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  Software agent

  In computer science, a software agent is a computer program that acts for a user or another program in a relationship of agency. The term agent is derived from the Latin agere (to do): an agreement to act on one's behalf. Such "action on behalf of" implies the authority to decide which, if any, action is appropriate. Some agents are colloquially known as bots, from robot. They may be embodied, as when execution is paired with a robot body, or as software such as a chatbot executing on a computer, such as a mobile device, e.g. Siri. Software agents may be autonomous or work together with other agents or people. Software agents interacting with people (e.g. chatbots, human-robot interaction environments) may possess human-like qualities such as natural language understanding and speech, personality or embody humanoid form (see Asimo). Related and derived concepts include intelligent agents (in particular exhibiting some aspects of artificial intelligence, such as reasoning), autonomous agents (capable of modifying the methods of achieving their objectives), distributed agents (being executed on physically distinct computers), multi-agent systems (distributed agents that work together to achieve an objective that could not be accomplished by a single agent acting alone), and mobile agents (agents that can relocate their execution onto different processors). == Concepts == The basic attributes of an autonomous software agent are that agents: are not strictly invoked for a task, but activate themselves, may reside in wait status on a host, perceiving context, may get to run status on a host upon starting conditions, do not require interaction of user, may invoke other tasks including communication. The concept of an agent provides a method of describing a complex software entity that is capable of acting with a certain degree of autonomy in order to accomplish tasks on behalf of its host. But unlike objects, which are defined in terms of methods and attributes, an agent is defined in terms of its behavior. Various authors have proposed different definitions of agents, these commonly include concepts such as: persistence: code is not executed on demand but runs continuously and decides for itself when it should perform some activity; autonomy: agents have capabilities of task selection, prioritization, goal-directed behavior, decision-making without human intervention; social ability: agents are able to engage other components through some sort of communication and coordination, they may collaborate on a task; reactivity: agents perceive the context in which they operate and react to it appropriately. === Distinguishing agents from programs === All agents are programs, but not all programs are agents. Contrasting the term with related concepts may help clarify its meaning. Franklin & Graesser (1997) discuss four key notions that distinguish agents from arbitrary programs: reaction to the environment, autonomy, goal-orientation and persistence. === Intuitive distinguishing agents from objects === Agents are more autonomous than objects. Agents have flexible behavior: reactive, proactive, social. Agents have at least one thread of control but may have more. === Distinguishing agents from expert systems === Expert systems are not coupled to their environment. Expert systems are not designed for reactive, proactive behavior. Expert systems do not consider social ability. === Distinguishing intelligent software agents from intelligent agents in AI === Intelligent agents (also known as rational agents) are not just computer programs: they may also be machines, human beings, communities of human beings (such as firms) or anything that is capable of goal-directed behavior. == Impact of software agents == Software agents may offer various benefits to their end users by automating complex or repetitive tasks. However, there are organizational and cultural impacts of this technology that need to be considered prior to implementing software agents. === Organizational impact === === Work contentment and job satisfaction impact === People like to perform easy tasks providing the sensation of success unless the repetition of the simple tasking is affecting the overall output. In general implementing software agents to perform administrative requirements provides a substantial increase in work contentment, as administering their own work does never please the worker. The effort freed up serves for a higher degree of engagement in the substantial tasks of individual work. Hence, software agents may provide the basics to implement self-controlled work, relieved from hierarchical controls and interference. Such conditions may be secured by application of software agents for required formal support. === Cultural impact === The cultural effects of the implementation of software agents include trust affliction, skills erosion, privacy attrition and social detachment. Some users may not feel entirely comfortable fully delegating important tasks to software applications. Those who start relying solely on intelligent agents may lose important skills, for example, relating to information literacy. In order to act on a user's behalf, a software agent needs to have a complete understanding of a user's profile, including his/her personal preferences. This, in turn, may lead to unpredictable privacy issues. When users start relying on their software agents more, especially for communication activities, they may lose contact with other human users and look at the world with the eyes of their agents. These consequences are what agent researchers and users must consider when dealing with intelligent agent technologies. === History === The concept of an agent can be traced back to Hewitt's Actor Model (Hewitt, 1977) - "A self-contained, interactive and concurrently-executing object, possessing internal state and communication capability." To be more academic, software agent systems are a direct evolution of Multi-Agent Systems (MAS). MAS evolved from Distributed Artificial Intelligence (DAI), Distributed Problem Solving (DPS) and Parallel AI (PAI), thus inheriting all characteristics (good and bad) from DAI and AI. John Sculley's 1987 "Knowledge Navigator" video portrayed an image of a relationship between end-users and agents. Being an ideal first, this field experienced a series of unsuccessful top-down implementations, instead of a piece-by-piece, bottom-up approach. The range of agent types is now (from 1990) broad: WWW, search engines, etc. == Examples of intelligent software agents == === Buyer agents (shopping bots) === Buyer agents travel around a network (e.g. the internet) retrieving information about goods and services. These agents, also known as 'shopping bots', work very efficiently for commodity products such as CDs, books, electronic components, and other one-size-fits-all products. Buyer agents are typically optimized to allow for digital payment services used in e-commerce and traditional businesses. === User agents (personal agents) === User agents, or personal agents, are intelligent agents that take action on your behalf. In this category belong those intelligent agents that already perform, or will shortly perform, the following tasks: Check your e-mail, sort it according to the user's order of preference, and alert you when important emails arrive. Play computer games as your opponent or patrol game areas for you. Assemble customized news reports for you. There are several versions of these, including CNN. Find information for you on the subject of your choice. Fill out forms on the Web automatically for you, storing your information for future reference Scan Web pages looking for and highlighting text that constitutes the "important" part of the information there Discuss topics with you ranging from your deepest fears to sports Facilitate with online job search duties by scanning known job boards and sending the resume to opportunities who meet the desired criteria Profile synchronization across heterogeneous social networks === Monitoring-and-surveillance (predictive) agents === Monitoring and surveillance agents are used to observe and report on equipment, usually computer systems. The agents may keep track of company inventory levels, observe competitors' prices and relay them back to the company, watch stock manipulation by insider trading and rumors, etc. For example, NASA's Jet Propulsion Laboratory has an agent that monitors inventory, planning, schedules equipment orders to keep costs down, and manages food storage facilities. These agents usually monitor complex computer networks that can keep track of the configuration of each computer connected to the network. A special case of monitoring-and-surveillance agents are organizations of agents used to automate decision-making process during tactical operations. The agents monitor the status of assets (ammunition, weapons available, platforms for transport, etc.) and receive goals from hi

