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  User-defined function

  A user-defined function (UDF) is a function provided by the user of a program or environment, in a context where the usual assumption is that functions are built into the program or environment. UDFs are usually written for the requirement of its creator. == BASIC language == In some old implementations of the BASIC programming language, user-defined functions are defined using the "DEF FN" syntax. More modern dialects of BASIC are influenced by the structured programming paradigm, where most or all of the code is written as user-defined functions or procedures, and the concept becomes practically redundant. == COBOL language == In the COBOL programming language, a user-defined function is an entity that is defined by the user by specifying a FUNCTION-ID paragraph. A user-defined function must return a value by specifying the RETURNING phrase of the procedure division header and they are invoked using the function-identifier syntax. See the ISO/IEC 1989:2014 Programming Language COBOL standard for details. As of May 2022, the IBM Enterprise COBOL for z/OS 6.4 (IBM COBOL) compiler contains support for user-defined functions. == Databases == In relational database management systems, a user-defined function provides a mechanism for extending the functionality of the database server by adding a function, that can be evaluated in standard query language (usually SQL) statements. The SQL standard distinguishes between scalar and table functions. A scalar function returns only a single value (or NULL), whereas a table function returns a (relational) table comprising zero or more rows, each row with one or more columns. User-defined functions in SQL are declared using the CREATE FUNCTION statement. For example, a user-defined function that converts Celsius to Fahrenheit (a temperature scale used in USA) might be declared like this: Once created, a user-defined function may be used in expressions in SQL statements. For example, it can be invoked where most other intrinsic functions are allowed. This also includes SELECT statements, where the function can be used against data stored in tables in the database. Conceptually, the function is evaluated once per row in such usage. For example, assume a table named Elements, with a row for each known chemical element. The table has a column named BoilingPoint for the boiling point of that element, in Celsius. The query would retrieve the name and the boiling point from each row. It invokes the CtoF user-defined function as declared above in order to convert the value in the column to a value in Fahrenheit. Each user-defined function carries certain properties or characteristics. The SQL standard defines the following properties: Language - defines the programming language in which the user-defined function is implemented; examples include SQL, C, C# and Java. Parameter style - defines the conventions that are used to pass the function parameters and results between the implementation of the function and the database system (only applicable if language is not SQL). Specific name - a name for the function that is unique within the database. Note that the function name does not have to be unique, considering overloaded functions. Some SQL implementations require that function names are unique within a database, and overloaded functions are not allowed. Determinism - specifies whether the function is deterministic or not. The determinism characteristic has an influence on the query optimizer when compiling a SQL statement. SQL-data access - tells the database management system whether the function contains no SQL statements (NO SQL), contains SQL statements but does not access any tables or views (CONTAINS SQL), reads data from tables or views (READS SQL DATA), or actually modifies data in the database (MODIFIES SQL DATA). User-defined functions should not be confused with stored procedures. Stored procedures allow the user to group a set of SQL commands. A procedure can accept parameters and execute its SQL statements depending on those parameters. A procedure is not an expression and, thus, cannot be used like user-defined functions. Some database management systems allow the creation of user defined functions in languages other than SQL. Microsoft SQL Server, for example, allows the user to use .NET languages including C# for this purpose. DB2 and Oracle support user-defined functions written in C or Java programming languages. === SQL Server 2000 === There are three types of UDF in Microsoft SQL Server 2000: scalar functions, inline table-valued functions, and multistatement table-valued functions. Scalar functions return a single data value (not a table) with RETURNS clause. Scalar functions can use all scalar data types, with exception of timestamp and user-defined data types. Inline table-valued functions return the result set of a single SELECT statement. Multistatement table-valued functions return a table, which was built with many TRANSACT-SQL statements. User-defined functions can be invoked from a query like built‑in functions such as OBJECT_ID, LEN, DATEDIFF, or can be executed through an EXECUTE statement like stored procedures. Performance Notes: User-defined functions are subroutines made of one or more Transact-SQL statements that can be used to encapsulate code for reuse. It takes zero or more arguments and evaluates a return value. Has both control-flow and DML statements in its body similar to stored procedures. Does not allow changes to any Global Session State, like modifications to database or external resource, such as a file or network. Does not support output parameter. DEFAULT keyword must be specified to pass the default value of parameter. Errors in UDF cause UDF to abort which, in turn, aborts the statement that invoked the UDF. === Apache Hive === Apache Hive defines, in addition to the regular user-defined functions (UDF), also user-defined aggregate functions (UDAF) and table-generating functions (UDTF). Hive enables developers to create their own custom functions with Java. === Apache Doris === Apache Doris, an open-source real-time analytical database, allows external users to contribute their own UDFs written in C++ to it.

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  Slopaganda

  Slopaganda is a portmanteau of "AI slop" and "propaganda", referring to AI-generated content designed to manipulate beliefs, emotions, and political decision-making at scale. The term is credited to Michał Klincewicz, an assistant professor in the Department of Computational Cognitive Science at Tilburg University, in 2025. == Definition == Slopaganda is distinguished from traditional propaganda by three features: scale, scope, and speed. Generative AI makes it possible to produce large volumes of content quickly and at low cost, allows for highly personalised and targeted messaging to specific sub-audiences, and leverages the hyper-connectivity of social networks to accelerate dissemination beyond what conventional media could achieve. Unlike traditional propaganda, which delivers a uniform message to all recipients, slopaganda can be micro-targeted — tailored to individuals based on estimated prior beliefs to reinforce political biases or emotional associations. The authors note that it need not aim at literal deception: much slopaganda is expressive rather than truth-apt, designed to create emotional associations rather than false factual beliefs. == Relation to AI slop == Slopaganda is a subset of AI slop — low-quality, mass-produced AI-generated content — distinguished by intent. Where AI slop may be produced indifferently for commercial or engagement-farming purposes, slopaganda is deployed with a deliberate political or ideological goal. == Notable examples == Examples discussed by the term's originators include Donald Trump's prolific use of AI in Truth Social posts and Iranian Lego-themed music videos. AI-generated videos posted by the White House mixing real military footage with clips from films and video games; and deepfake audio imitating political candidates during the 2024 US presidential campaign have also been given the label slopaganda.

