Ernst Dieter Dickmanns is a German pioneer of dynamic computer vision and of driverless cars. Dickmanns has been a professor at the University of the Bundeswehr Munich (1975–2001), and visiting professor to Caltech and to MIT, teaching courses on "dynamic vision". == Biography == Dickmanns was born in 1936. He studied aerospace and aeronautics at RWTH Aachen (1956–1961), and control engineering at Princeton University (1964/65); from 1961 to 1975 he was associated with the German Aero-Space Research Establishment (now DLR) Oberpfaffenhofen, working in the fields of flight dynamics and trajectory optimization. In 1971/72 he spent a Post-Doc Research Associateship with the NASA-Marshall Space Flight Center, Huntsville (orbiter re-entry). From 1975 to 2001 he was with UniBw Munich, where he initiated the 'Institut fuer Flugmechanik und Systemdynamik' (IFS), the Institut fuer die 'Technik Autonomer Systeme' (TAS), and the research activities in machine vision for vehicle guidance. == Pioneering work in autonomous driving == In the early 1980s his team equipped a Mercedes-Benz van with cameras and other sensors. The 5-ton van was re-engineered that it was possible to control steering wheel, throttle, and brakes through computer commands based on real-time evaluation of image sequences. Software was written that translated the sensory data into appropriate driving commands. For safety reasons, initial experiments in Bavaria took place on streets without traffic. In 1986 the Robot Car "VaMoRs" managed to drive all by itself and by 1987 was capable of driving itself at speeds up to 96 kilometres per hour (60 mph). One of the greatest challenges in high-speed autonomous driving arises through the rapidly changing visual street scenes. Back then, computers were much slower than they are today (~1% of 1%); therefore, sophisticated computer vision strategies were necessary to react in real time. The team of Dickmanns solved the problem through an innovative approach to dynamic vision. Spatiotemporal models were used right from the beginning, dubbed '4-D approach', which did not need storing previous images but nonetheless was able to yield estimates of all 3-D position and velocity components. Attention control including artificial saccadic movements of the platform carrying the cameras allowed the system to focus its attention on the most relevant details of the visual input. Kalman filters have been extended to perspective imaging and were used to achieve robust autonomous driving even in presence of noise and uncertainty. Feedback of prediction errors allowed bypassing the (ill-conditioned) inversion of perspective projection by least-squares parameter fits. When in 1986/83 the EUREKA-project 'PROgraMme for a European Traffic of Highest Efficiency and Unprecedented Safety' (PROMETHEUS) was initiated by the European car manufacturing industry (funding in the range of several hundred million Euros), the initially planned autonomous lateral guidance by buried cables was dropped and substituted by the much more flexible machine vision approach proposed by Dickmanns, and partially encouraged by his successes. Most of the major car companies participated; so did Dickmanns and his team in cooperation with the Daimler-Benz AG. Substantial progress was made in the following 7 years. In particular, Dickmanns' robot cars learned to drive in traffic under various conditions. An accompanying human driver with a "red button" made sure the robot vehicle could not get out of control and become a danger to the public. Since 1992, driving in public traffic was standard as final step in real-world testing. Several dozen Transputers, a special breed of parallel computers, were used to deal with the (by 1990s standards) enormous computational demands. Two culmination points were achieved in 1994/95, when Dickmanns´ re-engineered autonomous S-Class Mercedes-Benz performed international demonstrations. The first was the final presentation of the PROMETHEUS project in October 1994 on Autoroute 1 near the airport Charles-de-Gaulle in Paris. With guests on board, the twin vehicles of Daimler-Benz (VITA-2) and UniBwM (VaMP) drove more than 1,000 kilometres (620 mi) on the three-lane highway in standard heavy traffic at speeds up to 130 kilometres per hour (81 mph). Driving in free lanes, convoy driving with distance keeping depending on speed, and lane changes left and right with autonomous passing have been demonstrated; the latter required interpreting the road scene also in the rear hemisphere. Two cameras with different focal lengths for each hemisphere have been used in parallel for this purpose. The second culmination point was a 1,758 kilometres (1,092 mi) trip in the fall of 1995 from Munich in Bavaria to Odense in Denmark to a project meeting and back. Both longitudinal and lateral guidance were performed autonomously by vision. On highways, the robot achieved speeds exceeding 175 kilometres per hour (109 mph) (there is no general speed limit on the Autobahn). Publications from Dickmann's research group indicate a mean autonomously