Waveform graphics is a simple vector graphics system introduced by Digital Equipment Corporation (DEC) on the VT55 and VT105 terminals in the mid-1970s. It was used to produce graphics output from mainframes and minicomputers. DEC used the term "waveform graphics" to refer specifically to the hardware, but it was used more generally to describe the whole system. The system was designed to use as little computer memory as possible. At any given X location it could draw two dots at given Y locations, making it suitable for producing two superimposed waveforms, line charts or histograms. Text and graphics could be mixed, and there were additional tools for drawing axes and markers. The waveform graphics system was used only for a short period of time before it was replaced by the more sophisticated ReGIS system, first introduced on the VT125 in 1981. ReGIS allowed the construction of arbitrary vectors and other shapes. Whereas DEC normally provided a backward compatible solution in newer terminal models, they did not choose to do this when ReGIS was introduced, and waveform graphics disappeared from later terminals. == Description == Waveform graphics was introduced on the VT55 terminal in October 1975, an era when memory was extremely expensive. Although it was technically possible to produce a bitmap display using a framebuffer using technology of the era, the memory needed to do so at a reasonable resolution was typically beyond the price point that made it practical. All sorts of systems were used to replace computer memory with other concepts, like the storage tubes used in the Tektronix 4010 terminals, or the zero memory racing-the-beam system used in the Atari 2600. DEC chose to attack this problem through a clever use of a small buffer representing only the vertical positions on the screen. Such a system could not draw arbitrary shapes, but would allow the display of graph data. The system was based on a 512 by 236 pixel display, producing 512 vertical columns along the X-axis, and 236 horizontal rows on the Y-axis. Y locations were counted up from the bottom, so the coordinate 0,0 was in the lower left, and 511, 235 in the upper right. Had this been implemented using a framebuffer with each location represented by a single bit, 512 × 236 × 1 = 120,832 bits, or 15,104 bytes, would have been required. At the time, memory cost about $50 per kilobyte, so the buffer alone would cost over $700 (equivalent to $4,570 in 2025). Instead, the waveform graphic system used one byte of memory for each X axis location, with the byte's value representing the Y location. This required only 512 bytes for each graph, a total of 1024 bytes for the two graphs. Drawing a line required the programmer to construct a series of Y locations and send them as individual points, the terminal could not connect the dots itself. To make this easier, the terminal automatically incremented the X location every time an Y coordinate was received, so a graph line could be sent as a long string of numbers for subsequent Y locations instead of having to repeatedly send the X location every time. Drawing normally started by sending a single instruction to set the initial X location, often 0 on the left, and then sending in data for the entire curve. The system also included storage for up to 512 markers on both lines. These were always drawn centered on the Y value of the line they were associated with, meaning that a simple on/off indication for X locations was all that was needed, requiring only 1024 bits, or 128 bytes, in total. The markers extended 16 pixels vertically, and could only be aligned on 16-pixel boundaries, so they were not necessarily centered across the underlying graph. Markers were used to indicate important points on the graph, where a symbol of some sort would normally be used. The system also allowed a vertical line to be drawn for every horizontal location and a horizontal one at every vertical location. These were also stored as simple on/off bits, requiring another 128 bytes of memory. These lines were used to draw axes and scale lines, or could be used for a screen-spanning crosshair cursor. A separate set of two 7-bit registers held additional information about the drawing style and other settings. Although complex from the user's perspective, this system was easy to implement in hardware. A cathode ray tube produces a display by scanning the screen in a series of horizontal motions, moving down one vertical line after each horizontal scan. At any given instant during this process, the display hardware examines a few memory locations to see if anything needs to be displayed. For instance, it can determine whether to draw a marker on graph 0 by examining register 1 to see if markers are turned on, looking in the marker buffer to see if there is a 1 at the current X location, and then examining the Y location of graph 0 to see if it is within 16 pixels of the current scan line. If all of these are true, a spot is drawn to present that portion of the marker. As this will be true for 16 vertical locations during the scanning process, a 16-pixel high marker will be drawn. Sold alone, the VT55 was priced at $2,496 (equivalent to $16,295 in 2025),. Like other models of the VT50 series, the terminal could be equipped with an optional wet-paper printer in a panel on the right of the screen. This added $800 (equivalent to $5,223 in 2025) to the price. DEC also offered VT55 in a package with a small model of the PDP-11 to