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  Grammar systems theory

  Grammar systems theory is a field of theoretical computer science that studies systems of finite collections of formal grammars generating a formal language. Each grammar works on a string, a so-called sequential form that represents an environment. Grammar systems can thus be used as a formalization of decentralized or distributed systems of agents in artificial intelligence. Let A {\displaystyle \mathbb {A} } be a simple reactive agent moving on the table and trying not to fall down from the table with two reactions, t for turning and ƒ for moving forward. The set of possible behaviors of A {\displaystyle \mathbb {A} } can then be described as formal language L A = { ( f m t n f r ) + : 1 ≤ m ≤ k ; 1 ≤ n ≤ ℓ ; 1 ≤ r ≤ k } , {\displaystyle \mathbb {L_{A}} =\{(f^{m}t^{n}f^{r})^{+}:1\leq m\leq k;1\leq n\leq \ell ;1\leq r\leq k\},} where ƒ can be done maximally k times and t can be done maximally ℓ times considering the dimensions of the table. Let G A {\displaystyle \mathbb {G_{A}} } be a formal grammar which generates language L A {\displaystyle \mathbb {L_{A}} } . The behavior of A {\displaystyle \mathbb {A} } is then described by this grammar. Suppose the A {\displaystyle \mathbb {A} } has a subsumption architecture; each component of this architecture can be then represented as a formal grammar, too, and the final behavior of the agent is then described by this system of grammars. The schema on the right describes such a system of grammars which shares a common string representing an environment. The shared sequential form is sequentially rewritten by each grammar, which can represent either a component or generally an agent. If grammars communicate together and work on a shared sequential form, it is called a Cooperating Distributed (DC) grammar system. Shared sequential form is a similar concept to the blackboard approach in AI, which is inspired by an idea of experts solving some problem together while they share their proposals and ideas on a shared blackboard. Each grammar in a grammar system can also work on its own string and communicate with other grammars in a system by sending their sequential forms on request. Such a grammar system is then called a Parallel Communicating (PC) grammar system. PC and DC are inspired by distributed AI. If there is no communication between grammars, the system is close to the decentralized approaches in AI. These kinds of grammar systems are sometimes called colonies or Eco-Grammar systems, depending (besides others) on whether the environment is changing on its own (Eco-Grammar system) or not (colonies).

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  GOLOG

  GOLOG is a high-level logic programming language for the specification and execution of complex actions in dynamical domains. It is based on the situation calculus. It is a first-order logical language for reasoning about action and change. GOLOG was developed at the University of Toronto. == History == The concept of situation calculus on which the GOLOG programming language is based was first proposed by John McCarthy in 1963. == Description == A GOLOG interpreter automatically maintains a direct characterization of the dynamic world being modeled, on the basis of user supplied axioms about preconditions, effects of actions and the initial state of the world. This allows the application to reason about the condition of the world and consider the impacts of different potential actions before focusing on a specific action. Golog is a logic programming language and is very different from conventional programming languages. A procedural programming language like C defines the execution of statements in advance. The programmer creates a subroutine which consists of statements, and the computer executes each statement in a linear order. In contrast, fifth-generation programming languages like Golog work with an abstract model with which the interpreter can generate the sequence of actions. The source code defines the problem and it is up to the solver to find the next action. This approach can facilitate the management of complex problems from the domain of robotics. A Golog program defines the state space in which the agent is allowed to operate. A path in the symbolic domain is found with state space search. To speed up the process, Golog programs are realized as hierarchical task networks. Apart from the original Golog language, there are some extensions available. The ConGolog language provides concurrency and interrupts. Other dialects like IndiGolog and Readylog were created for real time applications in which sensor readings are updated on the fly. == Uses == Golog has been used to model the behavior of autonomous agents. In addition to a logic-based action formalism for describing the environment and the effects of basic actions, they enable the construction of complex actions using typical programming language constructs. It is also used for applications in high level control of robots and industrial processes, virtual agents, discrete event simulation etc. It can be also used to develop Belief Desire Intention-style agent systems. == Planning and scripting == In contrast to the Planning Domain Definition Language, Golog supports planning and scripting as well. Planning means that a goal state in the world model is defined, and the solver brings a logical system into this state. Behavior scripting implements reactive procedures, which are running as a computer program. For example, suppose the idea is to authoring a story. The user defines what should be true at the end of the plot. A solver gets started and applies possible actions to the current situation until the goal state is reached. The specification of a goal state and the possible actions are realized in the logical world model. In contrast, a hardwired reactive behavior doesn't need a solver but the action sequence is provided in a scripting language. The Golog interpreter, which is written in Prolog, executes the script and this will bring the story into the goal state.

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  Cloud-based design and manufacturing

