AI Chatbot Training

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  Robomart

  Robomart is an American technology company headquartered in Santa Monica, California that builds autonomous smart shops for cafes, ice cream parlors, and quick-service restaurants. The company’s white label platform gives retailers the option to expand their footprint at a significantly lower cost than traditional brick-and-mortar real-estate. Robomarts are equipped with a proprietary checkout-free system, temperature controlled compartments, sensors for autonomous operation, and external cameras for added security. The company licenses its technology and white label applications to retailers who manage their fleet of stores and deploy them to their consumers’ locations. After consumers have taken goods from the robomart, their order is automatically calculated, their card on file is charged and they are sent a receipt. The company has announced partnerships with Unilever, Mars, and Fatty Mart. == History == Robomart was founded by Ali Ahmed, Tigran Shahverdyan, and Emad Suhail Rahim. The company debuted at CES 2018 where it unveiled its concept of a self-driving store. At GITEX 2018 the company presented its first functional prototype of a fully driverless Robomart. At the 2019 Consumer Electronics Show the company demonstrated the technology behind its autonomous stores and checkout-free shopping experience. In January 2019, Robomart announced its first partnership with U.S. grocery chain Stop & Shop to test its driverless stores. In December 2020, Robomart deployed the Pharmacy Robomart in a trial in West Hollywood. In June 2021, the company launched its commercial service with a fleet of Pharmacy and Snacks Robomarts operating within West Hollywood and Central Hollywood. In August 2023, Robomart announced a $2 million seed round, putting its to-date funding at $3.4 million. == Partnerships == In September 2019, Robomart partnered with Avery Dennison to source the RFID tags used to enable its checkout-free shopping experience. In December 2020, Robomart partnered with Zeeba Vans to provide vehicles for its growing fleet. In June 2021, Robomart partnered with REEF Technology to provide inventory management and restocking services. In addition, REEF's Light Speed grocery division serves as the first merchant selling products through Robomart. == Products == The company currently offers three Robomart types. The frozen Robomart that stocks ice cream, the refrigerated Robomart that stocks perishable foods, and the ambient Robomart that stocks shelf-stable goods.

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  Aravind Joshi

  Aravind Krishna Joshi (August 5, 1929 – December 31, 2017) was the Henry Salvatori Professor of Computer and Cognitive Science in the computer science department of the University of Pennsylvania. Joshi defined the tree-adjoining grammar formalism which is often used in computational linguistics and natural language processing. Joshi studied at Pune University and the Indian Institute of Science, where he was awarded a BE in electrical engineering and a DIISc in communication engineering respectively. Joshi's graduate work was done in the electrical engineering department at the University of Pennsylvania, and he was awarded his PhD in 1960. He became a professor at Penn and was the co-founder and co-director of the Institute for Research in Cognitive Science. == Awards and recognitions == Guggenheim fellow, 1971–72 Fellow of the Institute of Electrical and Electronics Engineers (IEEE), 1976 Best Paper Award at the National Conference on Artificial Intelligence, 1987 Founding Fellow of the American Association for Artificial Intelligence (AAAI), 1990 IJCAI Award for Research Excellence, 1997 Fellow of the Association for Computing Machinery, 1998 Elected to the National Academy of Engineering, 1999 First to be awarded the Association for Computational Linguistics Lifetime Achievement Award at the 40th anniversary meeting of the ACL, 2002 Awarded the Rumelhart Prize, 2003 Benjamin Franklin Medal in Computer and Cognitive Science, 2005 Doctor honoris causa of mathematical and physical sciences, Charles University in Prague, October 30, 2013 S.-Y. Kuroda Prize of the SIG Mathematics of Language of the ACL, 2013 === Awarded history === On April 21, 2005, Joshi was awarded the Franklin Institute's Benjamin Franklin Medal in Computer and Cognitive Science. The Franklin Institute citation states that he was awarded the medal "for his fundamental contributions to our understanding of how language is represented in the mind, and for developing techniques that enable computers to process efficiently the wide range of human languages. These advances have led to new methods for computer translation."

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  Collocation

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

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  PROMT

  ProMT is a lead Russian developer of language translation software for businesses and private users since 1991. The company provides on-premises software based on neural technologies. == History == On March 6, 1998, ProMT launched a free online translation services, which is now known as PROMT.One. In 1997, ProMT and the French company Softissimo developed a line of products for the European company Reverso. == Technology == Historically, ProMT systems used rule-based machine translation (RBMT) technology. In 2011 a hybrid approach which combined rule-based and statistical MT was implemented. In 2019, ProMT introduced its new neural technology and flagship solution - PROMT Neural Translation Server. Since then all MT systems developed by ProMT are based on neural machine translation. The software can run on Microsoft Windows, Linux, MacOS, iOS and Android and works in offline mode providing secure machine translation. As of 2025, it translates 62 languages from and to English, German, and Russian.

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  Tertiary review

  In software engineering, a tertiary review is a systematic review of systematic reviews. It is also referred to as a tertiary study in the software engineering literature. However, Umbrella review is the term more commonly used in medicine. Kitchenham et al. suggest that methodologically there is no difference between a systematic review and a tertiary review. However, as the software engineering community has started performing tertiary reviews new concerns unique to tertiary reviews have surfaced. These include the challenge of quality assessment of systematic reviews, search validation and the additional risk of double counting. == Examples of Tertiary reviews in software engineering literature == Test quality Machine Learning Test-driven development