create one model of the DEClab 11/03 system. The DEClab normally sold for $14,000 (equivalent to $91,397 in 2025) with a DECwriter II (LA36) hard-copy terminal for $15,000 (equivalent to $97,925 in 2025), with the VT55. The system had I/O channels for up to 15 lab devices, and included libraries for FORTRAN and BASIC for reading the data and creating graphs. The fairly extensive VT55 Programmers Manual covered the latter in depth. == Commands and data == Data was sent to the terminal using an extended set of codes similar to those introduced on the VT52. VT52 codes generally started with the ESC character (octal 33, decimal 27) and was then followed by a single letter instruction. For instance, the string of four characters ESC H ESC J would reposition the cursor in the upper left (home) and then clear the screen from that point down. These codes were basically modeless; triggered by the ESC the resulting escape mode automatically exited again when the command was complete. Escape codes could be interspersed with display text anywhere in the stream of data. In contrast, the graphics system was entirely modal, with escape sequences being sent to cause the terminal to enter or exit graph drawing mode. Data sent between these two codes were interpreted by the graphics hardware, so text and graphics could not be mixed in a single stream of instructions. Graphics mode was entered by sending the string ESC 1, and exited again with the string ESC 2. Even the commands within the graphics mode were modal; characters were interpreted as being additional data for the previous load character (command) until another load character is seen. Ten load characters were available: @ - no operation, used to tell the terminal the last command is no longer active A - load data into register 0, selecting the drawing mode for the two graphs I - load data into register 1, selecting other drawing options H - load the starting X position (Horizontal) for the following commands B - load data for Y locations for graph 0 starting at the H position selected earlier J - load data for Y locations for graph 1 starting at the H position selected earlier C - store a marker on graph 0 at the following X location K - store a marker on graph 1 at the following X location D - draw a horizontal line at the given Y location L - draw a vertical line at the given X location X and Y locations were sent as 10-bit decimal numbers, encoded as ASCII characters, with 5 bits per character. This means that any number within the 1024 number space (210) can be stored as a string of two characters. To ensure the characters can be transmitted over 7-bit links, the pattern 01 is placed in front of both 5-bit numbers, producing 7-bit ASCII values that are always within the printable range. This results in a somewhat complex encoding algorithm. For instance, if one wanted to encode the decimal value 102, first you convert that to the 10-bit decimal pattern 0010010010. That is then split that into upper and lower 5-bit parts, 00100 and 10010. Then append 01 binary to produce 7-bit numbers 0100100 and 0110010. Individually convert back to decimal 40 and 50, and then look up those characters in an ASCII chart, finding ( and 2. These have to be sent to the terminal least significant character first. If these were being used to set the X coordinate, the complete string would be H2(. When used as X and Y locations for the graphs, extra digits were ignored. For instance, the 512 pixel X axis r
E-on Vue
Vue is a software tool for world generation by Bentley Systems, with support for many visual effects, animations, and various other features. The tool has been used in several feature-length films. In 2024, Bentley Systems announced that Vue would be discontinued, and be freely available to those that still wish to use it. == Versions == == Features == This is a list of features as of the 2023 release of Vue: === Terrains === Heightfield terrains Procedural terrains Infinite terrains Planetary terrains Real-world terrains 3D terrain sculpting Terrain export === EcoSystem Instancing Technology === Material-based EcoSystems Global EcoSystems Dynamic EcoSystems 360° EcoSystem Population Paint EcoSystem instances EcoParticles Export EcoSystem populations === Vegetation === Built-in Plant editor Compatible with PlantFactory Vegetation assets === Atmosphere, Skies and Clouds === Standard atmospheric model Spectral atmospheric model Photometric atmospheric model Atmosphere presets Procedural Volumetric 3D cloud layers Standalone 3D Metaclouds Convert meshes to Clouds Cloud morphing Import OpenVDB Export standalone and cloud layer zones to OpenVDB Export skies as HDRI === Modeling === Primitive and Feature modeling 3D Text edition tool Metablobbing Hyperblobs Export baked hyperblobs Splines Built in Road Construction toolkit Random rock generator Export rocks === Texturing and UVs === Material presets PBR Substance support Node-based procedural materials Volumetric materials and Hypertextures Stacked UVs Unwrapped UVs Ptex === Interoperability, Integration And Export === Export single assets to generic 3D formats Full scene export Integration plugins Import and Export Camera data as FBX and Nuke.chan Python API ZBrush GoZ bridge === Animation === Animate objects, materials, atmospheres, clouds, waves... Automatic wind and breeze Localized wind effects per plant / per EcoSystem population Omni and directional ventilators for local modifications of plants Time spline editor Automatic