driven distance without resets of ~9 kilometres (5.6 mi); the longest autonomously driven stretch reached 158 kilometres (98 mi). More than half of the resets required were achieved autonomously (no human intervention). This is particularly impressive considering that the system used black-and-white video-cameras and did not model situations like road construction sites with yellow lane markings; lane-changes at over 140 kilometres per hour (87 mph), and other traffic with more than 40 kilometres per hour (25 mph) relative speed have been handled. In total, 95% autonomous driving (by distance) was achieved. In the years 1994 to 2004 the elder 5-ton van 'VaMoRs' was used to develop the capabilities needed for driving on networks of minor (also unsealed) roads and for cross-country driving including avoidance of negative obstacles like ditches. Turning off onto crossroads of unknown width and intersection angles required a big effort, but has been achieved with "Expectation-based, Multi-focal, Saccadic vision" (EMS-vision). This vertebrate-type vision uses animation capabilities based on knowledge about subject classes (including the autonomous vehicle itself) and their potential behaviour in certain situations. This rich background is used for control of gaze and attention as well as for locomotion. Beside ground vehicle guidance, also applications of the 4-D approach to dynamic vision for unmanned air vehicles (conventional aircraft and helicopters) have been investigated. Autonomous visual landing approaches and landings have been demonstrated in hardware-in-the-loop simulations with visual/inertial data fusion. Real-world autonomous visual landing approaches till shortly before touchdown have been performed in 1992 with the twin-propeller aircraft Dornier 128 of the University of Brunswick at the airport there. Another success of this machine vision technology was the first ever visually controlled grasping experiment of a free-floating object in weightlessness on board the Space Shuttle Columbia D2-mission in 1993 as part of the 'Rotex'-experiment of DLR.
Teamwork (project management)
Teamwork.com is an Irish, privately owned, web-based software company headquartered in Cork, Ireland. Teamwork creates task management and team collaboration software. Founded in 2007, as of 2016 the company stated that its software was in use by over 370,000 organisations worldwide (including Disney, Spotify and HP), and that it had over 2.4m users. == History == Peter Coppinger and Dan Mackey founded a company, Digital Crew, in 2007. This company built websites, intranets and custom web-based solutions for clients in Cork, Ireland. Frustrated by whiteboards and software management tools, Coppinger wanted a software system that would help manage client projects and which would be easy to use and generic enough to be used by different types of companies. Originally 37signals Basecamp users themselves, Coppinger and Mackey were frustrated by the limited feature set, and by Basecamp's apparent inaction on their feedback. In October 2007, Coppinger and Mackey launched Teamwork Project Manager, nicknamed TeamworkPM. In March 2015, this was renamed as Teamwork Projects. In 2014, after two years of negotiations, TeamworkPM bought the domain name 'Teamwork.com' for US$675,000 (€500,000). At the time this was one of the most expensive domain name purchases by an Irish company, and involved the transfer of a domain name which had been dormant since it was first acquired by the original owner in 1999. In 2015, Teamwork.com was named by Gartner to be one of their "Cool Vendors" in the Program and Portfolio Management Category. This was followed by the launch of a new real-time messaging product, Teamwork Chat, in January 2015. In June 2015, the company announced a drive to recruit for 40 positions by the end of the year. This was followed by the announcement that the company was investing more than €1 million in a new office, and had leased office space in Park House, Blackpool. In June 2016, Teamwork.com undertook a further recruitment drive to entice developers to Cork. In July 2021, the company announced that it had raised an investment of $70 million (€59.1 million) from venture capital firm Bregal Milestone to fund further growth. == Products == Teamwork markets a number of cloud-based applications, including Teamwork, Teamwork Desk, Teamwork Spaces, Teamwork CRM and Teamwork Chat. Teamwork was launched on 4 October 2007, at which time it had time management, milestone management, file sharing, time tracking, and messaging features. Teamwork's platform reportedly integrates with martech software like HubSpot, as well as other productivity tools like Slack, G Suite, MS Teams, Zapier, Dropbox and QuickBooks. == Awards == In 2016, Teamwork was awarded Cork's Best SME in the Cork Chamber of Commerce "Company of the Year" awards. In 2016, Teamwork was named number 7 in Deloitte's Fast 50 tech companies hit €1.6bn turnover. In 2015, Teamwork was identified as a Gartner "Cool Vendor" in the Program and Portfolio Management Category.