  Cloud-based design and manufacturing (CBDM) refers to a service-oriented networked product development model in which service consumers are able to configure products or services and reconfigure manufacturing systems through Infrastructure-as-a-Service (IaaS), Platform-as-a-Service (PaaS), Hardware-as-a-Service (HaaS), and Software-as-a-Service (SaaS). Adapted from the original cloud computing paradigm and introduced into the realm of computer-aided product development, Cloud-Based Design and Manufacturing is gaining significant momentum and attention from both academia and industry. Cloud-based design and manufacturing includes two aspects: cloud-based design and cloud-based manufacturing. Another related concept is cloud manufacturing that is more general and popular. Cloud-Based Design (CBD) refers to a networked design model that leverages cloud computing, service-oriented architecture (SOA), Web 2.0 (e.g., social network sites), and semantic web technologies to support cloud-based engineering design services in distributed and collaborative environments. Cloud-Based Manufacturing (CBM) refers to a networked manufacturing model that exploits on-demand access to a shared collection of diversified and distributed manufacturing resources to form temporary, reconfigurable production lines which enhance efficiency, reduce product lifecycle costs, and allow for optimal resource allocation in response to variable-demand customer generated tasking. The enabling technologies for Cloud-Based Design and Manufacturing include cloud computing, Web 2.0, Internet of Things (IoT), and service-oriented architecture (SOA). == History == The term cloud-based design and manufacturing (CBDM) was initially coined by Dazhong Wu, David Rosen, and Dirk Schaefer at Georgia Tech in 2012 for the purpose of articulating a new paradigm for digital manufacturing and design innovation in distributed and collaborative settings. The main objective of CBDM is to further reduce time and cost associated with maintaining information and communication technology (ICT) infrastructures for design and manufacturing, enhancing digital manufacturing and design innovation in distributed and collaborative environments, and adapting to rapidly changing market demands. In 2014, the same research group also published the worldwide first two books on the subjects of Cloud-Based Design and Manufacturing (CBDM) and Social Product Development (SPD) with Springer, edited by Dirk Schaefer. == Characteristics == CBDM exhibits the following key characteristics: Cloud-based distributed file system High performance computing Cloud-based social collaboration Ubiquitous access to distributed big data Rapid manufacturing scalability Agility On-demand self-service Semantic Web Real-time request for quotation Pay-per-use pricing model Multi-tenancy CBDM differs from traditional collaborative and distributed design and manufacturing systems such as web-based systems and agent-based systems from a number of perspectives, including (1) computing architecture, (2) data storage, (3) sourcing process, (4) information and communication technology infrastructure, (5) business model, (6) programming model, and (7) communication. == Service models == Infrastructure as a service (IaaS) Platform as a service (PaaS) Hardware as a service (HaaS) Software as a service (SaaS) Similar to cloud computing, CBDM services can be categorized into four major deployment models: the public cloud, private cloud, hybrid cloud, and community cloud. == Research progress in Academia == The Defense Advanced Research Projects Agency (DARPA) MENTOR program Engineering and Physical Sciences Research Council cloud manufacturing program European Commission's Seventh Framework Program (EC FP7)

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  Principle of rationality

  The principle of rationality (or rationality principle) was coined by Karl R. Popper in his Harvard Lecture of 1963, and published in his book Myth of Framework. It is related to what he called the 'logic of the situation' in an Economica article of 1944/1945, published later in his book The Poverty of Historicism. According to Popper's rationality principle, agents act in the most adequate way according to the objective situation. It is an idealized conception of human behavior which he used to drive his model of situational analysis. Cognitive scientist Allen Newell elaborated on the principle in his account of knowledge level modeling. == Popper == Popper called for social science to be grounded in what he called situational analysis or situational logic. This requires building models of social situations which include individual actors and their relationship to social institutions, e.g. markets, legal codes, bureaucracies, etc. These models attribute certain aims and information to the actors. This forms the 'logic of the situation', the result of reconstructing meticulously all circumstances of an historical event. The 'principle of rationality' is the assumption that people are instrumental in trying to reach their goals, and this is what drives the model. Popper believed that this model could be continuously refined to approach the objective truth. Popper called his principle of rationality nearly empty (a technical term meaning without empirical content) and strictly speaking false, but nonetheless tremendously useful. These remarks earned him a lot of criticism because seemingly he had swerved from his famous Logic of Scientific Discovery. Among the many philosophers having discussed Popper's principle of rationality from the 1960s up to now are Noretta Koertge, R. Nadeau, Viktor J. Vanberg, Hans Albert, E. Matzner, Ian C. Jarvie, Mark A. Notturno, John Wettersten, Ian C. Böhm. == Newell == In the context of knowledge-based systems, Newell (in 1982) proposed the following principle of rationality: "If an agent has knowledge that one of its actions will lead to one of its goals, then the agent will select that action." This principle is employed by agents at the knowledge level to move closer to a desired goal. An important philosophical difference between Newell and Popper is that Newell argued that the knowledge level is real in the sense that it exists in nature and is not made up. This allowed Newell to treat the rationality principle as a way of understanding nature and avoid the problems Popper ran into by treating knowledge as non physical and therefore non empirical.

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  AIOps

  AIOps (Artificial Intelligence for IT Operations) refers to the use of artificial intelligence, machine learning, and big data analytics to automate and enhance data center management. It helps organizations manage complex IT environments by detecting, diagnosing, and resolving issues more efficiently than traditional methods. == History == AIOps was first defined by Gartner in 2016, combining "artificial intelligence" and "IT operations" to describe the application of AI and machine learning to enhance IT operations. This concept was introduced to address the increasing complexity and data volume in IT environments, aiming to automate processes such as event correlation, anomaly detection, and causality determination. == Definition == AIOps refers to multi-layered, complex technology platforms that enhance and automate IT operations by using machine learning and analytics to analyze the large amounts of data collected from various DevOps devices and tools, automatically identifying and responding to issues in real-time. AIOps represents a shift from isolated IT data to aggregated observational data (e.g., job logs and monitoring systems) and interaction data (such as ticketing, events, or incident records) within a big data platform. AIOps applies machine learning and analytics to this data, resulting in continuous visibility that, when combined with automation, can lead to ongoing improvements. AIOps connects three IT disciplines (automation, service management, and performance management) to achieve continuous visibility and improvement. This new approach in modern, accelerated, and hyper-scaled IT environments leverages advances in machine learning and big data to overcome previous limitations. == Components == AIOps includes, but is not limited to, the following processes and techniques: Anomaly Detection Log Analysis Root Cause Analysis Cohort Analysis Event Correlation Predictive Analytics Hardware Failure Prediction Automated Remediation Performance Prediction Incident Management Causality Determination Queue Management Resource Scheduling and Optimization Predictive Capacity Management Resource Allocation Service Quality Monitoring Deployment and Integration Testing System Configuration Auto-diagnosis and Problem Localization Efficient ML Training and Inferencing Using LLMs for Cloud Ops Auto Service Healing Data Center Management Customer Support Security and Privacy in Cloud Operations == Comparison with DevOps == AIOps is increasingly compared with DevOps in terms of impact on operational efficiency. While DevOps focuses on collaboration between development and operations teams to accelerate software delivery, AIOps integrates artificial intelligence to enhance monitoring, automation, and predictive capabilities. Various industry analyses have explored the similarities and differences between the two approaches, including discussions on how organizations can combine them to improve incident management and resource optimization. == Results == AI optimizes IT operations in five ways: First, intelligent monitoring powered by AI helps identify potential issues before they cause outages, improving metrics like Mean Time to Detect (MTTD) by 15-20%. Second, performance data analysis and insights enable quick decision-making by ingesting and analyzing large data sets in real time. Third, AI-driven automated infrastructure optimization efficiently allocates resources and thereby reducing cloud costs. Fourth, enhanced IT service management reduces critical incidents by over 50% through AI-driven end-to-end service management. Lastly, intelligent task automation accelerates problem resolution and automates remedial actions with minimal human intervention. In 2025, Atera Networks was identified as a leader in AIOps by the software review platform G2. == AIOps vs. MLOps == AIOps tools use big data analytics, machine learning algorithms, and predictive analytics to detect anomalies, correlate events, and provide proactive insights. This automation reduces the burden on IT teams, allowing them to focus on strategic tasks rather than routine operational issues. AIOps is widely used by IT operations teams, DevOps, network administrators, and IT service management (ITSM) teams to enhance visibility and enable quicker incident resolution in hybrid cloud environments, data centers, and other IT infrastructures. In contrast to MLOps (Machine Learning Operations), which focuses on the lifecycle management and operational aspects of machine learning models, AIOps focuses on optimizing IT operations using a variety of analytics and AI-driven techniques. While both disciplines rely on AI and data-driven methods, AIOps primarily targets IT operations, whereas MLOps is concerned with the deployment, monitoring, and maintenance of ML models. == Conferences == There are several conferences that are specific to AIOps: AIOps Summit AI Dev Summit IBM Think conference