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  Kalman filter

  In statistics and control theory, Kalman filtering (also known as linear quadratic estimation) is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unknown variables that tend to be more accurate than those based on a single measurement, by estimating a joint probability distribution over the variables for each time-step. The filter is constructed as a mean squared error minimiser, but an alternative derivation of the filter is also provided showing how the filter relates to maximum likelihood statistics. The filter is named after Rudolf E. Kálmán. Kalman filtering has numerous technological applications. A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and ships positioned dynamically. Furthermore, Kalman filtering is much applied in time series analysis tasks such as signal processing and econometrics. Kalman filtering is also important for robotic motion planning and control, and can be used for trajectory optimization. Kalman filtering also works for modeling the central nervous system's control of movement. Due to the time delay between issuing motor commands and receiving sensory feedback, the use of Kalman filters provides a realistic model for making estimates of the current state of a motor system and issuing updated commands. The algorithm works via a two-phase process: a prediction phase and an update phase. In the prediction phase, the Kalman filter produces estimates of the current state variables, including their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some error, including random noise) is observed, these estimates are updated using a weighted average, with more weight given to estimates with greater certainty. The algorithm is recursive. It can operate in real time, using only the present input measurements and the state calculated previously and its uncertainty matrix; no additional past information is required. Optimality of Kalman filtering assumes that errors have a normal (Gaussian) distribution. In the words of Rudolf E. Kálmán, "The following assumptions are made about random processes: Physical random phenomena may be thought of as due to primary random sources exciting dynamic systems. The primary sources are assumed to be independent gaussian random processes with zero mean; the dynamic systems will be linear." Regardless of Gaussianity, however, if the process and measurement covariances are known, then the Kalman filter is the best possible linear estimator in the minimum mean-square-error sense, although there may be better nonlinear estimators. It is a common misconception (perpetuated in the literature) that the Kalman filter cannot be rigorously applied unless all noise processes are assumed to be Gaussian. Extensions and generalizations of the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems. The basis is a hidden Markov model such that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions. Kalman filtering has been used successfully in multi-sensor fusion, and distributed sensor networks to develop distributed or consensus Kalman filtering. == History == The filtering method is named for Hungarian émigré Rudolf E. Kálmán, although Thorvald Nicolai Thiele and Peter Swerling developed a similar algorithm earlier. Richard S. Bucy of the Johns Hopkins Applied Physics Laboratory contributed to the theory, causing it to be known sometimes as Kalman–Bucy filtering. Kalman was inspired to derive the Kalman filter by applying state variables to the Wiener filtering problem. Stanley F. Schmidt is generally credited with developing the first implementation of a Kalman filter. He realized that the filter could be divided into two distinct parts, with one part for time periods between sensor outputs and another part for incorporating measurements. It was during a visit by Kálmán to the NASA Ames Research Center that Schmidt saw the applicability of Kálmán's ideas to the nonlinear problem of trajectory estimation for the Apollo program resulting in its incorporation in the Apollo navigation computer. This digital filter is sometimes termed the Stratonovich–Kalman–Bucy filter because it is a special case of a more general, nonlinear filter developed by the Soviet mathematician Ruslan Stratonovich. In fact, some of the special case linear filter's equations appeared in papers by Stratonovich that were published before the summer of 1961, when Kalman met with Stratonovich during a conference in Moscow. This Kalman filtering was first described and developed partially in technical papers by Swerling (1958), Kalman (1960) and Kalman and Bucy (1961). The Apollo computer used 2k of magnetic core RAM and 36k wire rope [...]. The CPU was built from ICs [...]. Clock speed was under 100 kHz [...]. The fact that the MIT engineers were able to pack such good software (one of the very first applications of the Kalman filter) into such a tiny computer is truly remarkable. Kalman filters have been vital in the implementation of the navigation systems of U.S. Navy nuclear ballistic missile submarines, and in the guidance and navigation systems of cruise missiles such as the U.S. Navy's Tomahawk missile and the U.S. Air Force's Air Launched Cruise Missile. They are also used in the guidance and navigation systems of reusable launch vehicles and the attitude control and navigation systems of spacecraft which dock at the International Space Station. == Overview of the calculation == Kalman filtering uses a system's dynamic model (e.g., physical laws of motion), known control inputs to that system, and multiple sequential measurements (such as from sensors) to form an estimate of the system's varying quantities (its state) that is better than the estimate obtained by using only one measurement alone. As such, it is a common sensor fusion and data fusion algorithm. Noisy sensor data, approximations in the equations that describe the system evolution, and external factors that are not accounted for, all limit how well it is possible to determine the system's state. The Kalman filter deals effectively with the uncertainty due to noisy sensor data and, to some extent, with random external factors. The Kalman filter produces an estimate of the state of the system as an average of the system's predicted state and of the new measurement using a weighted average. The purpose of the weights is that values with better (i.e., smaller) estimated uncertainty are "trusted" more. The weights are calculated from the covariance, a measure of the estimated uncertainty of the prediction of the system's state. The result of the weighted average is a new state estimate that lies between the predicted and measured state, and has a better estimated uncertainty than either alone. This process is repeated at every time step, with the new estimate and its covariance informing the prediction used in the following iteration. This means that Kalman filter works recursively and requires only the last "best guess", rather than the entire history, of a system's state to calculate a new state. The measurements' certainty-grading and current-state estimate are important considerations. It is common to discuss the filter's response in terms of the Kalman filter's gain. The Kalman gain is the weight given to the measurements and current-state estimate, and can be "tuned" to achieve a particular performance. With a high gain, the filter places more weight on the most recent measurements, and thus conforms to them more responsively. With a low gain, the filter conforms to the model predictions more closely. At the extremes, a high gain (close to one) will result in a more jumpy estimated trajectory, while a low gain (close to zero) will smooth out noise but decrease the responsiveness. When performing the actual calculations for the filter (as discussed below), the state estimate and covariances are coded into matrices because of the multiple dimensions involved in a single set of calculations. This allows for a representation of linear relationships between different state variables (such as position, velocity, and acceleration) in any of the transition models or covariances. == Example application == As an example application, consider the problem of determining the precise location of a truck. The truck can be equipped with a GPS unit that provides an estimate of the position within a few meters. The GPS estimate is likely to be noisy; readings 'jump around' rapidly, though remaining within a few meters of the real position. In addition, since the truck is expected to follow the laws of physics, its position can also be estimated by integrating its velocity over time, determined by keeping track of wheel revolutions and the

