AI Email Enhancer

AI Email Enhancer — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

  • Starlight Information Visualization System

    Starlight Information Visualization System

    Starlight is a software product originally developed at Pacific Northwest National Laboratory and now by Future Point Systems. It is an advanced visual analysis environment. In addition to using information visualization to show the importance of individual pieces of data by showing how they relate to one another, it also contains a small suite of tools useful for collaboration and data sharing, as well as data conversion, processing, augmentation and loading. The software, originally developed for the intelligence community, allows users to load data from XML files, databases, RSS feeds, web services, HTML files, Microsoft Word, PowerPoint, Excel, CSV, Adobe PDF, TXT files, etc. and analyze it with a variety of visualizations and tools. The system integrates structured, unstructured, geospatial, and multimedia data, offering comparisons of information at multiple levels of abstraction, simultaneously and in near real-time. In addition Starlight allows users to build their own named entity-extractors using a combination of algorithms, targeted normalization lists and regular expressions in the Starlight Data Engineer (SDE). As an example, Starlight might be used to look for correlations in a database containing records about chemical spills. An analyst could begin by grouping records according to the cause of the spill to reveal general trends. Sorting the data a second time, they could apply different colors based on related details such as the company responsible, age of equipment or geographic location. Maps and photographs could be integrated into the display, making it even easier to recognize connections among multiple variables. Starlight has been deployed to both the Iraq and Afghanistan wars and used on a number of large-scale projects. PNNL began developing Starlight in the mid-1990s, with funding from the Land Information Warfare Agency, a part of the Army Intelligence and Security Command and continued developed at the laboratory with funding from the NSA and the CIA. Starlight integrates visual representations of reports, radio transcripts, radar signals, maps and other information. The software system was recently honored with an R&D 100 Award for technical innovation. In 2006 Future Point Systems, a Silicon Valley startup, acquired rights to jointly develop and distribute the Starlight product in cooperation with the Pacific Northwest National Laboratory. The software is now also used outside of the military/intelligence communities in a number of commercial environments.

    Read more →
  • Public First Action

    Public First Action

    Public First Action is a 501(c)(4) nonprofit organization focused on United States public policy related to artificial intelligence. Public First Action is a bipartisan group that advocates for AI transparency, safeguards, and export controls on advanced AI chips. The organization is aligned with the political action committees Jobs and Democracy, Defending Our Values and Public First. == History == Public First Action was formed in 2025 by former Congressmen Brad Carson, a Democrat, and Chris Stewart, a Republican, to advocate for federal, state, and local regulations related to AI. The group's formation followed the founding of a super PAC network, Leading the Future, which advocates for deregulation of the AI industry and faster development of the new technology. Public First Action supports measures that would increase transparency at frontier AI companies and impose export controls on advanced AI chips, in addition to opposing the preemption of state-level AI laws. In February 2026, Public First Action received $20 million from the AI company Anthropic. That same month, the group announced plans to support 30 to 50 Democrats and Republicans in state and federal races, with Public First Action and aligned super PACs launching advertisements in Nebraska, Tennessee, and other states. In one ad, Public First Action touted Senator Marsha Blackburn for her work on child online safety. As of 2026, the group plans to raise between $50 and $75 million for public oversight of AI and related reforms. == Organization == === Leadership and funding === Public First Action is led by Carson and Stewart. The group has raised nearly $50 million in funding with a goal of raising $75 million during the 2026 midterms. Anthropic has contributed $20 million to the group. === Structure === Public First Action is aligned with three political action committees: "Jobs and Democracy", which supports Democratic candidates; "Defending Our Values", which supports Republican candidates; and "Public First", which supports both Republicans and Democrats.

    Read more →
  • NeOn Toolkit

    NeOn Toolkit

    The NeOn Toolkit is an open source, multi-platform ontology editor, which supports the development of ontologies in F-Logic and OWL/RDF. The editor is based on the Eclipse platform and provides a set of plug-ins (currently 20 plug-ins are available for the latest version, v2.4) covering a number of ontology engineering activities, including Annotation and Documentation, Modularization and Customization, Reuse, Ontology Evolution, translation and others. The NeOn Toolkit has been developed in the course of the EU-funded NeOn project and is currently maintained and distributed by the NeOn Technologies Foundation.

    Read more →
  • Stockfish (chess)

    Stockfish (chess)