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  Solomonoff's theory of inductive inference

  Solomonoff's theory of inductive inference proves that, under its common sense assumptions (axioms), the best possible scientific model is the shortest algorithm that generates the empirical data under consideration. In addition to the choice of data, other assumptions are that, to avoid the post-hoc fallacy, the programming language must be chosen prior to the data and that the environment being observed is generated by an unknown algorithm. This is also called a theory of induction. Due to its basis in the dynamical (state-space model) character of Algorithmic Information Theory, it encompasses statistical as well as dynamical information criteria for model selection. It was introduced by Ray Solomonoff, based on probability theory and theoretical computer science. In essence, Solomonoff's induction derives the posterior probability of any computable theory, given a sequence of observed data. This posterior probability is derived from Bayes' rule and some universal prior, that is, a prior that assigns a positive probability to any computable theory. Solomonoff proved that this induction is incomputable (or more precisely, lower semi-computable), but noted that "this incomputability is of a very benign kind", and that it "in no way inhibits its use for practical prediction" (as it can be approximated from below more accurately with more computational resources). It is only "incomputable" in the benign sense that no scientific consensus is able to prove that the best current scientific theory is the best of all possible theories. However, Solomonoff's theory does provide an objective criterion for deciding among the current scientific theories explaining a given set of observations. Solomonoff's induction naturally formalizes Occam's razor by assigning larger prior credences to theories that require a shorter algorithmic description. == Origin == === Philosophical === The theory is based in philosophical foundations, and was founded by Ray Solomonoff around 1960. It is a mathematically formalized combination of Occam's razor and the Principle of Multiple Explanations. All computable theories which perfectly describe previous observations are used to calculate the probability of the next observation, with more weight put on the shorter computable theories. Marcus Hutter's universal artificial intelligence builds upon this to calculate the expected value of an action. === Principle === Solomonoff's induction has been argued to be the computational formalization of pure Bayesianism. To understand, recall that Bayesianism derives the posterior probability P [ T | D ] {\displaystyle \mathbb {P} [T|D]} of a theory T {\displaystyle T} given data D {\displaystyle D} by applying Bayes rule, which yields P [ T | D ] = P [ D | T ] P [ T ] P [ D | T ] P [ T ] + ∑ A ≠ T P [ D | A ] P [ A ] {\displaystyle \mathbb {P} [T|D]={\frac {\mathbb {P} [D|T]\mathbb {P} [T]}{\mathbb {P} [D|T]\mathbb {P} [T]+\sum _{A\neq T}\mathbb {P} [D|A]\mathbb {P} [A]}}} where theories A {\displaystyle A} are alternatives to theory T {\displaystyle T} . For this equation to make sense, the quantities P [ D | T ] {\displaystyle \mathbb {P} [D|T]} and P [ D | A ] {\displaystyle \mathbb {P} [D|A]} must be well-defined for all theories T {\displaystyle T} and A {\displaystyle A} . In other words, any theory must define a probability distribution over observable data D {\displaystyle D} . Solomonoff's induction essentially boils down to demanding that all such probability distributions be computable. Interestingly, the set of computable probability distributions is a subset of the set of all programs, which is countable. Similarly, the sets of observable data considered by Solomonoff were finite. Without loss of generality, we can thus consider that any observable data is a finite bit string. As a result, Solomonoff's induction can be defined by only invoking discrete probability distributions. Solomonoff's induction then allows to make probabilistic predictions of future data F {\displaystyle F} , by simply obeying the laws of probability. Namely, we have P [ F | D ] = E T [ P [ F | T , D ] ] = ∑ T P [ F | T , D ] P [ T | D ] {\displaystyle \mathbb {P} [F|D]=\mathbb {E} _{T}[\mathbb {P} [F|T,D]]=\sum _{T}\mathbb {P} [F|T,D]\mathbb {P} [T|D]} . This quantity can be interpreted as the average predictions P [ F | T , D ] {\displaystyle \mathbb {P} [F|T,D]} of all theories T {\displaystyle T} given past data D {\displaystyle D} , weighted by their posterior credences P [ T | D ] {\displaystyle \mathbb {P} [T|D]} . === Mathematical === The proof of the "razor" is based on the known mathematical properties of a probability distribution over a countable set. These properties are relevant because the infinite set of all programs is a denumerable set. The sum S of the probabilities of all programs must be exactly equal to one (as per the definition of probability) thus the probabilities must roughly decrease as we enumerate the infinite set of all programs, otherwise S will be strictly greater than one. To be more precise, for every ϵ {\displaystyle \epsilon } > 0, there is some length l such that the probability of all programs longer than l is at most ϵ {\displaystyle \epsilon } . This does not, however, preclude very long programs from having very high probability. Fundamental ingredients of the theory are the concepts of algorithmic probability and Kolmogorov complexity. The universal prior probability of any prefix p of a computable sequence x is the sum of the probabilities of all programs (for a universal computer) that compute something starting with p. Given some p and any computable but unknown probability distribution from which x is sampled, the universal prior and Bayes' theorem can be used to predict the yet unseen parts of x in optimal fashion. == Mathematical guarantees == === Solomonoff's completeness === The remarkable property of Solomonoff's induction is its completeness. In essence, the completeness theorem guarantees that the expected cumulative errors made by the predictions based on Solomonoff's induction are upper-bounded by the Kolmogorov complexity of the (stochastic) data generating process. The errors can be measured using the Kullback–Leibler divergence or the square of the difference between the induction's prediction and the probability assigned by the (stochastic) data generating process. === Solomonoff's uncomputability === Unfortunately, Solomonoff also proved that Solomonoff's induction is uncomputable. In fact, he showed that computability and completeness are mutually exclusive: any complete theory must be uncomputable. The proof of this is derived from a game between the induction and the environment. Essentially, any computable induction can be tricked by a computable environment, by choosing the computable environment that negates the computable induction's prediction. This fact can be regarded as an instance of the no free lunch theorem. == Modern applications == === Artificial intelligence === Though Solomonoff's inductive inference is not computable, several AIXI-derived algorithms approximate it in order to make it run on a modern computer. The more computing power they are given, the closer their predictions are to the predictions of inductive inference (their mathematical limit is Solomonoff's inductive inference). Another direction of inductive inference is based on E. Mark Gold's model of learning in the limit from 1967 and has developed since then more and more models of learning. The general scenario is the following: Given a class S of computable functions, is there a learner (that is, recursive functional) which for any input of the form (f(0),f(1),...,f(n)) outputs a hypothesis (an index e with respect to a previously agreed on acceptable numbering of all computable functions; the indexed function may be required consistent with the given values of f). A learner M learns a function f if almost all its hypotheses are the same index e, which generates the function f; M learns S if M learns every f in S. Basic results are that all recursively enumerable classes of functions are learnable while the class REC of all computable functions is not learnable. Many related models have been considered and also the learning of classes of recursively enumerable sets from positive data is a topic studied from Gold's pioneering paper in 1967 onwards. A far reaching extension of the Gold’s approach is developed by Schmidhuber's theory of generalized Kolmogorov complexities, which are kinds of super-recursive algorithms.

