Xinhua News Agency and Sogou of China developed an artificial intelligence (AI) for news reporting purposes. The AI was unveiled in 2018. It is touted to be the "world's first AI news anchor". == History == The AI was unveiled at the 2018 World Internet Conference in Wuzhen, Zhejiang, China. The AI devises avatars patterned after real life Xinhua anchors. The AI patterned after Qiu Hao spoke in Chinese, while the one derived from the likeness of Zhang Zhao speaks in English. The unveiling of the AI raised concerns of its impact on employment. Xinhua and Sogou unveiled Xin Xiaomeng, an AI with a female avatar in 2019. People's Daily followed suit by unveiling its own AI newscaster in 2023.
Film recorder
A film recorder is a graphical output device for transferring images to photographic film from a digital source. In a typical film recorder, an image is passed from a host computer to a mechanism to expose film through a variety of methods, historically by direct photography of a high-resolution cathode-ray tube (CRT) display. The exposed film can then be developed using conventional developing techniques, and displayed with a slide or motion picture projector. The use of film recorders predates the current use of digital projectors, which eliminate the time and cost involved in the intermediate step of transferring computer images to film stock, instead directly displaying the image signal from a computer. Motion picture film scanners are the opposite of film recorders, copying content from film stock to a computer system. Film recorders can be thought of as modern versions of kinescopes. == Design == === Operation === All film recorders typically work in the same manner. The image is fed from a host computer as a raster stream over a digital interface. A film recorder exposes film through various mechanisms; flying spot (early recorders); photographing a high resolution video monitor; electron beam recorder (Sony HDVS); a CRT scanning dot (Celco); focused beam of light from a light valve technology (LVT) recorder; a scanning laser beam (Arrilaser); or recently, full-frame LCD array chips. For color image recording on a CRT film recorder, the red, green, and blue channels are sequentially displayed on a single gray scale CRT, and exposed to the same piece of film as a multiple exposure through a filter of the appropriate color. This approach yields better resolution and color quality than possible with a tri-phosphor color CRT. The three filters are usually mounted on a motor-driven wheel. The filter wheel, as well as the camera's shutter, aperture, and film motion mechanism are usually controlled by the recorder's electronics and/or the driving software. CRT film recorders are further divided into analog and digital types. The analog film recorder uses the native video signal from the computer, while the digital type uses a separate display board in the computer to produce a digital signal for a display in the recorder. Digital CRT recorders provide a higher resolution at a higher cost compared to analog recorders due to the additional specialized hardware. Typical resolutions for digital recorders were quoted as 2K and 4K, referring to 2048×1366 and 4096×2732 pixels, respectively, while analog recorders provided a resolution of 640×428 pixels in comparison. Higher-quality LVT film recorders use a focused beam of light to write the image directly onto a film loaded spinning drum, one pixel at a time. In one example, the light valve was a liquid-crystal shutter, the light beam was steered with a lens, and text was printed using a pre-cut optical mask. The LVT will record pixel beyond grain. Some machines can burn 120-res or 120 lines per millimeter. The LVT is basically a reverse drum scanner. The exposed film is developed and printed by regular photographic chemical processing. === Formats === Film recorders are available for a variety of film types and formats. The 35 mm negative film and transparencies are popular because they can be processed by any photo shop. Single-image 4×5 film and 8×10 are often used for high-quality, large format printing. Some models have detachable film holders to handle multiple formats with the same camera or with Polaroid backs to provide on-site review of output before exposing film. == Uses == Film recorders are used in digital printing to generate master negatives for offset and other bulk printing processes. For preview, archiving, and small-volume reproduction, film recorders have been rendered obsolete by modern printers that produce photographic-quality hardcopies directly on plain paper. They are also used to produce the master copies of movies that use computer animation or other special effects based on digital image processing. However, most cinemas nowadays use Digital Cinema Packages on hard drives instead of film stock. === Computer graphics === Film recorders were