    Stockfish is a free and open-source chess engine, available for various desktop and mobile platforms. It can be used in chess software through the Universal Chess Interface. Stockfish has been one of the strongest chess engines in the world for several years. It has won all main events of the Top Chess Engine Championship (TCEC) and the Chess.com Computer Chess Championship (CCC) since 2020 and, as of May 2026, is the strongest CPU chess engine in the world with an estimated Elo rating of 3653 in a time control of 40/15 (15 minutes to make 40 moves), according to CCRL. The Stockfish engine was developed by Tord Romstad, Marco Costalba, and Joona Kiiski, and was derived from Glaurung, an open-source engine by Tord Romstad released in 2004. It is now being developed and maintained by the Stockfish community. Stockfish historically used only a classical hand-crafted function to evaluate board positions, but with the introduction of the efficiently updatable neural network (NNUE) in August 2020, Stockfish 12 adopted a hybrid evaluation system that primarily used the neural network and occasionally relied on the hand-crafted evaluation. In July 2023, Stockfish removed the hand-crafted evaluation and transitioned to a fully neural network-based approach. == Features == Stockfish uses a tree-search algorithm based on alpha–beta search with several hand-designed heuristics. Stockfish represents positions using bitboards. Stockfish supports Chess960, a feature it inherited from Glaurung. Support for Syzygy tablebases, previously available in a fork maintained by Ronald de Man, was integrated into Stockfish in 2014. In 2018, support for the 7-man Syzygy was added, shortly after the tablebase was made available. Stockfish supports an unlimited number of CPU threads in multiprocessor systems, with a maximum transposition table size of 32 TB. Stockfish has been a very popular engine on various platforms. On desktop, it is the default chess engine bundled with the Internet Chess Club interface programs BlitzIn and Dasher. On mobile, it has been bundled with the Stockfish app, SmallFish and Droidfish. Other Stockfish-compatible graphical user interfaces (GUIs) include Fritz, Arena, Stockfish for Mac, and PyChess. Stockfish can be compiled to WebAssembly or JavaScript, allowing it to run in the browser. Both Chess.com and Lichess provide Stockfish in this form in addition to a server-side program. Release versions and development versions are available as C++ source code and as precompiled versions for Microsoft Windows, macOS, Linux 32-bit/64-bit and Android. == History == The program originated from Glaurung, an open-source chess engine created by Tord Romstad and first released in 2004. Four years later, Marco Costalba forked the project, naming it Stockfish because it was "produced in Norway and cooked in Italy" (Romstad is Norwegian and Costalba is Italian). The first version, Stockfish 1.0, was released in November 2008. For a while, new ideas and code changes were transferred between the two programs in both directions, until Romstad decided to discontinue Glaurung in favor of Stockfish, which was the stronger engine at the time. The last Glaurung version (2.2) was released in December 2008. Around 2011, Romstad decided to abandon his involvement with Stockfish in order to spend more time on his new iOS chess app. On 18 June 2014 Marco Costalba announced that he had "decided to step down as Stockfish maintainer" and asked that the community create a fork of the current version and continue its development. An official repository, managed by a volunteer group of core Stockfish developers, was created soon after and currently manages the development of the project. === Fishtest === Since 2013, Stockfish has been developed using a distributed testing framework named Fishtest, where volunteers can donate CPU time for testing improvements to the program. Changes to game-playing code are accepted or rejected based on results of playing of tens of thousands of games on the framework against an older "reference" version of the program, using sequential probability ratio testing. Tests on the framework are verified using the chi-squared test, and only if the results are statistically significant are they deemed reliable and used to revise the software code. After the inception of Fishtest, Stockfish gained 120 Elo points in 12 months, propelling it to the top of all major rating lists. As of May 2026, the framework has used a total of more than 20,100 years of CPU time to play over 10 billion chess games. === NNUE === In June 2020, Stockfish introduced the efficiently updatable neural network (NNUE) approach, based on earlier work by computer shogi programmers. Instead of using manually designed heuristics to evaluate the board, this approach introduced a neural network trained on millions of positions which could be evaluated quickly on CPU. On 2 September 2020, the twelfth version of Stockfish was released, incorporating NNUE, and reportedly winning ten times more game pairs than it loses when matched against version eleven. In July 2023, the classical evaluation was completely removed in favor of the NNUE evaluation. == Competition results == === Top Chess Engine Championship === Stockfish is a TCEC multiple-time champion and the current leader in trophy count. Ever since TCEC restarted in 2013, Stockfish has finished first or second in every season except one. Stockfish finished second in TCEC Season 4 and 5, with scores of 23–25 first against Houdini 3 and later against Komodo 1142 in the Superfinal event. Season 5 was notable for the winning Komodo team as they accepted the award posthumously for the program's creator Don Dailey, who succumbed to an illness during the final stage of the event. In his honor, the version of Stockfish that was released shortly after that season was named "Stockfish DD". On 30 May 2014, Stockfish 170514 (a development version of Stockfish 5 with tablebase support) convincingly won TCEC Season 6, scoring 35.5–28.5 against Komodo 7x in the Superfinal. Stockfish 5 was released the following day. In TCEC Season 7, Stockfish again made the Superfinal, but lost to Komodo with a score of 30.5–33.5. In TCEC Season 8, despite losses on time caused by buggy code, Stockfish nevertheless qualified once more for the Superfinal, but lost 46.5–53.5 to Komodo. In Season 9, Stockfish defeated Houdini 5 with a score of 54.5–45.5. Stockfish finished third during season 10 of TCEC, the only season since 2013 in which Stockfish had failed to qualify for the superfinal. It did not lose a game but was still eliminated because it was unable to score enough wins against lower-rated engines. After this technical elimination, Stockfish went on a long winning streak, winning seasons 11 (59–41 against Houdini 6.03), 12 (60–40 against Komodo 12.1.1), and 13 (55–45 against Komodo 2155.00) convincingly. In Season 14, Stockfish faced a new challenger in Leela Chess Zero, eking out a win by one point (50.5–49.5). Its winning streak was finally ended in Season 15, when Leela qualified again and won 53.5–46.5, but Stockfish promptly won Season 16, defeating AllieStein 54.5–45.5, after Leela failed to qualify for the Superfinal. In Season 17, Stockfish faced Leela again in the superfinal, losing 52.5–47.5. However, Stockfish has won every Superfinal since: beating Leela 53.5–46.5 in Season 18, 54.5–45.5 in Season 19, 53–47 in Season 20, and 56–44 in Season 21. In Season 22, Komodo Dragon beat out Leela to qualify for the Superfinal, losing to Stockfish by a large margin 59.5–40.5. Stockfish did not lose an opening pair in this match. Leela made the Superfinal in Seasons 23 and 24, but was crushed by Stockfish both times (58.5–41.5 and 58–42). In Season 25, Stockfish once again defeated Leela, but this time by a narrower margin of 52–48. Stockfish also took part in the TCEC cup, winning the first edition, but was surprisingly upset by Houdini in the semifinals of the second edition. Stockfish recovered to beat Komodo in the third-place playoff. In the third edition, Stockfish made it to the finals, but was defeated by Leela Chess Zero after blundering in a 7-man endgame tablebase draw. It turned this result around in the fourth edition, defeating Leela in the final 4.5–3.5. In TCEC Cup 6, Stockfish finished third after losing to AllieStein in the semifinals, the first time it had failed to make the finals. Since then, Stockfish has consistently won the tournament, with the exception of the 11th edition which Leela won 8.5–7.5. === Chess.com Computer Chess Championship === Ever since Chess.com hosted its first Chess.com Computer Chess Championship in 2018, Stockfish has been the most successful engine. It dominated the earlier championships, winning six consecutive titles before finishing second in CCC7. Since then, its dominance has come under threat from the neural-network engines Leelenstein and Leela Chess Zero, but it has continued to perform w

    Read more →
  • Fuse Services Framework

    Fuse Services Framework

    Fuse Services Framework is an open source SOAP and REST web services platform based on Apache CXF for use in enterprise IT organizations. It is productized and supported by the Fuse group at FuseSource Corp. Fuse Services Framework service-enables new and existing systems for use in enterprise SOA infrastructure. Fuse Services Framework is a pluggable, small-footprint engine that creates high performance, secure and robust services in minutes using front-end programming APIs like JAX-WS and JAX-RS. It supports multiple transports and bindings and is extensible so developers can add bindings for additional message formats so all systems can work together without having to communicate through a centralized server. Fuse Services Framework is now a part of Red Hat JBoss Fuse. Fabric8 is a free Apache 2.0 Licensed upstream community for the JBoss Fuse product from Red Hat.