keyframe creation Automatic synchronization of cameras and lights Animation export as AfterEffects Import motion tracking information === Lighting === Global illumination, Global Radiosity, Ambient occlusion Subsurface Scattering HDRI image based lighting Point light, Quadratic point light, Spotlight, Quadratic spotlight, Directional light Use IES distribution profiles on photometric lights Area lights, light panels, light portals Physically accurate caustics computation === Rendering === Render with Ray Tracer Render with Path Tracer Stereoscopic rendering 360/180 VR Panorama Render Option Spherical panoramic rendering Tone mapping options Multipass & G-Buffer Network rendering with HyperVue / RenderCows Network rendering with RenderNodes == Users == Blue Sky Studios Digital Domain DreamWorks Animation: Kung Fu Panda Industrial Light & Magic: Indiana Jones and the Kingdom of the Crystal Skull, Pirates of the Caribbean: Dead Man's Chest Sony Pictures Imageworks Warner Bros. Interactive Entertainment Weta Digital
Pulsar (social listening platform)
Pulsar is a software platform for social media monitoring, audience intelligence and social listening that allows organizations to monitor and analyze online conversations across social media, news, and other digital sources. The platform combines social media listening, media monitoring, trend analysis, and audience segmentation to help users understand public discussions and audience behavior in real time. The platform is a social listening platform, which aggregates data from networks such as X, Facebook, Instagram, and forums) and applies artificial intelligence for text and sentiment analysis. Pulsar is offered as a cloud-based Software as a Service (SaaS) tool and insights consultancy. It has been part of Pulsar Group (formerly Access Intelligence), a publicly listed group of communications software products, since 2019. As well as commercial uses, the platform has been used in peer-reviewed academic research analysing online discourse. The platform is listed on the UK government's G-Cloud 14 Digital Marketplace for the provision of social listening and audience intelligence services. == History == Pulsar originated in the early 2010s as a project within Face, a London-based innovation and market research consultancy. The platform's first product, Pulsar TRAC, launched in 2013 as a social media analytics tool. Pulsar TRAC was designed to measure the reach of conversations, mapping brand audiences, and tracking how content spreads through networks. The development was led by Dr Francesco D'Orazio, who created the Pulsar brand and led the development of the platform while serving as VP of Product and Innovation at Face. Face itself had been acquired by the Cello Group Plc (a UK-based advisory firm) in 2012, and Pulsar became part of Cello's portfolio of research and data tools. In January 2017, Cello Group made a significant investment to scale Pulsar and announced the merger of Face's qualitative research business into Pulsar, unifying both under the Pulsar brand for global expansion. In 2018, Pulsar opened an office in Los Angeles to better serve its growing U.S. client base in media, healthcare, and entertainment sectors and Francesco D'Orazio was appointed CEO. The company focused on developing new products amid a wave of consolidation in the social listening industry. In October 2019, Pulsar was acquired by Access Intelligence Plc (now Pulsar Group), an AIM-listed communications software company. The group, which also owns PR and media tools Isentia, Vuelio and ResponseSource, integrated Pulsar to their end-to-end marketing and communications insights offering. Pulsar established a new office in Sydney, Australia in 2022 as part of this global expansion, adding to its existing offices in London and Los Angeles. In 2023, Pulsar Group (then Access Intelligence) was recognised as one of Europe's fastest growing companies by the Financial Times. In May 2024, Access Intelligence PLC changed its name to Pulsar Group PLC. The company has since continued to develop its platform. In March 2025 it introduced new tool Narratives AI, described as a "search engine for public opinion" and the first of its kind for analyzing public narratives and their evolutions in both social media and the news. In October 2025, Pulsar launched Insight Agents, a set of AI agents embedded into the platform advertised to "proactively anticipate user needs or issues, carry out routine tasks, uncover anomalies in your datasets, and prompt responses at scale, 24/7." == Products == Pulsar's architecture integrates four main products into a single interface. The core product suite is often broken into three main components: Pulsar TRAC (for social listening and audience analysis), Pulsar TRENDS (for trend discovery and analysis), and Pulsar CORE (for owned-channel and web analytics). Pulsar's fourth product is Narratives AI. === Pulsar TRAC === Pulsar TRAC is a social listening and audience intelligence platform that allows users to configure searches that track public conversations and measure audience behaviour. Pulsar TRAC is focused on conversation insights and audience segmentations - the platform is reported to collect and analyse data from a wide range of sources, including major social networks, forums, news and review sites, and ecommerce platforms, with real-time visualisations and AI-supported analytics used to find patterns and communities of interest. Pulsar TRAC can be