among the earliest computer graphics output devices; for example, the IBM 740 CRT Recorder was announced in 1954. Film recorders were also commonly used to produce slides for slide projectors; but this need is now largely met by video projectors that project images directly from a computer to a screen. The terms "slide" and "slide deck" are still commonly used in presentation programs. === Current uses === Currently, film recorders are primarily used in the motion picture film-out process for the ever increasing amount of digital intermediate work being done. Although significant advances in large venue video projection alleviates the need to output to film, there remains a deadlock between the motion picture studios and theater owners over who should pay for the cost of these very costly projection systems. This, combined with the increase in international and independent film production, will keep the demand for film recording steady for at least a decade. == Key manufacturers == Traditional film recorder manufacturers have all but vanished from the scene or have evolved their product lines to cater to the motion picture industry. Dicomed was one such early provider of digital color film recorders. Polaroid, Management Graphics, Inc, MacDonald-Detwiler, Information International, Inc., and Agfa were other producers of film recorders. Arri is the only current major manufacturer of film recorders. Kodak Lightning I film recorder. One of the first laser recorders. Needed an engineering staff to set up. Kodak Lightning II film recorder used both gas and diode laser to record on to film. The last LVT machines produced by Kodak / Durst-Dice stopped production in 2002. There are no LVT film recorders currently being produced. LVT Saturn 1010 uses a LED exposure (RGB) to 8"x10" film at 1000-3000ppi. LUX Laser Cinema Recorder from Autologic/Information International in Thousand Oaks, California. Sales end in March 2000. Used on the 1997 film “Titanic”. Arri produces the Arrilaser line of laser-based motion picture film recorders. MGI produced the Solitaire line of CRT-based motion picture film recorders. Matrix, originally ImaPRO, a branch of Agfa Division, produced the QCR line of CRT-based motion picture film recorders. CCG, formerly Agfa film recorders, has been a steady manufacturer of film recorders based in Germany. In 2004 CCG introduced Definity, a motion picture film recorder utilizing LCD technology. In 2010 CCG introduced the first full LED LCD film recorder as a new step in film recording. Cinevator was made by Cinevation AS, in Drammen, Norway. The Cinevator was a real-time digital film recorder. It could record IN, IP and prints with and without sound Oxberry produced the Model 3100 film recorder camera system, with interchangeable pin-registered movements (shuttles) for 35 mm (full frame/Silent, 1.33:1) and 16 mm (regular 16, "2R"), and others have adapted the Oxberry movements for CinemaScope, 1.85:1, 1.75:1, 1.66:1, as well as Academy/Sound (1.37:1) in 35 mm and Super-16 in 16 mm ("1R"). For instance, the "Solitaire" and numerous others employed the Oxberry 3100 camera system. == History == Before video tape recorders or VTRs were invented, TV shows were either broadcast live or recorded to film for later showing, using the kinescope process. In 1967, CBS Laboratories introduced the Electronic Video Recording format, which used video and telecined-to-video film sources, which were then recorded with an electron-beam recorder at CBS' EVR mastering plant at the time to 35mm film stock in a rank of 4 strips on the film, which was then slit down to 4 8.75 mm (0.344 in) film copies, for playback in an EVR player. All types of CRT recorders were (and still are) used for film recording. Some early examples used for computer-output recording were the 1954 IBM 740 CRT Recorder, and the 1962 Stromberg-Carlson SC-4020, the latter using a Charactron CRT for text and vector graphic output to either 16 mm motion picture film, 16 mm microfilm, or hard-copy paper output. Later 1970 and 80s-era recording to B&W (and color, with 3 separate exposures for red, green, and blue)) 16 mm film was done with an EBR (Electron Beam Recorder), the most prominent examples made by 3M), for both video and COM (Computer Output Microfilm) applications. Image Transform in Universal City, California used specially modified 3M EBR film recorders that could perform color film-out recording on 16 mm by exposing three 16 mm frames in a row (one red, one green and one blue). The film was then printed to color 16 mm or 35 mm film. The video fed to the recorder could either be NTSC, PAL or SECAM. Later, Image Transform used specially modified VTRs to record 24 frame for their "Image Vision" system. The modified 1 inch type B videotape VTRs would record
Glottochronology