    Read more →
  • JaCoP (solver)

    JaCoP (solver)

    JaCoP is a constraint solver for constraint satisfaction problems. It is written in Java and it is provided as a Java library. JaCoP has an interface to the MiniZinc and AMPL modeling languages. Its main focus is on ease of use, modeling power, as well as efficiency. It has a large collection of global constraints implemented to facilitate problem modeling. JaCoP is actively developed since year 2001. Krzysztof Kuchcinski and Radoslaw Szymanek are the core developers of this Java library. There are number of people who have contributed to JaCoP development in addition to core developers. JaCoP development has been influenced by more than 20 research articles from Constraint Programming community. It has been used as a tool in more than 30 research articles. There are many different examples provided so it is easier to learn how to use JaCoP. The JaCoP project contains a wrapper for the Scala programming language, and a wrapper for Clojure is maintained as a separate project CloCoP.

    Read more →
  • STUDENT

    STUDENT

    STUDENT is an early artificial intelligence program that solves algebra word problems. It is written in Lisp by Daniel G. Bobrow as his PhD thesis in 1964 (Bobrow 1964). It was designed to read and solve the kind of word problems found in high school algebra books. The program is often cited as an early accomplishment of AI in natural language processing. == Technical description == Within Project MAC at MIT, the STUDENT system was an early example of a question answering software, which uniquely involved natural language processing and symbolic programming. Other early attempts for solving algebra story problems were realized with 1960s hardware and software as well: for example, the Philips, Baseball and Synthex systems. STUDENT accepts an algebra story written in the English language as input, and generates a number as output. This is realized with a layered pipeline that consists of heuristics for pattern transformation. At first, sentences in English are converted into kernel sentences, which each contain a single piece of information. Next, the kernel sentences are converted into mathematical expressions. The knowledge base that supports the transformation contains 52 facts. STUDENT uses a rule-based system with logic inference. The rules are pre-programmed by the software developer and are able to parse natural language. More powerful techniques for natural language processing, such as machine learning, came into use later as hardware grew more capable, and gained popularity over simpler rule-based systems.

    Read more →
  • Metaclass (knowledge representation)

    Metaclass (knowledge representation)

    In knowledge representation, particularly in the Semantic Web, a metaclass is a class whose instances can themselves be classes. Similar to their role in programming languages, metaclasses in ontology languages can have properties otherwise applicable only to individuals, while retaining the same class's ability to be classified in a concept hierarchy. This enables knowledge about instances of those metaclasses to be inferred by semantic reasoners using statements made in the metaclass. Metaclasses thus enhance the expressivity of knowledge representations in a way that can be intuitive for users. While classes are suitable to represent a population of individuals, metaclasses can, as one of their feature, be used to represent the conceptual dimension of an ontology. Metaclasses are supported in the Web Ontology Language (OWL) and the data-modeling vocabulary RDFS. Metaclasses are often modeled by setting them as the object of claims involving rdf:type and rdfs:subClassOf—built-in properties commonly referred to as instance of and subclass of. Instance of entails that the subject of the claim is an instance, i.e. an individual that is a member of a class. Subclass of entails that the subject is a class. In the context of instance of and subclass of, the key difference between metaclasses and ordinary classes is that metaclasses are the object of instance of claims used on a class, while ordinary classes are not objects of such claims. (e.g. in a claim Bob instance of Human, Bob is the subject and an Instance, while the object, Human, is an ordinary class; but a further claim that Human instance of Animal species makes "Animal species" a metaclass because it has a member, "Human", that is also a Class). OWL 2 DL supports metaclasses by a feature called punning, in which one entity is interpreted as two different types of thing—a class and an individual—depending on its syntactic context. For example, through punning, an ontology could have a concept hierarchy such as Harry the eagle instance of golden eagle, golden eagle subclass of bird, and golden eagle instance of species. In this case, the punned entity would be golden eagle, because it is represented as a class (second claim) and an instance (third claim); whereas the metaclass would be species, as it has an instance that is a class. Punning also enables other properties that would otherwise be applicable only to ordinary instances to be used directly on classes, for example "golden eagle conservation status least concern." Having arisen from the fields of knowledge representation, description logic and formal ontology, Semantic Web languages have a closer relationship to philosophical ontology than do conventional programming languages such as Java or Python. Accordingly, the nature of metaclasses is informed by philosophical notions such as abstract objects, the abstract and concrete, and type-token distinction. Metaclasses permit concepts to be construed as tokens of other concepts while retaining their ontological status as types. This enables types to be enumerated over, while preserving the ability to inherit from types. For example, metaclasses could allow a machine reasoner to infer from a human-friendly ontology how many elements are in the periodic table, or, given that number of protons is a property of chemical element and isotopes are a subclass of elements, how many protons exist in the isotope hydrogen-2. Metaclasses are sometime organized by levels, in a similar way to the simple Theory of types where classes that are not metaclasses are assigned the first level, classes of classes in the first level are in the second level, classes of classes in the second level on the next and so on. == Examples == Following the type-token distinction, real world objects such as Abraham Lincoln or the planet Mars are regrouped into classes of similar objects. Abraham Lincoln is said to be an instance of human, and Mars is an instance of planet. This is a kind of is-a relationship. Metaclasses are class of classes, such as for example the nuclide concept. In chemistry, atoms are often classified as elements and, more specifically, isotopes. The glass of water one last drank has many hydrogen atoms, each of which is an instance of hydrogen. Hydrogen itself, a class of atoms, is an instance of nuclide. Nuclide is a class of classes, hence a metaclass. == Implementations == === RDF and RDFS === In RDF, the rdf:type property is used to state that a resource is an instance of a class. This enables metaclasses to be easily created by using rdf:type in a chain-like fashion. For example, in the two triples the resource species is a metaclass, because golden eagle is used as a class in the first statement and the class golden eagle is said to be an instance of the class species in the second statement. This way of doing allows :species to have non-class instances. RDF also provides rdf:Property as a way to create properties beyond those defined in the built-in vocabulary. Properties can be used directly on metaclasses, for example "species quantity 8.7 million", where quantity is a property defined via rdf:Property and species is a metaclass per the preceding example above. RDFS, an extension of RDF, introduced rdfs:Class and rdfs:subClassOf and enriched how vocabularies can classify concepts. Whereas rdf:type enables vocabularies to represent instantiation, the property rdfs:subClassOf enables vocabularies to represent subsumption. RDFS thus makes it possible for vocabularies to represent taxonomies, also known as subsumption hierarchies or concept hierarchies, which is an important addition to the type–token distinction made possible by RDF. Notably, the resource rdfs:Class is an instance of itself, demonstrating both the use of metaclasses in the language's internal implementation and a reflexive usage of rdf:type. RDFS is its own metamodel. This allows a second way to express that a resource is a metaclass. A triple to instantiate rdfs:Class, for example :golden_eagle rdf:type rdfs:Class will declare :golden_eagle as a class. It's also possible to subclass the rdfs:Class resource to declare a meta-class resource, for example :species rdfs:SubclassOf. By deduction, any instance of :species is then a class, so it is a class with class-instances, a meta-class.. This second way does not allows non-class instances of species and explicitly declares :tpecies as a meta-class. === OWL === In some OWL flavors like OWL1-DL, entities can be either classes or instances, but cannot be both. This limitations forbids metaclasses and metamodeling. This is not the case in the OWL1 full flavor, but this allows the model to be computationally undecidable. In OWL2, metaclasses can implemented with punning, that is a way to treat classes as if they were individuals. Other approaches have also been proposed and used to check the properties of ontologies at a meta level. ==== Punning ==== OWL 2 supports metaclasses through a feature called punning. In metaclasses implemented by punning, the same subject is interpreted as two fundamentally different types of thing—a class and an individual—depending on its syntactic context. This is similar to a pun in natural language, where different senses of the same word are emphasized to illustrate a point. Unlike in natural language, where puns are typically used for comedic or rhetorical effect, the main goal of punning in Semantic Web technologies is to make concepts easier to represent, closer to how they are discussed in everyday speech or academic literature. Although OWL 2 permits the same symbol to assume different roles, its standard semantics (known as Direct Semantics) still interprets the symbol differently depending on whether it is used as an individual, a class, or a property. === Protégé === In the ontology editor Protégé, metaclasses are templates for other classes who are their instances. == Classification == Some ontologies like the Cyc AI project's classifies classes and metaclasses. Classes are divided into fixed-order classes and variable-order classes. In the case of fixed-order classes, an order is attributed for metaclasses by measuring the distance to individuals with respect to the number of "instance of" triples that are necessary to find an individual. Classes that are not metaclasses are classes of individuals, so their order is "1" (first-order classes). Metaclasses that are classes of first-order classes' order is "2" (second-order classes), and so on. Variable-order metaclasses, on the other hand, can have instances; one example of variable-order metaclass is the class of all fixed-order classes.