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  Magnetic ink character recognition

  Magnetic ink character recognition code, known in short as MICR code, is a character recognition technology used mainly by the banking industry to streamline the processing and clearance of cheques and other documents. MICR encoding, called the MICR line, is at the bottom of cheques and other vouchers and typically includes the document-type indicator, bank code, bank account number, cheque number, cheque amount (usually added after a cheque is presented for payment), and a control indicator. The format for the bank code and bank account number is country-specific. The technology allows MICR readers to scan and read the information directly into a data-collection device. Unlike barcode and similar technologies, MICR characters can be read easily by humans. MICR encoded documents can be processed much faster and more accurately than conventional OCR encoded documents. == Pre-Unicode standard representation == The ISO standard ISO 2033:1983, and the corresponding Japanese Industrial Standard JIS X 9010:1984 (originally JIS C 6229–1984), define character encodings for OCR-A, OCR-B and E-13B. == International spread == There are two major MICR fonts in use: E-13B and CMC-7. There is no particular international agreement on which countries use which font. In practice, this does not create particular problems as cheques and other vouchers do not usually flow out of a particular jurisdiction. The E-13B font has been adopted as an international standard in ISO 1004-1:2013, and is the standard in Australia, Canada, the United Kingdom, the United States, as well as Central America and much of Asia, besides other countries. The CMC-7 font has been adopted as an international standard in ISO 1004-2:2013, and is widely used in Europe, including France and Italy, Mexico, and South America, including Argentina, Brazil, Chile, besides other countries. Israel is the only country that can use both fonts simultaneously, though the practice makes the system significantly less efficient. This situation is the product of the Israelis adopting CMC-7, while the Palestinians opted for E-13B. == Fonts == === E-13B === E-13B is a 14-character set, comprising the 10 decimal digits, and the following symbols: ⑆ (transit: used to delimit a bank code); ⑈ (on-us: used to delimit a customer account number); ⑇ (amount: used to delimit a transaction amount); ⑉ (dash: used to delimit parts of numbers—e.g., routing numbers or account numbers). In the check printing and banking industries the E-13B MICR line is also commonly referred to as the TOAD line. This reference comes from the 4 characters: Transit, On-us, Amount, and Dash. Compared to CMC-7, some pairs of E-13B characters (notably 2 and 5) can produce relatively similar results when magnetically scanned; however, as a fallback if magnetic reading fails, E-13B also performs well under optical character recognition. The E-13B repertoire can be represented in Unicode (see below). The official Unicode names contain misnomers. For example, the ⑈ on-us symbol is official titled "OCR Dash". Prior to Unicode, it could be encoded according to ISO 2033:1983, which encodes digits in their usual ASCII locations, transit as 0x3A, on-us as 0x3C, amount as 0x3B, and dash as 0x3D. For EBCDIC, IBM code page 1001 encodes digits in their usual EBCDIC locations, transit as 0xDB, on-us as 0xEB, amount as 0xCB, and dash as 0xFB. IBM code page 1032 extends code page 1001 by adding alternative encodings for transit at 0x5C, 0x7A and 0xC1, on-us at 0x4C, 0x61 and 0xC3, amount at 0x5B, 0x5E and 0xC2 and dash at 0x60, 0x7E and 0xC4, in addition to a zero-width space at 0x5A. These alternative representations were added for interoperability with Siemens and Océ printers. === CMC-7 === CMC-7 includes 10 numeric digits, 26 capital letters, and 5 control characters: S I (internal), S II (terminator), S III (amount), S IV (an unused character), and S V (routing). CMC-7 has a barcode format, with every character having two distinct large gaps in different places, as well as distinct patterns in between, to minimize any chance for character confusion while reading magnetically; however, these bars are too close and narrow to be reliably recognised at a typical scan resolution if falling back to optical scanning. CMC-7 can also produce superficially successful, but incorrect, scans of upside-down MICR lines. Unicode does not include support for the CMC-7 control symbols. IBM code page 1033 encodes: Digits and capitals in their usual EBCDIC locations S I (internal) as 0x5E, 0x61 or 0xCB; S II (terminator) as 0x4C, 0x5B or 0xEB; S III (amount) as 0x60, 0x7E or 0xFB; S IV as 0x50, 0x7A or 0xDB; S V (routing) as 0x5C, 0x6E or 0xBB. == MICR reader == MICR characters are printed on documents in one of the two MICR fonts, using magnetizable (commonly known as magnetic) ink or toner, usually containing iron oxide. In scanning, the document is passed through a MICR reader, which performs two functions: magnetization of the ink, and detection of the characters. The characters are read by a MICR reader head, a device similar to the playback head of a tape recorder. As each character passes over the head, it produces a unique waveform that can be easily identified by the system. MICR readers are the primary tool for cheque sorting and are used across the cheque distribution network at multiple stages. For example, a merchant will use a MICR reader to sort cheques by bank and send the sorted cheques to a clearing house for redistribution to those banks. Upon receipt, the banks perform another MICR sort to determine which customer's account is charged and to which branch the cheque should be sent on its way back to the customer. However, many banks no longer offer this last step of returning the cheque to the customer. Instead, cheques are scanned and stored digitally. Sorting of cheques is done as per the geographical coverage of banks in a nation. == Unicode == OCR and MICR characters have been included in the Unicode Standard since at least version 1.1 (June 1993). Since the Unicode Character Database only tracks characters starting with version 1.1, they may also have been present in Unicode 1.0 or 1.0.1. The Unicode block that includes OCR and MICR characters is called Optical Character Recognition and covers U+2440–U+245F. Of the characters in this block, four are from the MICR E-13B font: U+2446 ⑆ OCR BRANCH BANK IDENTIFICATION U+2447 ⑇ OCR AMOUNT OF CHECK U+2448 ⑈ OCR DASH (corrected alias MICR ON US SYMBOL) U+2449 ⑉ OCR CUSTOMER ACCOUNT NUMBER (corrected alias MICR DASH SYMBOL) The names of the latter two characters were inadvertently switched when they were named in ISO/IEC 10646:1993, and they have been assigned accurate names as formal aliases. Per the Unicode Stability Policy, the existing names remain, allowing their use as stable identifiers. Additionally, all four characters have informative (non-formal) aliases in the Unicode charts: "transit", "amount", "on-us", and "dash" respectively. Prior to Unicode, these symbols had been encoded by the ISO-IR-98 encoding defined by ISO 2033:1983, in which they were simply named SYMBOL ONE through SYMBOL FOUR. They were encoded immediately following the digits, which were encoded at their ASCII locations. Although ISO 2033 also specifies encoding for OCR-A and OCR-B, its encoding for E-13B is known simply as ISO_2033-1983 by the IANA. == History == Before the mid-1940s, cheques were processed manually using the Sort-A-Matic or Top Tab Key method. The processing and cheque clearing was very time-consuming and was a significant cost in cheque clearance and bank operations. As the number of cheques increased, ways were sought for automating the process. Standards were developed to ensure uniformity in financial institutions. By the mid-1950s, the Stanford Research Institute and General Electric Computer Laboratory had developed the first automated system to process cheques using MICR. The same team also developed the E-13B MICR font. "E" refers to the font being the fifth considered, and "B" to the fact that it was the second version. The "13" refers to the 0.013-inch character grid. The trial of MICR E-13B font was shown to the American Bankers Association (ABA) in July 1956, which adopted it in 1958 as the MICR standard for negotiable documents in the United States. ABA adopted MICR as its standard because machines could read MICR accurately, and MICR could be printed using existing technology. In addition, MICR remained machine readable, even through overstamping, marking, mutilation and more. The first cheques using MICR were printed by the end of 1959. Although compliance with MICR standards was voluntary in the United States, it had been almost universally adopted in the United States by 1963. In 1963, ANSI adopted the ABA's E-13B font as the American standard for MICR printing, and E-13B was also standardized as ISO 1004:1995. Other countries set their own standards, though the MICR readers and m