Glottochronology (from Attic Greek γλῶττα 'tongue, language' and χρόνος 'time') is the part of lexicostatistics which involves comparative linguistics and deals with the chronological relationship between languages. The idea was developed by Morris Swadesh in the 1950s in his article on Salish internal relationships. He developed the idea under two assumptions: there indeed exists a relatively stable basic vocabulary (referred to as Swadesh lists) in all languages of the world; and, any replacements happen in a way analogous to radioactive decay in a constant percentage per time elapsed. Using mathematics and statistics, Swadesh developed an equation to determine when languages separated and give an approximate time of when the separation occurred. His methods aimed to aid linguistic anthropologists by giving them a definitive way to determine a separation date between two languages. The formula provides an approximate number of centuries since two languages were supposed to have separated from a singular common ancestor. His methods also purported to provide information on when ancient languages may have existed. Despite multiple studies and literature containing the information of glottochronology, it is not widely used today and is surrounded with controversy. Glottochronology tracks language separation from thousands of years ago but many linguists are skeptical of the concept because it is more of a 'probability' rather than a 'certainty.' On the other hand, some linguists may say that glottochronology is gaining traction because of its relatedness to archaeological dates. Glottochronology is not as accurate as archaeological data, but some linguists still believe that it can provide a solid estimate. Over time many different extensions of the Swadesh method evolved; however, Swadesh's original method is so well known that 'glottochronology' is usually associated with him. == Methodology == The original method of glottochronology presumed that the core vocabulary of a language is replaced at a constant (or constant average) rate across all languages and cultures and so can be used to measure the passage of time. The process makes use of a list of lexical terms and morphemes which are similar to multiple languages. Lists were compiled by Morris Swadesh and assumed to be resistant against borrowing (originally designed in 1952 as a list of 200 items, but the refined 100-word list in Swadesh (1955) is much more common among modern day linguists). The core vocabulary was designed to encompass concepts common to every human language such as personal pronouns, body parts, heavenly bodies and living beings, verbs of basic actions, numerals, basic adjectives, kin terms, and natural occurrences and events. Through a basic word list, one eliminates concepts that are specific to a particular culture or time period. It has been found through differentiating word lists that the ideal is really impossible and that the meaning set may need to be tailored to the languages being compared. Word lists are not homogenous throughout studies and they are often changed and designed to suit both languages being studied. Linguists find that it is difficult to find a word list where all words used are culturally unbiased. Many alternative word lists have been compiled by other linguists and often use fewer meaning slots. The percentage of cognates (words with a common origin) in the word lists is then measured. The larger the percentage of cognates, the more recently the two languages being compared are presumed to have separated. === Glottochronologic constant === Determining word lists rely on morpheme decay or change in vocabulary. Morpheme decay must stay at a constant rate for glottochronology to be applied to a language. This leads to a critique of the glottochronologic formula because some linguists argue that the morpheme decay rate is not guaranteed to stay the same throughout history. American Linguist Robert Lees obtained a value for the "glottochronological constant" (r) of words by considering the known changes in 13 pairs of languages using the 200 word list. He obtained a value of 0.8048 ± 0.0176 with 90% confidence. For his 100-word list Swadesh obtained a value of 0.86, the higher value reflecting the elimination of semantically unstable words. === Divergence time === The basic formula of glottochronology proposed by Morris Swadesh is: t = − ln ( c ) 2 ln ( r ) {\displaystyle t=-{\frac {\ln(c)}{2\ln(r)}}} t = a given period of time from one stage of the language to another (measured in millennia), c = proportion of wordlist items retained at the end of that period and r = rate of replacement for that word list. By testing historically verifiable cases in which t is known by nonlinguistic data (such as the approximate distance from Classical Latin to modern Romance languages), Swadesh arrived at the empirical value of approximately 0.14 for L, (c?) which means that the rate of replacement