    Read more →
  • Incremental heuristic search

    Incremental heuristic search

    Incremental heuristic search algorithms combine both incremental and heuristic search to speed up searches of sequences of similar search problems, which is important in domains that are only incompletely known or change dynamically. Incremental search has been studied at least since the late 1960s. Incremental search algorithms reuse information from previous searches to speed up the current search and solve search problems potentially much faster than solving them repeatedly from scratch. Similarly, heuristic search has also been studied at least since the late 1960s. Heuristic search algorithms, often based on A, use heuristic knowledge in the form of approximations of the goal distances to focus the search and solve search problems potentially much faster than uninformed search algorithms. The resulting search problems, sometimes called dynamic path planning problems, are graph search problems where paths have to be found repeatedly because the topology of the graph, its edge costs, the start vertex or the goal vertices change over time. So far, three main classes of incremental heuristic search algorithms have been developed: The first class restarts A at the point where its current search deviates from the previous one (example: Fringe Saving A). The second class updates the h-values (heuristic, i.e. approximate distance to goal) from the previous search during the current search to make them more informed (example: Generalized Adaptive A). The third class updates the g-values (distance from start) from the previous search during the current search to correct them when necessary, which can be interpreted as transforming the A search tree from the previous search into the A search tree for the current search (examples: Lifelong Planning A, D, D Lite). All three classes of incremental heuristic search algorithms are different from other replanning algorithms, such as planning by analogy, in that their plan quality does not deteriorate with the number of replanning episodes. == Applications == Incremental heuristic search has been extensively used in robotics, where a larger number of path planning systems are based on either D (typically earlier systems) or D Lite (current systems), two different incremental heuristic search algorithms.