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  Iterative Viterbi decoding

  Iterative Viterbi decoding is an algorithm that spots the subsequence S of an observation O = {o1, ..., on} having the highest average probability (i.e., probability scaled by the length of S) of being generated by a given hidden Markov model M with m states. The algorithm uses a modified Viterbi algorithm as an internal step. The scaled probability measure was first proposed by John S. Bridle. An early algorithm to solve this problem, sliding window, was proposed by Jay G. Wilpon et al., 1989, with constant cost T = mn2/2. A faster algorithm consists of an iteration of calls to the Viterbi algorithm, reestimating a filler score until convergence. == The algorithm == A basic (non-optimized) version, finding the sequence s with the smallest normalized distance from some subsequence of t is: // input is placed in observation s[1..n], template t[1..m], // and [[distance matrix]] d[1..n,1..m] // remaining elements in matrices are solely for internal computations (int, int, int) AverageSubmatchDistance(char s[0..(n+1)], char t[0..(m+1)], int d[1..n,0..(m+1)]) { // score, subsequence start, subsequence end declare int e, B, E t'[0] := t'[m+1] := s'[0] := s'[n+1] := 'e' e := random() do e' := e for i := 1 to n do d'[i,0] := d'[i,m+1] := e (e, B, E) := ViterbiDistance(s', t', d') e := e/(E-B+1) until (e == e') return (e, B, E) } The ViterbiDistance() procedure returns the tuple (e, B, E), i.e., the Viterbi score "e" for the match of t and the selected entry (B) and exit (E) points from it. "B" and "E" have to be recorded using a simple modification to Viterbi. A modification that can be applied to CYK tables, proposed by Antoine Rozenknop, consists in subtracting e from all elements of the initial matrix d.