constitutes around 14 words from the 100-wordlist per millennium. This is represented in the table below. === Results === Glottochronology was applied to a range of language families, including Salishan, Indo-European, Japonic, Afro-Asiatic, Chinese and Mayan and other American languages. For Amerind, correlations have been obtained with radiocarbon dating and blood groups as well as archaeology. === Example Wordlist === Below is an example of a basic word list composed of basic Turkish words and their English translations. == Discussion == The concept of language change is old, and its history is reviewed in Hymes (1973) and Wells (1973). In some sense, glottochronology is a reconstruction of history and can often be closely related to archaeology. Many linguistic studies find the success of glottochronology to be found alongside archaeological data. Glottochronology itself dates back to the mid-20th century. An introduction to the subject is given in Embleton (1986) and in McMahon and McMahon (2005). Glottochronology has been controversial ever since, partly because of issues of accuracy but also because of the question of whether its basis is sound (for example, Bergsland 1958; Bergsland and Vogt 1962; Fodor 1961; Chrétien 1962; Guy 1980). The concerns have been addressed by Dobson et al. (1972), Dyen (1973) and Kruskal, Dyen and Black (1973). The assumption of a single-word replacement rate can distort the divergence-time estimate when borrowed words are included (Thomason and Kaufman 1988). The presentations vary from "Why linguists don't do dates" to the one by Starostin discussed below. Since its original inception, glottochronology has been rejected by many linguists, mostly Indo-Europeanists of the school of the traditional comparative method. Criticisms have been answered in particular around three points of discussion: Criticism levelled against the higher stability of lexemes in Swadesh lists alone (Haarmann 1990) misses the point because a certain amount of losses only enables the computations (Sankoff 1970). The non-homogeneity of word lists often leads to lack of understanding between linguists. Linguists also have difficulties finding a completely unbiased list of basic cultural words. it can take a long time for linguists to find a viable word list which can take several test lists to find a usable list. Traditional glottochronology presumes that language changes at a stable rate. Thus, in Bergsland & Vogt (1962), the authors make an impressive demonstration, on the basis of actual language data verifiable by extralinguistic sources, that the "rate of change" for Icelandic constituted around 4% per millennium, but for closely connected Riksmal (Literary Norwegian), it would amount to as much as 20% (Swadesh's proposed "constant rate" was supposed to be around 14% per millennium). That and several other similar examples effectively proved that Swadesh's formula would not work on all available material, which is a serious accusation since evidence that can be used to "calibrate" the meaning of L (language history recorded during prolonged periods of time) is not overwhelmingly large in the first place. It is highly likely that the chance of replacement is different for every word or feature ("each word has its own history", among hundreds of other sources:). That global assumption has been modified and downgraded to single words, even in single languages, in many newer attempts (see below). There is a lack of understanding of Swadesh's mathematical/statistical methods. Some linguists reject the methods in full because the statistics lead to 'probabilities' when linguists trust 'certainties' more. A serious argument is that language change arises from socio-historical events that are, of course, unforeseeable and, therefore, uncomputable. == Modifications == Somewhere in between the original concept of Swadesh and the rejection of glottochronology in its entirety lies the idea that glottochronology as a formal method of linguistic
Pietro Perona
Pietro Perona (born 3 September 1961) is an Italian-American educator and computer scientist. He is the Allan E. Puckett Professor of Electrical Engineering and Computation and Neural Systems at the California Institute of Technology and director of the National Science Foundation Engineering Research Center in Neuromorphic Systems Engineering. He is known for his research in computer vision and is the director of the Caltech Computational Vision Group. == Academic biography == Perona obtained his D.Eng. in electrical engineering cum laude from the University of Padua in 1985 and completed his Ph.D. at the University of California, Berkeley in 1990. His dissertation was titled Finding Texture and Brightness Boundaries in Images, and his adviser was Jitendra Malik. In 1990, Perona was a postdoctoral fellow at the International Computer Science Institute