Reference Software International, Inc. (RSI), was an American software developer active from 1985 to 1993 and based in Albuquerque, New Mexico, and San Francisco, California. The company released several productivity and reference software packages, including the Grammatik grammar checker, for MS-DOS. The company was acquired by WordPerfect Corporation in 1993. == History == === Background (1980–1985) === Reference Software International, Inc., was founded by Donald "Don" Emery and Bruce Wampler in 1985 in San Francisco, California. Both Wampler and Emery were college professors when they founded RSI: Wampler at the University of New Mexico as a professor of computer science and Emery a professor of marketing at San Francisco State University. After graduating from the University of Utah in around 1978, Wampler founded his first software company, Aspen Software, in Tijeras, New Mexico, in 1979. Wampler founded Aspen to develop an early spell checker software package, called Proofreader, for the TRS-80, licensing Random House's Webster's Unabridged Dictionary for the package's lexicon. In 1980, he began development on a grammar checker inspired by Writer's Workbench, a pioneering grammar checker for Unix systems. Wampler used Writer's Workbench heavily during the writer of his doctoral dissertation but disliked having to jump between the Apple II on which he composed the dissertation and the mainframe on which Writer's Workbench ran, and so wanted to develop a version of the latter for microcomputers. Wampler's work came to fruition as Grammatik in 1981, eventually ported to several other microcomputer platforms in the early 1980s. In 1983, by which point the company had 12 employees and sold a combined 80,000 units of Grammatik and Proofreader, Wampler sold Aspen to Dictronics, a software company best known for developing the Electronic Thesaurus, an early thesaurus program for microcomputers. Dictronics was in turn purchased by Wang Laboratories; according to Wampler, "Wang bought [Aspen] and sat on it. They did nothing with it". Wampler moved on to teach for the University of New Mexico, but, frustrated by Wang's inaction, got the urge to resurrect his work. In 1985, he was able to license back Grammatik and Proofreader from a small California-based software firm that had grandfathered rights to a forked version of both. In the same year, he met Emery, who, impressed by Wampler's, founded Reference Software International to market his software. RSI's research and development headquarters were based in Albuquerque, while the company's sales and marketing department was based in Walnut Creek, California. === Success (1985–1992) === In August 1985, RSI released their first product: the Random House Reference Set, a new version of Proofreader for the IBM Personal Computer and compatibles, revised to be a terminate-and-stay-resident program that ran atop other word processors such as WordStar or WordPerfect. At the time, Reference Set was the only such program on the market that functioned like this. RSI netted $114,000 from sales of Reference Set by the end of 1985. In June 1986, they released version 2.0 of Grammatik as Grammatik II for the PC. The latter was a breakout hit for RSI, receiving praise in the press (including technology journals such as PC Magazine) and RSI selling 1,000 units a month. In spring 1987, they released Reference Set II, which allowed users to import their own words into the built-in dictionary and added a thesaurus of 300,000 words. In November 1987, they released version 3.0 of Reference Set, which comprised two new field-specific dictionaries for the medical and legal professions. As well as the general Random House dictionary and thesaurus, it included Stedman's Medical Dictionary and Black's Law Dictionary. Emery consulted Paul Brest and Bob Jackson—professors of law at Stanford Law School and San Francisco State respectively—for the curation of the law dictionary; and Burton Grebin—at the time the executive director of Mount Saint Mary's Hospital—for the curation of the medical dictionary. In fall 1988, the company released Grammatik III, a total rewrite that made use of artificial intelligence to more accurately judge the grammar of sentences by breaking them down into a syntactic hierarchy. Grammatik III received universal acclaim, with Gloria Morris of InfoWorld calling it the apparent leader in the grammar checking field and Sandra Anderson of Mac Home Journal calling it "hands down ... the best of the industry" six years after its release. By 1989, the product had competitors in Correct Grammar by Lifetree Software and RightWriter by Rightsoft, Inc. By 1990, RSI achieved annual sales of $9.7 million. In the same year they released Grammatik IV, which was the first to offer direct integration with WordPerfect on both MS-DOS and Windows. In March 1992—by which point RSI had sold 1.5 million copies of Grammatik across all versions—the company released version 5 of the program, another rewrite that updated the lexicon further and added new functions such as word redundancy detection. Around the same time, the company introduced Easy Proof, a pared-down version of Grammatik intended for novice writers, students, and family computers. In 1991, the company was engaged in a trademark dispute with Systems Compatibility Corporation (SCC) of Chicago, Illinois, over the rights to the Software Toolkit title. Both companies had published software bundles bearing the name in the turn of the 1990s; SCC had published theirs first in 1988 and registered the trademark with the USPTO. SCC was granted a restraining order against RSI in January 1991. The following month, RSI agreed to rename their product, preventing a protracted legal battle. === Decline and acquisition (1992–1993) === By early 1992, RSI achieved annual sales of more than $13 million, employed 120 people, and had opened international offices in London, Belgium, and Antwerp to sell foreign versions of Reference Set and Grammatik. The company reached peak employment in the middle of 1992, with 140 employees. However, RSI's launch of six disparate titles in the year proved problematic for the company when they failed to sell as well as they had projected, and the company laid off employees by the dozens. By December 1992, only 71 employees were left, 32 from their San Francisco office. On the last day of 1992, RSI received an acquisition offer from WordPerfect Corporation, makers of the namesake word processor based in Orem, Utah. The deal was inked in January 1993, RSI's stakeholders receiving $19 million. The company's remaining employees were absorbed into WordPerfect in Orem. WordPerfect continued selling Grammatik as a standalone product for several years.