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  List of security hacking incidents

  This list of security hacking incidents covers important or noteworthy events in the history of security hacking and cracking. == 1900 == === 1903 === Magician and inventor Nevil Maskelyne disrupts John Ambrose Fleming's public demonstration of Guglielmo Marconi's purportedly secure wireless telegraphy technology, sending insulting Morse code messages through the auditorium's projector. == 1930s == === 1932 === Polish cryptologists Marian Rejewski, Henryk Zygalski and Jerzy Różycki broke the Enigma machine code. === 1939 === Alan Turing, Gordon Welchman and Harold Keen worked together to develop the codebreaking device Bombe (based off of Rejewski's work on Bomba). The Enigma machine's use of a reliably small key space makes it vulnerable to brute force attacks. == 1940s == === 1943 === René Carmille, comptroller general of the Vichy French Army, hacked the punch card system used by the Nazis to locate Jews. === 1949 === The theory that underlies computer viruses was first made public in 1949, when computer pioneer John von Neumann presented a paper titled "Theory and Organization of Complicated Automata". In the paper, von Neumann speculated that computer programs could reproduce themselves. == 1950s == === 1955 === At MIT, "hack" first came to mean playing with machines. An April 1955 meeting of the Tech Model Railroad Club has one say that "Mr. Eccles requests that anyone working or hacking on the electrical system turn the power off to avoid fuse blowing." === 1957 === Joe "Joybubbles" Engressia, a blind seven-year-old boy with perfect pitch, discovered that whistling the fourth E above middle C (a frequency of 2600 Hz) would interfere with AT&T's automated telephone systems, thereby inadvertently opening the door for phreaking. == 1960s == Various phreaking boxes are used to interact with automated telephone systems. === 1963 === The first ever reference to malicious hacking is 'phreaking' in MIT's student newspaper, The Tech, containing hackers tying up the lines with Harvard, configuring the PDP-1 to make free calls, war dialing and accumulating large phone bills. === 1965 === William D. Mathews from MIT finds a vulnerability in a CTSS running on an IBM 7094. The standard text editor on the system was designed to be used by one user at a time, working in one directory, and so it created a temporary file with a constant name for all instances of the editor. The flaw was discovered when two system programmers were editing at the same time and the temporary files for the message of the day and the password file became swapped, causing the contents of the system CTSS password file to display to any user logging into the system. === 1967 === The first known incidence of network penetration hacking took place when members of a computer club at a suburban Chicago high school were provided access to IBM's APL network. In the fall of 1967, IBM (through Science Research Associates) approached Evanston Township High School with the offer of four 2741 Selectric teletypewriter-based terminals with dial-up modem connectivity to an experimental computer system which implemented an early version of the APL programming language. The APL network system was structured into workspaces which were assigned to various clients using the system. Working independently, the students quickly learned the language and the system. They were free to explore the system, often using existing code available in public workspaces as models for their own creations. Eventually, curiosity drove the students to explore the system's wider context. This first informal network penetration effort was later acknowledged as helping harden the security of one of the first publicly accessible networks:Science Research Associates undertook to write a full APL system for the IBM 1500. They modeled their system after APL/360, which had by that time been developed and seen substantial use inside of IBM, using code borrowed from MAT/1500 where possible. In their documentation, they acknowledge their gratitude to "a number of high school students for their compulsion to bomb the system". This was an early example of a kind of sportive, but very effective, debugging that was often repeated in the evolution of APL systems. == 1970s == === 1971 === John T. Draper (later nicknamed Captain Crunch), his friend Joe Engressia (also known as Joybubbles), and blue box phone phreaking hit the news with an Esquire magazine feature story. === 1979 === Kevin Mitnick breaks into his first major computer system, the Ark, which was the computer system Digital Equipment Corporation (DEC) used for developing their RSTS/E operating system software. == 1980s == === 1980 === The FBI investigates a breach of security at National CSS (NCSS). The New York Times, reporting on the incident in 1981, describes hackers as: Technical experts, skilled, often young, computer programmers who almost whimsically probe the defenses of a computer system, searching out the limits and the possibilities of the machine. Despite their seemingly subversive role, hackers are a recognized asset in the computer industry, often highly prized. The newspaper describes white hat activities as part of a "mischievous but perversely positive 'hacker' tradition". When a National CSS employee revealed the existence of his password cracker, which he had used on customer accounts, the company chastised him not for writing the software but for not disclosing it sooner. The letter of reprimand stated that "The Company realizes the benefit to NCSS and in fact encourages the efforts of employees to identify security weaknesses to the VP, the directory, and other sensitive software in files". === 1981 === Chaos Computer Club forms in Germany. Ian Murphy, aka Captain Zap, was the first cracker to be tried and convicted as a felon. Murphy broke into AT&T's computers in 1981 and changed the internal clocks that metered billing rates. People were getting late-night discount rates when they called at midday. Of course, the bargain-seekers who waited until midnight to call long distance were hit with high bills. === 1983 === The 414s break into 60 computer systems at institutions ranging from the Los Alamos National Laboratory to Manhattan's Memorial Sloan-Kettering Cancer Center. The incident appeared as the cover story of Newsweek with the title "Beware: Hackers at play". As a result, the U.S. House of Representatives held hearings on computer security and passed several laws. The group KILOBAUD is formed in February, kicking off a series of other hacker groups that formed soon after. The movie WarGames introduces the wider public to the phenomenon of hacking and creates a degree of mass paranoia about hackers and their supposed abilities to bring the world to a screeching halt by launching nuclear ICBMs. The U.S. House of Representatives begins hearings on computer security hacking. In his Turing Award lecture, Ken Thompson mentions "hacking" and describes a security exploit that he calls a "Trojan horse". === 1984 === Someone calling himself Lex Luthor founds the Legion of Doom. Named after a Saturday morning cartoon, the LOD had the reputation of attracting "the best of the best"—until one of the most talented members called Phiber Optik feuded with Legion of Doomer Erik Bloodaxe and got 'tossed out of the clubhouse'. Phiber's friends formed a rival group, the Masters of Deception. The Comprehensive Crime Control Act gives the Secret Service jurisdiction over computer fraud. The Cult of the Dead Cow forms in Lubbock, Texas, and begins publishing its underground ezine. The hacker magazine 2600 begins regular publication, right when TAP was putting out its final issue. The editor of 2600, "Emmanuel Goldstein" (whose real name is Eric Corley), takes his handle from the leader of the resistance in George Orwell's Nineteen Eighty-Four. The publication provides tips for would-be hackers and phone phreaks, as well as commentary on the hacker issues of the day. Today, copies of 2600 are sold at most large retail bookstores. The Chaos Communication Congress, the annual European hacker conference organized by the Chaos Computer Club, is held in Hamburg, Germany. William Gibson's groundbreaking science fiction novel Neuromancer, about "Case", a futuristic computer hacker, is published. Considered the first major cyberpunk novel, it brought into hacker jargon such terms as "cyberspace", "the matrix", "simstim", and "ICE". === 1985 === KILOBAUD is re-organized into P.H.I.R.M. and begins sysopping hundreds of bulletin board systems (BBSs) throughout the United States, Canada, and Europe. The online 'zine Phrack is established. The Hacker's Handbook is published in the UK. The FBI, Secret Service, Middlesex County NJ Prosecutor's Office and various local law enforcement agencies execute seven search warrants concurrently across New Jersey on July 12, 1985, seizing equipment from BBS operators and users alike for "complicity in computer theft", under a n