incorporated into workflows with other audience tools, such as an integration with Audiense that connects TRAC's conversation insights to external audience-segmentation datasets. === Pulsar CORE === Pulsar CORE centres on the analysis of owned-channel data, such as brand social media profiles, website interaction and other in-house digital assets, to generate audience and content insights. CORE can monitor published content, evaluate competitors, and extract demographic and behavioural segmentation from owned channels. === Narratives AI === Narratives AI is a tool within the Pulsar audience intelligence platform that uses artificial intelligence to detect, cluster and analyse narratives forming across social and news media. It was launched in March 2025 as a standalone search interface that processes real-time and historical data to find cultural trends, behaviours and beliefs. It uses clustering algorithms and visualisation to show how conversations form and spread online, and their relative importance within wider discourse. == Notable features == === Insight Agents === Pulsar's Insight Agents are AI-powered agents within the Pulsar platform designed to automate and augment common tasks in media, social, audience and narrative intelligence. Branded as TeamMates, these agents are grouped into four functional types: Sentinels for real-time monitoring, anomaly detection and alerting Oracles for forecasting and scenario planning Custodians for governance, compliance and policy enforcement Analysts for research, reporting and recommendations Each agent is trained on Pulsar's multi-source data and domain-specific workflows. In February 2026, Pulsar introduced 'Crisis Oracle,' an AI-driven system designed to quantify narrative momentum and predict reputational risk. == Academic research == Pulsar has been used as a data collection and analysis tool in peer-reviewed academic research across public health, infodemiology, veterinary science, and policy research. Published uses include a World Health Organization report on infodemic management, a Journal of Medical Internet Research study on headache and migraine discourse across Japan, Germany, and France, a Frontiers in Big Data study of Long COVID narratives, and Frontiers in Veterinary Science studies on canine chronic kidney disease and oral medication administration in dogs.
Artificial Intelligence Cold War
The Artificial Intelligence Cold War (AI Cold War) is a narrative in which geopolitical tensions between the United States of America (USA) and the People's Republic of China (PRC) could lead to a Second Cold War waged in the area of artificial intelligence technology rather than in the areas of nuclear capabilities or ideology. The context of the AI Cold War narrative is the AI arms race, which involves a build-up of military capabilities using AI technology by the US and China and the usage of increasingly advanced semiconductors which power those capabilities. According to a February 2019 publication by the Center for a New American Security, General Secretary of the Chinese Communist Party Xi Jinping – believes that being at the forefront of AI technology will be critical to the future of China's global military and economic power competition. == Origins of the term == The term AI Cold War first appeared in 2018 in an article in Wired magazine by Nicholas Thompson and Ian Bremmer. The two authors trace the emergence of the AI Cold War narrative to 2017, when China published its AI Development Plan, which included a strategy aimed at becoming the global leader in AI by 2030. While the authors acknowledge the use of AI by China to strengthen its authoritarian (totalitarian) rule, they warn against the perils for the US of engaging in an AI Cold War strategy. Thompson and Bremmer rather advocate for a technological cooperation between the US and China to encourage global standards in privacy and ethical use of AI. Shortly after the publication of the article in Wired magazine, the former U.S. Treasury Secretary Hank Paulson referred to the emergence of an ‘Economic Iron Curtain’ between the US and China, reinforcing the new AI Cold War narrative. == Proponents of the AI Cold War narrative == Politico contributed to reinforcing the AI Cold War narrative. In 2020, the paper argued that because of the increasing AI capabilities of China, the US and other democratic countries have to create an alliance to stay ahead of China. Former Google chief executive Eric Schmidt, together with Graham T. Allison alleged in an article in Project Syndicate that, in the context of the COVID-19 pandemic, the AI capabilities of China are ahead of the US in most critical areas. Scientists who have immigrated to the U.S. play an outsize role in the country's development of AI technology. Many of them were educated in China, prompting debates about national security concerns amid worsening relations between the two countries. Policy and technology experts have pointed to concerns about unethical use of AI which would be primarily associated with China. Ethics would therefore constitute a major ideological divide in the upcoming AI Cold War. Fears around disrupting supply chains and a global semiconductor shortage are linked to Taiwan's critical role in the production of semiconductors. 