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
Cloud testing
Cloud testing is a form of software testing in which web applications use cloud computing environments (a "cloud") to simulate real-world user traffic. == Steps == Companies simulate real world Web users by using cloud testing services that are provided by cloud service vendors such as Advaltis, Compuware, HP, Keynote Systems, Neotys, RadView and SOASTA. Once user scenarios are developed and the test is designed, these service providers leverage cloud servers (provided by cloud platform vendors such as Amazon.com, Google, Rackspace, Microsoft, etc.) to generate web traffic that originates from around the world. Once the test is complete, the cloud service providers deliver results and analytics back to corporate IT professionals through real-time dashboards for a complete analysis of how their applications and the internet will perform during peak volumes. == Applications == Cloud testing is often seen as only performance or load tests, however, as discussed earlier it covers many other types of testing. Cloud computing itself is often referred to as the marriage of software as a service (SaaS) and utility computing. In regard to test execution, the software offered as a service may be a transaction generator and the cloud provider's infrastructure software, or may just be the latter. Distributed Systems and Parallel Systems mainly use this approach for testing, because of their inherent complex nature. D-Cloud is an example of such a software testing environment. == Tools == Leading cloud computing service providers include, among others, Amazon, Microsoft, Google, RadView, Skytap, HP and SOASTA. == Benefits == The ability and cost to simulate web traffic for software testing purposes has been an inhibitor to overall web reliability. The low cost and accessibility of the cloud's extremely large computing resources provides the ability to replicate real world usage of these systems by geographically distributed users, executing wide varieties of user scenarios, at scales previously unattainable in traditional testing environments. Minimal start-up time along with quality assurance can be achieved by cloud testing. Following are some of the key benefits: Reduction in capital expenditure Highly scalable
Data preservation
Data preservation is the act of conserving and maintaining both the safety and integrity of data. Preservation is done through formal activities that are governed by policies, regulations and strategies directed towards protecting and prolonging the existence and authenticity of data and its metadata. Data can be described as the elements or units in which knowledge and information is created, and metadata are the summarizing subsets of the elements of data; or the data about the data. The main goal of data preservation is to protect data from being lost or destroyed and to contribute to the reuse and progression of the data. == History == Most historical data collected over time has been lost or destroyed. War and natural disasters combined with the lack of materials and necessary practices to preserve and protect data has caused this. Usually, only the most important data sets were saved, such as government records and statistics, legal contracts and economic transactions. Scientific research and doctoral theses data have mostly been destroyed from improper storage and lack of data preservation awareness and execution. Over time, data preservation has evolved and has generated importance and awareness. We now have many different ways to preserve data and many different important organizations involved in doing so. The first digital data preservation storage solutions appeared in the 1950s, which were usually flat or hierarchically structured. While there were still issues with these solutions, it made storing data much cheaper, and more easily accessible. In the 1970s relational databases as well as spreadsheets appeared. Relational data bases structure data into tables using structured query languages which made them more efficient than the preceding storage solutions, and spreadsheets hold high volumes of numeric data which can be applied to these relational databases to produce derivative data. More recently, non-relational (non-structured query language) databases have appeared as complements to relational databases which hold high volumes of unstructured or semi-structured data. == Importance == The scope of data preservation is vast. Everything from governmental to business records to art essentially can be represented as data, and is amenable to be lost. This then leads to loss of human history, for perpetuity. Data can be lost on a small or independent scale whether it's personal data loss, or data loss within businesses and organizations, as well as on a larger or national or global scale which can negatively and potentially permanently affect things such as environmental protection, medical research, homeland security, public health and safety, economic development and culture. The mechanisms of data loss are also as many as they are varied, spanning from disaster, wars, data breaches, negligence, all the way through simple forgetting to natural decay. Ways in which data collections can be used when preserved and stored properly can be seen through the U.S. Geological Survey, which stores data collections on natural hazards, natural resources, and landscapes. The data collected by the Survey is used by federal and state land management agencies towards land use planning and management, and continually needs access to historical reference data. == Related Concepts == In contrast, data holdings are collections of gathered data that are informally kept, and not necessarily prepared for long-term preservation. For example, a collection or back-up of personal files. Data holdings are generally the storage methods used in