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  Inauthentic text

  An inauthentic text is a computer-generated expository document meant to appear as genuine, but which is actually meaningless. Frequently they are created in order to be intermixed with genuine documents and thus manipulate the results of search engines, as with Spam blogs. They are also carried along in email in order to fool spam filters by giving the spam the superficial characteristics of legitimate text. Sometimes nonsensical documents are created with computer assistance for humorous effect, as with Dissociated press or Flarf poetry. They have also been used to challenge the veracity of a publication—MIT students submitted papers generated by a computer program called SCIgen to a conference, where they were initially accepted. This led the students to claim that the bar for submissions was too low. With the amount of computer generated text outpacing the ability of people to humans to curate it, there needs some means of distinguishing between the two. Yet automated approaches to determining absolutely whether a text is authentic or not face intrinsic challenges of semantics. Noam Chomsky coined the phrase "Colorless green ideas sleep furiously" giving an example of grammatically correct, but semantically incoherent sentence; some will point out that in certain contexts one could give this sentence (or any phrase) meaning. The first group to use the expression in this regard can be found below from Indiana University. Their work explains in detail an attempt to detect inauthentic texts and identify pernicious problems of inauthentic texts in cyberspace. The site has a means of submitting text that assesses, based on supervised learning, whether a corpus is inauthentic or not. Many users have submitted incorrect types of data and have correspondingly commented on the scores. This application is meant for a specific kind of data; therefore, submitting, say, an email, will not return a meaningful score.

