24SevenOffice

24SevenOffice is a Norwegian software company headquartered in Oslo, Norway, with offices in Stockholm, Sweden and London, United Kingdom. Founded in 1997, the company specializes in web-based (SaaS) ERP and CRM systems. == Company history == 24SevenOffice was founded in 1997 in Porsgrunn, Norway, as IKT Interactive AS and marketed as kontorplassen.no. The name "24SevenOffice" was introduced for the company's London branch when the company entered the British market in 2003. The company changed its name to 24SevenOffice in February 2005. Originally based in Skien, the company later moved to Oslo Innovation Center, then to Tjuvholmen in the waterfront Fjord City of Oslo, and now the headquarters are located in Inkognitogaten 33, Solli plass, Oslo. The idea for the company's product was developed in 1996, and 24SevenOffice was an early innovator in the Scandinavian market in web-based enterprise resource planning solutions (ERP). A British office was established at Surrey Business Park in May 2003, with the company launching its web-based (SaaS) utility computing system to the UK SME market in 2004. An office in Chennai, India, was established in 2005, and 24SevenOffice entered the Swedish market when they acquired the leading competitor and ERP-provider Start & Run in a cash deal. In August 2005, the company had an initial public offering that raised NOK 15 million, and the company entered The Norwegian Over the Counter Market list as of 5 October 2005 (the ticker was 24SO), reaching a market value of NOK 175 million, with 5000 customers in Norway. In 2006, the company signed a deal to sponsor rally driver Petter Solberg, at the time the largest private sponsorship in Norwegian sport. Instead of receiving NOK 5 million in cash, Solberg received a 2.9 per cent ownership in the company. The company entered the German-speaking market in April 2006 when an office in Frankfurt am Main was opened. In late August/early September, they established an office with ten sales agents plus a general manager in Stockholm for the Swedish market. 24SevenOffice initiated strategic cooperation with Active 24 in early 2006 to develop a common platform. During the summer, Active 24 was bought by 24SevenOffice's ERP/CRM competitor Mamut (company), and 24SevenOffice terminated the contract with Active 24 in October demanding NOK 200 million in compensation for lost revenue. After a breakdown of settlement negotiations in the Forliksråd in January 2007, 24SevenOffice filed a case against Active 24 for breach of agreement in the Oslo District Court in March. 24SevenOffice lost on all counts in the District Court in December 2007. In January 2008, 24SevenOffice appealed the case to the Borgarting Court of Appeal, reducing the cause of action from NOK 250 to 30 million. 24SevenOffice lost on all counts in the Court of Appeal in December 2008, and was ordered to cover the costs incurred by Active 24 in connection with the dispute totaling NOK 6.91 million. 24SevenOffice appealed the case to the Supreme Court of Norway, but the Supreme Court Appeals Committee in March 2008 unanimously rejected the appeal from 24SevenOffice over the Borgarting Appeal Court's unanimous judgment of December 2008. On a counterclaim from Active 24 and Mamut against 24SevenOffice, the Oslo District Court in May 2010 found, that 24SevenOffice should pay Active 24 NOK 12 million in compensation for wrongfully having terminated the agreement, and a further NOK 360.000 of the opponent's legal costs. 24SevenOffice disagreed with the court ruling, and appealed once again. The Borgarting Court of Appeal in November 2011, ruled to reduce the amount of damages to NOK 4.4 million plus NOK 900.000 in penal interest. With several scrip issues, 24SevenOffice raised 25 million NOK (about $4 million at the time) between October 2005 and July 2006. They entered into a strategic partnership with Bluegarden, who for 30 years had delivered digital services for payroll, human resource planning, recruitment and training, in March 2006, and they made a large-scale agreement in April 2006, with US telecommunications software company Webex, a competitor to Norwegian Tandberg videoconferencing equipment manufacturer. In September 2006, 24SevenOffice signed an agreement with Fokus Bank to provide their customers with extended functionality in Internet banking. 24SevenOffice had by 2007 reportedly 9000 customers, joined the OpenAjax Alliance, and entered into a strategic partnership with Dun & Bradstreet in May 2007, but despite getting listed on Oslo Axess on 22 June (ticker: TFSO), reaching a market capitalization of NOK 120 million, the company was still losing money. The company ended 2007 with a revenue of NOK 21.7 million. In 2008, 24SevenOffice bought 50% of the stocks in telecommunication company Oyatel, partnered with Nets Group to facilitate invoicing for businesses, and telecommunications company Telipol chose 24SevenOffice's second-generation Internet platform for its 8,000 users. They announced an increase in revenues in Q2 to 11.1 million, up from 4.7 million in the same period the year before. 