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  Generalized nondeterministic finite automaton

  In the theory of computation, a generalized nondeterministic finite automaton (GNFA), also known as an expression automaton or a generalized nondeterministic finite state machine, is a variation of a nondeterministic finite automaton (NFA) where each transition is labeled with any regular expression. The GNFA reads blocks of symbols from the input which constitute a string as defined by the regular expression on the transition. There are several differences between a standard finite state machine and a generalized nondeterministic finite state machine. A GNFA must have only one start state and one accept state, and these cannot be the same state, whereas an NFA or DFA both may have several accept states, and the start state can be an accept state. A GNFA must have only one transition between any two states, whereas a NFA or DFA both allow for numerous transitions between states. In a GNFA, a state has a single transition to every state in the machine, although often it is a convention to ignore the transitions that are labelled with the empty set when drawing generalized nondeterministic finite state machines. == Formal definition == A GNFA can be defined as a 5-tuple, (S, Σ, T, s, a), consisting of a finite set of states (S); a finite set called the alphabet (Σ); a transition function (T : (S ∖ {\displaystyle \setminus } {a}) × (S ∖ {\displaystyle \setminus } {s}) → R); a start state (s ∈ S); an accept state (a ∈ S); where R is the collection of all regular expressions over the alphabet Σ. The transition function takes as its argument a pair of two states and outputs a regular expression (the label of the transition). This differs from other finite state machines, which take as input a single state and an input from the alphabet (or the empty string in the case of nondeterministic finite state machines) and outputs the next state (or the set of possible states in the case of nondeterministic finite state machines). A DFA or NFA can easily be converted into a GNFA and then the GNFA can be easily converted into a regular expression by repeatedly collapsing parts of it to single edges until S = {s, a}. Similarly, GNFAs can be reduced to NFAs by changing regular expression operators into new edges until each edge is labelled with a regular expression matching a single string of length at most 1. NFAs, in turn, can be reduced to DFAs using the powerset construction. This shows that GNFAs recognize the same set of formal languages as DFAs and NFAs.

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  Hierarchical navigable small world