70% of semiconductors are either produced in Taiwan or transfer through Taiwan, where TSMC, world's largest chipmaker is headquartered. The PRC does not recognize the sovereignty of Taiwan and trade restrictions by the US on companies selling semiconductors to the PRC have disrupted in the past the commercial relationships between TSMC and Huawei. == Reactions to the AI Cold War == === Review of the validity of the AI Cold War narrative === Academics and observers expressed concerns about the validity and soundness of the AI Cold War narrative. Denise Garzia expressed concern in Nature that the AI Cold War narrative will undermine the efforts by the US to establish global rules for AI ethics. Researchers have warned in MIT Technology Review that the breakdown in international collaboration in the area of science because of the threat of the alleged AI Cold War would be detrimental to progress. Additionally, the AI Cold War narrative impacts on many more areas including the planning of supply chains and the proliferation of AI. The dissemination of the AI Cold War narrative could therefore be costly and destructive and exacerbate existing tensions. Joanna Bryson and Helena Malikova have pointed to Big Tech's potential interest in promoting the AI Cold War narrative, as technology companies lobby for less onerous regulation of AI in the US and the EU. A factual assessment of the existing AI capabilities of different countries shows a less binary reality than portrayed by the AI Cold War narrative. The AI Cold War started as a narrative but it could turn into a self-fulfilling prophecy and fuel an arms race, not only because of corporate interests but also because of the existing interests at different national security departments. Regarding cyber power, the International Institute for Strategic Studies published a study in June 2021, which argued that the online capabilities of China have been exaggerated and that Chinese cyber power is at least a decade behind the US, largely due to lingering security issues. === Restrictions to trading with China === US politicians and European industry players have invoked the looming AI Cold War as a reason to ban procurement by public authorities in Europe of Huawei 5G technology due to concerns over the Chinese state-sponsored surveillance industry. In 2019, the Trump administration successfully lobbied the Dutch government into stopping the Netherlands-based company ASML from exporting equipment to China. ASML manufactures a machine called an extreme ultraviolet lithography system used by semiconductor producers, including TSMC and Intel to produce state-of the-art microchips. The Biden administration adopted the same course of action as the Trump administration and requested the Netherlands to restrict sales by ASML to China, invoking national-security concerns. The trade restrictions imposed by the Trump administration affected semiconductors imports from China to the US and raised concerns by the US industry that supply chains will be disrupted in case of an AI Cold War. This prompted US technology companies to develop mitigation strategies including hoarding semiconductors and trying to set up local semiconductor production facilities, with the support of government subsidies. === Industrial policy initiatives === ==== United States ==== In June 2021, the US Senate approved the U.S. Innovation and Competition Act providing around 250 billion US dollars public money support to the US technological and manufacturing industry. The alleged Chinese threat in the area of technology helped secure a strong bipartisan support for the new legislation, amounting to the largest industrial policy move by the US in decades. Chinese authorities reproached to the US that the bill was “full of cold war zero-sum thinking”. The legislative bill is aimed at strengthening capabilities in the area of technology, such as quantum computing and AI specifically to face the competitive threat from China perceived as urgent. Senator Chuck Schumer, the leader of the Senate majority and one of the sponsors of the industrial policy bill invoked the threat of authoritarian regimes that want “grab the mantle of global economic leadership and own the innovations”. In 2022, U.S. Innovation and Competition Act was amended and turned into the Chips and Science Act with planned spending of 280 billion US dollars, 53 billion thereof are allocated directly to subsidies for semiconductors manufacturing. Commentators identified possible positive effects on innovation from the US attempts to compete with China in a perceived rivalry. Among the main beneficiaries of the US CHIPS Act are the semiconductor producers Intel, TSMC and Micron Technology. ==== European Chips Act ==== In February 2022, the European Union introduced its own European Chips Act initiative. The background of the initiative would be the objective of European strategic autonomy. The EU's initiative puts forward subsidies of 30 billion euros to encourage manufacturing of semiconductors in the EU. The US company Intel is one beneficiary of the initiative. The US and European chips acts raise concerns of protectionism and a risk of a subsidies "race to the bottom." === New world order === The AI Cold War heralds a new world order in geopolitics, according to Hemant Taneja and Fareed Zakaria. This new world order is a departure from the unipolar system dominated by the US. It is characterized by existence of two parallel digital ecosystems, ran by China and the US. In order to succeed countries that consider themselves as democracies are to align their technological ecosystems to that of the US, in a process labelled re-globalization.