24SevenOffice had a turnover of NOK 37 million in the first half of 2009, a doubling compared to the same period the previous year, and presented its first positive EBITDA in Q2. The Norwegian Association of Auditors signed an agreement with 24SevenOffice in 2011, whereby they only recommend 24SevenOffice as a system for their members to use. On 27 June 2013, the shareholders of 24SevenOffice took off from the stock exchange and privatized the company. In recent years, the company has invested heavily in finance and accounting – and got leading auditing companies such as PwC and KPMG on the customer list. == Products == 24SevenOffice is a web-based (SaaS) ERP system. It includes modules for CRM, accounting, invoicing, e-mail, file/document management and project management. == Awards == 24SevenOffice won the Seal of Excellence in Multimedia Award at the 2004 CeBIT, became Norwegian Gazelle Company of the year 2004, chosen by Dagens Næringsliv and Dun & Bradstreet, won Product of the Year in the Norwegian finance magazine Kapital, and the IKT Grenland Innovation Award in 2008.

Film-out

Film-out is the process in the computer graphics, video production and filmmaking disciplines of transferring images or animation from videotape or digital files to a traditional film print. Film-out is a broad term that encompasses the conversion of frame rates, color correction, as well as the actual printing, also called scannior recording. The film-out process is different depending on the regional standard of the master videotape in question – NTSC, PAL, or SECAM – or likewise on the several emerging region-independent formats of high definition video (HD video); thus each type is covered separately, taking into account regional film-out industries, methods and technical considerations. == Live action video == Many modern documentaries and low-budget films are shot on videotape or other digital video media, instead of film stock, and completed as digital video. Video production means substantially lower costs than 16 mm or 35 mm film production on all levels. Until recently, the relatively low cost of video ended when the issue of a theatrical presentation was raised, which required a print for film projection. With the growing presence of digital projection, this is becoming less of a factor. === Standard definition (SD) video === Film-out of standard-definition video – or any source that has an incompatible frame rate – is the up-conversion of video media to film for theatrical viewing. The video-to-film conversion process consists of two major steps: first, the conversion of video into digital film frames which are then stored on a computer or on HD videotape; and secondly, the printing of these digital film frames onto actual film. To understand these two steps, it is important to understand how video and film differ. Film (sound film, at least) has remained unchanged for almost a century and creates the illusion of moving images through the rapid projection of still images, frames, upon a screen, typically 24 per second. Traditional interlaced SD video has no real frame rate, (though the term frame is applied to video, it has a different meaning). Instead, video consists of a very fast succession of horizontal lines that continually cascade down the television screen – streaming top to bottom, before jumping back to the top and then streaming down to the bottom again, repeatedly, almost 60 alternating screen-fulls every second for NTSC, or exactly 50 such screen-fulls per second for PAL and SECAM. Since visual movement in video is infused in this continuous cascade of scan lines, there is no discrete image or real frame that can be identified at any one time. Therefore, when transferring video to film, it is necessary to invent individual film frames, 24 for every second of elapsed time. The bulk of the work done by a film-out company is this first step, creating film frames out of the stream of interlaced video. Each company employs its own (often proprietary) technology for turning interlaced video into high-resolution digital video files of 24 discrete images every second, called 24 progressive video or 24p. The technology must filter out all the visually unappealing artifacting that results from the inherent mismatch between video and film movement. Moreover, the conversion process usually requires human intervention at every edit point of a video program, so that each type of scene can be calibrated for maximum visual quality. The use of archival footage in video especially calls for extra attention. Step two, the scanning to film, is the rote part of the process. This is the mechanical step where lasers print each of the newly created frames of the 24p video, stored on computer files or HD videotape, onto rolls of film. Most companies that do film-out, do all the stages of the process themselves for a lump sum. The job includes converting interlaced video into 24p and often a color