  Hierarchical navigable small world (HNSW) is an algorithm for approximate nearest neighbor search. It is used to find items that are similar to a query item in a large collection, without comparing the query with every item one by one. The algorithm is commonly used for searching vector data. In these systems, an item such as a document, image, song, or user profile is represented by a list of numbers called a vector. Items with similar vectors are treated as similar according to the model that produced the vectors. HNSW provides a way to search these vectors quickly, especially in large datasets. HNSW stores vectors in a graph. Each vector is a node, and links connect it to some nearby vectors. The graph has several layers: upper layers contain fewer nodes and act like a rough map, while the bottom layer contains all nodes and gives a more detailed view. A search starts in an upper layer, follows links toward nodes that are closer to the query, and then repeats the process in lower layers until it finds a set of likely nearest neighbors. == Background == The nearest neighbor search problem asks which items in a dataset are closest to a query item. A direct search can compare the query with every item in the dataset, but this becomes slow when the dataset is large. Exact search methods based on spatial trees, such as the k-d tree and R-tree, can also become less effective for high-dimensional data, a problem often associated with the curse of dimensionality. Approximate nearest neighbor methods trade some exactness for speed or lower resource use. Instead of always guaranteeing the exact closest item, they try to return close items quickly. Other approximate methods include locality-sensitive hashing and product quantization. HNSW builds on research into small-world networks and navigable graphs. In a small-world graph, most nodes can be reached from other nodes through a short chain of links. In a navigable graph, a search procedure can use local information to move toward a target. Jon Kleinberg's work on navigation in small-world networks is an important example of this research area. Later work studied ways to add links that make graphs easier to navigate greedily. The HNSW algorithm extends earlier navigable small world methods for similarity search by adding a hierarchy of graph layers. This hierarchy helps the algorithm find a good region of the graph before doing a more detailed search in the bottom layer. == Algorithm == HNSW is based on a proximity graph. In this graph, nearby vectors are connected by edges. The algorithm uses these edges to move through the dataset, rather than scanning every vector. The graph is hierarchical. Every vector appears in the bottom layer. Some vectors are also placed in higher layers, with fewer vectors appearing as the layers go upward. The upper layers allow long-range movement across the dataset, while the lower layers allow a more detailed search near promising candidates. A typical search proceeds as follows: The search begins from an entry point in the highest layer. At each step, the algorithm looks at neighboring nodes and moves to a neighbor that is closer to the query. When it cannot find a closer neighbor in that layer, it moves down to the next layer. In the bottom layer, it explores a wider set of candidate nodes and returns the nearest candidates found. This search strategy is often described as greedy navigation. The algorithm repeatedly chooses locally better nodes, using the graph structure to approach the query point. == Construction and parameters == The HNSW graph is built incrementally. When a new vector is inserted, the algorithm assigns it a maximum layer, searches for nearby existing nodes, and connects the new node to selected neighbors in each layer where it appears. Implementations usually expose parameters that control the trade-off between speed, accuracy, memory use, and construction time. A higher number of graph connections can improve recall but requires more memory. A larger search candidate list can improve accuracy but makes queries slower. A larger construction candidate list can improve the quality of the graph but makes index building slower. Because HNSW is approximate, its results are not always identical to a full exact search. Its practical performance depends on the dataset, distance measure, implementation, and parameter settings. Benchmarking studies have found HNSW-based libraries to be strong performers among approximate nearest neighbor methods, although worst-case performance can differ from performance on common benchmark datasets. == Use in vector search systems == HNSW is used as an index in systems that store and search high-dimensional vectors. These systems include vector databases, search engines, and database extensions. Typical uses include semantic search, recommender systems, image similarity search, and retrieval-augmented generation. Several software projects implement or support HNSW. Libraries include hnswlib, which is associated with the original HNSW authors, and FAISS. Database and search systems that document HNSW support include Apache Lucene, Chroma, ClickHouse, DuckDB, MariaDB, Milvus, pgvector, Qdrant, and Redis.

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  Karsten Borgwardt

  Karsten Borgwardt (born 1980) is a German computer scientist and biologist specializing in machine learning and computational biology. Since February 2023, he has been a director at the Max Planck Institute of Biochemistry in Martinsried, Germany, where he leads the Department of Machine Learning and Systems Biology. == Education and career == Borgwardt was born in Kaiserslautern. He obtained a Diplom (equivalent to a master’s degree) in computer science from LMU Munich in 2004 and a Master of Science in biology from the University of Oxford in 2003. In 2007, he obtained his PhD from LMU Munich in computer science. Following a postdoctoral position at the University of Cambridge, he became a research group leader for machine learning and computational biology at the Max Planck Institute for Biological Cybernetics and the former Max Planck Institute for Developmental Biology in Tübingen in 2008. In 2011, Borgwardt was appointed professor of data mining in the life sciences at the University of Tübingen. In 2014, he joined ETH Zurich as an associate professor in the Department of Biosystems Science and Engineering (D-BSSE) and was promoted to full professor in 2017. During his tenure at ETH Zurich, he coordinated significant research programs, including two Marie Curie Innovative Training Networks and the Personalized Swiss Sepsis Study, focusing on the prediction of sepsis using machine learning. In 2023, he was appointed as Scientific Member of the Max Planck Society and as Director at the Max Planck Institute of Biochemistry in Martinsried. == Research contributions == Borgwardt’s research integrates big data analysis with biomedical research. He develops novel machine learning algorithms to detect patterns and statistical dependencies in large biological and medical datasets. His work aims to enable the automatic generation of new knowledge from big data and to understand the relationship between the function of biological systems and their molecular properties, which is fundamental for personalized medicine. == Awards and honors == During his studies, he was a scholar of the Stiftung Maximilianeum, and the Bavarian Foundation for the Promotion of the Gifted. Borgwardt received scholarships from the Studienstiftung des deutschen Volkes in 2002 and 2007. His PhD dissertation received the Heinz Schwärtzel Dissertation Award for Foundations of Computer Science in 2007. As a professor in Tübingen, he was awarded the Alfried-Krupp-Förderpreis for Young Professors in 2013. In 2015, he received an SNSF Starting Grant. In 2014, 2015 and 2016, he was listed in “Top 40 under 40” in Germany rankings selected by Capital magazine. In 2018, Borgwardt was named among “25 individuals who have the potential to shape the next 25 years” by Focus magazine. In 2023, Borgwardt received an honorary professorship from LMU Munich by the Faculty of Chemistry and Pharmacy. Publications from Borgwardt's group have received the Outstanding Student Paper Award in NIPS in 2009, the SIB Graduate Paper Award in 2020 and SIB Remarkable Output Awards in 2020 and 2021 from the Swiss Institute of Bioinformatics (SIB). == Selected publications == Weisfeiler-Lehman Graph Kernels (’‘Journal of Machine Learning Research’’, 2011): Introduced an efficient graph kernel based on the Weisfeiler-Lehman algorithm. “Direct antimicrobial resistance prediction from clinical MALDI-TOF mass spectra using machine learning” (’‘Nature Medicine’’, 2022): showcased the feasibility of predicting antimicrobial resistance from readily collected mass spectrometry data in the hospital. The new method is able to identify antibiotic resistance 24 hours earlier than previous methods.