Seismological Facility for the Advancement of Geoscience
The U.S. National Science Foundation's Seismological Facility for the Advancement of Geoscience (NSF SAGE) is a distributed, multi-user national facility that provides support for state of-the-art seismic research. It is operated by EarthScope Consortium. Its previous operator was the Incorporated Research Institutions for Seismology (IRIS), until its merger with UNAVCO to become EarthScope Consortium. NSF SAGE is one of the two premier geophysical facilities in support of geoscience and geoscience education of the National Science Foundation. The other premiere geophysical facility is NSF GAGE, the Geodetic Facility for the Advancement of Geoscience. The services of the facility include support for the Global Seismographic Network (GSN), Data Services, and instrument support via the EarthScope Primary Instrument Center (EPIC), including magnetotelluric (MT) geophysical research. == Global Seismographic Network (GSN) == NSF SAGE manages 40 stations of the 152-station Global Seismographic Network (GSN) for basic global seismicity and Earth structure research. The GSN also enables earthquake hazard mission-related data operations such as: Earthquake location and characterization Tsunami warning Nuclear explosion monitoring == Data Services == SAGE Data Services (DS) is the largest facility for the archiving, curation, and distribution of seismological and other geophysical data in the world. == EarthScope Primary Instrument Center (EPIC) == The EPIC facility maintains the largest open access, shared-use pool of portable seismic sensors in the world. It is located on the campus of New Mexico Tech. == MT == NSF SAGE provides instruments for magnetotelluric (MT) or electromagnetic geophysical research for the recording of our planet's ambient electric and magnetic fields, which allow for the characterization of the conductivity of the area consisting of the shallow crust to upper mantle. This helps with analysis of results obtained from seismic imaging methodologies. The NSF SAGE facility is: Developing open source MT data formatting and processing software. Providing access to proprietary software products.
Controlled vocabulary
A controlled vocabulary provides a way to organize knowledge for subsequent retrieval. Controlled vocabularies are used in subject indexing schemes, subject headings, thesauri, taxonomies and other knowledge organization systems. Controlled vocabulary schemes mandate the use of predefined, preferred terms that have been preselected by the designers of the schemes, in contrast to natural language vocabularies, which have no such restriction. == In library and information science == In library and information science, controlled vocabulary is a carefully selected list of words and phrases, which are used to tag units of information (document or work) so that they may be more easily retrieved by a search. Controlled vocabularies solve the problems of homographs, synonyms and polysemes by a bijection between concepts and preferred terms. In short, controlled vocabularies reduce unwanted ambiguity inherent in normal human languages where the same concept can be given different names and ensure consistency. For example, in the Library of Congress Subject Headings (a subject heading system that uses a controlled vocabulary), preferred terms—subject headings in this case—have to be chosen to handle choices between variant spellings of the same word (American versus British), choice among scientific and popular terms (cockroach versus Periplaneta americana), and choices between synonyms (automobile versus car), among other difficult issues. Choices of preferred terms are based on the principles of user warrant (what terms users are likely to use), literary warrant (what terms are generally used in the literature and documents), and structural warrant (terms chosen by considering the structure, scope of the controlled vocabulary). Controlled vocabularies also typically handle the problem of homographs with qualifiers. For example, the term pool has to be qualified to refer to either swimming pool or the game pool to ensure that each preferred term or heading refers to only one concept. === Types used in libraries === There are two main kinds of controlled vocabulary tools used in libraries: subject headings and thesauri. While the differences between the two are diminishing, there are still some minor differences: Historically, subject headings were designed to describe books in library catalogs by catalogers while thesauri were used by indexers to apply index terms to documents and articles. Subject headings tend to be broader in scope describing whole books, while thesauri tend to be more specialized covering very