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  Dilek Hakkani-Tür

  Dilek Z. Hakkani-Tür is a Turkish-American computer scientist focusing on speech processing, speech recognition, and dialogue systems. She is a professor of computer science at the University of Illinois Urbana-Champaign. == Education and career == Hakkani-Tür is a 1994 graduate of Middle East Technical University in Ankara, Turkey. She continued her studies at Bilkent University, also in Ankara, where she earned a master's degree in 1996 and completed her Ph.D. in 2000. She worked as a researcher at AT&T Labs from 2001 to 2005, at the International Computer Science Institute from 2006 to 2010, at Microsoft Research from 2010 to 2016, at Google Research from 2016 to 2018, and at Amazon Alexa from 2018 to 2023. At Microsoft, she was in the team of scientists that built the first prototype of the Cortana virtual assistant. While working for Amazon Alexa, she also taught at the University of California, Santa Cruz as a distinguished visiting instructor. She joined the University of Illinois Urbana-Champaign faculty in 2023. She was editor-in-chief of IEEE/ACM Transactions on Audio, Speech and Language Processing from 2019 to 2021, and is president of the Special Interest Group on Discourse and Dialogue of the Association for Computational Linguistics for the 2023–2025 term. She has served as co-editor-in-chief of Transactions of the Association for Computational Linguistics since 2024. == Recognition == In 2014, Hakkani-Tür was elected as an IEEE Fellow "for contributions to spoken language processing", and as a Fellow of the International Speech Communication Association "for contributions to advancing the state-of-the-art in spoken language processing, especially for human/human and human/machine conversational understanding". In 2024, she was elected as a Fellow of the Association for Computational Linguistics for her contributions to spoken dialogue systems.

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  Kurt Keutzer

  Kurt Keutzer (born November 9, 1955) is an American computer scientist. == Early life and education == Kurt Keutzer grew up in Indianapolis, Indiana. He earned a bachelor's degree in mathematics from Maharishi University of Management (formerly Mararishi International University) in 1978, and a PhD in computer science from Indiana University Bloomington in 1984. == Career == Keutzer joined Bell Labs in 1984, where he worked on logic synthesis. In 1991, he joined the electronic design automation company Synopsys, where he was promoted to chief technology officer. He subsequently joined the University of California, Berkeley as a professor in 1998. His research at Berkeley has focused on the intersection of high performance computing and machine learning. Working with a number of graduate students at Berkeley, Keutzer developed FireCaffe, which scaled the training of deep neural networks to over 100 GPUs. Later, with LARS and LAMB optimizers, they scaled it to over 1000 servers. Keutzer and his students also developed deep neural networks such as SqueezeNet, SqueezeDet, and SqueezeSeg, which can run efficiently on mobile devices. Keutzer co-founded DeepScale with his PhD student Forrest Iandola in 2015, and Keutzer served as the company's chief strategy officer. The firm was focused on developing deep neural networks for advanced driver assistance systems in passenger cars. On October 1, 2019, electric vehicle manufacturer Tesla, Inc. purchased DeepScale to augment and accelerate its self-driving vehicle work. == Honors and awards == Keutzer was named a Fellow of the IEEE in 1996. Recipient of DAC Most Influential Paper (MIP) award (24th DAC, 1987) for his "Dagon: technology binding and local optimization by DAG matching” publication. == Books by Keutzer == 1988. Dwight Hill, Don Shugard, John Fishburn, and Kurt Keutzer. Algorithms and Techniques for VLSI Layout Synthesis. Springer. 1994. Srinivas Devadas, Abhijit Ghosh, and Kurt Keutzer. Logic Synthesis. McGraw-Hill. 2002. David Chinnery and Kurt Keutzer. Closing the Gap Between ASIC & Custom: Tools and Techniques for High-Performance ASIC Design. Springer. (2nd edition appeared in 2007.) 2004. Pinhong Chen, Desmond A. Kirkpatrick, and Kurt Keutzer. Static Crosstalk-Noise Analysis: For Deep Sub-Micron Digital Designs. Springer. 2005. Matthias Gries and Kurt Keutzer. Building ASIPs: The Mescal Methodology. Springer.