correction session – (calibrating the image for theatrical projection), before scanning to physical film, (possibly followed by color correction of the film print made from the digital intermediary) – is offered. At the very least, film-out can be understood as the process of converting interlaced video to 24p and then scanning it to film. ==== NTSC video ==== NTSC is the most challenging of the formats when it comes to standards conversion and, specifically, converting to film prints. NTSC runs at the approximate rate of 29.97 video frames (consisting of two interlaced screen-fulls of scan lines, called fields, per frame) per second. In this way, NTSC resolves actual live action movement at almost – but not quite – 60 alternating half-resolution images every second. Because of this 29.97 rate, no direct correlation to film frames at 24 frames per second can be achieved. NTSC is hardest to reconcile with film, thus motivating its own unique processes. ==== PAL and SECAM video ==== PAL and SECAM run at 25 interlaced video frames per second, which can be slowed down or frame-dropped, then deinterlaced, to correlate frame for frame with film running at 24 actual frames per second. PAL and SECAM are less complex and demanding than NTSC for film-out. PAL and SECAM conversions do agitate, though, with the unpleasant choice between slowing down video (and audio pitch, noticeably) by four percent, from 25 to 24 frames per second, in order to maintain a 1:1 frame match, slightly changing the rhythm and feel of the program; or maintaining original speed by periodically dropping frames, thereby creating jerkiness and possible loss of vital detail in fast-moving action or precise edits. === High definition (HD) digital video === High definition digital video can be shot at a variety of frame rates, including 29.97 interlaced (like NTSC) or progressive; or 25 interlaced (like PAL) or progressive; or even 24-progressive (just like film). HD, if shot in 24-progressive, scans nearly perfectly to film without the need for a frame or field conversion process. Other issues remain though, based on the different resolutions, color spaces, and compression schemes that exist in the high-definition video world. == Computer graphics and animation == Artists working with CGI-Computer-generated imagery animation computers create pictures frame by frame. Once the finished product is done, the frames are outputted, normally in a DPX file. These picture data files can then be put on to film using a film recorder for film out. SGI computers started the high-end CGI-Computer-generated imagery animation systems, but with faster computers and the growth of Linux-based systems, many others are on the market now. Movies fully rendered and animated in CGI such as Toy Story, and Antz utilize the film-out method to produce 35mm copies for archival and release prints. Most CGI work is done in 2K Display resolution files (about the size of QXGA) and then output to the Film-out device for creation of 35 mm elements. With 4K Display resolution digital intermediates on the rise, newer types of film-out recorders are being developed to accept 4k resolution files. A 2K movie requires a Storage Area Network storage several terabytes in size to be properly stored and played out. Computer graphics files are handled the same way but in single frames and may use DPX, TIFF or other file formats. == Digital intermediates == Film-out-recording is the last step of digital intermediate workflow. DPX files that were scanned on a motion picture film scanner are stored on a storage area network (often abbreviated as SAN). The scanned DPX footage is edited and composited-FX on workstations, then mastered back on film. Film restoration is also done this way. A "film intermediate" is an analog variation of a digital intermediate, where a project shot on digital video is printed onto film stock and transferred back to digital video to emulate film. The term was coined after it was used on the Oscar-winning 2012 short film "Curfew". The process was also used on the films Dune (2021) and The Batman (2022). == Images for graphic design and print industries == The days of newspapers and magazines shooting 35mm film are almost gone. Digital cameras can now shoot all the images needed, storing them as files (e.g. JPEG, DPX or another format) that are readily edited prior to use. Once the final copy is approved, it can be filmed out for publishing. Digital stills are not the only way to get pictures used in the graphic design and print industries. Film scanners and computer graphics programs are also common sources for graphic design and print industries. == Types of devices == The following devices are used in film-out processes: CRT recorder. Camera and a special TV display Kinescope – early type Electronic Video Recording or EVR – early type EBR Electron Beam Film Recorder 16 mm by 3M Laser film recorder, like Kodak's high-end Lightning II recorder and Arri's Arrilaser. DLP Film recorder, like Cinevation's real-time Cinevator. == History == Lately it has become possible to transfer video images, inclu