specific disciplines. Because of the card catalog system, subject headings tend to have terms that are in indirect order (though with the rise of automated systems this is being removed), while thesaurus terms are always in direct order. Subject headings tend to use more pre-coordination of terms such that the designer of the controlled vocabulary will combine various concepts together to form one preferred subject heading. (e.g., children and terrorism) while thesauri tend to use singular direct terms. Thesauri list not only equivalent terms but also narrower, broader terms and related terms among various preferred and non-preferred (but potentially synonymous) terms, while historically most subject headings did not. For example, the Library of Congress Subject Heading itself did not have much syndetic structure until 1943, and it was not until 1985 when it began to adopt the thesauri type term "Broader term" and "Narrow term". The terms are chosen and organized by trained professionals (including librarians and information scientists) who possess expertise in the subject area. Controlled vocabulary terms can accurately describe what a given document is actually about, even if the terms themselves do not occur within the document's text. Well known subject heading systems include the Library of Congress system, Medical Subject Headings (MeSH) created by the United States National Library of Medicine, and Sears. Well known thesauri include the Art and Architecture Thesaurus and the ERIC Thesaurus. When selecting terms for a controlled vocabulary, the designer has to consider the specificity of the term chosen, whether to use direct entry, inter consistency and stability of the language. Lastly the amount of pre-coordination (in which case the degree of enumeration versus synthesis becomes an issue) and post-coordination in the system is another important issue. Controlled vocabulary elements (terms/phrases) employed as tags, to aid in the content identification process of documents, or other information system entities (e.g. DBMS, Web Services) qualifies as metadata. == Indexing languages == There are three main types of indexing languages. Controlled indexing language – only approved terms can be used by the indexer to describe the document Natural language indexing language – any term from the document in question can be used to describe the document Free indexing language – any term (not only from the document) can be used to describe the document When indexing a document, the indexer also has to choose the level of indexing exhaustivity, the level of detail in which the document is described. For example, using low indexing exhaustivity, minor aspects of the work will not be described with index terms. In general the higher the indexing exhaustivity, the more terms indexed for each document. In recent years free text search as a means of access to documents has become popular. This involves using natural language indexing with an indexing exhaustively set to maximum (every word in the text is indexed). These methods have been compared in some studies, such as the 2007 article, "A Comparative Evaluation of Full-text, Concept-based, and Context-sensitive Search". === Advantages === Controlled vocabularies are often claimed to improve the accuracy of free text searching, such as to reduce irrelevant items in the retrieval list. These irrelevant items (false positives) are often caused by the inherent ambiguity of natural language. Take the English word football for example. Football is the name given to a number of different team sports. Worldwide the most popular of these team sports is association football, which also happens to be called soccer in several countries. The word football is also applied to rugby football (rugby union and rugby league), American football, Australian rules football, Gaelic football, and Canadian football. A search for football therefore will retrieve documents that are about several completely different sports. Controlled vocabulary solves this problem by tagging the documents in such a way that the ambiguities are eliminated. Compared to free text searching, the use of a controlled vocabulary can dramatically increase the performance of an information retrieval system, if performance is measured by precision (the percentage of documents in the retrieval list that are actually relevant to the search topic). In some cases controlled vocabulary can enhance recall as well, because unlike natural language schemes, once the correct preferred term is searched, there is no need to search for other terms that might be synonyms of that term. === Disadvantages === A controlled vocabulary search may lead to unsatisfactory recall, in that it will fail to retrieve