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  Sepp Hochreiter

  Josef "Sepp" Hochreiter (born 14 February 1967) is a German computer scientist. Since 2018 he has led the Institute for Machine Learning at the Johannes Kepler University of Linz after having led the Institute of Bioinformatics from 2006 to 2018. In 2017 he became the head of the Linz Institute of Technology (LIT) AI Lab. Hochreiter is also a founding director of the Institute of Advanced Research in Artificial Intelligence (IARAI). Previously, he was at Technische Universität Berlin, at University of Colorado Boulder, and at the Technical University of Munich. He is a chair of the Critical Assessment of Massive Data Analysis (CAMDA) conference. Hochreiter has made contributions in the fields of machine learning, deep learning and bioinformatics, most notably the development of the long short-term memory (LSTM) neural network architecture, but also in meta-learning, reinforcement learning and biclustering with application to bioinformatics data. == Scientific career == === Long short-term memory (LSTM) === Hochreiter developed the long short-term memory (LSTM) neural network architecture in his diploma thesis in 1991 leading to the main publication in 1997. LSTM overcomes the problem of numerical instability in training recurrent neural networks (RNNs) that prevents them from learning from long sequences (vanishing or exploding gradient). In 2007, Hochreiter and others successfully applied LSTM with an optimized architecture to very fast protein homology detection without requiring a sequence alignment. LSTM networks have also been used in Google Voice for transcription and search, and in the Google Allo chat app for generating response suggestion with low latency. === Other machine learning contributions === Beyond LSTM, Hochreiter has developed "Flat Minimum Search" to increase the generalization of neural networks and introduced rectified factor networks (RFNs) for sparse coding which have been applied in bioinformatics and genetics. Hochreiter introduced modern Hopfield networks with continuous states and applied them to the task of immune repertoire classification. Hochreiter worked with Jürgen Schmidhuber in the field of reinforcement learning on actor-critic systems that learn by "backpropagation through a model". Hochreiter has been involved in the development of factor analysis methods with application to bioinformatics, including FABIA for biclustering, HapFABIA for detecting short segments of identity by descent and FARMS for preprocessing and summarizing high-density oligonucleotide DNA microarrays to analyze RNA gene expression. In 2006, Hochreiter and others proposed an extension of the support vector machine (SVM), the "Potential Support Vector Machine" (PSVM), which can be applied to non-square kernel matrices and can be used with kernels that are not positive definite. Hochreiter and his collaborators have applied PSVM to feature selection, including gene selection for microarray data. == Awards == Hochreiter was awarded the IEEE CIS Neural Networks Pioneer Prize in 2021 for his work on LSTM.

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  Noémie Elhadad

  Noémie Elhadad is an American data scientist who is an associate professor of biomedical informatics at the Columbia University Vagelos College of Physicians and Surgeons. As of 2022, she serves as the chair of the Department of Biomedical Informatics. Her research considers machine learning in bioinformatics, natural language processing and medicine. == Early life and education == Elhadad studied computer software engineering at École nationale supérieure d'électronique, informatique, télécommunications, mathématique et mécanique de Bordeaux (ENSEIRB). She completed her doctoral research at Columbia University. She was based in the Department of Computer Science, where she developed patient-focused text summaries of clinical literature. == Research and career == Elhadad joined the faculty at the City College of New York. In 2007 she joined the Department of Biomedical Informatics at Columbia University. She was made Chair of the Health Analytics Center at the Columbia Data Science Institute in 2013. Her research considers how clinical data, electronic health records and patient-generated data can enhance access to information for researchers, patients and physicians. She developed an artificial intelligence tool that supported patients in the NewYork-Presbyterian Hospital. Elhadad is interested in using data to advance women's health. She led the Citizen Endo Project that looks to comprehensively describe how patients experience endometriosis. It was built using principles of citizen science, using patient testimonials from focus groups in New York City and data aggregation. She created the app, Phendo, which asks patients about their experience of the disease. The name Phendo is a portmanteau of phenotyping endometriosis. Elhadad was announced as chair of the Department of Biomedical Informatics in December 2022. == Selected publications == Caruana, Rich; Lou, Yin; Gehrke, Johannes; Koch, Paul; Sturm, Marc; Elhadad, Noemie (August 10, 2015). "Intelligible Models for HealthCare". Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. New York, NY, USA: ACM. pp. 1721–1730. doi:10.1145/2783258.2788613. ISBN 9781450336642. S2CID 14190268. Chaitanya Shivade; Preethi Raghavan; Eric Fosler-Lussier; Peter J Embi; Noemie Elhadad; Stephen B Johnson; Albert M Lai (November 7, 2013). "A review of approaches to identifying patient phenotype cohorts using electronic health records". Journal of the American Medical Informatics Association. 21 (2): 221–230. doi:10.1136/AMIAJNL-2013-001935. ISSN 1067-5027. PMC 3932460. PMID 24201027. Wikidata Q37598951. Shivade, Chaitanya; Raghavan, Preethi; Fosler-Lussier, Eric; Embi, Peter J; Elhadad, Noemie; Johnson, Stephen B; Lai, Albert M (March 2014). "A review of approaches to identifying patient phenotype cohorts using electronic health records". Journal of the American Medical Informatics Association. 21 (2): 221–230. doi:10.1136/amiajnl-2013-001935. ISSN 1067-5027. PMC 3932460. PMID 24201027. == Personal life == Elhadad suffers from endometriosis.