at Berkeley. From 1990 to 1991, he was a postdoctoral fellow at the Massachusetts Institute of Technology in the Laboratory for Information and Decision Systems. He has been on the faculty of the California Institute of Technology since 1991, and he was named Allan E. Puckett Professor in 2008. == Research == Perona’s research focuses on the computational aspects of vision and learning. He developed the anisotropic diffusion equation, a partial differential equation that reduces noise in images while enhancing region boundaries. He is currently interested in visual recognition and in visual analysis of behavior. Perona and Serge Belongie lead the Visipedia project, which facilitates research on visual knowledge representation, visual search, and human-in-the-loop machine learning systems. Perona pioneered the study of visual categorization (including the publication of the Caltech 101 dataset) for which he was awarded the Longuet-Higgins Prize in 2013. He is also the recipient of the 2010 Koenderink Prize for Fundamental Contributions in Computer Vision, the 2003 Conference on Computer Vision and Pattern Recognition best paper award, and a 1996 NSF Presidential Young Investigator Award. == Media coverage == Perona has been quoted or had his research featured in various national media outlets, including the New York Times, Science Friday, The New Yorker, and the Los Angeles Times. In 2003, Perona and Stephen Nowlin organized the NEURO art exhibition, which brought together contemporary artists and scientists to explore neuromorphic engineering.
Deepset
deepset is an enterprise software vendor that provides developers with the tools to build production-ready Artificial Intelligence (AI) and natural language processing (NLP) systems, using architectures such as agents, retrieval augmented generation (RAG) and multimodal AI. It was founded in 2018 in Berlin by Milos Rusic, Malte Pietsch, and Timo Möller. deepset authored and maintains the open source software Haystack and its commercial SaaS and self-hosted (VPC, on-prem, air gapped) offering, Haystack Enterprise Platform. (formerly known as deepset Cloud and deepset AI Platform) == History == In June 2018, Milos Rusic, Malte Pietsch, and Timo Möller co-founded deepset in Berlin, Germany. In the same year, the company served first customers who wanted to implement NLP services by tailoring BERT language models to their domain. In July 2019, the company released the initial version of the open source software FARM. In November 2019, the company released the initial version of the open source software Haystack. Throughout 2020 and 2021 deepset published several applied research papers at EMNLP, COLING and ACL, the leading conferences in the area of NLP. In 2020, the research contributions comprised German language models named GBERT and GELECTRA, and a question answering dataset addressing the COVID-19 pandemic called COVID-QA, which was created in collaboration with Intel and has been annotated by biomedical experts. In 2021, the research contributions comprised German models and datasets for question answering and passage retrieval named GermanQuAD and GermanDPR, a semantic answer similarity metric, and an approach for multimodal retrieval of texts and tables to enable question answering on tabular data. Haystack contains implementations of all three contributions, enabling the use of the research through the open source framework. In November 2021, the development of the FARM framework was discontinued and its main features were integrated into the Haystack framework. In April 2022, the company announced its commercial SaaS offering deepset Cloud, which was rebranded in 2025 as Haystack Enterprise Platform supporting SaaS and on-premise deployment options. As of August 2023, the most popular finetuned language model created by deepset was downloaded more than 52 million times. In 2024, deepset was named a Gartner Cool Vendor in AI Engineering. In 2025, deepset was recognized for its growth by WirtschaftsWoche and Sifted and shared partnership integrations and announcements with Meta Llama Stack, MongoDB, NVIDIA, Amazon Web Services (AWS), and PwC. As of September 2025, the Haystack open source AI orchestration framework has more than 24,000 GitHub stars. == Products and applications == Haystack is an open source Python AI Orchestration framework for building custom AI agents and applications with large language models. With its modular building block components, software developers and AI engineers can implement pipelines to build and customize various AI architectures over large document and multimodal data collections, such as agents, retrieval augmented generation (RAG), intelligent document processing (IDP), text-to-SQL as well as document retrieval, semantic search, text generation, question answering, or summarization. Haystack emphasizes context engineering, an