some documents that are actually relevant to the search question. This is particularly problematic when the search question involves terms that are sufficiently tangential to the subject area such that the indexer might have decided to tag it using a different term (but the searcher might consider the same). Essentially, this can be avoided only by an experienced user of controlled vocabulary whose understanding of the vocabulary coincides with that of the indexer. Another possibility is that the article is just not tagged by the indexer because indexing exhaustivity is low. For example, an article might mention football as a secondary focus, and the indexer might decide not to tag it with "football" because it is not important enough compared to the main focus. But it turns out that for the searcher that article is relevant and hence recall fails. A free text search would automatically pick up that article regardless. On the other hand, free text searches have high exhaustivity (every word is searched) so although it has much lower precision, it has potential for high recall as long as the searcher overcome the problem of synonyms by entering every combination. Controlled vocabularies may become outdated rapidly in fast developing fields of knowledge, unless the preferred terms are updated regularly. Even in an ideal scenario, a controlled vocabulary is often less specific than the words of the text itself. Indexers trying to choose the appropriate index terms might misinterpret the author, while this precise problem is not a factor in a free text, as it uses the author's own words. The use of controlled vocabularies can be costly compared to free
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 In computer graphics, clipmapping is a method of clipping a mipmap to a subset of data pertinent to the geometry being displayed. This is useful for loading as little data as possible when memory is limited, such as on a graphics processing unit. The technique is used for LODing in NVIDIA’s implementation of voxel cone tracing. The high-resolution levels of the mipmapped scene representation are clipped to a region near the camera, while lower resolution levels are clipped further away. == MegaTexture == MegaTexture is a clipmap implementation developed by id Software. It was introduced in their id Tech 4 engine and also appeared in id Tech 5 and id Tech 6 before being removed in id Tech 7. MegaTexture is a texture allocation technique that uses a single, extremely large texture rather than repeating multiple smaller textures. It is also featured in Splash Damage's game Enemy Territory: Quake Wars, and was developed by id Software former technical director John Carmack. MegaTexture employs a single large texture space for static terrain. The texture is stored on removable media or a computer's hard drive and streamed as needed, allowing large amounts of detail and variation over a large area with comparatively little RAM usage. Depending on the pixel resolution per square meter, covering a large area could require several gigabytes of memory. However, RAM is also filled by the rest of the game and the underlying operating system, limiting the amount available for texturing. As the player moves around the game, different sections of the MegaTexture are loaded into memory. They are then scaled to the correct size and applied to the 3D models of the terrain. Id has presented a more advanced technique that builds upon the MegaTexture idea and virtualizes both the geometry and the textures to obtain unique geometry down to the equivalent of the texel: the sparse voxel octree (SVO). It works by raycasting the geometry represented by voxels (instead of triangles) stored in an octree. The goal is to stream parts of the octree into video memory, going further down along the tree for nearby objects to give them more details, and to use higher level, larger voxels for farther objects, which give an automatic level of detail (LOD) system for both geometry and textures at the same time. The geometric detail that can be obtained using this method is nearly infinite, which removes the need for faking 3-dimensional details with techniques such as normal mapping. Despite that most voxel rendering tests use very large amounts of memory (up to several GB), Jon Olick of id Software claimed the technology is able to compress such SVO to 1.15 bits per voxel of position data. == Virtual texturing == Unlike clipmaps, which clip each mip level around a viewpoint-dependent clipcenter and therefore work best for terrain, virtual texturing preprocesses texture data into equally sized tiles that can be streamed for arbitrary textured geometry. Rage, powered by the id Tech 5 engine, uses a more advanced technique