Education by algorithm

Education by algorithm refers to automated solutions that algorithmic agents or social bots offer to education, to assist with mundane educational tasks. These are often instrumentalist “educational reforms” or “curriculum transformations”, which have been implemented by policy makers and are supported by proprietary education technologies. New educational policies, mandated by transnational governance forums (like the OECD), have manufactured a connection between economies and education. Governments, schools and universities are expected to introduce or prepare students for an “unknown future”, to “future proof” them against an identified issue or to mitigate a national crisis. Technologies are seen as a catalyst to effect these changes. However, these policies mask a deeper problem, which include the assetization of education and the use of technologies as a means for surveillance and behavior modification. The traces that students and leave, through cookies, logins learning activities, assignments and tests, are collected, facetted, and shared with commercial organizations by these agents, to both predict future behavior and shape it. Techno solutionist thinking has led to managers adopting educational policies and reforms, and looking towards technologies to act as disrupters, liberators or agents to improve efficiency. During the COVID-19 pandemic, many more students had to modify their learning and working circumstances to protect themselves. Academics shifted their assessment practices from the dominant assessment of learning paradigm to an orientation that saw value in "assessment for learning". Big tech assisted, and teaching infrastructure became further privatized, and unbundling of education provision went a step further. Following the return to class, this assessment paradigm became rationalised in education. Leaving the space for algorithmic agents to step in. Academics work was increasingly driven by learning experience platforms and student understanding was extended through interleaving, behavior modification nudges and rewards and scheduled high stakes assessments. This data collection may also be construed as surveillance., or perceived as evidence of a Fourth Industrial Revolution

Vinberg's algorithm

In mathematics, Vinberg's algorithm is an algorithm, introduced by Ernest Borisovich Vinberg, for finding a fundamental domain of a hyperbolic reflection group. Conway (1983) used Vinberg's algorithm to describe the automorphism group of the 26-dimensional even unimodular Lorentzian lattice II25,1 in terms of the Leech lattice. == Description of the algorithm == Let Γ < I s o m ( H n ) {\displaystyle \Gamma <\mathrm {Isom} (\mathbb {H} ^{n})} be a hyperbolic reflection group. Choose any point v 0 ∈ H n {\displaystyle v_{0}\in \mathbb {H} ^{n}} ; we shall call it the basic (or initial) point. The fundamental domain P 0 {\displaystyle P_{0}} of its stabilizer Γ v 0 {\displaystyle \Gamma _{v_{0}}} is a polyhedral cone in H n {\displaystyle \mathbb {H} ^{n}} . Let H 1 , . . . , H m {\displaystyle H_{1},...,H_{m}} be the faces of this cone, and let a 1 , . . . , a m {\displaystyle a_{1},...,a_{m}} be outer normal vectors to it. Consider the half-spaces H k − = { x ∈ R n , 1 | ( x , a k ) ≤ 0 } . {\displaystyle H_{k}^{-}=\{x\in \mathbb {R} ^{n,1}|(x,a_{k})\leq 0\}.} There exists a unique fundamental polyhedron P {\displaystyle P} of Γ {\displaystyle \Gamma } contained in P 0 {\displaystyle P_{0}} and containing the point v 0 {\displaystyle v_{0}} . Its faces containing v 0 {\displaystyle v_{0}} are formed by faces H 1 , . . . , H m {\displaystyle H_{1},...,H_{m}} of the cone P 0 {\displaystyle P_{0}} . The other faces H m + 1 , . . . {\displaystyle H_{m+1},...} and the corresponding outward normals a m + 1 , . . . {\displaystyle a_{m+1},...} are constructed by induction. Namely, for H j {\displaystyle H_{j}} we take a mirror such that the root a j {\displaystyle a_{j}} orthogonal to it satisfies the conditions (1) ( v 0 , a j ) < 0 {\displaystyle (v_{0},a_{j})<0} ; (2) ( a i , a j ) ≤ 0 {\displaystyle (a_{i},a_{j})\leq 0} for all i < j {\displaystyle i

Bottom-up and top-down approaches

Bottom-up and top-down are strategies of composition and decomposition in fields as diverse as information processing and ordering knowledge, software, humanistic and scientific theories (see systemics), time management, and organization. In practice they can be seen as a style of thinking, teaching, or leadership. A top-down approach (also known as stepwise design and stepwise refinement and in some cases used as a synonym of decomposition) is essentially the breaking down of a system to gain insight into its compositional subsystems in a reverse engineering fashion. In a top-down approach an overview of the system is formulated, specifying, but not detailing, any first-level subsystems. Each subsystem is then refined in yet greater detail, sometimes in many additional subsystem