Polyworld
Polyworld is a cross-platform (Linux, Mac OS X) program written by Larry Yaeger to evolve Artificial Intelligence through natural selection and evolutionary algorithms. It uses the Qt graphics toolkit and OpenGL to display a graphical environment in which a population of trapezoid agents search for food, mate, have offspring, and prey on each other. The population is typically only in the hundreds, as each individual is rather complex and the environment consumes considerable computer resources. The graphical environment is necessary since the individuals actually move around the 2-D plane and must be able to "see." Since some basic abilities, like eating carcasses or randomly generated food, seeing other individuals, mating or fighting with them, etc., are possible, a number of interesting behaviours have been observed to spontaneously arise after prolonged evolution, such as cannibalism, predators and prey, and mimicry. Each individual makes decisions based on a neural net using Hebbian learning; the neural net is derived from each individual's genome. The genome does not merely specify the wiring of the neural nets, but also determines their size, speed, color, mutation rate and a number of other factors. The genome is randomly mutated at a set probability, which are also changed in descendant organisms.
Phrase structure grammar
The term phrase structure grammar was originally introduced by Noam Chomsky as the term for grammar studied previously by Emil Post and Axel Thue (Post canonical systems). Some authors, however, reserve the term for more restricted grammars in the Chomsky hierarchy: context-sensitive grammars or context-free grammars. In a broader sense, phrase structure grammars are also known as constituency grammars. The defining character of phrase structure grammars is thus their adherence to the constituency relation, as opposed to the dependency relation of dependency grammars. == History == In 1956, Chomsky wrote, "A phrase-structure grammar is defined by a finite vocabulary (alphabet) Vp, and a finite set Σ of initial strings in Vp, and a finite set F of rules of the form: X → Y, where X and Y are strings in Vp." == Constituency relation == In linguistics, phrase structure grammars are all those grammars that are based on the constituency relation, as opposed to the dependency relation associated with dependency grammars; hence, phrase structure grammars are also known as constituency grammars. Any of several related theories for the parsing of natural language qualify as constituency grammars, and most of them have been developed from Chomsky's work, including Government and binding theory Generalized phrase structure grammar Head-driven phrase structure grammar Lexical functional grammar The minimalist program Nanosyntax Further grammar frameworks and formalisms also qualify as constituency-based, although they may not think of themselves as having spawned from Chomsky's work, e.g. Arc pair grammar, and Categorial grammar.