approach to AI system design that focuses on explicit control over how contextual information is retrieved, structured, routed to language models, and evaluated after generation. This allows developers to build AI systems with transparent data flow, tool usage, and configurable reasoning processes. Haystack integrates with 90+ model and technology providers including Hugging Face Transformers, Elasticsearch, OpenSearch, OpenAI, Cohere, Anthropic, Mistral and others. Developers can extend these integrations with their own custom components. The framework has an active community on Discord with more than 4k members and GitHub, where so far more than 300 people have contributed to its continuous development, and engage on Meetup. Thousands of organizations use the framework, including public sector leaders like the European Commission and Global 500 enterprises like Airbus, Intel, NVIDIA, Lufthansa, Netflix, Apple, Infineon, Alcatel-Lucent Enterprise, BetterUp, Etalab, Sooth.ai, and Lego. On top of the Haystack open source framework, deepset offers two enterprise offerings to organizations. Haystack Enterprise Starter provides enterprise support on the open source framework from the Haystack engineering team as well as a private GitHub repository with production use case templates and Kubernetes deployment guides. The Haystack Enterprise Platform supports customers at building scalable AI applications by covering the entire process of prototyping, experimentation, deployment, monitoring, and governance. It is built on the Haystack open source framework and is available for hosting in the cloud and self-hosted via VPC, on-premise, or air gapped environments. deepset's enterprise tools are used by organizations including The European Commission, The Economist, Oxford University Press, the German Federal Ministry of Research, Technology, and Space (BMFTR), Manz Verlag, and the German Armed Forces. FARM was an earlier framework for adapting representation models. One of its core concepts was the implementation of adaptive models, which comprised language models and an arbitrary number of prediction heads. FARM supported domain-adaptation and finetuning of these models with advanced options, for example gradient accumulation, cross-validation or automatic mixed-precision training. Its main features were integrated into Haystack in November 2021, and its development was discontinued at that time. == Funding == On August 9, 2023, deepset announced a Series B investment round of $30 million led by Balderton Capital and including participation from existing investors GV, System.One, Lunar Ventures and Harpoon Ventures. On April 28, 2022, deepset announced a Series A investment round of $14 million led by GV, with the participation of Harpoon Ventures, Acequia Capital and a team of experienced commercial open source software and machine learning founders, such as Alex Ratner (Snorkel AI), Mustafa Suleyman (Deepmind), Spencer Kimball (Cockroach Labs), Jeff Hammerbacher (Cloudera) and Emil Eifrem (Neo4j). A previous pre-seed investment round of $1.6 million on March 8, 2021, was led by System.One and Lunar Ventures, who also participated in the subsequent Series A round.
Instance (computer science)
In computer science, an instance or token (from metalogic and metamathematics) is a specific occurrence of a software element that is based on a type definition. When created, an occurrence is said to have been instantiated, and both the creation process and the result of creation are called instantiation. == Examples == Chat AI instance In chat-based AI systems, an assistant can be invoked across many independent conversation sessions (often called a thread), each with its own message history. A specific execution of the assistant over that session may be represented as a run (an execution on a thread). Class instance In object-oriented programming, an object created from a class type. Each instance of a class shares the class-defined structure and behavior but has its own identity and state. Procedural instance In some contexts (including Simula), each procedure call can be viewed as an instance of that procedure—an activation with its own parameters and local variables. Computer instance In cloud computing and virtualization, an instance commonly refers to a provisioned virtual machine or virtual server with an allocated combination of compute, memory, network, and storage resources. Polygonal model In computer graphics, a model may be instanced so it can be drawn multiple times with different transforms and parameters, improving performance by reusing shared geometry data. Program instance In a POSIX-oriented operating system, a running process is an instance of a program. It can be instantiated via system calls such as fork() and exec(). Each executing process is an instance of a program it has been instantiated from.