levels, until the entire specification is reduced to base elements. A top-down model is often specified with the assistance of black boxes, which makes it easier to manipulate. However, black boxes may fail to clarify elementary mechanisms or be detailed enough to realistically validate the model. A top-down approach starts with the big picture, then breaks down into smaller segments. A bottom-up approach is the piecing together of systems to give rise to more complex systems, thus making the original systems subsystems of the emergent system. Bottom-up processing is a type of information processing based on incoming data from the environment to form a perception. From a cognitive psychology perspective, information enters the eyes in one direction (sensory input, or the "bottom"), and is then turned into an image by the brain that can be interpreted and recognized as a perception (output that is "built up" from processing to final cognition). In a bottom-up approach the individual base elements of the system are first specified in great detail. These elements are then linked together to form larger subsystems, which then in turn are linked, sometimes in many levels, until a complete top-level system is formed. This strategy often resembles a "seed" model, by which the beginnings are small but eventually grow in complexity and completeness. But "organic strategies" may result in a tangle of elements and subsystems, developed in isolation and subject to local optimization as opposed to meeting a global purpose. == Computer science == === Software development === In the software development process, the top-down and bottom-up approaches play a key role. Top-down approaches emphasize planning and a complete understanding of the system. It is inherent that no coding can begin until a sufficient level of detail has been reached in the design of at least some part of the system. Top-down approaches are implemented by attaching the stubs in place of the module. But these delay testing of the ultimate functional units of a system until significant design is complete. Bottom-up emphasizes coding and early testing, which can begin as soon as the first module has been specified. But this approach runs the risk that modules may be coded without having a clear idea of how they link to other parts of the system, and that such linking may not be as easy as first thought. Re-usability of code is one of the main benefits of a bottom-up approach. Top-down design was promoted in the 1970s by IBM researchers Harlan Mills and Niklaus Wirth. Mills developed structured programming concepts for practical use and tested them in a 1969 project to automate the New York Times morgue index. The engineering and management success of this project led to the spread of the top-down approach through IBM and the rest of the computer industry. Among other achievements, Niklaus Wirth, the developer of Pascal programming language, wrote the influential paper Program Development by Stepwise Refinement. Since Niklaus Wirth went on to develop languages such as Modula and Oberon (where one could define a module before knowing about the entire program specification), one can infer that top-down programming was not strictly what he promoted. Top-down methods were favored in software engineering until the late 1980s, and object-oriented programming assisted in demonstrating the idea that both aspects of top-down and bottom-up programming could be used. Modern software design approaches usually combine top-down and bottom-up approaches. Although an understanding of the complete system is usually considered necessary for good design—leading theoretically to a top-down approach—most software projects attempt to make use of existing code to some degree. Pre-existing modules give designs a bottom-up flavor. === Programming === Top-down is a programming style, the mainstay of traditional procedural languages, in which design begins by specifying complex pieces and then dividing them into successively smaller pieces. The technique for writing a program using top-down methods is to write a main procedure that names all the major functions it will need. Later, the programming team looks at the requirements of each of those functions and the process is repeated. These compartmentalized subroutines eventually will perform actions so simple they can be easily and concisely coded. When all the various subroutines have been coded the program is ready for testing. By defining how the application comes together at a high level, lower-level work can be self-contained. In a bottom-up approach the individual base elements of the system are first specified in great detail. These elements are then linked together to form larger subsystems, which in turn are linked, sometimes at many levels, until a complete top-level system is formed. This strategy often resembles a "seed" model, by which the beginnings are small, but eventually grow in complexity and completeness. Object-oriented programming (OOP) is a paradigm that uses "objects" to