    Read more →
  • Leela Chess Zero

    Leela Chess Zero

    Leela Chess Zero (abbreviated as LCZero, lc0) is a free, open-source chess engine and volunteer computing project based on Google's AlphaZero engine. It was spearheaded by Gary Linscott, a developer for the Stockfish chess engine, and adapted from the Leela Zero Go engine. Like Leela Zero and AlphaGo Zero, early iterations of Leela Chess Zero started with no intrinsic chess-specific knowledge other than the basic rules of the game. It learned how to play chess through reinforcement learning from repeated self-play, using a distributed computing network coordinated at the Leela Chess Zero website. However, as of November 2024 most models used by the engine are trained through supervised learning on data generated by previous reinforcement learning runs. As of June 2025, Leela Chess Zero has played over 2.5 billion games against itself, playing around 1 million games every day, and is capable of play at a level that is comparable with Stockfish, the leading conventional chess program. == History == The Leela Chess Zero project was first announced on TalkChess.com on January 9, 2018, as an open-source, self-learning chess engine attempting to recreate the success of AlphaZero. Within the first few months of training, Leela Chess Zero had already reached the Grandmaster level, surpassing the strength of early releases of Rybka, Stockfish, and Komodo, despite evaluating orders of magnitude fewer positions due to the size of the deep neural network it uses as its evaluation function. In December 2018, the AlphaZero team published a paper in Science magazine revealing previously undisclosed details of the architecture and training parameters used for AlphaZero. These changes were soon incorporated into Leela Chess Zero and increased both its strength and training efficiency. Work on Leela Chess Zero has informed the AobaZero project for shogi. The engine has been rewritten and carefully iterated upon since its inception, and since 2019 has run on multiple backends, allowing it to run on both CPU and GPU. The engine can be configured to use different weights, including even different architectures. This same mechanism of substitutable weights can also be used for alternative chess rules, such as for the Fischer Random Chess variant, which was done in 2019. == Neural network == Like AlphaZero, Leela Chess Zero employs neural networks which output both a policy vector, a distribution over subsequent moves used to guide search, and a position evaluation. These neural networks are designed to run on GPU, unlike traditional engines. It originally used residual neural networks, but in 2022 switched to using a transformer-based architecture designed by Daniel Monroe and Philip Chalmers. These models represent a chessboard as a sequence of 64 tokens and apply a trunk consisting of a stack of Post-LN encoder layers, outputting a sequence of 64 encoded tokens which is used to generate a position evaluation and a distribution over subsequent moves. They use a custom domain-specific position encoding called smolgen to improve the self-attention layer. As of November 2024, the models used by the engine are significantly larger and more efficient than the residual network used by AlphaZero, reportedly achieving grandmaster-level strength at one position evaluation per move. These models are able to detect and exploit positional features like trapped pieces and fortresses to outmaneuver traditional engines, giving Leela a unique playstyle. There is also evidence that they are able to perform look-ahead. == Program and use == Like AlphaZero, Leela Chess Zero learns through reinforcement learning, continually training on data generated through self-play. However, unlike AlphaZero, Leela Chess Zero decentralizes its data generation through distributed computing, with volunteers generating self-play data on local hardware which is fed to the reinforcement algorithm. In order to contribute training games, volunteers must download the latest non-release candidate (non-rc) version of the engine and the client. The client connects to the Leela Chess Zero server and iteratively receives the latest neural network version and produces self-play games which are sent back to the server and use to train the network. In order to run the Leela Chess Zero engine, two components are needed: the engine binary used to perform search, and a network used to evaluate positions. The client, which is used to contribute training data to the project, is not needed for this purpose. Older networks can also be downloaded and used by placing those networks in the folder with the Lc0 binary. == Spinoffs == In season 15 of the Top Chess Engine Championship, the engine AllieStein competed alongside Leela. AllieStein is a combination of two different spinoffs from Leela: Allie, which uses the same neural network as Leela, but has a unique search algorithm for exploring different lines of play, and Stein, a network which was trained using supervised learning on existing game data from games between other engines. While neither of these projects were admitted to TCEC separately due to their similarity to Leela, the combination of Allie's search algorithm with the Stein network, called AllieStein, was deemed unique enough to warrant its inclusion in the competition. In early 2021, the LcZero blog announced Ceres, a transliteration of the engine to C# which introduced several algorithmic improvements. The engine has performed competitively in tournaments, achieving third place in the TCEC Swiss 7 and fourth place in the TCEC Cup 14. In 2024, the CeresTrain framework was announced to support training deep neural networks for chess in PyTorch. == Competition results == In April 2018, Leela Chess Zero became the first engine using a deep neural network to enter the Top Chess Engine Championship (TCEC), during Season 12 in the lowest division, Division 4. Out of 28 games, it won one, drew two, and lost the remainder; its sole victory came from a position in which its opponent, Scorpio 2.82, crashed in three moves. However, it improved quickly. In July 2018, Leela placed seventh out of eight competitors at the 2018 World Computer Chess Championship. In August 2018, it won division 4 of TCEC season 13 with a record of 14 wins, 12 draws, and 2 losses. In Division 3, Leela scored 16/28 points, finishing third behind Ethereal, which scored 22.5/28 points, and Arasan on tiebreak. By September 2018, Leela had become competitive with the strongest engines in the world. In the 2018 Chess.com Computer Chess Championship (CCCC), Leela placed fifth out of 24 entrants. The top eight engines advanced to round 2, where Leela placed fourth. Leela then won the 30-game match against Komodo to secure third place in the tournament. Leela participated in the "TCEC Cup", an event in which engines from different TCEC divisions can play matches against one another. Leela defeated higher-division engines Laser, Ethereal and Fire before finally being eliminated by Stockfish in the semi-finals. In December 2018, Leela participated in Season 14 of the Top Chess Engine Championship. Leela dominated divisions 3, 2, and 1, easily finishing first in all of them. In the premier division, Stockfish dominated while Houdini, Komodo and Leela competed for second place. It came down to a final-round game where Leela needed to hold Stockfish to a draw with black to finish second ahead of Komodo. Leela managed this and therefore met Stockfish in the superfinal. In a back and forth match, first Stockfish and then Leela took three game leads before Stockfish won by the narrow margin of 50.5–49.5. In February 2019, Leela scored its first major tournament win when it defeated Houdini in the final of the second TCEC cup. Leela did not lose a game the entire tournament. In April 2019, Leela won the Chess.com Computer Chess Championship 7: Blitz Bonanza, becoming the first neural-network project to take the title. In the season 15 of the Top Chess Engine Championship (May 2019), Leela defended its TCEC Cup title, this time defeating Stockfish with a score of 5.5–4.5 (+2 =7 −1) in the final after Stockfish blundered a seven-man tablebase draw. Leela also won the Superfinal for the first time, scoring 53.5–46.5 (+14 −7 =79) versus Stockfish, including winning as both white and black in the same predetermined opening in games 61 and 62. Season 16 of TCEC saw Leela finish in third place in premier division, missing qualification for the Superfinal to Stockfish and the new deep neural network engine AllieStein. Leela was the only engine not to suffer any losses in the Premier division, and defeated Stockfish in one of the six games they played. However, Leela only managed to score nine wins, while AllieStein and Stockfish both scored 14 wins. This inability to defeat weaker engines led to Leela finishing third, half a point behind AllieStein and a point behind Stockfish. In the fourth TCEC Cup, Leela was seeded first as the defending champion,

    Read more →
  • Interim Measures for the Management of Anthropomorphic AI Interactive Services

    Interim Measures for the Management of Anthropomorphic AI Interactive Services

    The Interim Measures for the Management of Anthropomorphic AI Interactive Services (Chinese: 人工智能拟人化互动服务管理暂行办法) is a document proposed by the Cyberspace Administration of China to regulate anthropomorphic artificial intelligence systems. The draft was released on December 27, 2026 for public comment period until January 25, 2026. The proposed document would prohibit AI companies and users of AI services from generating certain types of content deemed harmful to national interests or the social order, and impose various regulatory and safety requirements on providers of AI systems. The proposed regulation is motivated by concerns about the psychological and social effects of AI systems that are perceived as personalities by their users, including addiction, encouragement of self-harm, or generation of illegal content. == Description == === Scope === The regulation would apply to AI systems that are offered to the general public within China. They would not apply to company-internal or research use, or to products that are only available outside of China. For the purpose of the regulation, anthropomorphic Ai systems are defined as those that "simulate human personality traits, modes of thinking, and communication styles, and that engage in emotional interaction with humans through text, images, audio, video, or other means". === Requirements === The regulation would require AI providers to monitor users for signs of harmful use and to take various interventions when indications of harmful use are detected. It would also prohibit AI systems from certain types of behaviors and generation of certain types of content. In some circumstances where a user appears to be at risk of self harm, the system would be required to hand over control to a human operator who would manually intervene. The regulation would also require more rigorous practices for managing the provenance of training data used to develop these systems, and would require explicit opt-in consent from users before their interactions with an AI system were used as training data. Data used to train the regulated systems would be required to reflect core socialist values and traditional Chinese culture.