Krohn–Rhodes theory
In mathematics and computer science, the Krohn–Rhodes theory (or algebraic automata theory) is an approach to the study of finite semigroups and automata that seeks to decompose them in terms of elementary components. These components correspond to finite aperiodic semigroups and finite simple groups that are combined in a feedback-free manner (called a "wreath product" or "cascade"). Krohn and Rhodes found a general decomposition for finite automata. The authors discovered and proved an unexpected major result in finite semigroup theory, revealing a deep connection between finite automata and semigroups. Decidability of Krohn-Rhodes complexity long motivated much work in semigroup theory. In June 2024, Stuart Margolis, John Rhodes, and Anne Schilling announced a proof that the complexity is decidable. == Definitions and description of the Krohn–Rhodes theorem == Let T {\displaystyle T} be a semigroup. A semigroup S {\displaystyle S} that is a homomorphic image of a subsemigroup of T {\displaystyle T} is said to be a divisor of T {\displaystyle T} . The Krohn–Rhodes theorem for finite semigroups states that every finite semigroup S {\displaystyle S} is a divisor of a finite alternating wreath product of finite simple groups, each a divisor of S {\displaystyle S} , and finite aperiodic semigroups (which contain no nontrivial subgroups). In the automata formulation, the Krohn–Rhodes theorem for finite automata states that given a finite automaton A {\displaystyle A} with states Q {\displaystyle Q} and input alphabet I {\displaystyle I} , output alphabet U {\displaystyle U} , then one can expand the states to Q ′ {\displaystyle Q'} such that the new automaton A ′ {\displaystyle A'} embeds into a cascade of "simple", irreducible automata: In particular, A {\displaystyle A} is emulated by a feed-forward cascade of (1) automata whose transformation semigroups are finite simple groups and (2) automata that are banks of flip-flops running in parallel. The new automaton A ′ {\displaystyle A'} has the same input and output symbols as A {\displaystyle A} . Here, both the states and inputs of the cascaded automata have a very special hierarchical coordinate form. Moreover, each simple group (prime) or non-group irreducible semigroup (subsemigroup of the flip-flop monoid) that divides the transformation semigroup of A {\displaystyle A} must divide the transformation semigroup of some component of the cascade, and only the primes that must occur as divisors of the components are those that divide A {\displaystyle A} 's transformation semigroup. == Group complexity == The Krohn–Rhodes complexity (also called group complexity or just complexity) of a finite semigroup S is the least number of groups in a wreath product of finite groups and finite aperiodic semigroups of which S is a divisor. All finite aperiodic semigroups have complexity 0, while non-trivial finite groups have complexity 1. In fact, there are semigroups of every non-negative integer complexity. For example, for any n greater than 1, the multiplicative semigroup of all (n+1) × (n+1) upper-triangular matrices over any fixed finite field has complexity n (Kambites, 2007). A major open problem in finite semigroup theory is the decidability of complexity: is there an algorithm that will compute the Krohn–Rhodes complexity of a finite semigroup, given its multiplication table? Upper bounds and ever more precise lower bounds on complexity have been obtained (see, e.g. Rhodes & Steinberg, 2009). Rhodes has conjectured that the problem is decidable. In June 2024, Stuart Margolis, John Rhodes, and Anne Schilling announced a proof in the affirmative of the conjecture, though as of 2025 the result has yet to be confirmed. == History and applications == At a conference in 1962, Kenneth Krohn and John Rhodes announced a method for decomposing a (deterministic) finite automaton into "simple" components that are themselves finite automata. This joint work, which has implications for philosophy, comprised both Krohn's doctoral thesis at Harvard University and Rhodes' doctoral thesis at MIT. Simpler proofs, and generalizations of the theorem to