design applications and computer programs. In mechanical engineering with software programs such as Pro/ENGINEER, Solidworks, and Autodesk Inventor users can design products as pieces not part of the whole and later add those pieces together to form assemblies like building with Lego. Engineers call this "piece part design". === Parsing === Parsing is the process of analyzing an input sequence (such as that read from a file or a keyboard) in order to determine its grammatical structure. This method is used in the analysis of both natural languages and computer languages, as in a compiler. Bottom-up parsing is parsing strategy that recognizes the text's lowest-level small details first, before its mid-level structures, and leaves the highest-level overall structure to last. In top-down parsing, on the other hand, one first looks at the highest level of the parse tree and works down the parse tree by using the rewriting rules of a formal grammar. == Natural sciences == === Nanotechnology === Top-down and bottom-up are two approaches for the manufacture of products. These terms were first applied to the field of nanotechnology by the Foresight Institute in 1989 to distinguish between molecular manufacturing (to mass-produce large atomically precise objects) and conventional manufacturing (which can mass-produce large objects that are not atomically precise). Bottom-up approaches seek to have smaller (usually molecular) components built up into more complex assemblies, while top-down approaches seek to create nanoscale devices by using larger, externally controlled ones to direct their assembly. Certain valuable nanostructures, such as Silicon nanowires, can be fabricated using either approach, with processing methods selected on the basis of targeted applications. A top-down approach often uses the traditional workshop or microfabrication methods where externally controlled tools are used to cut, mill, and shape materials into the desired shape and order. Micropatterning techniques, such as photolithography and inkjet printing belong to this category. Vapor treatment can be regarded as a new top-down secondary approaches to engineer nanostructures. Bottom-up approaches, in contrast, use the chemical properties of single molecules to cause single-molecule components to (a) self-organize or self-assemble into some useful conformation, or (b) rely on positional assembly. These approaches use the concepts of molecular self-assembly and/or molecular recognition. See also Supramolecular chemistry. Such bottom-up approaches should, broadly speaking, be able to produce devices in parallel and much cheaper than top-down methods but could potentially be overwhelmed as the size and complexity of the desired assembly increases. === Neuroscience and psychology === These terms are also employed in cognitive sciences including neuroscience, cognitive neuroscience and cognitive psychology to discuss the flow of information in processing. Typically, sensory input is considered bottom-up, and higher cognitive processes, which have more information from other sources, are considered top-down. A bottom-up proc

ISLRN

The ISLRN or International Standard Language Resource Number is Persistent Unique Identifier for Language Resources. == Context == On November 18, 2013, 12 major organisations (see list below) from the fields Language Resources and Technologies, Computational Linguistics, and Digital Humanities held a cooperation meeting in Paris (France) and agreed to announce the establishment of the International Standard Language Resource Number (ISLRN), to be assigned to each Language Resource. Among the 12 organisations, 4 institutions constitute the ISLRN Steering Committee (ST) ADHO ACL Asian Federation of Natural Language Processing ST COCOSDA, International Committee for the Coordination & Standardisation of Speech Databases and Assessment Techniques ICCL (COLING) European Data Forum ELRA ST IAMT, International Association for Machine Translation Archived 2010-06-24 at the Wayback Machine ISCA LDC ST Oriental COCOSDA ST RMA, Language Resource Management Agency == Size and Content == The Joint Research Centre(JRC), the [European Commission]'s in-house science service, was the first organisation to adopt the ISLRN initiative and requested. 2500 resources and tools have already been allocated an ISLRN. These resources include written data (Annotated corpus, Annotated text, List of misspelled word, Terminological database, Treebank, Wordnet, etc.) and speech corpora (Synthesised Speech, Transcripts and Audiovisual Recordings, Conversational Speech, Folk Sayings, etc.) == Objectives == Providing Language Resources with unique names and identifiers using a standardized nomenclature ensures the identification of each Language Resources and streamlines the citation with proper references in activities within Human Language Technology as well as in documents and scientific publications. Such unique identifier also enhances the reproducibility, an essential feature of scientific work.