the past when data has been lost due to environmental and other historical disasters. Furthermore, data retention differs from data preservation in the sense that by definition, to retain an object (data) is to hold or keep possession or use of the object. To preserve an object is to protect, maintain and keep up for future use. Retention policies often circle around when data should be deleted on purpose as well, and held from public access, while preservation prioritizes permanence and more widely shared access. Thus, data preservation exceeds the concept of having or possessing data or back up copies of data. Data preservation ensures reliable access to data by including back-up and recovery mechanisms that precede the event of a disaster or technological change. == Methods == === Digital === Digital preservation, is similar to data preservation, but is mainly concerned with technological threats, and solely digital data. Essentially digital data is a set of formal activities to enable ongoing or persistent use and access of digital data exceeding the occurrence of technological malfunction or change. Digital preservation is aware of the inevitable change in technology and protocols, and prepares for data that will need to be accessible across new types of technologies and platforms while the integrity of the data and metadata are being conserved. Technology, while providing great process in conserving data that may not have been possible in the past, is also changing at such a quick rate that digital data may not be accessible anymore due to the format being incompatible with new software. Without the use of data preservation much of our existing digital data is at risk. The majority of methods used towards data preservation today are digital methods, which are so far the most effective methods that exist. === Archives === Archives are a collection of historical documents and records. Archives contribute and work towards the preservation of data by collecting data that is well organized, while providing the appropriate metadata to confirm it. An example of an important data archive is The LONI Image Data Archive, which is an archive that collects data regarding clinical trials and clinical research studies. === Catalogues, directories and portals === Catalogues, directories and portals are consolidated resources which are kept by individual institutions, and are associated with data archives and holdings. In other words, the data is not presented on the site, but instead might act as metadata and aggregators, and may administer thorough inventories. === Repositories === Repositories are places where data archives and holdings can be accessed and stored. The goal of repositories is to make sure that all requirements and protocols of archives and holdings are being met, and data is being certified to ensure data integrity and user trust. Single-site Repositories A repository that holds all data sets on a single site. An example of a major single-site repository the Data Archiving and Networking Services which is a repository which provides ongoing access to digital research resources for the Netherlands. Multi-Site Repositories A repository that hosts data set on multiple institutional sites. An example of a well known multi-site repository is OpenAIRE which is a repository that hosts research data and publications collaborating all of the EU countries and more. OpenAIRE promotes open scholarship and seeks to improves discover-ability and re-usability of data. Trusted Digital Repository A repository that seeks to provide reliable, trusted access over a long period of time. The repository can be single or multi-sited but must cooperate with the Reference Model for an Open Archival Information System, as well as adhere to a set of rules or attributes that contribute to its trust such as having persistent financial responsibility, organizational buoyancy, administrative responsibility security and safety. An example of a trusted digital repository is The Digital Repository of Ireland (DRI) which is a multi-site repository that hosts Ireland's humanity and social science data sets. === Cyber Infrastructures === Cyber infrastructures which consists of archive collections which are made available through the system of hardware, technologies, software, policies, services and tools. Cyber infrastructures are geared towards the sharing of data supporting peer-to-peer collaborations and a cultural community. An example of a major cyber-infrastructure is The Canadian Geo-spatial Data Infrastructure which provides access to spatial data in Canada.
Squeaky Dolphin
Squeaky Dolphin is a program developed by the Government Communications Headquarters (GCHQ), a British intelligence and security organization, to collect and analyze data from social media networks. The program was first revealed to the general public on NBC on 27 January 2014 based on documents previously leaked by Edward Snowden. == Scope of surveillance == According to a document of the GCHQ dated August 2012, the program enables broad, real-time surveillance of the following items: YouTube video views The Like button on Facebook. Facebook has since then encrypted the data. Blogspot/Blogger visits Twitter, which has however encrypted its communications since this presentation was made The program can be supplemented with commercially available analytic software to determine which videos are popular among residents of specific cities. The dashboard software chosen was made by Splunk. The presentation, which was originally shown to an NSA audience and was made public by the NBC, contains a note saying the program was "Not interested in individuals just broad trends!". However, "according to other Snowden documents" obtained by NBC, in 2010, "GCHQ exploited unencrypted data from Twitter to identify specific users around the world and target them with propaganda."