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  Inductive bias

  The inductive bias (also known as learning bias) of a learning algorithm is the set of assumptions that the learner uses to predict outputs of given inputs that it has not encountered. Inductive bias is anything which makes the algorithm learn one pattern instead of another pattern (e.g., step-functions in decision trees instead of continuous functions in linear regression models). Learning involves searching a space of solutions for a solution that provides a good explanation of the data. However, in many cases, there may be multiple equally appropriate solutions. An inductive bias allows a learning algorithm to prioritize one solution (or interpretation) over another, independently of the observed data. In machine learning, the aim is to construct algorithms that are able to learn to predict a certain target output. To achieve this, the learning algorithm is presented some training examples that demonstrate the intended relation of input and output values. Then the learner is supposed to approximate the correct output, even for examples that have not been shown during training. Without any additional assumptions, this problem cannot be solved since unseen situations might have an arbitrary output value. The kind of necessary assumptions about the nature of the target function are subsumed in the phrase inductive bias. A classical example of an inductive bias is Occam's razor, assuming that the simplest consistent hypothesis about the target function is actually the best. Here, consistent means that the hypothesis of the learner yields correct outputs for all of the examples that have been given to the algorithm. Approaches to a more formal definition of inductive bias are based on mathematical logic. Here, the inductive bias is a logical formula that, together with the training data, logically entails the hypothesis generated by the learner. However, this strict formalism fails in many practical cases in which the inductive bias can only be given as a rough description (e.g., in the case of artificial neural networks), or not at all. == Types == The following is a list of common inductive biases in machine learning algorithms. Maximum conditional independence: if the hypothesis can be cast in a Bayesian framework, try to maximize conditional independence. This is the bias used in the Naive Bayes classifier. Minimum cross-validation error: when trying to choose among hypotheses, select the hypothesis with the lowest cross-validation error. Although cross-validation may seem to be free of bias, the "no free lunch" theorems show that cross-validation must be biased, for example assuming that there is no information encoded in the ordering of the data. Maximum margin: when drawing a boundary between two classes, attempt to maximize the width of the boundary. This is the bias used in support vector machines. The assumption is that distinct classes tend to be separated by wide boundaries. Minimum description length: when forming a hypothesis, attempt to minimize the length of the description of the hypothesis. Minimum features: unless there is good evidence that a feature is useful, it should be deleted. This is the assumption behind feature selection algorithms. Nearest neighbors: assume that most of the cases in a small neighborhood in feature space belong to the same class. Given a case for which the class is unknown, guess that it belongs to the same class as the majority in its immediate neighborhood. This is the bias used in the k-nearest neighbors algorithm. The assumption is that cases that are near each other tend to belong to the same class. == Shift of bias == Although most learning algorithms have a static bias, some algorithms are designed to shift their bias as they acquire more data. This does not avoid bias, since the bias shifting process itself must have a bias.

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  Spreading activation

  Spreading activation is a method for searching associative networks, biological and artificial neural networks, or semantic networks. The search process is initiated by labeling a set of source nodes (e.g. concepts in a semantic network) with weights or "activation" and then iteratively propagating or "spreading" that activation out to other nodes linked to the source nodes. Most often these "weights" are real values that decay as activation propagates through the network. When the weights are discrete this process is often referred to as marker passing. Activation may originate from alternate paths, identified by distinct markers, and terminate when two alternate paths reach the same node. However brain studies show that several different brain areas play an important role in semantic processing. Spreading activation in semantic networks as a model were invented in cognitive psychology to model the fan out effect. Spreading activation can also be applied in information retrieval, by means of a network of nodes representing documents and terms contained in those documents. == Cognitive psychology == As it relates to cognitive psychology, spreading activation is the theory of how the brain iterates through a network of associated ideas to retrieve specific information. The spreading activation theory presents the array of concepts within our memory as cognitive units, each consisting of a node and its associated elements or characteristics, all connected together by edges. A spreading activation network can be represented schematically, in a sort of web diagram with shorter lines between two nodes meaning the ideas are more closely related and will typically be associated more quickly to the original concept. In memory psychology, the spreading activation model holds that people organize their knowledge of the world based on their personal experiences, which in turn form the network of ideas that is the person's knowledge of the world. When a word (the target) is preceded by an associated word (the prime) in word recognition tasks, participants seem to perform better in the amount of time that it takes them to respond. For instance, subjects respond faster to the word "doctor" when it is preceded by "nurse" than when it is preceded by an unrelated word like "carrot". This semantic priming effect with words that are close in meaning within the cognitive network has been seen in a wide range of tasks given by experimenters, ranging from sentence verification to lexical decision and naming. As another example, if the original concept is "red" and the concept "vehicles" is primed, they are much more likely to say "fire engine" instead of something unrelated to vehicles, such as "cherries". If instead "fruits" was primed, they would likely name "cherries" and continue on from there. The activation of pathways in the network has everything to do with how closely linked two concepts are by meaning, as well as how a subject is primed. == Algorithm == A directed graph is populated by Nodes[ 1...N ] each having an associated activation value A [ i ] which is a real number in the range [0.0 ... 1.0]. A Link[ i, j ] connects source node[ i ] with target node[ j ]. Each edge has an associated weight W [ i, j ] usually a real number in the range [0.0 ... 1.0]. Parameters: Firing threshold F, a real number in the range [0.0 ... 1.0] Decay factor D, a real number in the range [0.0 ... 1.0] Steps: Initialize the graph setting all activation values A [ i ] to zero. Set one or more origin nodes to an initial activation value greater than the firing threshold F. A typical initial value is 1.0. For each unfired node [ i ] in the graph having an activation value A [ i ] greater than the node firing threshold F: For each Link [ i, j ] connecting the source node [ i ] with target node [ j ], adjust A [ j ] = A [ j ] + (A [ i ] W [ i, j ] D) where D is the decay factor. If a target node receives an adjustment to its activation value so that it would exceed 1.0, then set its new activation value to 1.0. Likewise maintain 0.0 as a lower bound on the target node's activation value should it receive an adjustment to below 0.0. Once a node has fired it may not fire again, although variations of the basic algorithm permit repeated firings and loops through the graph. Nodes receiving a new activation value that exceeds the firing threshold F are marked for firing on the next spreading activation cycle. If activation originates from more than one node, a variation of the algorithm permits marker passing to distinguish the paths by which activation is spread over the graph The procedure terminates when either there are no more nodes to fire or in the case of marker passing from multiple origins, when a node is reached from more than one path. Variations of the algorithm that permit repeated node firings and activation loops in the graph, terminate after a steady activation state, with respect to some delta, is reached, or when a maximum number of iterations is exceeded. == Examples ==

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  Wadhwani Institute for Artificial Intelligence