Data event
A data event is a relevant state transition defined in an event schema. Typically, event schemata are described by pre- and post condition for a single or a set of data items. In contrast to ECA (Event condition action), which considers an event to be a signal, the data event not only refers to the change (signal), but describes specific state transitions, which are referred to in ECA as conditions. Considering data events as relevant data item state transitions allows defining complex event-reaction schemata for a database. Defining data event schemata for relational databases is limited to attribute and instance events. Object-oriented databases also support collection properties, which allows defining changes in collections as data events, too.
Fitness function
A fitness function is a particular type of objective or cost function that is used to summarize, as a single figure of merit, how close a given candidate solution is to achieving the set aims. It is an important component of evolutionary algorithms (EA), such as genetic programming, evolution strategies or genetic algorithms. An EA is a metaheuristic that reproduces the basic principles of biological evolution as a computer algorithm in order to solve challenging optimization or planning tasks, at least approximately. For this purpose, many candidate solutions are generated, which are evaluated using a fitness function in order to guide the evolutionary development towards the desired goal. Similar quality functions are also used in other metaheuristics, such as ant colony optimization or particle swarm optimization. In the field of EAs, each candidate solution, also called an individual, is commonly represented as a string of numbers (referred to as a chromosome). After each round of testing or simulation the idea is to delete the n worst individuals, and to breed n new ones from the best solutions. Each individual must therefore to be assigned a quality number indicating how close it has come to the overall specification, and this is generated by applying the fitness function to the test or simulation results obtained from that candidate solution. Two main classes of fitness functions exist: one where the fitness function does not change, as in optimizing a fixed function or testing with a fixed set of test cases; and one where the fitness function is mutable, as in niche differentiation or co-evolving the set of test cases. Another way of looking at fitness functions is in terms of a fitness landscape, which shows the fitness for each possible chromosome. In the following, it is assumed that the fitness is determined based on an evaluation that remains unchanged during an optimization run. A fitness function does not necessarily have to be able to calculate an absolute value, as it is sometimes sufficient to compare candidates in order to select the better one. A relative indication of fitness (candidate a is better than b) is sufficient in some cases, such as tournament selection or Pareto optimization. == Requirements of evaluation and fitness function == The quality of the evaluation and calculation of a fitness function is fundamental to the success of an EA optimisation. It implements Darwin's principle of "survival of the fittest". Without fitness-based selection mechanisms for mate selection and offspring acceptance, EA search would be blind and hardly distinguishable from the Monte Carlo method. When setting up a fitness function, one must always be aware that it is about more than just describing the desired target state. Rather, the evolutionary search on the way to the optimum should also be supported as much as possible (see also section on auxiliary objectives), if and insofar as this is not already done by the fitness function alone. If the fitness function is designed badly, the algorithm will either converge on an inappropriate solution, or will have difficulty converging at all. Definition of the fitness function is not straightforward in many cases and often is performed iteratively if the fittest solutions produced by an EA is not what is desired. Interactive genetic algorithms address this difficulty by outsourcing evaluation to external agents which are normally humans. == Computational efficiency == The fitness function should not only closely align with the designer's goal, but also be computationally efficient. Execution speed is crucial, as a typical evolutionary algorithm must be iterated many times in order to produce a usable result for a non-trivial problem. Fitness approximation may be appropriate, especially in the following cases: Fitness computation time of a single solution is extremely high Precise model for fitness computation is missing The fitness function is uncertain or noisy. Alternatively or also in addition to the fitness approximation, the fitness calculations can also be distributed to a parallel computer in order to reduce the execution times. Depending on the population model of the EA used, both the EA itself and the fitness calculations of all offspring of one generation can be executed in parallel. == Multi-objective optimization == Practical applications usually aim at optimizing multiple and at least partially conflicting objectives. Two fundamentally different approaches are often used for this purpose, Pareto optimization and optimization based on fitness calculated using the weighted sum. === Weighted sum and penalty functions === When optimizing with the weighted sum, the single values of the O {\displaystyle O} objectives are first normalized so that they can be compared. This can be done with the help of costs or by specifying target values and determining the current value as the degree of fulfillment. Costs or degrees of fulfillment can then be compared with each other and, if required, can also be mapped to a uniform fitness scale. Without loss of generality, fitness is assumed to represent a value to be maximized. Each objective o i {\displaystyle o_{i}} is assigned a weight w i {\displaystyle w_{i}} in the form of a percentage value so that the overall raw fitness f r a w {\displaystyle f_{raw}} can be calculated as a weighted sum: f r a w = ∑ i = 1 O o i ⋅ w i w i t h ∑ i = 1 O w i = 1 {\displaystyle f_{raw}=\sum _{i=1}^{O}{o_{i}\cdot w_{i}}\quad {\mathsf {with}}\quad \sum _{i=1}^{O}{w_{i}}=1} A violation of R {\displaystyle R} restrictions r j {\displaystyle r_{j}} can be included in the fitness determined in this way in the form of penalty functions. For this purpose, a