Chunked transfer encoding
Chunked transfer encoding is a streaming data transfer mechanism available in Hypertext Transfer Protocol (HTTP) version 1.1, defined in RFC 9112 §7.1. In chunked transfer encoding, the data stream is divided into a series of non-overlapping "chunks". The chunks are sent out and received independently of one another. At any given time, no knowledge of the data stream outside the currently-being-processed chunk is necessary for either the sender or the receiver. Each chunk is preceded by its size in bytes and transmission ends when a zero-length chunk is received. The chunked keyword in the Transfer-Encoding header is used to indicate chunked transfer. Chunked transfer encoding is not supported in HTTP/2, which provides its own mechanisms for data streaming. == Rationale == The introduction of chunked encoding provided various benefits: Chunked transfer encoding allows a server to maintain an HTTP persistent connection for dynamically generated content. In this case, the HTTP Content-Length header cannot be used to delimit the content and the next HTTP request/response, as the content size is not yet known. Chunked encoding has the benefit that it is not necessary to generate the full content before writing the header, as it allows streaming of content as chunks and explicitly signaling the end of the content, making the connection available for the next HTTP request/response. Chunked encoding allows the sender to send additional header fields after the message body. This is important in cases where values of a field cannot be known until the content has been produced, such as when the content of the message must be digitally signed. Without chunked encoding, the sender would have to buffer the content until it was complete in order to calculate a field value and send it before the content. == Applicability == For version 1.1 of the HTTP protocol, the chunked transfer mechanism is considered to be always and anyway acceptable, even if not listed in the Transfer-Encoding (TE) request header field, and when used with other transfer mechanisms, should always be applied last to the transferred data and never more than one time. This transfer encoding method also allows additional entity header fields to be sent after the last chunk if the client specified the "trailers" parameter as an argument of the TE request field. The origin server of the response can also decide to send additional entity trailers even if the client did not specify the "trailers" parameter, but only if the metadata is optional (i.e. the client can use the received entity without them). Whenever the trailers are used, the server should list their names in the Trailer header field; three header field types are specifically prohibited from appearing as a trailer field: Content-Length, Trailer, and Transfer-Encoding. == Format == If a Transfer-Encoding field with a value of "chunked" is specified in an HTTP message (either a request sent by a client or the response from the server), the body of the message consists of one or more chunks and one terminating chunk with an optional trailer before the final ␍␊ sequence (i.e. carriage return followed by line feed). Each chunk starts with the number of octets of the data it embeds expressed as a hexadecimal number in ASCII followed by optional parameters (chunk extension) and a terminating ␍␊ sequence, followed by the chunk data. The chunk is terminated by ␍␊. If chunk extensions are provided, the chunk size is terminated by a semicolon and followed by the parameters, each also delimited by semicolons. Each parameter is encoded as an extension name followed by an optional equal sign and value. These parameters could be used for a running message digest or digital signature, or to indicate an estimated transfer progress, for instance. The terminating chunk is a special chunk of zero length. It may contain a trailer, which consists of a (possibly empty) sequence of entity header fields. Normally, such header fields would be sent in the message's header; however, it may be more efficient to determine them after processing the entire message entity. In that case, it is useful to send those headers in the trailer. Header fields that regulate the use of trailers are Transfer-Encoding with the "trailers" parameter (used in requests) and Trailer (used in responses). == Use with compression == HTTP servers often use compression to optimize transmission, for example with Content-Encoding: gzip or Content-Encoding: deflate. If both compression and chunked encoding are enabled, then the content stream is first compressed, then chunked; so the chunk encoding itself is not compressed, and the data in each chunk is compressed holistically (i.e. based on the whole content). The remote endpoint then decodes the stream by concatenating the chunks and uncompressing the result. == Example == === Encoded data === The following example contains three chunks of size 4, 7, and 11 (hexadecimal "B") octets of data. 4␍␊Wiki␍␊7␍␊pedia i␍␊B␍␊n ␍␊chunks.␍␊0␍␊␍␊ Below is an annotated version of the encoded data. 4␍␊ (chunk size is four octets) Wiki (four octets of data) ␍␊ (end of chunk) 7␍␊ (chunk size is seven octets) pedia i (seven octets of data) ␍␊ (end of chunk) B␍␊ (chunk size is eleven octets) n ␍␊chunks. (eleven octets of data) ␍␊ (end of chunk) 0␍␊ (chunk size is zero octets, no more chunks) ␍␊ (end of final chunk with zero data octets) Note: Each chunk's size excludes the two ␍␊ bytes that terminate the data of each chunk. === Decoded data === Decoding the above example produces the following octets: Wikipedia in ␍␊chunks. The bytes above are typically displayed as Wikipedia in chunks.