    Read more →
  • Connectionism

    Connectionism

    Connectionism is an approach to the study of human mental processes and cognition that utilizes mathematical models known as connectionist networks or artificial neural networks. Connectionism has had many "waves" since its beginnings. The first wave appeared 1943 with Warren Sturgis McCulloch and Walter Pitts both focusing on comprehending neural circuitry through a formal and mathematical approach, and Frank Rosenblatt who published the 1958 paper "The Perceptron: A Probabilistic Model For Information Storage and Organization in the Brain" in Psychological Review, while working at the Cornell Aeronautical Laboratory. The first wave ended with the 1969 book Perceptrons about limitations of the original perceptron idea, written by Marvin Minsky and Seymour Papert, which contributed to discouraging major funding agencies in the US from investing in connectionist research. With a few noteworthy deviations, most connectionist research entered a period of inactivity until the mid-1980s. The term connectionist model was reintroduced in a 1982 paper in the journal Cognitive Science by Jerome Feldman and Dana Ballard. The second wave blossomed in the late 1980s, following a 1987 book Parallel Distributed Processing by James L. McClelland, David E. Rumelhart, et al., which introduced a couple of improvements to the simple perceptron idea, such as intermediate processors (now known as "hidden layers") alongside input and output units, and used a sigmoid activation function instead of the old "all-or-nothing" function. Their work built upon that of John Hopfield, who was a key figure investigating the mathematical characteristics of sigmoid activation functions. From the late 1980s to the mid-1990s, connectionism took on an almost revolutionary tone when Schneider, Terence Horgan and Tienson posed the question of whether connectionism represented a fundamental shift in psychology and so-called "good old-fashioned AI", or GOFAI. Some advantages of the second wave connectionist approach included its applicability to a broad array of functions, structural approximation to biological neurons, low requirements for innate structure, and capacity for graceful degradation. Its disadvantages included the difficulty in deciphering how ANNs process information or account for the compositionality of mental representations, and a resultant difficulty explaining phenomena at a higher level. The current (third) wave has been marked by advances in deep learning, which have made possible the creation of large language models. The success of deep-learning networks in the past decade has greatly increased the popularity of this approach, but the complexity and scale of such networks has brought with them increased interpretability problems. == Basic principle == The central connectionist principle is that mental phenomena can be described by interconnected networks of simple and often uniform units. The form of the connections and the units can vary from model to model. For example, units in the network could represent neurons and the connections could represent synapses, as in the human brain. This principle has been seen as an alternative to GOFAI and the classical theories of mind based on symbolic computation, but the extent to which the two approaches are compatible has been the subject of much debate since their inception. === Activation function === Internal states of any network change over time due to neurons sending a signal to a succeeding layer of neurons in the case of a feedforward network, or to a previous layer in the case of a recurrent network. Discovery of non-linear activation functions has enabled the second wave of connectionism. === Memory and learning === Neural networks follow two basic principles: Any mental state can be described as a n-dimensional vector of numeric activation values over neural units in a network. Memory and learning are created by modifying the 'weights' of the connections between neural units, generally represented as an n×m matrix. The weights are adjusted according to some learning rule or algorithm, such as Hebbian learning. Most of the variety among the models comes from: Interpretation of units: Units can be interpreted as neurons or groups of neurons. Definition of activation: Activation can be defined in a variety of ways. For example, in a Boltzmann machine, the activation is interpreted as the probability of generating an action potential spike, and is determined via a logistic function on the sum of the inputs to a unit. Learning algorithm: Different networks modify their connections differently. In general, any mathematically defined change in connection weights over time is referred to as the "learning algorithm". === Biological realism === Connectionist work in general does not need to be biologically realistic. One area where connectionist models are thought to be biologically implausible is with respect to error-propagation networks that are needed to support learning, but error propagation can explain some of the biologically-generated electrical activity seen at the scalp in event-related potentials such as the N400 and P600, and this provides some biological support for one of the key assumptions of connectionist learning procedures. Many recurrent connectionist models also incorporate dynamical systems theory. Many researchers, such as the connectionist Paul Smolensky, have argued that connectionist models will evolve toward fully continuous, high-dimensional, non-linear, dynamic systems approaches. == Precursors == Precursors of the connectionist principles can be traced to early work in psychology, such as that of William James. Psychological theories based on knowledge about the human brain were fashionable in the late 19th century. As early as 1869, the neurologist John Hughlings Jackson argued for multi-level, distributed systems. Following from this lead, Herbert Spencer's Principles of Psychology, 3rd edition (1872), and Sigmund Freud's Project for a Scientific Psychology (composed 1895) propounded connectionist or proto-connectionist theories. These tended to be speculative theories. But by the early 20th century, Edward Thorndike was writing about human learning that posited a connectionist type network. Hopfield networks had precursors in the Ising model due to Wilhelm Lenz (1920) and Ernst Ising (1925), though the Ising model conceived by them did not involve time. Monte Carlo simulations of Ising model required the advent of computers in the 1950s. == The first wave == The first wave begun in 1943 with Warren Sturgis McCulloch and Walter Pitts both focusing on comprehending neural circuitry through a formal and mathematical approach. McCulloch and Pitts showed how neural systems could implement first-order logic: Their classic paper "A Logical Calculus of Ideas Immanent in Nervous Activity" (1943) is important in this development here. They were influenced by the work of Nicolas Rashevsky in the 1930s and symbolic logic in the style of Principia Mathematica. Hebb contributed greatly to speculations about neural functioning, and proposed a learning principle, Hebbian learning. Lashley argued for distributed representations as a result of his failure to find anything like a localized engram in years of lesion experiments. Friedrich Hayek independently conceived the model, first in a brief unpublished manuscript in 1920, then expanded into a book in 1952. The Perceptron machines were proposed and built by Frank Rosenblatt, who published the 1958 paper “The Perceptron: A Probabilistic Model For Information Storage and Organization in the Brain” in Psychological Review, while working at the Cornell Aeronautical Laboratory. He cited Hebb, Hayek, Uttley, and Ashby as main influences. Another form of connectionist model was the relational network framework developed by the linguist Sydney Lamb in the 1960s. The research group led by Widrow empirically searched for methods to train two-layered ADALINE networks (MADALINE), with limited success. A method to train multilayered perceptrons with arbitrary levels of trainable weights was published by Alexey Grigorevich Ivakhnenko and Valentin Lapa in 1965, called the Group Method of Data Handling. This method employs incremental layer by layer training based on regression analysis, where useless units in hidden layers are pruned with the help of a validation set. The first multilayered perceptrons trained by stochastic gradient descent was published in 1967 by Shun'ichi Amari. In computer experiments conducted by Amari's student Saito, a five layer MLP with two modifiable layers learned useful internal representations to classify non-linearily separable pattern classes. In 1972, Shun'ichi Amari produced an early example of self-organizing network. == The neural network winter == There was some conflict among artificial intelligence researchers as to what neural networks are useful for. Around late 1960s, there was a widespread lull in research a