function p f j ( r j ) {\displaystyle pf_{j}(r_{j})} can be defined for each restriction which returns a value between 0 {\displaystyle 0} and 1 {\displaystyle 1} depending on the degree of violation, with the result being 1 {\displaystyle 1} if there is no violation. The previously determined raw fitness is multiplied by the penalty function(s) and the result is then the final fitness f f i n a l {\displaystyle f_{final}} : f f i n a l = f r a w ⋅ ∏ j = 1 R p f j ( r j ) = ∑ i = 1 O ( o i ⋅ w i ) ⋅ ∏ j = 1 R p f j ( r j ) {\displaystyle f_{final}=f_{raw}\cdot \prod _{j=1}^{R}{pf_{j}(r_{j})}=\sum _{i=1}^{O}{(o_{i}\cdot w_{i})}\cdot \prod _{j=1}^{R}{pf_{j}(r_{j})}} This approach is simple and has the advantage of being able to combine any number of objectives and restrictions. The disadvantage is that different objectives can compensate each other and that the weights have to be defined before the optimization. This means that the compromise lines must be defined before optimization, which is why optimization with the weighted sum is also referred to as the a priori method. In addition, certain solutions may not be obtained, see the section on the comparison of both types of optimization. === Pareto optimization === A solution is called Pareto-optimal if the improvement of one objective is only possible with a deterioration of at least one other objective. The set of all Pareto-optimal solutions, also called Pareto set, represents the set of all optimal compromises between the objectives. The figure below on the right shows an example of the Pareto set of two objectives f 1 {\displaystyle f_{1}} and f 2 {\displaystyle f_{2}} to be maximized. The elements of the set form the Pareto front (green line). From this set, a human decision maker must subsequently select the desired compromise solution. Constraints are included in Pareto optimization in that solutions without constraint violations are per se better than those with violations. If two solutions to be compared each have constraint violations, the respective extent of the violations decides. It was recognized early on that EAs with their simultaneously considered solution set are well suited to finding solutions in one run that cover the Pareto front sufficiently well. They are therefore well suited as a-posteriori methods for multi-objective optimization, in which the final decision is made by a human decision maker after optimization and determination of the Pareto front. Besides the SPEA2, the NSGA-II and NSGA-III have established themselves as standard methods. The advantage of Pareto optimization is that, in contrast to the weighted sum, it provides all alternatives that are equivalent in terms of the objectives as an overall solution. The disadvantage is that a visualization of the alternatives becomes problematic or even impossible from four objectives on. Furthermore, the effort increases exponentially with the number of objectives. If there are more than three or four objectives, some have to be combined using the weighted sum or other aggregation methods. === Comparison of both types of assessment === With the help of the weighted sum, the total Pareto front can be obtained by a suitable choice of weights, provided that it is convex
Julia (programming language)
Julia is a dynamic general-purpose programming language. As a high-level language, distinctive aspects of Julia's design include a type system with parametric polymorphism, the use of multiple dispatch as a core programming paradigm, just-in-time compilation and a parallel garbage collection implementation. Notably, Julia does not support classes with encapsulated methods but instead relies on the types of all of a function's arguments to determine which method will be called. By default, Julia is run similarly to scripting languages, using its runtime, and allows for interactions, but Julia programs can also be compiled to small binary standalone executables (or to small libraries for e.g. Python), with e.g. the JuliaC.jl compiler. Julia programs can reuse libraries from other languages, and vice versa. Julia has interoperability with C, C++, Fortran, Rust, Python, and R. Additionally, some Julia packages have bindings to be used from Python and R as libraries. Julia is supported by programmer tools like IDEs (see below) and by notebooks like Pluto.jl, Jupyter, and since 2025, Google Colab officially supports Julia natively. Julia is sometimes used in embedded systems (e.g. has been used in a satellite in space on a Raspberry Pi Compute Module 4; 64-bit Pis work best with Julia, and Julia is supported in Raspbian). == History == Work on Julia began in 2009, when Jeff Bezanson, Stefan Karpinski, Viral B. Shah, and Alan Edelman set out to create a free language that was both high-level and fast. On 14 February 2012, the team launched a website with a blog post explaining the language's mission. In an interview with InfoWorld in April 2012, Karpinski said about the name of the language, Julia: "There's no good reason, really. It just seemed like a pretty name." Bezanson said he chose the name on the recommendation of a friend, then years later wrote: Maybe julia stands for "Jeff's uncommon lisp is automated"? Julia's syntax is stable, since version 1.0 in 2018, and Julia has a backward compatibility guarantee for 1.x and also a stability promise for the documented (stable) API, while in the years before in the early development prior to 0.7 the syntax (and semantics) was changed in new versions. All of the (registered package) ecosystem uses the new and improved syntax, and in most cases relies on new APIs that have been added regularly, and in some cases minor additional syntax added in a forward compatible way e.g. in Julia 1.7. In the 10 years since the 2012 launch of pre-1.0 Julia, the community has grown. The Julia package ecosystem has over 11.8 million lines of code (including docs and tests). The JuliaCon academic conference for Julia users and developers has been held annually since 2014 with JuliaCon2020 welcoming over 28,900 unique viewers, and then JuliaCon2021 breaking all previous records (with more than 300 JuliaCon2021 presentations available for free on YouTube, up from 162 the year before), and 43,000 unique viewers during the conference. Three of the Julia co-creators are the recipients of the 2019 James H. Wilkinson Prize for Numerical Software (awarded every four years) "for the creation of Julia, an innovative environment for the creation of high-performance tools that