,q'_{A})\in \Delta _{A}{\text{ and }}(q_{B},a,q'_{B})\in \Delta _{B}{\text{ and if }}q_{B}\in F_{B}{\text{ then }}i=1{\text{ else }}i=2\}} I ′ = I A × I B × { 1 } {\displaystyle I'=I_{A}\times I_{B}\times \{1\}} F ′ = { ( q A , q B , 2 ) | q B ∈ F B } {\displaystyle F'=\{(q_{A},q_{B},2)|q_{B}\in F_{B}\}} By construction, r ′ = ( q A 0 , q B 0 , i 0 ) , ( q A 1 , q B 1 , i 1 ) , … {\displaystyle r'=(q_{A}^{0},q_{B}^{0},i^{0}),(q_{A}^{1},q_{B}^{1},i^{1}),\dots } is a run of automaton A' on input word w {\textstyle w} if r A = q A 0 , q A 1 , … {\displaystyle r_{A}=q_{A}^{0},q_{A}^{1},\dots } is run of A {\textstyle A} on w {\textstyle w} and r B = q B 0 , q B 1 , … {\displaystyle r_{B}=q_{B}^{0},q_{B}^{1},\dots } is run of B {\textstyle B} on w {\textstyle w} . r A {\textstyle r_{A}} is accepting and r B {\textstyle r_{B}} is accepting if r ′ {\textstyle r'} is concatenation of an infinite series of finite segments of 1-states (states with third component 1) and 2-states (states with third component 2) alternatively. There is such a series of segments of r ′ {\textstyle r'} if r ′ {\textstyle r'} is accepted by A ′ {\textstyle A'} . Concatenation: There is a Büchi automaton that recognizes the language L ( C ) ⋅ L ( A ) . {\displaystyle L(C)\cdot L(A).} Proof: If we assume, w.l.o.g., Q C ∩ Q A {\displaystyle Q_{C}\cap Q_{A}} is empty then the Büchi automaton A ′ = ( Q C ∪ Q A , Σ , Δ ′ , I ′ , F A ) {\displaystyle A'=(Q_{C}\cup Q_{A},\Sigma ,\Delta ',I',F_{A})} recognizes L ( C ) ⋅ L ( A ) {\displaystyle L(C)\cdot L(A)} , where Δ ′ = Δ A ∪ Δ C ∪ { ( q , a , q ′ ) | q ′ ∈ I A and ∃ f ∈ F C . ( q , a , f ) ∈ Δ C } {\displaystyle \Delta '=\Delta _{A}\cup \Delta _{C}\cup \{(q,a,q')|q'\in I_{A}{\text{ and }}\exists f\in F_{C}.(q,a,f)\in \Delta _{C}\}} if I C ∩ F C is empty then I ′ = I C otherwise I ′ = I C ∪ I A {\displaystyle {\text{ if }}I_{C}\cap F_{C}{\text{ is empty then }}I'=I_{C}{\text{ otherwise }}I'=I_{C}\cup I_{A}} ω-closure: If L ( C ) {\displaystyle L(C)} does not contain the empty word then there is a Büchi automaton that recognizes the language L ( C ) ω . {\displaystyle L(C)^{\omega }.} Proof: The Büchi automaton that recognizes L ( C ) ω {\displaystyle L(C)^{\omega }} is constructed in two stages. First, we construct a finite automaton A ′ {\textstyle A'} such that A ′ {\textstyle A'} also recognizes L ( C ) {\displaystyle L(C)} but there are no incoming transitions to initial states of A ′ {\textstyle A'} . So, A ′ = ( Q C ∪ { q new } , Σ , Δ ′ , { q new } , F C ) , {\displaystyle A'=(Q_{C}\cup \{q_{\text{new}}\},\Sigma ,\Delta ',\{q_{\text{new}}\},F_{C}),} where Δ ′ = Δ C ∪ { ( q new , a , q ′ ) | ∃ q ∈ I C . ( q , a , q ′ ) ∈ Δ C } . {\displaystyle \Delta '=\Delta _{C}\cup \{(q_{\text{new}},a,q')|\exists q\in I_{C}.(q,a,q')\in \Delta _{C}\}.} Note that L ( C ) = L ( A ′ ) {\displaystyle L(C)=L(A')} because L ( C ) {\displaystyle L(C)} does not contain the empty string. Second, we will construct the Büchi automaton A ″ {\textstyle A''} that recognize L ( C ) ω {\displaystyle L(C)^{\omega }} by adding a loop back to the initial state of A ′ {\textstyle A'} . So, A ″ = ( Q C ∪ { q new } , Σ , Δ ″ , { q new } , { q new } ) {\displaystyle A''=(Q_{C}\cup \{q_{\text{new}}\},\Sigma ,\Delta '',\{q_{\text{new}}\},\{q_{\text{new}}\})} , where Δ ″ = Δ ′ ∪ { ( q , a , q new ) | ∃ q ′ ∈ F C . ( q , a , q ′ ) ∈ Δ ′ } . {\displaystyle \Delta ''=\Delta '\cup \{(q,a,q_{\text{new}})|\exists q'\in F_{C}.(q,a,q')\in \Delta '\}.} Complementation:

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  • AI Sales Assistants: Free vs Paid (2026)

    AI Sales Assistants: Free vs Paid (2026)

    Trying to pick the best AI sales assistant? An AI sales assistant 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 sales 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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