    Read more →
  • Software diversity

    Software diversity

    Software diversity is a research field about the comprehension and engineering of diversity in the context of software. == Areas == The different areas of software diversity are discussed in surveys on diversity for fault-tolerance or for security. The main areas are: design diversity, n-version programming, data diversity for fault tolerance randomization software variability == Techniques == === Code transformations === It is possible to amplify software diversity through automated transformation processes that create synthetic diversity. A "multicompiler" is compiler embedding a diversification engine. A multi-variant execution environment (MVEE) is responsible for selecting the variant to execute and compare the output. Fred Cohen was among the very early promoters of such an approach. He proposed a series of rewriting and code reordering transformations that aim at producing massive quantities of different versions of operating systems functions. These ideas have been developed over the years and have led to the construction of integrated obfuscation schemes to protect key functions in large software systems. Another approach to increase software diversity of protection consists in adding randomness in certain core processes, such as memory loading. Randomness implies that all versions of the same program run differently from each other, which in turn creates a diversity of program behaviors. This idea was initially proposed and experimented by Stephanie Forrest and her colleagues. Recent work on automatic software diversity explores different forms of program transformations that slightly vary the behavior of programs. The goal is to evolve one program into a population of diverse programs that all provide similar services to users, but with a different code. This diversity of code enhances the protection of users against one single attack that could crash all programs at the same time. Transformation operators include: code layout randomization: reorder functions in code globals layout randomization: reorder and pad globals stack variable randomization: reorder variables in each stack frame heap layout randomization === Natural software diversity === It is known that some functionalities are available in multiple interchangeable implementations. This natural diversity can be exploited, for example it has been shown valuable to increase security in cloud systems.

    Read more →
  • Metaclass (knowledge representation)

    Metaclass (knowledge representation)

    In knowledge representation, particularly in the Semantic Web, a metaclass is a class whose instances can themselves be classes. Similar to their role in programming languages, metaclasses in ontology languages can have properties otherwise applicable only to individuals, while retaining the same class's ability to be classified in a concept hierarchy. This enables knowledge about instances of those metaclasses to be inferred by semantic reasoners using statements made in the metaclass. Metaclasses thus enhance the expressivity of knowledge representations in a way that can be intuitive for users. While classes are suitable to represent a population of individuals, metaclasses can, as one of their feature, be used to represent the conceptual dimension of an ontology. Metaclasses are supported in the Web Ontology Language (OWL) and the data-modeling vocabulary RDFS. Metaclasses are often modeled by setting them as the object of claims involving rdf:type and rdfs:subClassOf—built-in properties commonly referred to as instance of and subclass of. Instance of entails that the subject of the claim is an instance, i.e. an individual that is a member of a class. Subclass of entails that the subject is a class. In the context of instance of and subclass of, the key difference between metaclasses and ordinary classes is that metaclasses are the object of instance of claims used on a class, while ordinary classes are not objects of such claims. (e.g. in a claim Bob instance of Human, Bob is the subject and an Instance, while the object, Human, is an ordinary class; but a further claim that Human instance of Animal species makes "Animal species" a metaclass because it has a member, "Human", that is also a Class). OWL 2 DL supports metaclasses by a feature called punning, in which one entity is interpreted as two different types of thing—a class and an individual—depending on its syntactic context. For example, through punning, an ontology could have a concept hierarchy such as Harry the eagle instance of golden eagle, golden eagle subclass of bird, and golden eagle instance of species. In this case, the punned entity would be golden eagle, because it is represented as a class (second claim) and an instance (third claim); whereas the metaclass would be species, as it has an instance that is a class. Punning also enables other properties that would otherwise be applicable only to ordinary instances to be used directly on classes, for example "golden eagle conservation status least concern." Having arisen from the fields of knowledge representation, description logic and formal ontology, Semantic Web languages have a closer relationship to philosophical ontology than do conventional programming languages such as Java or Python. Accordingly, the nature of metaclasses is informed by philosophical notions such as abstract objects, the abstract and concrete, and type-token distinction. Metaclasses permit concepts to be construed as tokens of other concepts while retaining their ontological status as types. This enables types to be enumerated over, while preserving the ability to inherit from types. For example, metaclasses could allow a machine reasoner to infer from a human-friendly ontology how many elements are in the periodic table, or, given that number of protons is a property of chemical element and isotopes are a subclass of elements, how many protons exist in the isotope hydrogen-2. Metaclasses are sometime organized by levels, in a similar way to the simple Theory of types where classes that are not metaclasses are assigned the first level, classes of classes in the first level are in the second level, classes of classes in the second level on the next and so on. == Examples == Following the type-token distinction, real world objects such as Abraham Lincoln or the planet Mars are regrouped into classes of similar objects. Abraham Lincoln is said to be an instance of human, and Mars is an instance of planet. This is a kind of is-a relationship. Metaclasses are class of classes, such as for example the nuclide concept. In chemistry, atoms are often classified as elements and, more specifically, isotopes. The glass of water one last drank has many hydrogen atoms, each of which is an instance of hydrogen. Hydrogen itself, a class of atoms, is an instance of nuclide. Nuclide is a class of classes, hence a metaclass. == Implementations == === RDF and RDFS === In RDF, the rdf:type property is used to state that a resource is an instance of a class. This enables metaclasses to be easily created by using rdf:type in a chain-like fashion. For example, in the two triples the resource species is a metaclass, because golden eagle is used as a class in the first statement and the class golden eagle is said to be an instance of the class species in the second statement. This way of doing allows :species to have non-class instances. RDF also provides rdf:Property as a way to create properties beyond those defined in the built-in vocabulary. Properties can be used directly on metaclasses, for example "species quantity 8.7 million", where quantity is a property defined via rdf:Property and species is a metaclass per the preceding example above. RDFS, an extension of RDF, introduced rdfs:Class and rdfs:subClassOf and enriched how vocabularies can classify concepts. Whereas rdf:type enables vocabularies to represent instantiation, the property rdfs:subClassOf enables vocabularies to represent subsumption. RDFS thus makes it possible for vocabularies to represent taxonomies, also known as subsumption hierarchies or concept hierarchies, which is an important addition to the type–token distinction made possible by RDF. Notably, the resource rdfs:Class is an instance of itself, demonstrating both the use of metaclasses in the language's internal implementation and a reflexive usage of rdf:type. RDFS is its own metamodel. This allows a second way to express that a resource is a metaclass. A triple to instantiate rdfs:Class, for example :golden_eagle rdf:type rdfs:Class will declare :golden_eagle as a class. It's also possible to subclass the rdfs:Class resource to declare a meta-class resource, for example :species rdfs:SubclassOf. By deduction, any instance of :species is then a class, so it is a class with class-instances, a meta-class.. This second way does not allows non-class instances of species and explicitly declares :tpecies as a meta-class. === OWL === In some OWL flavors like OWL1-DL, entities can be either classes or instances, but cannot be both. This limitations forbids metaclasses and metamodeling. This is not the case in the OWL1 full flavor, but this allows the model to be computationally undecidable. In OWL2, metaclasses can implemented with punning, that is a way to treat classes as if they were individuals. Other approaches have also been proposed and used to check the properties of ontologies at a meta level. ==== Punning ==== OWL 2 supports metaclasses through a feature called punning. In metaclasses implemented by punning, the same subject is interpreted as two fundamentally different types of thing—a class and an individual—depending on its syntactic context. This is similar to a pun in natural language, where different senses of the same word are emphasized to illustrate a point. Unlike in natural language, where puns are typically used for comedic or rhetorical effect, the main goal of punning in Semantic Web technologies is to make concepts easier to represent, closer to how they are discussed in everyday speech or academic literature. Although OWL 2 permits the same symbol to assume different roles, its standard semantics (known as Direct Semantics) still interprets the symbol differently depending on whether it is used as an individual, a class, or a property. === Protégé === In the ontology editor Protégé, metaclasses are templates for other classes who are their instances. == Classification == Some ontologies like the Cyc AI project's classifies classes and metaclasses. Classes are divided into fixed-order classes and variable-order classes. In the case of fixed-order classes, an order is attributed for metaclasses by measuring the distance to individuals with respect to the number of "instance of" triples that are necessary to find an individual. Classes that are not metaclasses are classes of individuals, so their order is "1" (first-order classes). Metaclasses that are classes of first-order classes' order is "2" (second-order classes), and so on. Variable-order metaclasses, on the other hand, can have instances; one example of variable-order metaclass is the class of all fixed-order classes.