enable the analysis and solution of computational science problems." Also, Alan Edelman, professor of applied mathematics at MIT, has been selected to receive the 2019 IEEE Computer Society Sidney Fernbach Award "for outstanding breakthroughs in high-performance computing, linear algebra, and computational science and for contributions to the Julia programming language." Version 0.3 was released in August 2014. Both Julia 0.7 and version 1.0 were released on 8 August 2018. Julia 1.4 added syntax for generic array indexing to handle e.g. 0-based arrays. The memory model was also changed. Julia 1.5 released in August 2020 added record and replay debugging support, for Mozilla's rr tool. The release changed the behavior in the REPL (to soft scope) to the one used in Jupyter, but keeps full compatible with non-REPL code (that retains hard scope). Julia 1.6 was the largest release since 1.0, and it was the long-term support (LTS) version for the longest time. Since Julia 1.7 development is back to time-based releases, and it was released in November 2021 with e.g. a new default random-number generator and Julia 1.7.3 fixed at least one security issue. Julia 1.8 added options for hiding source code when compiling Julia source code to executables. Julia 1.9 has added the ability to precompile packages to native machine code, done automatically; to improve precompilation of packages a new package PrecompileTools.jl was introduced, for use by package developers. Julia 1.10 was released on 25 December 2023 with new features such as parallel garbage collection. Julia 1.11 was released on 7 October 2024, and with it 1.10.5 became the next long-term support (LTS) version (i.e. those became the only two supported versions), since replaced by 1.10.10 released on 27 June, and 1.6 is no longer an LTS version. Julia 1.11 adds e.g. the new public keyword to signal safe public API (Julia users are advised to use such API, not internals, of Julia or packages, and package authors advised to use the keyword, generally indirectly, e.g. prefixed with the @compat macro, from Compat.jl, to also support older Julia versions, at least the LTS version). Julia 1.12 was released on 7 October 2025 (and 1.12.5 on 9 February 2026), and with it a JuliaC.jl package including the juliac compiler that works with it, for making rather small binary executables (much smaller than was possible before; through the use of new so-called trimming feature). Julia 1.10 LTS is an officially still-supported branch, but the 1.11 branch has also been maintained after 1.12 release, with 1.11.8 released and then 1.11.9 released on 8 February 2026. === JuliaCon === Since 2014, the Julia Community has hosted an annual Julia Conference focused on developers and users. The first JuliaCon took place in Chicago and kickstarted the annual occurrence of the conference. Since 2014, the conference has taken place across a number of locations including MIT and the University of Maryland, Baltimore. The event audience has grown from a few dozen people to over 28,900 unique attendees during JuliaCon 2020, which took place virtually. JuliaCon 2021 also took place virtually with keynote addresses from professors William Kahan, the primary architect of the IEEE 754 floating-point standard (which virtually all CPUs and languages, including Julia, use), Jan Vitek, Xiaoye Sherry Li, and Soumith Chintala, a co-creator of PyTorch. JuliaCon grew to 43,000 unique attendees and more than 300 presentations (still freely accessible, plus for older years). JuliaCon 2022 will also be virtual held between July 27 and July 29, 2022, for the first time in several languages, not just in English. === Sponsors === The Julia language became a NumFOCUS fiscally sponsored project in 2014 in an effort to ensure the project's long-term sustainability. Jeremy Kepner at MIT Lincoln Laboratory was the founding sponsor of the Julia project in its early days. In addition, funds from the Gordon and Betty Moore Foundation, the Alfred P. Sloan Foundation, Intel, and agencies such as NSF, DARPA, NIH, NASA, and FAA have been essential to the development of Julia. Mozilla, the maker of Firefox web browser, with its research grants for H1 2019, sponsored "a member of the official Julia team" for the project "Bringing Julia to the Browser", meaning to Firefox and other web browsers. The Julia language is also supported by individual donors on GitHub. === The Julia company === JuliaHub, Inc. was founded in 2015 as Julia Computing, Inc. by Viral B. Shah, Deepak Vinchhi, Alan Edelman, Jeff Bezanson, Stefan Karpinski and Keno Fischer. In June 2017, Julia Computing raised US$4.6 million in seed funding from General Catalyst and Founder Collective, the same month was "granted $910,000 by the Alfred P. Sloan Foundation to support open-source Julia development, including $160,000 to promote diversity in the Julia community", and in December 2019 the company got $1.1 million funding from the US government to "develop a neural component machine learning tool to reduce the total energy consumption of heating, ventilation, and air conditioning (HVAC) systems in buildings". In July 2021, Julia Computing announced they raised a $24 million Series A round led by Dorilton Ventures, which also owns Formula One team Williams Racing, that partnered with Julia Computing. Williams' Commercial Director said: "Investing in companies building best-in-class cloud technology is a strategic focus for Dorilton and Julia's versatile platform, with revolutionary capabilities in simulation and modelling, is hugely relevant to our business. We look forward to embedding Julia Computing in the world's most technologically advanced sport". In June 2023, JuliaHub received (again, now
Evolutionary algorithm