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  ActivityPub

  ActivityPub is a protocol and open standard for decentralized social networking. It provides a client-to-server (C2S) API for creating and modifying content, as well as a federated server-to-server (S2S) protocol for delivering notifications and content to other servers. ActivityPub is the defining standard of the Fediverse, a decentralised social network of various social interaction models, and content types, which consists of independently managed instances of software such as Mastodon, Pixelfed and PeerTube, among others. ActivityPub is considered to be an update to the ActivityPump protocol used in pump.io, and the official W3C repository for ActivityPub is identified as a fork of ActivityPump. The creation of a new standard for decentralized social networking was prompted by the complexity of OStatus, the most commonly used protocol at the time. OStatus was built using a multitude of technologies (such as Atom, Salmon, WebSub and WebFinger), a product of the infrastructure used in GNU social (the originator and largest user of the OStatus protocol), which made it difficult to implement the protocol into new software. OStatus was also only designed to work with microblogging services, with little flexibility to the types of data that it could hold. The standard was first published by the World Wide Web Consortium (W3C) as a W3C Recommendation in January 2018 by the Social Web Working Group (SocialWG), a working group chartered to build the protocols and vocabularies needed to create a standard for social functionality. Shortly after, further development was moved to the Social Web Community Group (SocialCG), the successor to the SocialWG. == Design == ActivityPub uses the ActivityStreams 2.0 format for building its content, which itself uses JSON-LD. The three main data types used in ActivityPub are Objects, Activities and Actors. Objects are the most common data type, and can be images, videos, or more abstract items such as locations or events. Activities are actions that create and modify objects, for example a Create activity creates an object. Actors are representative of an individual, a group, an application or a service, and are the owners of objects. Every actor type contains an inbox and outbox stream, which sends and receives activities for a user. In order to publish data (for example liking an article), a user creates an activity that declares that they liked an Article object and publishes it to their outbox, where it is then delivered by the ActivityPub server via a POST request to the inboxes listed in the activity's to, bto, cc and bcc fields. The receiving servers then account for the newly received activity and update the article by adding the like action to it. === Example data === An example actor object that represents a user account: An example activity that likes an article object: An example article object: == Project status == The SocialCG previously organized a yearly free conference called ActivityPub Conf about the future of ActivityPub. Triages are held regularly to review issues pertaining to the ActivityPub and ActivityStreams 2.0 specifications as part of the SocialCG. In 2023, Germany's Sovereign Tech Fund donated €152,000 to socialweb.coop with the goal of building a new suite for testing various ActivityPub implementations and their compliance with the specification. === Adoption === The initial wave of adoption for ActivityPub (circa 2016–2018) came from software that was already using OStatus as their federation protocol, such as Mastodon, GNU social and Pleroma. Following the acquisition of Twitter by Elon Musk in 2022, many groups of users that were critical of the acquisition migrated to Mastodon, bringing new attention to the ActivityPub protocol with it. Various major social media platforms and corporations have since pledged to implement ActivityPub support, including Tumblr, Flipboard and Meta Platforms' Threads. Threads introduced crossposting to ActivityPub in 2024 for users outside of the European Economic Area, however full 2-way compatibility remains incomplete as of 2025. == Criticism == === Accidental denial-of-service attacks === Poorly optimized ActivityPub implementations can cause unintentional distributed denial-of-service (DDOS) attacks on other websites and servers, due to the decentralized nature of the network. An example would be Mastodon's implementation of OpenGraph link previews, wherein every instance that receives a post that contains a link with OpenGraph metadata will download the associated data, such as a thumbnail, in a very short timeframe, which can slow down or crash servers as a result of the sudden burst of requests. === Account migration === ActivityPub has been criticized for not natively supporting moving accounts from one server to another, forcing implementations to build their own solutions. While there has been work on building a standardized system for migrating accounts using the Move activity via the Fediverse Enhancement Proposal organization, the current proposal only allows for basic follower migration, with all other data remaining linked to the original account. === Missing content and data === ActivityPub implementations have been criticized for missing replies and parts of reply threads from remote posts, and presenting outdated statistics (e.g. likes and reposts) about remote posts. However, this isn't a problem with the ActivityPub protocol itself, but with implementations not refreshing their content for updated data when needed. == Software using ActivityPub == === Future implementations === Flarum, an internet forum software Forgejo, a Git forge and development platform === Uncertain future implementations === GitLab, a Git forge and development platform which had previously had an open issue discussing the topic, but was later closed due to the development team moving focus to other areas. Tumblr, a microblogging platform. Despite previous statements from Automattic CEO Matt Mullenweg, ActivityPub integration has been delayed indefinitely. The integration would have been implemented with its WordPress migration, as the first-party plugin for interoperability would have been used for federation. Flickr, an image and video hosting site.

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  Top 10 AI Video Editors Compared (2026)

  Looking for the best AI video editor? An AI video editor 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 video editor 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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  Laws of Form