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  Contextual image classification

  Contextual image classification, a topic of pattern recognition in computer vision, is an approach of classification based on contextual information in images. "Contextual" means this approach is focusing on the relationship of the nearby pixels, which is also called neighbourhood. The goal of this approach is to classify the images by using the contextual information. == Introduction == Similar as processing language, a single word may have multiple meanings unless the context is provided, and the patterns within the sentences are the only informative segments we care about. For images, the principle is same. Find out the patterns and associate proper meanings to them. As the image illustrated below, if only a small portion of the image is shown, it is very difficult to tell what the image is about. Even try another portion of the image, it is still difficult to classify the image. However, if we increase the contextual of the image, then it makes more sense to recognize. As the full images shows below, almost everyone can classify it easily. During the procedure of segmentation, the methods which do not use the contextual information are sensitive to noise and variations, thus the result of segmentation will contain a great deal of misclassified regions, and often these regions are small (e.g., one pixel). Compared to other techniques, this approach is robust to noise and substantial variations for it takes the continuity of the segments into account. Several methods of this approach will be described below. == Applications == === Functioning as a post-processing filter to a labelled image === This approach is very effective against small regions caused by noise. And these small regions are usually formed by few pixels or one pixel. The most probable label is assigned to these regions. However, there is a drawback of this method. The small regions also can be formed by correct regions rather than noise, and in this case the method is actually making the classification worse. This approach is widely used in remote sensing applications. === Improving the post-processing classification === This is a two-stage classification process: For each pixel, label the pixel and form a new feature vector for it. Use the new feature vector and combine the contextual information to assign the final label to the === Merging the pixels in earlier stages === Instead of using single pixels, the neighbour pixels can be merged into homogeneous regions benefiting from contextual information. And provide these regions to classifier. === Acquiring pixel feature from neighbourhood === The original spectral data can be enriched by adding the contextual information carried by the neighbour pixels, or even replaced in some occasions. This kind of pre-processing methods are widely used in textured image recognition. The typical approaches include mean values, variances, texture description, etc. === Combining spectral and spatial information === The classifier uses the grey level and pixel neighbourhood (contextual information) to assign labels to pixels. In such case the information is a combination of spectral and spatial information. === Powered by the Bayes minimum error classifier === Contextual classification of image data is based on the Bayes minimum error classifier (also known as a naive Bayes classifier). Present the pixel: A pixel is denoted as x 0 {\displaystyle x_{0}} . The neighbourhood of each pixel x 0 {\displaystyle x_{0}} is a vector and denoted as N ( x 0 ) {\displaystyle N(x_{0})} . The values in the neighbourhood vector is denoted as f ( x i ) {\displaystyle f(x_{i})} . Each pixel is presented by the vector ξ = ( f ( x 0 ) , f ( x 1 ) , … , f ( x k ) ) {\displaystyle \xi =\left(f(x_{0}),f(x_{1}),\ldots ,f(x_{k})\right)} x i ∈ N ( x 0 ) ; i = 1 , … , k {\displaystyle x_{i}\in N(x_{0});\quad i=1,\ldots ,k} The labels (classification) of pixels in the neighbourhood N ( x 0 ) {\displaystyle N(x_{0})} are presented as a vector η = ( θ 0 , θ 1 , … , θ k ) {\displaystyle \eta =\left(\theta _{0},\theta _{1},\ldots ,\theta _{k}\right)} θ i ∈ { ω 0 , ω 1 , … , ω k } {\displaystyle \theta _{i}\in \left\{\omega _{0},\omega _{1},\ldots ,\omega _{k}\right\}} ω s {\displaystyle \omega _{s}} here denotes the assigned class. A vector presents the labels in the neighbourhood N ( x 0 ) {\displaystyle N(x_{0})} without the pixel x 0 {\displaystyle x_{0}} η ^ = ( θ 1 , θ 2 , … , θ k ) {\displaystyle {\hat {\eta }}=\left(\theta _{1},\theta _{2},\ldots ,\theta _{k}\right)} The neighbourhood: Size of the neighbourhood. There is no limitation of the size, but it is considered to be relatively small for each pixel x 0 {\displaystyle x_{0}} . A reasonable size of neighbourhood would be 3 × 3 {\displaystyle 3\times 3} of 4-connectivity or 8-connectivity ( x 0 {\displaystyle x_{0}} is marked as red and placed in the centre). The calculation: Apply the minimum error classification on a pixel x 0 {\displaystyle x_{0}} , if the probability of a class ω r {\displaystyle \omega _{r}} being presenting the pixel x 0 {\displaystyle x_{0}} is the highest among all, then assign ω r {\displaystyle \omega _{r}} as its class. θ 0 = ω r if P ( ω r ∣ f ( x 0 ) ) = max s = 1 , 2 , … , R P ( ω s ∣ f ( x 0 ) ) {\displaystyle \theta _{0}=\omega _{r}\quad {\text{ if }}\quad P(\omega _{r}\mid f(x_{0}))=\max _{s=1,2,\ldots ,R}P(\omega _{s}\mid f(x_{0}))} The contextual classification rule is described as below, it uses the feature vector x 1 {\displaystyle x_{1}} rather than x 0 {\displaystyle x_{0}} . θ 0 = ω r if P ( ω r ∣ ξ ) = max s = 1 , 2 , … , R P ( ω s ∣ ξ ) {\displaystyle \theta _{0}=\omega _{r}\quad {\text{ if }}\quad P(\omega _{r}\mid \xi )=\max _{s=1,2,\ldots ,R}P(\omega _{s}\mid \xi )} Use the Bayes formula to calculate the posteriori probability P ( ω s ∣ ξ ) {\displaystyle P(\omega _{s}\mid \xi )} P ( ω s ∣ ξ ) = p ( ξ ∣ ω s ) P ( ω s ) p ( ξ ) {\displaystyle P(\omega _{s}\mid \xi )={\frac {p(\xi \mid \omega _{s})P(\omega _{s})}{p\left(\xi \right)}}} The number of vectors is the same as the number of pixels in the image. For the classifier uses a vector corresponding to each pixel x i {\displaystyle x_{i}} , and the vector is generated from the pixel's neighbourhood. The basic steps of contextual image classification: Calculate the feature vector ξ {\displaystyle \xi } for each pixel. Calculate the parameters of probability distribution p ( ξ ∣ ω s ) {\displaystyle p(\xi \mid \omega _{s})} and P ( ω s ) {\displaystyle P(\omega _{s})} Calculate the posterior probabilities P ( ω r ∣ ξ ) {\displaystyle P(\omega _{r}\mid \xi )} and all labels θ 0 {\displaystyle \theta _{0}} . Get the image classification result. == Algorithms == === Template matching === The template matching is a "brute force" implementation of this approach. The concept is first create a set of templates, and then look for small parts in the image match with a template. This method is computationally high and inefficient. It keeps an entire templates list during the whole process and the number of combinations is extremely high. For a m × n {\displaystyle m\times n} pixel image, there could be a maximum of 2 m × n {\displaystyle 2^{m\times n}} combinations, which leads to high computation. This method is a top down method and often called table look-up or dictionary look-up. === Lower-order Markov chain === The Markov chain also can be applied in pattern recognition. The pixels in an image can be recognised as a set of random variables, then use the lower order Markov chain to find the relationship among the pixels. The image is treated as a virtual line, and the method uses conditional probability. === Hilbert space-filling curves === The Hilbert curve runs in a unique pattern through the whole image, it traverses every pixel without visiting any of them twice and keeps a continuous curve. It is fast and efficient. === Markov meshes === The lower-order Markov chain and Hilbert space-filling curves mentioned above are treating the image as a line structure. The Markov meshes however will take the two dimensional information into account. === Dependency tree === The dependency tree is a method using tree dependency to approximate probability distributions.