Evolutionary algorithms (EA) reproduce essential elements of biological evolution in a computer algorithm in order to solve "difficult" problems, at least approximately, for which no exact or satisfactory solution methods are known. They are metaheuristics and population-based bio-inspired algorithms and evolutionary computation, which itself are part of the field of computational intelligence. The mechanisms of biological evolution that an EA mainly imitates are reproduction, mutation, recombination and selection. Candidate solutions to the optimization problem play the role of individuals in a population, and the fitness function determines the quality of the solutions (see also loss function). Evolution of the population then takes place after the repeated application of the above operators. Evolutionary algorithms often perform well approximating solutions to all types of problems because they ideally do not make any assumption about the underlying fitness landscape. Techniques from evolutionary algorithms applied to the modeling of biological evolution are generally limited to explorations of microevolution (microevolutionary processes) and planning models based upon cellular processes. In most real applications of EAs, computational complexity is a prohibiting factor. In fact, this computational complexity is due to fitness function evaluation. Fitness approximation is one of the solutions to overcome this difficulty. However, seemingly simple EA can solve often complex problems; therefore, there may be no direct link between algorithm complexity and problem complexity. == Generic definition == The following is an example of a generic evolutionary algorithm: Randomly generate the initial population of individuals, the first generation. Evaluate the fitness of each individual in the population. Check, if the goal is reached and the algorithm can be terminated. Select individuals as parents, preferably of higher fitness. Produce offspring with optional crossover (mimicking reproduction). Apply mutation operations on the offspring. Select individuals preferably of lower fitness for replacement with new individuals (mimicking natural selection). Return to 2 == Types == Similar techniques differ in genetic representation and other implementation details, and the nature of the particular applied problem. Genetic algorithm – This is the most popular type of EA. One seeks the solution of a problem in the form of strings of numbers (traditionally binary, although the best representations are usually those that reflect something about the problem being solved), by applying operators such as recombination and mutation (sometimes one, sometimes both). This type of EA is often used in optimization problems. Genetic programming – Here the solutions are in the form of computer programs, and their fitness is determined by their ability to solve a computational problem. There are many variants of Genetic Programming: Cartesian genetic programming Gene expression programming Grammatical evolution Linear genetic programming Multi expression programming Evolutionary programming – Similar to evolution strategy, but with a deterministic selection of all parents. Evolution strategy (ES) – Works with vectors of real numbers as representations of solutions, and typically uses self-adaptive mutation rates. The method is mainly used for numerical optimization, although there are also variants for combinatorial tasks. CMA-ES Natural evolution strategy Differential evolution – Based on vector differences and is therefore primarily suited for numerical optimization problems. Coevolutionary algorithm – Similar to genetic algorithms and evolution strategies, but the created solutions are compared on the basis of their outcomes from interactions with other solutions. Solutions can either compete or cooperate during the search process. Coevolutionary algorithms are often used in scenarios where the fitness landscape is dynamic, complex, or involves competitive interactions. Neuroevolution – Similar to genetic programming but the genomes represent artificial neural networks by describing structure and connection weights. The genome encoding can be direct or indirect. Learning classifier system – Here the solution is a set of classifiers (rules or conditions). A Michigan-LCS evolves at the level of individual classifiers whereas a Pittsburgh-LCS uses populations of classifier-sets. Initially, classifiers were only binary, but now include real, neural net, or S-expression types. Fitness is typically determined with either a strength or accuracy based reinforcement learning or supervised learning approach. Quality–Diversity algorithms – QD algorithms simultaneously aim for high-quality and diverse solutions. Unlike traditional optimization algorithms that solely focus on finding the best solution to a problem, QD algorithms explore a wide variety of solutions across a problem space and keep those that are not just high performing, but also diverse and unique. == Theoretical background == The following theoretical principles apply to all or almost all EAs. === No free lunch theorem === The no free lunch theorem of optimization states that all optimization strategies are equally effective when the set of all optimization problems is considered. Under the same condition, no evolutionary algorithm is fundamentally better than another. This can only be the case if the set of all problems is restricted. This is exactly what is inevitably done in practice. Therefore, to improve an EA, it must exploit problem knowledge in some form (e.g. by choosing a certain mutation strength or a problem-adapted coding). Thus, if two EAs are compared, this constraint is implied. In addition, an EA can use problem specific knowledge by, for example, not randomly generating the entire start population, but creating some individuals through heuristics or other procedures. Another possibility to tailor an EA to a given problem domain is to involve suitable heuristics, local search procedures or other problem-related procedures in the process of generating the offspring. This form of extension of an EA is also known as a memetic algorithm. Both extensions play a major role in practical applications, as they can speed up the search process and make it more robust. === Convergence === For EAs in which, in addition to the offspring, at least the best individual of the parent generation is used to form the subsequent generation (so-called elitist EAs), there is a general proof of convergence under the condition that an optimum exists. Without loss of generality, a maximum search is assumed for the proof: From the property of elitist offspring acceptance and the existence of the optimum it follows that per generation k {\displaystyle k} an improvement of the fitness F {\displaystyle F} of the respective best individual x ′ {\displaystyle x'} will occur with a probability P > 0 {\displaystyle P>0} . Thus: F ( x 1 ′ ) ≤ F ( x 2 ′ ) ≤ F ( x 3 ′ ) ≤ ⋯ ≤ F ( x k ′ ) ≤ ⋯ {\displaystyle F(x'_{1})\leq F(x'_{2})\leq F(x'_{3})\leq \cdots \leq F(x'_{k})\leq \cdots } I.e., the fitness values represent a monotonically non-decreasing sequence, which is bounded due to the existence of the optimum. From this follows the convergence