Algorithm characterizations

Algorithm characterizations are attempts to formalize the word algorithm. Algorithm does not have a generally accepted formal definition. Researchers are actively working on this problem. This article will present some of the "characterizations" of the notion of "algorithm" in more detail. == The problem of definition == Over the last 200 years, the definition of the algorithm has become more complicated and detailed as researchers have tried to pin down the term. Indeed, there may be more than one type of "algorithm". But most agree that algorithm has something to do with defining generalized processes for the creation of "output" integers from other "input" integers – "input parameters" arbitrary and infinite in extent, or limited in extent but still variable—by the manipulation of distinguishable symbols (counting numbers) with finite collections of rules that a person can perform with paper and pencil. The most common number-manipulation schemes—both in formal mathematics and in routine life—are: (1) the recursive functions calculated by a person with paper and pencil, and (2) the Turing machine or its Turing equivalents—the primitive register-machine or "counter-machine" model, the random-access machine model (RAM), the random-access stored-program machine model (RASP) and its functional equivalent "the computer". When we are doing "arithmetic" we are really calculating by the use of "recursive functions" in the shorthand algorithms we learned in grade school, for example, adding and subtracting. The proofs that every "recursive function" we can calculate by hand we can compute by machine and vice versa—note the usage of the words calculate versus compute—is remarkable. But this equivalence together with the thesis (unproven assertion) that this includes every calculation/computation indicates why so much emphasis has been placed upon the use of Turing-equivalent machines in the definition of specific algorithms, and why the definition of "algorithm" itself often refers back to "the Turing machine". This is discussed in more detail under Stephen Kleene's characterization. The following are summaries of the more famous characterizations (Kleene, Markov, Knuth) together with those that introduce novel elements—elements that further expand the definition or contribute to a more precise definition. [ A mathematical problem and its result can be considered as two points in a space, and the solution consists of a sequence of steps or a path linking them. Quality of the solution is a function of the path. There might be more than one attribute defined for the path, e.g. length, complexity of shape, an ease of generalizing, difficulty, and so on. ] == Chomsky hierarchy == There is more consensus on the "characterization" of the notion of "simple algorithm". All algorithms need to be specified in a formal language, and the "simplicity notion" arises from the simplicity of the language. The Chomsky (1956) hierarchy is a containment hierarchy of classes of formal grammars that generate formal languages. It is used for classifying of programming languages and abstract machines. From the Chomsky hierarchy perspective, if the algorithm can be specified on a simpler language (than unrestricted), it can be characterized by this kind of language, else it is a typical "unrestricted algorithm". Examples: a "general purpose" macro language, like M4 is unrestricted (Turing complete), but the C preprocessor macro language is not, so any algorithm expressed in C preprocessor is a "simple algorithm". See also Relationships between complexity classes. == Features of a good algorithm == The following are desirable features of a well-defined algorithm, as discussed in Scheider and Gersting (1995): Unambiguous Operations: an algorithm must have specific, outlined steps. The steps should be exact enough to precisely specify what to do at each step. Well-Ordered: The exact order of operations performed in an algorithm should be concretely defined. Feasibility: All steps of an algorithm should be possible (also known as effectively computable). Input: an algorithm should be able to accept a well-defined set of inputs. Output: an algorithm should produce some result as an output, so that its correctness can be reasoned about. Finiteness: an algorithm should terminate after a finite number of instructions. Properties of specific algorithms that may be desirable include space and time efficiency, generality (i.e. being able to