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  Collostructional analysis

  Collostructional analysis is a family of methods developed by (in alphabetical order) Stefan Th. Gries (University of California, Santa Barbara) and Anatol Stefanowitsch (Free University of Berlin). Collostructional analysis aims at measuring the degree of attraction or repulsion that words exhibit to constructions, where the notion of construction has so far been that of Goldberg's construction grammar. == Collostructional methods == Collostructional analysis so far comprises three different methods: collexeme analysis, to measure the degree of attraction/repulsion of a lemma to a slot in one particular construction; distinctive collexeme analysis, to measure the preference of a lemma to one particular construction over another, functionally similar construction; multiple distinctive collexeme analysis extends this approach to more than two alternative constructions; covarying collexeme analysis, to measure the degree of attraction of lemmas in one slot of a construction to lemmas in another slot of the same construction. == Input frequencies == Collostructional analysis requires frequencies of words and constructions and is similar to a wide variety of collocation statistics. It differs from raw frequency counts by providing not only observed co-occurrence frequencies of words and constructions, but also (i) a comparison of the observed frequency to the one expected by chance; thus, collostructional analysis can distinguish attraction and repulsion of words and constructions; (ii) a measure of the strength of the attraction or repulsion; this is usually the log-transformed p-value of a Fisher-Yates exact test. == Versus other collocation statistics == Collostructional analysis differs from most collocation statistics such that (i) it measures not the association of words to words, but of words to syntactic patterns or constructions; thus, it takes syntactic structure more seriously than most collocation-based analyses; (ii) it has so far only used the most precise statistics, namely the Fisher-Yates exact test based on the hypergeometric distribution; thus, unlike t-scores, z-scores, chi-square tests etc., the analysis is not based on, and does not violate, any distributional assumptions.

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  Best AI Sales Assistants in 2026

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

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  How to Choose an AI Presentation Maker

  Comparing the best AI presentation maker? An AI presentation maker 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 presentation maker slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.

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  I-MSCP

  i-MSCP (internet Multi Server Control Panel) was a free and open-source software for shared hosting environments management on Linux servers. It comes with a large choice of modules for various services such as Apache2, ProFTPd, Dovecot, Courier, Bind9, and can be easily extended through plugins, or listener files using its events-based API. Latest stable is the 1.5.3 version (build 2018120800) which has been released on 8 December 2018. The i-MSCP is no longer under development, although the developer has repeatedly claimed to be working on a new version, which has never has been published or even shown in any possible way. Whether development occurs or not, the current version of the software is not installable, as it only supports outdated versions of systems for which some of the necessary software to install i-MSCP cannot be installed. == Licensing == i-MSCP has a dual license. A part of the base code is licensed under the Mozilla Public License. All new code, and submissions to i-MSCP are licensed under the GNU Lesser General Public License Version 2.1 (LGPLv2). To solve this license conflict there is work on a complete rewrite for a completely LGPLv2 licensed i-MSCP. == Features == === Supported Linux Distributions === Debian Jessie (8.x), Stretch (9.x), Buster (10.x) Devuan Jessie (1.0), ASCII (2.x) Ubuntu Trusty Thar (14.04 LTS), Bionic Beaver (18.04 LTS) === Supported Daemons / Services === Web server: Apache (ITK, Fcgid and FastCGI/PHP-FPM), Nginx Name server: Bind9 MTA (Mail Transport Agent): Postfix MDA (Mail Delivery Agent): Courier, Dovecot Database: MySQL, MariaDB, Percona FTP-Server: ProFTPD, vsftpd Web statistics: AWStats === Addons === PhpMyAdmin Pydio, formerly AjaXplorer Net2ftp Roundcube Rainloop == Competing software == cPanel DTC Froxlor ISPConfig ispCP OpenPanel hestiacp Plesk SysCP Virtualmin