    Read more →
  • Mathematical model

    Mathematical model

    A mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in many fields, including applied mathematics, natural sciences, social sciences and engineering. In particular, the field of operations research studies the use of mathematical modelling and related tools to solve problems in business or military operations. A model may help to characterize a system by studying the effects of different components, which may be used to make predictions about behavior or solve specific problems. == Elements of a mathematical model == Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed. In the physical sciences, a traditional mathematical model contains most of the following elements: Governing equations Supplementary sub-models Defining equations Constitutive equations Assumptions and constraints Initial and boundary conditions Classical constraints and kinematic equations == Classifications == Mathematical models are of different types: === Linear vs. nonlinear === If all the operators in a mathematical model exhibit linearity, the resulting mathematical model is defined as linear. All other models are considered nonlinear. The definition of linearity and nonlinearity is dependent on context, and linear models may have nonlinear expressions in them. For example, in a statistical linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, a differential equation is said to be linear if it can be written with linear differential operators, but it can still have nonlinear expressions in it. In a mathematical programming model, if the objective functions and constraints are represented entirely by linear equations, then the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear equation, then the model is known as a nonlinear model. Linear structure implies that a problem can be decomposed into simpler parts that can be treated independently or analyzed at a different scale, and therefore that the results will remain valid if the initial is recomposed or rescaled. Nonlinearity, even in fairly simple systems, is often associated with phenomena such as chaos and irreversibility. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is linearization, but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity. === Static vs. dynamic === A dynamic model accounts for time-dependent changes in the state of the system, while a static (or steady-state) model calculates the system in equilibrium, and thus is time-invariant. Dynamic models are typically represented by differential equations or difference equations. === Explicit vs. implicit === If all of the input parameters of the overall model are known, and the output parameters can be calculated by a finite series of computations, the model is said to be explicit. But sometimes it is the output parameters which are known, and the corresponding inputs must be solved for by an iterative procedure, such as Newton's method or Broyden's method. In such a case the model is said to be implicit. For example, a jet engine's physical properties such as turbine and nozzle throat areas can be explicitly calculated given a design thermodynamic cycle (air and fuel flow rates, pressures, and temperatures) at a specific flight condition and power setting, but the engine's operating cycles at other flight conditions and power settings cannot be explicitly calculated from the constant physical properties. === Discrete vs. continuous === A discrete model treats objects as discrete, such as the particles in a molecular model or the states in a statistical model; while a continuous model represents the objects in a continuous manner, such as the velocity field of fluid in pipe flows, temperatures and stresses in a solid, and electric field that applies continuously over the entire model due to a point charge. === Deterministic vs. probabilistic (stochastic) === A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables; therefore, a deterministic model always performs the same way for a given set of initial conditions. Conversely, in a stochastic model—usually called a "statistical model"—randomness is present, and variable states are not described by unique values, but rather by probability distributions. === Deductive, inductive, or floating === A deductive model is a logical structure based on a theory. An inductive model arises from empirical findings and generalization from them. If a model rests on neither theory nor observation, it may be described as a 'floating' model. Application of mathematics in social sciences outside of economics has been criticized for unfounded models. Application of catastrophe theory in science has been characterized as a floating model. === Strategic vs. non-strategic === Models used in game theory are distinct in the sense that they model agents with incompatible incentives, such as competing species or bidders in an auction. Strategic models assume that players are autonomous decision makers who rationally choose actions that maximize their objective function. A key challenge of using strategic models is defining and computing solution concepts such as the Nash equilibrium. An interesting property of strategic models is that they separate reasoning about rules of the game from reasoning about behavior of the players. == Construction == In business and engineering, mathematical models may be used to maximize a certain output. The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: decision variables, state variables, exogenous variables, and random variables. Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables). Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance, as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally) as the number increases. For example, economists often apply linear algebra when using input–output models. Complicated mathematical models that have many variables may be consolidated by use of vectors where one symbol represents several variables. === A priori information === Mathematical modeling problems are often classified into black box or white box models, according to how much a priori information on the system is available. A black-box model is a system of which there is no a priori information available. A white-box model (also called glass box or clear box) is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take. Usually, it is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying function, but we are still left with several unknown parameters; how

    Read more →