of the sequence against the optimum. Since the proof makes no statement about the speed of convergence, it is of little help in practical applications of EAs. But it does justify the recommendation to use elitist EAs. However, when using the usual panmictic population model, elitist EAs tend to converge prematurely more than non-elitist ones. In a panmictic population model, mate selection (see step 4 of the generic definition) is such that every individual in the entire population is eligible as a mate. In non-panmictic populations, selection is suitably restricted, so that the dispersal speed of better individuals is reduced compared to panmictic ones. Thus, the general risk of premature convergence of elitist EAs can be significantly reduced by suitable population models that restrict mate selection. === Virtual alphabets === With the theory of virtual alphabets, David E. Goldberg showed in 1990 that by using a representation with real numbers, an EA that uses classical recombination operators (e.g. uniform or n-point crossover) cannot reach certain areas of the search space, in contrast to a coding with binary numbers. This results in the recommendation for EAs with real representation to use arithmetic operators for recombination (e.g. arithmetic mean or intermediate recombination). With suitable operators, real-valued representations are more effective than binary ones, contrary to earlier opinion. == Comparison to other concepts == === Biological processes === A possible limitation of many evolutionary algorithms is their lack of a clear genotype–phenotype distinction. In nature, the fertilized egg cell undergoes a complex process known as embryogenesis to become a mature p
List of monochrome and RGB color formats
This list of monochrome and RGB palettes includes generic repertoires of colors (color palettes) to produce black-and-white and RGB color pictures by a computer's display hardware. RGB is the most common method to produce colors for displays; so these complete RGB color repertoires have every possible combination of R-G-B triplets within any given maximum number of levels per component. Each palette is represented by a series of color patches. When the number of colors is low, a 1-pixel-size version of the palette appears below it, for easily comparing relative palette sizes. Huge palettes are given directly in one-color-per-pixel color patches. For each unique palette, an image color test chart and sample image (truecolor original follows) rendered with that palette (without dithering) are given. The test chart shows the full 256 levels of the red, green, and blue (RGB) primary colors and cyan, magenta, and yellow complementary colors, along with a full 256-level grayscale. Gradients of RGB intermediate colors (orange, lime green, sea green, sky blue, violet, and fuchsia), and a full hue spectrum are also present. Color charts are not gamma corrected. These elements illustrate the color depth and distribution of the colors of any given palette, and the sample image indicates how the color selection of such palettes could represent real-life images. These images are not necessarily representative of how the image would be displayed on the original graphics hardware, as the hardware may have additional limitations regarding the maximum display resolution, pixel aspect ratio and color placement. Implementation of these formats is specific to each machine. Therefore, the number of colors that can be simultaneously displayed in a given text or graphic mode might be different. Also, the actual displayed colors are subject to the output format used - PAL or NTSC, composite or component video, etc. - and might be slightly different. For simulated images and specific hardware and alternate methods to produce colors other than RGB (ex: composite), see the List of 8-bit computer hardware palettes, the List of 16-bit computer hardware palettes and the List of video game console palettes. For various software arrangements and sorts of colors, including other possible full RGB arrangements within 8-bit color depth displays, see the List of software palettes. == Monochrome palettes == These palettes only have some shades of gray, from black to white (considered the darkest and lightest "grays", respectively). The general rule is that those palettes have 2n different shades of gray, where n is the number of bits needed to represent a single pixel. === Monochrome (1-bit grayscale) === Monochrome graphics displays typically have a black background with a white or light gray image, though green and amber monochrome monitors were also common. Such a palette requires only one bit per pixel. Where photo-realism was desired, these early computer systems had a heavy reliance on dithering to make up for the limits of the technology. In some systems, as Hercules and CGA graphic cards for the IBM PC, a bit value of 1 represents white pixels (light on) and a value of 0 the black ones (light off); others, like the Playdate and Atari ST and Apple Macintosh with monochrome monitors, a bit value of 0 means a white pixel (no ink) and a value of 1 means a black pixel (dot of ink), which it approximates to the printing logic. === 2-bit Grayscale === In a 2-bit color palette each pixel's value is represented by 2 bits resulting in a 4-value palette (22 = 4). 2-bit dithering: It has black, white and two intermediate levels of gray as follows: A monochrome 2-bit palette is used on: The Monochrome Display Adapter for the IBM PC NeXT Computer, NeXTcube and NeXTstation monochrome graphic displays. Original Game Boy system portable video game console. Macintosh PowerBook 150 monochrome LC displays. Amiga with A2024 monochrome monitor in high-resolution mode. The original Amazon Kindle The original WonderSwan The Tiger Electronics Game.com portable video game console The original Neo Geo Pocket. === 4-bit Grayscale === In a 4-bit color palette each pixel's value is represented by 4 bits resulting in a 16-value palette (24 = 16): 4-bit grayscale dithering does a fairly good job of reducing visible banding of the level changes: A monochrome 4-bit palette is used on: MOS Technology VDC (on the Commodore 128 with monochrome monitor) Amstrad CPC series with a GT64/GT65 Green Monitor (16 unique green shades) Amstrad CPC Plus series with the MM12 Monochrome monitor (16 shades of grey) Some Apple PowerBooks equipped with monochrome displays like the PowerBook 5300 The original VideoNow === 8-bit Grayscale === In an 8-bit color palette each