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  DFA minimization

  In automata theory (a branch of theoretical computer science), DFA minimization is the task of transforming a given deterministic finite automaton (DFA) into an equivalent DFA that has a minimum number of states. Here, two DFAs are called equivalent if they recognize the same regular language. Several different algorithms accomplishing this task are known and described in standard textbooks on automata theory. == Minimal DFA == For each regular language, there also exists a minimal automaton that accepts it, that is, a DFA with a minimum number of states and this DFA is unique (except that states can be given different names). The minimal DFA ensures minimal computational cost for tasks such as pattern matching. There are three classes of states that can be removed or merged from the original DFA without affecting the language it accepts. Unreachable states are the states that are not reachable from the initial state of the DFA, for any input string. These states can be removed. Dead states are the states from which no final state is reachable. These states can be removed unless the automaton is required to be complete. Nondistinguishable states are those that cannot be distinguished from one another for any input string. These states can be merged. DFA minimization is usually done in three steps: remove dead and unreachable states (this will accelerate the following step), merge nondistinguishable states, optionally, re-create a single dead state ("sink" state) if the resulting DFA is required to be complete. == Unreachable states == The state p {\displaystyle p} of a deterministic finite automaton M = ( Q , Σ , δ , q 0 , F ) {\displaystyle M=(Q,\Sigma ,\delta ,q_{0},F)} is unreachable if no string w {\displaystyle w} in Σ ∗ {\displaystyle \Sigma ^{}} exists for which p = δ ∗ ( q 0 , w ) {\displaystyle p=\delta ^{}(q_{0},w)} . In this definition, Q {\displaystyle Q} is the set of states, Σ {\displaystyle \Sigma } is the set of input symbols, δ {\displaystyle \delta } is the transition function (mapping a state and an input symbol to a set of states), δ ∗ {\displaystyle \delta ^{}} is its extension to strings (also known as extended transition function), q 0 {\displaystyle q_{0}} is the initial state, and F {\displaystyle F} is the set of accepting (also known as final) states. Reachable states can be obtained with the following algorithm: Assuming an efficient implementation of the state sets (e.g. new_states) and operations on them (such as adding a state or checking whether it is present), this algorithm can be implemented with time complexity O ( n + m ) {\displaystyle O(n+m)} , where n {\displaystyle n} is the number of states and m {\displaystyle m} is the number of transitions of the input automaton. Unreachable states can be removed from the DFA without affecting the language that it accepts. == Nondistinguishable states == The following algorithms present various approaches to merging nondistinguishable states. === Hopcroft's algorithm === One algorithm for merging the nondistinguishable states of a DFA, due to Hopcroft (1971), is based on partition refinement, partitioning the DFA states into groups by their behavior. These groups represent equivalence classes of the Nerode congruence, whereby every two states are equivalent if they have the same behavior for every input sequence. That is, for every two states p1 and p2 that belong to the same block of the partition P, and every input word w, the transitions determined by w should always take states p1 and p2 to either states that both accept or states that both reject. It should not be possible for w to take p1 to an accepting state and p2 to a rejecting state or vice versa. The following pseudocode describes the form of the algorithm as given by Xu. Alternative forms have also been presented. The algorithm starts with a partition that is too coarse: every pair of states that are equivalent according to the Nerode congruence belong to the same set in the partition, but pairs that are inequivalent might also belong to the same set. It gradually refines the partition into a larger number of smaller sets, at each step splitting sets of states into pairs of subsets that are necessarily inequivalent. The initial partition is a separation of the states into two subsets of states that clearly do not have the same behavior as each other: the accepting states and the rejecting states. The algorithm then repeatedly chooses a set A from the current partition