BL (logic)
In mathematical logic, basic fuzzy logic (or shortly BL), the logic of the continuous t-norms, is one of the t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices; it extends the logic MTL of all left-continuous t-norms. == Syntax == === Language === The language of the propositional logic BL consists of countably many propositional variables and the following primitive logical connectives: Implication → {\displaystyle \rightarrow } (binary) Strong conjunction ⊗ {\displaystyle \otimes } (binary). The sign & is a more traditional notation for strong conjunction in the literature on fuzzy logic, while the notation ⊗ {\displaystyle \otimes } follows the tradition of substructural logics. Bottom ⊥ {\displaystyle \bot } (nullary — a propositional constant); 0 {\displaystyle 0} or 0 ¯ {\displaystyle {\overline {0}}} are common alternative signs and zero a common alternative name for the propositional constant (as the constants bottom and zero of substructural logics coincide in MTL). The following are the most common defined logical connectives: Weak conjunction ∧ {\displaystyle \wedge } (binary), also called lattice conjunction (as it is always realized by the lattice operation of meet in algebraic semantics). Unlike MTL and weaker substructural logics, weak conjunction is definable in BL as A ∧ B ≡ A ⊗ ( A → B ) {\displaystyle A\wedge B\equiv A\otimes (A\rightarrow B)} Negation ¬ {\displaystyle \neg } (unary), defined as ¬ A ≡ A → ⊥ {\displaystyle \neg A\equiv A\rightarrow \bot } Equivalence ↔ {\displaystyle \leftrightarrow } (binary), defined as A ↔ B ≡ ( A → B ) ∧ ( B → A ) {\displaystyle A\leftrightarrow B\equiv (A\rightarrow B)\wedge (B\rightarrow A)} As in MTL, the definition is equivalent to ( A → B ) ⊗ ( B → A ) . {\displaystyle (A\rightarrow B)\otimes (B\rightarrow A).} (Weak) disjunction ∨ {\displaystyle \vee } (binary), also called lattice disjunction (as it is always realized by the lattice operation of join in algebraic semantics), defined as A ∨ B ≡ ( ( A → B ) → B ) ∧ ( ( B → A ) → A ) {\displaystyle A\vee B\equiv ((A\rightarrow B)\rightarrow B)\wedge ((B\rightarrow A)\rightarrow A)} Top ⊤ {\displaystyle \top } (nullary), also called one and denoted by 1 {\displaystyle 1} or 1 ¯ {\displaystyle {\overline {1}}} (as the constants top and zero of substructural logics coincide in MTL), defined as ⊤ ≡ ⊥ → ⊥ {\displaystyle \top \equiv \bot \rightarrow \bot } Well-formed formulae of BL are defined as usual in propositional logics. In order to save parentheses, it is common to use the following order of precedence: Unary connectives (bind most closely) Binary connectives other than implication and equivalence Implication and equivalence (bind most loosely) === Axioms === A Hilbert-style deduction system for BL has been introduced by Petr Hájek (1998). Its single derivation rule is modus ponens: from A {\displaystyle A} and A → B {\displaystyle A\rightarrow B} derive B . {\displaystyle B.} The following are its axiom schemata: ( B L 1 ) : ( A → B ) → ( ( B → C ) → ( A → C ) ) ( B L 2 ) : A ⊗ B → A ( B L 3 ) : A ⊗ B → B ⊗ A ( B L 4 ) : A ⊗ ( A → B ) → B ⊗ ( B → A ) ( B L 5 a ) : ( A → ( B → C ) ) → ( A ⊗ B → C ) ( B L 5 b ) : ( A ⊗ B → C ) → ( A → ( B → C ) ) ( B L 6 ) : ( ( A → B ) → C ) → ( ( ( B → A ) → C ) → C ) ( B L 7 ) : ⊥ → A {\displaystyle {\begin{array}{ll}{\rm {(BL1)}}\colon &(A\rightarrow B)\rightarrow ((B\rightarrow C)\rightarrow (A\rightarrow C))\\{\rm {(BL2)}}\colon &A\otimes B\rightarrow A\\{\rm {(BL3)}}\colon &A\otimes B\rightarrow B\otimes A\\{\rm {(BL4)}}\colon &A\otimes (A\rightarrow B)\rightarrow B\otimes (B\rightarrow A)\\{\rm {(BL5a)}}\colon &(A\rightarrow (B\rightarrow C))\rightarrow (A\otimes B\rightarrow C)\\{\rm {(BL5b)}}\colon &(A\otimes B\rightarrow C)\rightarrow (A\rightarrow (B\rightarrow C))\\{\rm {(BL6)}}\colon &((A\rightarrow B)\rightarrow C)\rightarrow (((B\rightarrow A)\rightarrow C)\rightarrow C)\\{\rm {(BL7)}}\colon &\bot \rightarrow A\end{array}}} The axioms (BL2) and (BL3) of the original axiomatic system were shown to be redundant (Chvalovský, 2012) and (Cintula, 2005). All the other axioms were shown to be independent (Chvalovský, 2012). == Semantics == Like in other propositional t-norm fuzzy logics, algebraic semantics is predominantly used for BL, with three main classes of algebras with respect to which the logic is complete: General semantics, formed of all BL-algebras — that is, all algebras for which the logic is sound Linear semantics, formed of all linear BL-algebras — that is, all BL-algebras whose lattice order is linear Standard semantics, formed of all standard BL-algebras — that is, all BL-algebras whose lattice reduct is the real unit interval [0, 1] with the usual order; they are uniquely determined by the function that interprets strong conjunction, which can be any continuous t-norm.