  Laws of Form (hereinafter LoF) is a book by G. Spencer-Brown, written by August 1967 and published in 1969. The book straddles the boundary between mathematics and philosophy. LoF describes three distinct logical systems: The primary arithmetic (described in Chapter 4 of LoF), whose models include Boolean arithmetic; The primary algebra (Chapter 6 of LoF), whose models include the two-element Boolean algebra (hereinafter abbreviated 2), Boolean logic, and the classical propositional calculus; Equations of the second degree (Chapter 11), whose interpretations include finite automata and Alonzo Church's Restricted Recursive Arithmetic (RRA). "Boundary algebra" is a Meguire (2011) term for the union of the primary algebra and the primary arithmetic. Laws of Form sometimes loosely refers to the "primary algebra" as well as to LoF. == Contents == The preface states that the work was first explored in 1959, and Spencer Brown cites Bertrand Russell as being supportive of his endeavour. He also thanks J. C. P. Miller of University College London for helping with the proofreading and offering other guidance. In 1963 Spencer Brown was invited by Harry Frost, staff lecturer in the physical sciences at the department of Extra-Mural Studies of the University of London, to deliver a course on the mathematics of logic. LoF emerged from work in electronic engineering its author did around 1960. Key ideas of the LOF were first outlined in his 1961 manuscript Design with the Nor, which remained unpublished until 2021, and further refined during subsequent lectures on mathematical logic he gave under the auspices of the University of London's extension program. LoF has appeared in several editions. The second series of editions appeared in 1972 with the "Preface to the First American Edition", which emphasised the use of self-referential paradoxes, and the most recent being a 1997 German translation. LoF has never gone out of print. LoF's mystical and declamatory prose and its love of paradox make it a challenging read for all. Spencer-Brown was influenced by Ludwig Wittgenstein and R. D. Laing. LoF also echoes a number of themes from the writings of Charles Sanders Peirce, Bertrand Russell, and Alfred North Whitehead. The work has had curious effects on some classes of its readership; for example, on obscure grounds, it has been claimed that the entire book is written in an operational way, giving instructions to the reader instead of telling them what "is", and that in accordance with G. Spencer-Brown's interest in paradoxes, the only sentence that makes a statement that something is, is the statement which says no such statements are used in this book. Furthermore, the claim asserts that except for this one sentence the book can be seen as an example of E-Prime. What prompted such a claim, is obscure, either in terms of incentive, logical merit, or as a matter of fact, because the book routinely and naturally uses the verb to be throughout, and in all its grammatical forms, as may be seen both in the original and in quotes shown below. == Reception == Ostensibly a work of formal mathematics and philosophy, LoF became something of a cult classic: it was praised by Heinz von Foerster when he reviewed it for the Whole Earth Catalog. Those who agree point to LoF as embodying an enigmatic "mathematics of consciousness", its algebraic symbolism capturing an (perhaps even "the") implicit root of cognition: the ability to "distinguish". LoF argues that primary algebra reveals striking connections among logic, Boolean algebra, and arithmetic, and the philosophy of language and mind. Stafford Beer wrote in a review for Nature in 1969, "When one thinks of all that Russell went through sixty years ago, to write the Principia, and all we his readers underwent in wrestling with those three vast volumes, it is almost sad". Banaschewski (1977) argues that the primary algebra is nothing but new notation for Boolean algebra. Indeed, the two-element Boolean algebra 2 can be seen as the intended interpretation of the primary algebra. Yet the notation of the primary algebra: Fully exploits the duality characterizing not just Boolean algebras but all lattices; Highlights how syntactically distinct statements in logic and 2 can have identical semantics; Dramatically simplifies Boolean algebra calculations, and proofs in sentential and syllogistic logic. Moreover, the syntax of the primary algebra can be extended to formal systems other than 2 and sentential logic, resulting in boundary mathematics. LoF has influenced, among others, Heinz von Foerster, Louis Kauffman, Niklas Luhmann, Humberto Maturana, Francisco Varela and William Bricken. Some of these authors have modified the primary algebra in a variety of interesting ways. LoF claimed that certain well-known mathematical conjectures of very long standing, such as the four color theorem, Fermat's Last Theorem, and the Goldbach conjecture, are provable using extensions of the primary algebra. Spencer-Brown eventually circulated a purported proof of the four color theorem, but it was met with skepticism. == The form (Chapter 1) == The symbol: Also called the "mark" or "cross", is the essential feature of the Laws of Form. In Spencer-Brown's inimitable and enigmatic fashion, the Mark symbolizes the root of cognition, i.e., the dualistic Mark indicates the capability of differentiating a "this" from "everything else but this". In LoF, a Cross denotes the drawing of a "distinction", and can be thought of as signifying the following, all at once: The act of drawing a boundary around something, thus separating it from everything else; That which becomes distinct from everything by drawing the boundary; Crossing from one side of the boundary to the other. All three ways imply an action on the part of the cognitive entity (e.g., person) making the distinction. As LoF puts it: "The first command: Draw a distinction can well be expressed in such ways as: Let there be a distinction, Find a distinction, See a distinction, Describe a distinction, Define a distinction, Or: Let a distinction be drawn". (LoF, Notes to chapter 2) The counterpoint to the Marked state is the Unmarked state, which is simply nothing, the void, or the un-expressable infinite represented by a blank space. It is simply the absence of a Cross. No distinction has been made and nothing has been crossed. The Marked state and the void are the two primitive values of the Laws of Form. The Cross can be seen as denoting the distinction between two states, one "considered as a symbol" and another not so considered. From this fact arises a curious resonance with some theories of consciousness and language. Paradoxically, the Form is at once Observer and Observed, and is also the creative act of making an observation. LoF (excluding back matter) closes with the words: ...the first distinction, the Mark and the observer are not only interchangeable, but, in the form, identical. C. S. Peirce came to a related insight in the 1890s; see § Related work. == The primary arithmetic (Chapter 4) == The syntax of the primary arithmetic goes as follows. There are just two atomic expressions: The empty Cross ; All or part of the blank page (the "void"). There are two inductive rules: A Cross may be written over any expression; Any two expressions may be concatenated. The semantics of the primary arithmetic are perhaps nothing more than the sole explicit definition in LoF: "Distinction is perfect continence". Let the "unmarked state" be a synonym for the void. Let an empty Cross denote the "marked state". To cross is to move from one value, the unmarked or marked state, to the other. We can now state the "arithmetical" axioms A1 and A2, which ground the primary arithmetic (and hence all of the Laws of Form): "A1. The law of Calling". Calling twice from a state is indistinguishable from calling once. To make a distinction twice has the same effect as making it once. For example, saying "Let there be light" and then saying "Let there be light" again, is the same as saying it once. Formally: = {\displaystyle \ =} "A2. The law of Crossing". After crossing from the unmarked to the marked state, crossing again ("recrossing") starting from the marked state returns one to the unmarked state. Hence recrossing annuls crossing. Formally: = {\displaystyle \ =} In both A1 and A2, the expression to the right of '=' has fewer symbols than the expression to the left of '='. This suggests that every primary arithmetic expression can, by repeated application of A1 and A2, be simplified to one of two states: the marked or the unmarked state. This is indeed the case, and the result is the expression's "simplification". The two fundamental metatheorems of the primary arithmetic state that: Every finite expression has a unique simplification. (T3 in LoF); Starting from an initial marked or unmarked state, "complicating" an expression by a finite number of repeated application of A1 and A2 cannot yield

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