called virtual texturing. Textures can measure up to 128000×128000 pixels and are also used for in-game models and sprites, etc. and not just the terrain. Wolfenstein: The New Order and the 2016 version of Doom also use these. Carmageddon: Reincarnation also uses virtual texturing, though unlike id's virtual texturing system, which is designed for unique texture-mapping everywhere, their system is designed to use storage space sparingly while still offering good blend of texture variation and resolution. A Hindley–Milner (HM) type system is a classical type system for the lambda calculus with parametric polymorphism. It is also known as Damas–Milner or Damas–Hindley–Milner. It was first described by J. Roger Hindley and later rediscovered by Robin Milner. Luis Damas contributed a close formal analysis and proof of the method in his PhD thesis. Among HM's more notable properties are its completeness and its ability to infer the most general type of a given program without programmer-supplied type annotations or other hints. Algorithm W is an efficient type inference method in practice and has been successfully applied on large code bases, although it has a high theoretical complexity. HM is preferably used for functional programming languages. It was first implemented as part of the type system of the programming language ML. Since then, HM has been extended in various ways, most notably with type class constraints like those in Haskell. == Introduction == As a type inference method, Hindley–Milner is able to deduce the types of variables, expressions and functions from programs written in an entirely untyped style. Being scope sensitive, it is not limited to deriving the types only from a small portion of source code, but rather from complete programs or modules. Being able to cope with parametric types, too, it is core to the type systems of many functional programming languages. It was first applied in this manner in the ML programming language. The origin is the type inference algorithm for the simply typed lambda calculus that was devised by Haskell Curry and Robert Feys in 1958. In 1969, J. Roger Hindley extended this work and proved that their algorithm always inferred the most general type. In 1978, Robin Milner, independently of Hindley's work, provided an equivalent algorithm, Algorithm W. In 1982, Luis Damas finally proved that Milner's algorithm is complete and extended it to support systems with polymorphic references. === Monomorphism vs. polymorphism === In the simply typed lambda calculus, types T are either atomic type constants or function types of form T → T {\displaystyle T\rightarrow T} . Such types are monomorphic. Typical examples are the types used in arithmetic values: 3 : N u m b e r a d d 3 4 : N u m b e r a d d : N u m b e r → N u m b e r → N u m b e r {\displaystyle {\begin{array}{ll}3&:{\mathtt {Number}}\\{\mathtt {add}}\ 3\ 4&:{\mathtt {Number}}\\{\mathtt {add}}&:{\mathtt {Number}}\rightarrow {\mathtt {Number}}\rightarrow {\mathtt {Number}}\end{array}}} Contrary to this, the untyped lambda calculus is neutral to typing at all, and many of its functions can be meaningfully applied to all type of arguments. The trivial example is the identity function i d ≡ λ x . x {\displaystyle {\mathtt {id}}\equiv \lambda x.x} which simply returns whatever value it is applied to. Less trivial examples include parametric types like lists. While polymorphism in general means that operations accept values of more than one type, the polymorphism used here is parametric. One finds the notation of type schemes in the literature, too, emphasizing the parametric nature of the polymorphism. Additionally, constants may be typed with (quantified) type variables. For example, the following type schemes quantify universally over α {\displaystyle \alpha } , meaning that they are true for all possible α {\displaystyle \alpha } : c o n s : ∀ α . α → L i s t α → L i s t α n i l : ∀ α . L i s t α i d : ∀ α . α → α {\displaystyle {\begin{array}{ll}{\mathtt {cons}}&:\forall \alpha .\alpha \rightarrow {\mathtt {List}}\ \alpha \rightarrow {\mathtt {List}}\ \alpha \\{\mathtt {nil}}&:\forall \alpha .{\mathtt {List}}\ \alpha \\{\mathtt {id}}&:\forall \alpha .