handle many inputs), or determinism. == 1881 John Venn's negative reaction to W. Stanley Jevons's Logical Machine of 1870 == In early 1870 W. Stanley Jevons presented a "Logical Machine" (Jevons 1880:200) for analyzing a syllogism or other logical form e.g. an argument reduced to a Boolean equation. By means of what Couturat (1914) called a "sort of logical piano [,] ... the equalities which represent the premises ... are "played" on a keyboard like that of a typewriter. ... When all the premises have been "played", the panel shows only those constituents whose sum is equal to 1, that is, ... its logical whole. This mechanical method has the advantage over VENN's geometrical method..." (Couturat 1914:75). For his part John Venn, a logician contemporary to Jevons, was less than thrilled, opining that "it does not seem to me that any contrivances at present known or likely to be discovered really deserve the name of logical machines" (italics added, Venn 1881:120). But of historical use to the developing notion of "algorithm" is his explanation for his negative reaction with respect to a machine that "may subserve a really valuable purpose by enabling us to avoid otherwise inevitable labor": (1) "There is, first, the statement of our data in accurate logical language", (2) "Then secondly, we have to throw these statements into a form fit for the engine to work with – in this case the reduction of each proposition to its elementary denials", (3) "Thirdly, there is the combination or further treatment of our premises after such reduction," (4) "Finally, the results have to be interpreted or read off. This last generally gives rise to much opening for skill and sagacity." He concludes that "I cannot see that any machine can hope to help us except in the third of these steps; so that it seems very doubtful whether any thing of this sort really deserves the name of a logical engine."(Venn 1881:119–121). == 1943, 1952 Stephen Kleene's characterization == This section is longer and more detailed than the others because of its importance to the topic: Kleene was the first to propose that all calculations/computations—of every sort, the totality of—can equivalently be (i) calculated by use of five "primitive recursive operators" plus one special operator called the mu-operator, or be (ii) computed by the actions of a Turing machine or an equivalent model. Furthermore, he opined that either of these would stand as a definition of algorithm. A reader first confronting the words that follow may well be confused, so a brief explanation is in order. Calculation means done by hand, computation means done by Turing machine (or equivalent). (Sometimes an author slips and interchanges the words). A "function" can be thought of as an "input-output box" into which a person puts natural numbers called "arguments" or "parameters" (but only the counting numbers including 0—the nonnegative integers) and gets out a single nonnegative integer (conventionally called "the answer"). Think of the "function-box" as a little man either calculating by hand using "general recursion" or computing by Turing machine (or an equivalent machine). "Effectively calculable/computable" is more generic and means "calculable/computable by some procedure, method, technique ... whatever...". "General recursive" was Kleene's way of writing what today is called just "recursion"; however, "primitive recursion"—calculation by use of the five recursive operators—is a lesser form of recursion that lacks access to the sixth, additional, mu-operator that is needed only in rare instances. Thus most of life goes on requiring only the "primitive recursive functions." === 1943 "Thesis I", 1952 "Church's Thesis" === In 1943 Kleene proposed what has come to be known as Church's thesis: "Thesis I. Every effectively calculable function (effectively decidable predicate) is general recursive" (First stated by Kleene in 1943 (reprinted page 274 in Davis, ed. The Undecidable; appears also verbatim in Kleene (1952) p.300) In a nutshell: to calculate any function the only operations a person needs (technically, formally) are the 6 primitive operators of "general" recursion (nowadays called the operators of the mu recursive functions). Kleene's first statement of this was under the section title "12. Algorithmic theories". He would later amplify it in his text (1952) as follows: "Thesis I and its converse provide the exact definition of the notion of a calculation (decision) procedure or algorithm, for the