pixel's value is represented by 8 bits resulting in a 256-value palette (28 = 256). This is usually the maximum number of grays in ordinary monochrome systems; each image pixel occupies a single memory byte. Most scanners can capture images in 8-bit grayscale, and image file formats like TIFF and JPEG natively support this monochrome palette size. Alpha channels employed for video overlay also use (conceptually) this palette. The gray level indicates the opacity of the blended image pixel over the background image pixel. == Dichrome palettes == === 16-bit RG palette === The RG or red–green color space is a color space that uses only two primary colors: red and green. It was used on early color processes for films. It was used as an additive format, similar to the RGB color model but without a blue channel, on processes such as Kinemacolor, Prizma, Technicolor I, Raycol, etc., producing shades of black, red, green and yellow. Alternatively, it was used as a subtractive format on Brewster Color I, Kodachrome I, Prizma II, Technicolor II, etc., producing shades of transparent, red, green and black. Until recently, its primary use was in low-cost light-emitting diode displays in which red and green tended to be far more common than the still nascent blue LED technology, but full-color LEDs with blue have become more common in recent years. ColorCode 3-D, a anaglyph stereoscopic color scheme, uses the RG color space to simulate a broad spectrum of color in one eye, while the blue portion of the spectrum transmits a black-and-white (black-and-blue) image to the other eye to give depth perception. === 16-bit RB palette === === 16-bit GB palette === == Regular RGB palettes == Here are grouped those full RGB hardware palettes that have the same number of binary levels (i.e., the same number of bits) for every red, green and blue components using the full RGB color model. Thus, the total number of colors are always the number of possible levels by component, n, raised to a power of 3: n×n×n = n3. === 3-bit RGB === 3-bit RGB dithering: Systems with a 3-bit RGB palette use 1 bit for each of the red, green and blue color components. That is, each component is either "on" or "off" with no intermediate states. This results in an 8-color palette ((21)3 = 23 = 8) that has black, white, the three RGB primary colors red, green and blue and their correspondent complementary colors cyan, magenta and yellow as follows: The color indices vary between implementations; therefore, index numbers are not given. The 3-bit RGB palette is used by: Text terminals following the ECMA-48 standard (sometimes known as the "ANSI standard", although ANSI X3.128 does not define colors) World System Teletext Level 1/1.5 Videotex Oric computers BBC Micro PC-8801 (up to the MkII) PC-9801 (with original 8086 CPU, before the VM/VX models) Sharp X1 (models before the X1 Turbo Z) Sharp MZ 700 FM-7, FM New 7, FM 77 (before the FM77AV) Sinclair QL Space Invaders Part II (arcade hardware) Macintosh SE (with a color printer or external monitor) Atari 2600 (SECAM version) Color Maximite (PIC32 based microcomputer) Arcadia 2001 PV-1000 Monkey Magic (arcade hardware) VIC-20 (high-res mode) Mouse Trap (arcade hardware) Sanyo MBC-550 series Windows 1.0 (includes dithering) === 6-bit RGB === Systems with a 6-bit RGB palette use 2 bits for each of the red, green, and blue color components. This results in a (22)3 = 43 = 64-color palette as follows: 6-bit RGB systems include the following: Enhanced Graphics Adapter (EGA) for IBM PC/AT (16 colors at once) Sega Master System video game console (32 colors at once) GIME for TRS-80 Color Computer 3 (16 colors at once) Pebble Time smartwatch which has a 6-bit (64 color) e-paper display Parallax Propeller using the reference VGA circuit === 9-bit RGB === Systems with a 9-bit RGB palette use 3 bits for each of the red, green, and blue color components. This results in a (23)3 = 83 = 512-color palette as follows: 9-bit RGB systems include the following: Atari ST (Normally 4 to 16 at once without tricks) MSX2 computers (up to 16 at once) Sega Genesis video game console, (64 colors at once) Sega Nomad TurboGrafx-16 (NEC PC-Engine) ZX Spectrum Next The NEC PC-88
Discrete diffusion model
In machine learning, discrete diffusion models are a class of diffusion models, which themselves are a class of latent variable generative models. Each discrete diffusion model consists of two major components: the forward jump diffusion process, and the reverse jump diffusion process. The goal of diffusion modeling is, given a given dataset and a forward process, to learn a model for the reverse process, such that the reverse process can generate new elements that are distributed similarly as the original dataset. A trained discrete diffusion model can be sampled in many ways, which trades off computational efficiency and sample quality. In general, higher quality data can be obtained, but at the price of higher computational cost. In standard diffusion modeling, the diffusion process takes place over a state space that is continuous space of R n {\displaystyle \mathbb {R} ^{n}} , but over a discrete set S {\displaystyle S} . A discrete set is simply a set where one cannot speak of "infinitesimally close" points. Points can be more or less separated from each other, but the separation is always a finite number. This in particular means the standard framework of continuous diffusion does not apply, since it uses gaussian noise, which is continuous. Nevertheless, an analogous theory can be produced. Discrete diffusion is usually used for language modeling. In practice, the state space S {\displaystyle S} is not only discrete, but finite, so this is what we will assume from now on. == Continuous time Markov process == In the case of continuous state space, during the forward discrete diffusion process, at each step t → t + d t {\displaystyle t\to t+dt} , we mix in an infinitesimal amount of gaussian noise d x t = − 1 2 β ( t ) x t d t + β ( t ) d W t {\displaystyle dx_{t}=-{\frac {1}{2}}\beta (t)x_{t}dt+{\sqrt {\beta (t)}}dW_{t}} . This changes the probability density function, by first a convolution with the density of a gaussian, followed by a scaling. In the case of discrete state space, the gaussian noise must be replaced by a noise that takes values over a finite set. For example, if the noise is the uniform distribution over S {\displaystyle S} , then the probability distribution at time t + d t {\displaystyle t+dt} satisfies q t + d t ( x ) = ( 1 − d t ) q t ( x ) + d t ( 1 | S | ∑ y ∈ S q t ( y ) ) {\displaystyle q_{t+dt}(x)=(1-dt)q_{t}(x)+dt\left({\frac {1}{|S|}}\sum _{y\in S}q_{t}(y)\right)} More succinctly, ∂ t q t ( x ) = − ( 1 − 1 | S | ) q t ( x ) + ∑ y ∈ S , y ≠ x 1 | S | q t ( y ) {\displaystyle \partial _{t}q_{t}(x)=-\left(1-{\frac {1}{|S|}}\right)q_{t}(x)+\sum _{y\in S,y\neq x}{\frac {1}{|S|}}q_{t}(y)} In general, we do not need to convolve with a uniformly distributed noise, but with an arbitrary noise process. That is, we use an arbitrary matrix Q t {\displaystyle Q_{t}} such that ∂ t q t ( y ) = ∑ x ∈ S Q t ( y , x ) q t ( x ) {\displaystyle \partial _{t}q_{t}(y)=\sum _{x\in S}Q_{t}(y,x)q_{t}(x)} where Q t {\displaystyle Q_{t}} is called the rate matrix. Any matrix may be used as a rate matrix if it has non-negative off-diagonals, and each column sums to 0: Q t ( y , x ) ≥ 0 ∀ y ≠ x , ∑ y ∈ S Q t ( y , x ) = 0 ∀ x {\displaystyle Q_{t}(y,x)\geq 0\quad \forall y\neq x,\quad \sum _{y\in S}Q_{t}(y,x)=0\quad \forall x} A continuous time Markov chain (CTMC) is defined by a continuous function Q {\displaystyle Q} that maps any time t ∈ [ 0 , T ) {\displaystyle t\in [0,T)} to a rate matrix Q t {\displaystyle Q_{t}} . Given the function Q {\displaystyle Q} , time-evolution under the CTMC is done as follows: Given state x t {\displaystyle x_{t}} at time t {\displaystyle t} , and given an infinitesimal d t {\displaystyle dt} , the state at t + d t {\displaystyle t+dt} is x t + d t {\displaystyle x_{t+dt}} , such that Pr ( x t + d t | x t ) = { 1 + Q t ( x t + d t , x t ) d t if x t + d t = x t Q t ( x t + d t , x t ) d t else {\displaystyle \Pr(x_{t+dt}|x_{t})={\begin{cases}1+Q_{t}(x_{t+dt},x_{t})dt&{\text{if }}x_{t+dt}=x_{t}\\Q_{t}(x_{t+dt},x_{t})dt&{\text{else}}\end{cases}}} This implies that the probability distribution function evolves according to ∂ t q t ( y ) = ∑ x ∈ S Q t ( y , x ) q t ( x ) {\displaystyle \partial _{t}q_{t}(y)=\sum _{x\in S}Q_{t}(y,x)q_{t}(x)} which is what we previously specified. === Backward process === Similarly to the case of continuous diffusion, in discrete diffusion, there exists a backward diffusion process Q ¯ t {\displaystyle {\bar {Q}}_{t}} : s ( x , t ) y := q t ( y ) q t ( x ) , Q ¯ t ( y , x ) := { s ( x , t ) y Q t ( x , y ) if y ≠ x − ∑ y : y ≠ x Q ¯ t ( y , x ) if y = x {\displaystyle s(x,t)_{y}:={\frac {q_{t}(y)}{q_{t}(x)}},\quad {\bar {Q}}_{t}(y,x):={\begin{cases}s(x,t)_{y}Q_{t}(x,y)&{\text{if }}y\neq x\\-\sum _{y:y\neq x}{\bar {Q}}_{t}(y,x)&{\text{if }}y=x\end{cases}}} where s ( x , t ) y {\displaystyle s(x,t)_{y}} should be interpreted as the discrete score or concrete score, since, abusing notation a bit, the score function is ∇ ln ρ t ( x ) = 1 d x ( ρ t ( x + d x ) ρ t ( x ) − 1 ) {\displaystyle \nabla \ln \rho _{t}(x)={\frac {1}{dx}}\left({\frac {\rho _{t}(x+dx)}{\rho _{t}(x)}}-1\right)} . If we picture the distribution q t {\displaystyle q_{t}} as a bunch of point-masses, one per state x ∈ S {\displaystyle x\in S} , then the forward diffusion from time t {\displaystyle t} to t + d t {\displaystyle t+dt} is performed by removing Q t ( x , y ) q t ( y ) d t {\displaystyle Q_{t}(x,y)q_{t}(y)dt} from the mass at y {\displaystyle y} and moving it to the mass at x {\displaystyle x} , for each pair x ≠ y {\displaystyle x\neq y} . Thus, the process is reversed in detail by the CTMC defined by Q ¯ {\displaystyle {\bar {Q}}} , since Q ¯ t ( y , x ) q t ( x ) = Q t ( x , y ) q t ( y ) {\displaystyle {\bar {Q}}_{t}(y,x)q_{t}(x)=Q_{t}(x,y)q_{t}(y)} . Given Q ¯ t {\displaystyle {\bar {Q}}_{t}} , if we have a way to sample from q t {\displaystyle q_{t}} , then we can sample from q t − d t {\displaystyle q_{t-dt}} by first sampling x t ∼ q t {\displaystyle x_{t}\sim q_{t}} , then sampling x t − d t {\displaystyle x_{t-dt}} according to Pr ( x t − d t | x t ) = { 1 + Q ¯ t ( x t − d t , x t ) d t if x t − d t = x t Q ¯ t ( x t − d t , x t ) d t else {\displaystyle \Pr(x_{t-dt}|x_{t})={\begin{cases}1+{\bar {Q}}_{t}(x_{t-dt},x_{t})dt&{\text{if }}x_{t-dt}=x_{t}\\{\bar {Q}}_{t}(x_{t-dt},x_{t})dt&{\text{else}}\end{cases}}} === Overall plan of score-matching discrete diffusion modeling === Similar to score-matching continuous diffusion, score-matching discrete diffusion is a method to sample an initial distribution. If we have a certain function s θ {\displaystyle s_{\theta }} that approximates the true score function s θ ( x , t ) y ≈ s ( x , t ) y {\displaystyle s_{\theta }(x,t)_{y}\approx s(x,t)_{y}} , then it allows a corresponding Q ¯ θ {\displaystyle {\bar {Q}}^{\theta }} to be defined in the same way. If we also have a base distribution q base {\displaystyle q_{\text{base}}} such that it is easy to sample from, and approximately equal to the true terminal distribution q base ≈ q T {\displaystyle q_{\text{base}}\approx q_{T}} , then we can perform the backward CTMC with Q ¯ θ {\displaystyle {\bar {Q}}^{\theta }} and q T θ := q terminal {\displaystyle q_{T}^{\theta }:=q_{\text{terminal}}} . When both approximations are good, the backward CTMC would give q 0 θ ≈ q 0 {\displaystyle q_{0}^{\theta }\approx q_{0}} . This is the idea of score-matching discrete diffusion modeling. If q data {\displaystyle q_{\text{data}}} is sharp, in the sense that for some x , x ′ {\displaystyle x,x'} , we have q data ( x ) ≫ q data ( x ′ ) {\displaystyle q_{\text{data}}(x)\gg q_{\text{data}}(x')} , then the score function would diverge as 1 / t {\displaystyle 1/t} at the t → 0 {\displaystyle t\to 0} limit. To avoid this in practice, it is common to use early stopping, which is to stop the backward process at some time δ > 0 {\displaystyle \delta >0} , and sample from q δ θ {\displaystyle q_{\delta }^{\theta }} instead of q 0 θ {\displaystyle q_{0}^{\theta }} . === Tractable forward processes === The theory of CTMC works for any continuous choice of rate matrices Q {\displaystyle Q} . However, most choices are computationally expensive and cannot be used in practice. In the case of continuous diffusion, the gaussian noise is used for the simple reason that the sum of any number of gaussians is still a gaussian. This allows one to sample any x t ∼ ρ t {\displaystyle x_{t}\sim \rho _{t}} by sampling a single x 0 ∼ ρ 0 {\displaystyle x_{0}\sim \rho _{0}} , followed by a single gaussian noise z ∼ N ( 0 , I ) {\displaystyle z\sim {\mathcal {N}}(0,I)} , and let x t = α ¯ t x 0 + σ t z {\displaystyle x_{t}={\sqrt {{\bar {\alpha }}_{t}}}x_{0}+\sigma _{t}z} , without needing any x s {\displaystyle x_{s}} for any 0 < s < t {\displaystyle 0