Final Cut Express
Final Cut Express was a video editing software suite created by Apple Inc. It was the consumer version of Final Cut Pro and was designed for advanced editing of digital video as well as high-definition video, which was used by many amateur and professional videographers. Final Cut Express was considered a step above iMovie in terms of capabilities, but a step underneath Final Cut Pro and its suite of applications. As of June 21, 2011, Final Cut Express was discontinued in favor of Final Cut Pro X. == History == Final Cut Express 1.0, based on Final Cut Pro 3, was released at Macworld Conference and Expo in San Francisco in 2003. The second version, based on Final Cut Pro 4, was released at Macworld San Francisco in 2004. The third version, capable of editing high definition video, was also announced at Macworld San Francisco a year later, and was released as Final Cut Express HD in February 2005. It was based on Final Cut Pro HD (version 4.5) and included LiveType 1.2 and Soundtrack 1.2. Final Cut Express version 3.5 was released with little fanfare in May 2006 as a Universal Binary. In addition to improving real-time rendering with Dynamic RT, version 3.5 upgraded LiveType to version 2.0 and Soundtrack to version 1.5. In November 2007, Apple released Final Cut Express 4, which although it did not support real-time editing in the AVCHD format (it only allowed for transcoding AVCHD to Apple Intermediate Codec (AIC) provided that the camera was actually attached to the computer - it did not convert AVCHD files stored elsewhere and is currently for Intel processors only), imported iMovie '08 projects and included 50 new filters. It did not include Soundtrack 1.5, but it still included LiveType which enables users to create advanced text for the movies they created in Final Cut. The price was dropped from $299 for version 3.5 to $199 for version 4.0. In June 2011, Final Cut Express was officially discontinued, in favor of Final Cut Pro X. == Features == Final Cut Express' interface was identical to that of Final Cut Pro, but lacks some film-specific features, including Cinema Tools, multi-cam editing, batch capture, and a time code view. The program performed 32 undo operations, while Final Cut Pro did 99 [2]. Features the program did include were: The ability to keyframe filters Dynamic RT, which changes real-time settings on-the-fly Motion path keyframing Opacity keyframing Ripple, roll, slip, slide and blade edits Picture-in-picture and split-screen effects Up to 99 video tracks and 12 compositing modes Up to 99 audio tracks Motion project import Two-way color correction. Chroma key One feature of Final Cut Express that was not available in Final Cut Pro is the ability to import iMovie '08 projects (though transitions are not preserved). === RT Extreme === Inherited from Final Cut Pro, Final Cut Express features RT Extreme, which allows previews of some video filters and transitions without rendering. Audio that is not in the native AIFF file format needs rendering before it can be played back. RT Extreme has three modes: 'Safe', for seeing multiple video layers at a quality that more or less guarantees a smooth playback; 'Unlimited', which allows the maximum number of composited video layers to be viewed at the same time; and 'Dynamic', which alternates between these settings depending on how many simultaneous video tracks are present. Frame dropping may result from using 'Unlimited' on low-resource machines. === Boris Calligraphy === Like Final Cut Pro, Express also comes with Boris Calligraphy, a plugin for advanced titling and scrolling/crawling titles more sophisticated than the ones that can be created with the built-in title overlays. Calligraphy has a WYSIWYG interface and features wrapping, alignment, leading, kerning and tracking features, as well as allowing up to five custom outlines and five custom drop shadows to be defined for a selected portion of the title. == Soundtrack == Prior to version 4, Final Cut Express included Soundtrack 1.5, a music program similar to the consumer-level GarageBand, but designed for videographers who wish to add music to their films. Soundtrack comes with around 4,000 professionally recorded instrument loops and sound effects that can be arranged in multiple tracks beneath the video track. To use Soundtrack, users export their Final Cut Express sequence, or a marked portion thereof, as a reference file, which can include scoring markers defined in the timeline. This reference file can be imported as the video track in Soundtrack. Soundtrack is functionally and visually identical to Soundtrack Pro's multitrack editing mode, but includes fewer Logic plugins and lacks the highly regarded noise removal tool. Soundtrack was removed from Final Cut Express 4, which lowered its price and may have encouraged people to buy Logic Express.