\alpha \rightarrow \alpha \end{array}}} Polymorphic types can become monomorphic by consistent substitution of their variables. Examples of monomorphic instances are: i d ′ : S t r i n g → S t r i n g n i l ′ : L i s t N u m b e r {\displaystyle {\begin{array}{ll}{\mathtt {id}}'&:{\mathtt {String}}\rightarrow {\mathtt {String}}\\{\mathtt {nil}}'&:{\mathtt {List}}\ {\mathtt {Number}}\end{array}}} More generally, types are polymorphic when they contain type variables, while types without them are monomorphic. Contrary to the type systems used for example in Pascal (1970) or C (1972), which only support monomorphic types, HM is designed with emphasis on parametric polymorphism. The successors of the languages mentioned, like C++ (1985), focused on different types of polymorphism, namely subtyping in connection with object-oriented programming and overloading. While subtyping is incompatible with HM, a variant of systematic overloading is available in the HM-based type system of Haskell. === Let-polymorphism === When extending the type inference for the simply-typed lambda calculus towards polymorphism, one has to decide whether assigning a polymorphic type not only as type of an expression, but also as the type of a λ-bound variable is admissible. This would allow the generic identity type to be assigned to the variable 'id' in: (λ id . ... (id 3) ... (id "text") ... ) (λ x . x) Allowing this gives rise to the polymorphic lambda calculus; however, type inference in this system is not decidable. Instead, HM distinguishes variables that are immediately bound to an expression from more general λ-bound variables, calling the former let-bound variables, and allows polymorphic types to be assigned only to these. This leads to let-polymorphism where the above example takes the form let id = λ x . x in ... (id 3) ... (id "text") ... which can be typed with a polymorphic type for 'id'. As indicated, the expression syntax is extended to make the let-bound variables explicit, and by restricting the type system to allow only let-bound variable to have polymorphic types, while the parameters in lambda-abstractions must get a monomorphic type, type inference becomes decidable. == Overview == The remainder of this article proceeds as follows: The HM type system is defined. This is done by describing a deduction system that makes precise what expressions have what type, if any. From there, it works towards an implementation of the type inference method. After introducing a syntax-driven variant of the above deductive system, it sketches an efficient implementation (algorithm J), appealing mostly to the reader's metalogical intuition. Because it remains open whether algorithm J indeed realises the initial deduction system, a less efficient implementation (algorithm W), is introduced and its use in a proof is hinted. Finally, further topics related to the algorithm are discussed. The same description of the deduction system is used throughout, even for the two algorithms, to make the various forms in which the HM method is presented directly comparable. == The Hindley–Milner type system == The type system can be formally described by syntax rules that fix a language for the expressions, types, etc. The presentation here of such a syntax is not too formal, in that it is written down not to study the surface grammar, but rather the depth grammar, and leaves some syntactical details open. This form of presentation is usual. Building on this, typing rules are used to define how expressions and types are related. As before, the form used is a bit liberal. === Syntax === The expressions to be typed are exactly those of the lambda calculus extended with a let-expression as shown in the adjacent table. Parentheses can be used to disambiguate an expression. The application is left-binding and binds stronger than abstraction or the let-in construct. Types are syntactically split into two groups, monotypes and polytypes. ==== Monotypes ==== Monotypes always designate a particular type. Monotypes τ {\displaystyle \tau } are syntactically represented as terms. Examples of monotypes include type constants like i n t {\displaystyle {\mathtt {int}}} or s t r i n g {\displaystyle {\mathtt {string}}} , and parametric types like M a p ( S e t s t r i n g ) i n t {\displaystyle {\mathtt {Map\ (Set\ string)\ int}}} . The latter types are examples of applications of type functions, for example, from the set { M a p 2 , S e t 1 , s t r i n g 0 , i n t 0 , → 2 } {\displaystyle \{{\mathtt {Map^{2},\ Set^{1},\ string^{0},\ int^{0}}},\ \rightarrow ^{2}\}} , where the superscript indicates the number of type parameters. The complete set of type functions C {\displaystyle C} is arbitrary in HM, except that it must contain at least → 2 {\displaystyle \rightarrow ^{2}} , the type of functions. It is often written in infix notation for convenience. For example, a function mapping integers to strings has type i n t → s t r i n g {\displaystyle {\mathtt {int}}\rightarrow {\mathtt {string}}} . Again, parentheses can be used to disambiguate a type expression. The application binds stronger than the infix arrow, which is right-binding. Type variables are admitted as monotypes. Monotypes are not to be confused with monomorphic types, which excClipmap
Hindley–Milner type system