and an input symbol c, and splits each of the sets of the partition into two (possibly empty) subsets: the subset of states that lead to A on input symbol c, and the subset of states that do not lead to A. Since A is already known to have different behavior than the other sets of the partition, the subsets that lead to A also have different behavior than the subsets that do not lead to A. When no more splits of this type can be found, the algorithm terminates. Lemma. Given a fixed character c and an equivalence class Y that splits into equivalence classes B and C, only one of B or C is necessary to refine the whole partition. Example: Suppose we have an equivalence class Y that splits into equivalence classes B and C. Suppose we also have classes D, E, and F; D and E have states with transitions into B on character c, while F has transitions into C on character c. By the Lemma, we can choose either B or C as the distinguisher, let's say B. Then the states of D and E are split by their transitions into B. But F, which doesn't point into B, simply doesn't split during the current iteration of the algorithm; it will be refined by other distinguisher(s). Observation. All of B or C is necessary to split referring classes like D, E, and F correctly—subsets won't do. The purpose of the outermost if statement (if Y is in W) is to patch up W, the set of distinguishers. We see in the previous statement in the algorithm that Y has just been split. If Y is in W, it has just become obsolete as a means to split classes in future iterations. So Y must be replaced by both splits because of the Observation above. If Y is not in W, however, only one of the two splits, not both, needs to be added to W because of the Lemma above. Choosing the smaller of the two splits guarantees that the new addition to W is no more than half the size of Y; this is the core of the Hopcroft algorithm: how it gets its speed, as explained in the next paragraph. The worst case running time of this algorithm is O(ns log n), where n is the number of states and s is the size of the alphabet. This bound follows from the fact that, for each of the ns transitions of the automaton, the sets drawn from Q that contain the target state of the transition have sizes that decrease relative to each other by a factor of two or more, so each transition participates in O(log n) of the splitting steps in the algorithm. The partition refinement data structure allows each splitting step to be performed in time proportional to the number of transitions that participate in it. This remains the most efficient algorithm known for solving the problem, and for certain distributions of inputs its average-case complexity is even better, O(n log log n). Once Hopcroft's algorithm has been used to group the states of the input DFA into equivalence classes, the minimum DFA can be constructed by forming one state for each equivalence class. If S is a set of states in P, s is a state in S, and c is an input character, then the transition in the minimum DFA from the state for S, on input c, goes to the set containing the state that the input automaton would go to from state s on input c. The initial state of the minimum DFA is the one containing the initial state of the input DFA, and the accepting states of the minimum DFA are the ones whose members are accepting states of the input DFA. === Moore's algorithm === Moore's algorithm for DFA minimization is due to Edward F. Moore (1956). Like Hopcroft's algorithm, it maintains a partition that starts off separating the accepting from the rejecting states, and repeatedly refines the partition until no more refinements can be made. At each step, it replaces the current partition with the coarsest common refinement of s + 1 partitions, one of which is the current one and the rest of which are the preimages of the current partition under the transition functions for each of the input symbols. The algorithm terminates when this replacement does not change the current partition. Its worst-case time complexity is O(n2s): each step of the algorithm may be performed in time O(ns) using a variant of radix sort to reorder the states so that states in the same set of the new partition are consecutive in the ordering, and there are at most n steps since each one but the last increases the number of sets in the partition. The instances of the DFA minimization problem that cause the worst-case behavior are the same as for Hopcroft's algorithm. The number of steps th

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