Symmetric Boolean function
In mathematics, a symmetric Boolean function is a Boolean function whose value does not depend on the order of its input bits, i.e., it depends only on the number of ones (or zeros) in the input. For this reason they are also known as Boolean counting functions. There are 2n+1 symmetric n-ary Boolean functions. Instead of the truth table, traditionally used to represent Boolean functions, one may use a more compact representation for an n-variable symmetric Boolean function: the (n + 1)-vector, whose i-th entry (i = 0, ..., n) is the value of the function on an input vector with i ones. Mathematically, the symmetric Boolean functions correspond one-to-one with the functions that map n+1 elements to two elements, f : { 0 , 1 , . . . , n } → { 0 , 1 } {\displaystyle f:\{0,1,...,n\}\rightarrow \{0,1\}} . Symmetric Boolean functions are used to classify Boolean satisfiability problems. == Special cases == A number of special cases are recognized: Majority function: their value is 1 on input vectors with more than n/2 ones Threshold functions: their value is 1 on input vectors with k or more ones for a fixed k All-equal and not-all-equal function: their values is 1 when the inputs do (not) all have the same value Exact-count functions: their value is 1 on input vectors with k ones for a fixed k One-hot or 1-in-n function: their value is 1 on input vectors with exactly one one One-cold function: their value is 1 on input vectors with exactly one zero Congruence functions: their value is 1 on input vectors with the number of ones congruent to k mod m for fixed k, m Parity function: their value is 1 if the input vector has odd number of ones The n-ary versions of AND, OR, XOR, NAND, NOR and XNOR are also symmetric Boolean functions. == Properties == In the following, f k {\displaystyle f_{k}} denotes the value of the function f : { 0 , 1 } n → { 0 , 1 } {\displaystyle f:\{0,1\}^{n}\rightarrow \{0,1\}} when applied to an input vector of weight k {\displaystyle k} . === Weight === The weight of the function can be calculated from its value vector: | f | = ∑ k = 0 n ( n k ) f k {\displaystyle |f|=\sum _{k=0}^{n}{\binom {n}{k}}f_{k}} === Algebraic normal form === The algebraic normal form either contains all monomials of certain order m {\displaystyle m} , or none of them; i.e. the Möbius transform f ^ {\displaystyle {\hat {f}}} of the function is also a symmetric function. It can thus also be described by a simple (n+1) bit vector, the ANF vector f ^ m {\displaystyle {\hat {f}}_{m}} . The ANF and value vectors are related by a Möbius relation: f ^ m = ⨁ k 2 ⊆ m 2 f k {\displaystyle {\hat {f}}_{m}=\bigoplus _{k_{2}\subseteq m_{2}}f_{k}} where k 2 ⊆ m 2 {\displaystyle k_{2}\subseteq m_{2}} denotes all the weights k whose base-2 representation is covered by the base-2 representation of m (a consequence of Lucas’ theorem). Effectively, an n-variable symmetric Boolean function corresponds to a log(n)-variable ordinary Boolean function acting on the base-2 representation of the input weight. For example, for three-variable functions: f ^ 0 = f 0 f ^ 1 = f 0 ⊕ f 1 f ^ 2 = f 0 ⊕ f 2 f ^ 3 = f 0 ⊕ f 1 ⊕ f 2 ⊕ f 3 {\displaystyle {\begin{array}{lcl}{\hat {f}}_{0}&=&f_{0}\\{\hat {f}}_{1}&=&f_{0}\oplus f_{1}\\{\hat {f}}_{2}&=&f_{0}\oplus f_{2}\\{\hat {f}}_{3}&=&f_{0}\oplus f_{1}\oplus f_{2}\oplus f_{3}\end{array}}} So the three variable majority function with value vector (0, 0, 1, 1) has ANF vector (0, 0, 1, 0), i.e.: Maj ( x , y , z ) = x y ⊕ x z ⊕ y z {\displaystyle {\text{Maj}}(x,y,z)=xy\oplus xz\oplus yz} === Unit hypercube polynomial === The coefficients of the real polynomial agreeing with the function on { 0 , 1 } n {\displaystyle \{0,1\}^{n}} are given by: f m ∗ = ∑ k = 0 m ( − 1 ) | k | + | m | ( m k ) f k {\displaystyle f_{m}^{}=\sum _{k=0}^{m}(-1)^{|k|+|m|}{\binom {m}{k}}f_{k}} For example, the three variable majority function polynomial has coefficients (0, 0, 1, -2): Maj ( x , y , z ) = ( x y + x z + y z ) − 2 ( x y z ) {\displaystyle {\text{Maj}}(x,y,z)=(xy+xz+yz)-2(xyz)} == Examples ==