infinite structures, have been published since then (see Chapter 4 of Rhodes and Steinberg's 2009 book The q-Theory of Finite Semigroups for an overview). In the 1965 paper by Krohn and Rhodes, the proof of the theorem on the decomposition of finite automata (or, equivalently sequential machines) made extensive use of the algebraic semigroup structure. Later proofs contained major simplifications using finite wreath products of finite transformation semigroups. The theorem generalizes the Jordan–Hölder decomposition for finite groups (in which the primes are the finite simple groups), to all finite transformation semigroups (for which the primes are again the finite simple groups plus all subsemigroups of the "flip-flop" (see above)). Both the group and more general finite automata decomposition require expanding the state-set of the general, but allow for the same number of input symbols. In the general case, these are embedded in a larger structure with a hierarchical "coordinate system". One must be careful in understanding the notion of "prime" as Krohn and Rhodes explicitly refer to their theorem as a "prime decomposition theorem" for automata. The components in the decomposition, however, are not prime automata (with prime defined in a naïve way); rather, the notion of prime is more sophisticated and algebraic: the semigroups and groups associated to the constituent automata of the decomposition are prime (or irreducible) in a strict and natural algebraic sense with respect to the wreath product (Eilenberg, 1976). Also, unlike earlier decomposition theorems, the Krohn–Rhodes decompositions usually require expansion of the state-set, so that the expanded automaton covers (emulates) the one being decomposed. These facts have made the theorem difficult to understand and challenging to apply in a practical way—until recently, when computational implementations became available (Egri-Nagy & Nehaniv 2005, 2008). H.P. Zeiger (1967) proved an important variant called the holonomy decomposition (Eilenberg 1976). The holonomy method appears to be relatively efficient and has been implemented computationally by A. Egri-Nagy (Egri-Nagy & Nehaniv 2005). Meyer and Thompson (1969) give a version of Krohn–Rhodes decomposition for finite automata that is equivalent to the decomposition previously developed by Hartmanis and Stearns, but for useful decompositions, the notion of expanding the state-set of the original automaton is essential (for the non-permutation automata case). Many proofs and constructions now exist of Krohn–Rhodes decompositions (e.g., [Krohn, Rhodes & Tilson 1968], [Ésik 2000], [Diekert et al. 2012]), with the holonomy method the most popular and efficient in general (although not in all cases). [Zimmermann 2010] gives an elementary proof of the theorem. Owing to the close relation between monoids and categories, a version of the Krohn–Rhodes theorem is applicable to category theory. This observation and a proof of an analogous result were offered by Wells (1980). The Krohn–Rhodes theorem for semigroups/monoids is an analogue of the Jordan–Hölder theorem for finite groups (for semigroups/monoids rather than groups). As such, the theorem is a deep and important result in semigroup/monoid theory. The theorem was also surprising to many mathematicians and computer scientists since it had previously been widely believed that the semigroup/monoid axioms were too weak to admit a structure theorem of any strength, and prior work (Hartmanis & Stearns) was only able to show much more rigid and less general decomposition results for finite automata. Work by Egri-Nagy and Nehaniv (2005, 2008–) continues to further automate the holonomy version of the Krohn–Rhodes decomposition extended with the related decomposition for finite groups (so-called Frobenius–Lagrange coordinates) using the computer algebra system GAP. Applications outside of the semigroup and monoid theories are now computationally feasible. They include computations in biology and biochemical systems (e.g. Egri-Nagy & Nehaniv 2008), artificial intelligence, finite-state physics, psychology, and game theory (see, for example, Rhodes 2009).