  Wadhwani AI, based in Mumbai, Maharashtra, is an independent, non-profit institute. Founded in 2018, it is dedicated to developing Artificial intelligence solutions for social good. Their mission is to build AI-based innovations and solutions for underserved communities in developing countries, for a wide range of domains including agriculture, education, financial inclusion, healthcare, and infrastructure. == History and funding == The institute was founded with a $30 million philanthropic effort by the Wadhwani brothers, Romesh Wadhwani and Sunil Wadhwani. The institute was inaugurated and dedicated to the nation by Narendra Modi, the 14th Prime Minister of India. In 2019, the institute received a $2 million grant from Google.org to create technologies to help reduce crop losses in cotton farming, through integrated pest management. The United States Agency for International Development awarded $2 million to the institute in 2020 to develop tools, using mathematical modeling techniques and digital technologies such as artificial intelligence and machine learning, to forecast COVID-19 disease patterns, estimate resources needed, and plan interventions. == Collaboration == With assistance from Google, the Ministry of Agriculture and Farmers' Welfare and the Wadhwani AI developed Krishi 24/7, the first AI-powered automated agricultural news monitoring and analysis tool. Through better decision-making, Krishi 24/7 will support the identification of valuable news, provide timely notifications, and respond quickly to safeguard farmers' interests and advance sustainable agricultural growth. The application converts news articles into English after scanning them in several languages. It ensures that the ministry is informed in a timely manner about pertinent occurrences that are published online by extracting key information from news items, including the headline, crop name, event type, date, location, severity, summary, and source link. The National Center for Disease Control has effectively implemented a comparable automated surveillance and analysis tool for disease outbreaks.

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  The 2028 Global Intelligence Crisis

  The 2028 Global Intelligence Crisis is a report authored by James van Geelen and Alap Shah and published by Citrini Research in February 2026, on the impact of artificial intelligence on humanity's future. Written in the form of a scenario analysis, it was viewed millions of times online and reportedly caused a fall in the stock market prices of major tech and financial firms. It also received criticism among others, for its allegedly flawed economic logic. The 'thought exercise', as the authors called it, painted a gloomy picture for the near future, where outputs keep growing while consumer's ability to spend collapses. "...driven by ai agents that don’t sleep, take sick days or require health insurance”, "outputs that are shown in national accounts increases, "but never circulates through the real economy"(which the report calls 'Ghost GDP'), the authors argued. In other words, the authors predict a scenario where the owners of the AI firms will accumulate a vast fortune but there will be scant demand from consumers as AI would cause massive unemployment. The authors caution the reader that what they make is a scenario and not a prediction. In the scenario they visualise, any service whose value proposition is “I will navigate complexity that you find tedious” is getting disrupted. The reports argues that the unique ability of human beings to analyse, decide, create, persuade, and coordinate was “the thing that could not be replicated at scale,” and call the historical scarcity of this precious entity 'friction'. When this friction becomes zero, a gamut of changes occur which then triggers a cascading of changes across the economy. ”Travel booking platforms are an early casualty; Financial advice. tax prep., and routine legal work follow suit. National unemployment rate go as high 10.2% and the S&P 500 goes for a massive 38% peak-to-trough crash. In contrast to the previous technological revolutions the high-earning professionals suffers more and get forced to take up roles in the gig economy. Labour supply becomes abundant and this cuts wages all across the economy. The dent in income for the employees then affects other sectors of the economy such as the residential mortgage market. The losses for the software companies triggers loan defaults and heralds peril for the private credit sector.

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  Granular computing

  Granular computing is an emerging computing paradigm of information processing that concerns the processing of complex information entities called "information granules", which arise in the process of data abstraction and derivation of knowledge from information or data. Generally speaking, information granules are collections of entities that usually originate at the numeric level and are arranged together due to their similarity, functional or physical adjacency, indistinguishability, coherency, or the like. At present, granular computing is more a theoretical perspective than a coherent set of methods or principles. As a theoretical perspective, it encourages an approach to data that recognizes and exploits the knowledge present in data at various levels of resolution or scales. In this sense, it encompasses all methods which provide flexibility and adaptability in the resolution at which knowledge or information is extracted and represented. == Types of granulation == As mentioned above, granular computing is not an algorithm or process; there is no particular method that is called "granular computing". It is rather an approach to looking at data that recognizes how different and interesting regularities in the data can appear at different levels of granularity, much as different features become salient in satellite images of greater or lesser resolution. On a low-resolution satellite image, for example, one might notice interesting cloud patterns representing cyclones or other large-scale weather phenomena, while in a higher-resolution image, one misses these large-scale atmospheric phenomena but instead notices smaller-scale phenomena, such as the interesting pattern that is the streets of Manhattan. The same is generally true of all data: At different resolutions or granularities, different features and relationships emerge. The aim of granular computing is to try to take advantage of this fact in designing more effective machine-learning and reasoning systems. There are several types of granularity that are often encountered in data mining and machine learning, and we review them below: === Value granulation (discretization/quantization) === One type of granulation is the quantization of variables. It is very common that in data mining or machine-learning applications the resolution of variables needs to be decreased in order to extract meaningful regularities. An example of this would be a variable such as "outside temperature" (temp), which in a given application might be recorded to several decimal places of precision (depending on the sensing apparatus). However, for purposes of extracting relationships between "outside temperature" and, say, "number of health-club applications" (club), it will generally be advantageous to quantize "outside temperature" into a smaller number of intervals. ==== Motivations ==== There are several interrelated reasons for granulating variables in this fashion: Based on prior domain knowledge, there is no expectation that minute variations in temperature (e.g., the difference between 80–80.7 °F (26.7–27.1 °C)) could have an influence on behaviors driving the number of health-club applications. For this reason, any "regularity" which our learning algorithms might detect at this level of resolution would have to be spurious, as an artifact of overfitting. By coarsening the temperature variable into intervals the difference between which we do anticipate (based on prior domain knowledge) might influence number of health-club applications, we eliminate the possibility of detecting these spurious patterns. Thus, in this case, reducing resolution is a method of controlling overfitting. By reducing the number of intervals in the temperature variable (i.e., increasing its grain size), we increase the amount of sample data indexed by each interval designation. Thus, by coarsening the variable, we increase sample sizes and achieve better statistical estimation. In this sense, increasing granularity provides an antidote to the so-called curse of dimensionality, which relates to the exponential decrease in statistical power with increase in number of dimensions or variable cardinality. Independent of prior domain knowledge, it is often the case that meaningful regularities (i.e., which can be detected by a given learning methodology, representational language, etc.) may exist at one level of resolution and not at another. For example, a simple learner or pattern recognition system may seek to extract regularities satisfying a conditional probability threshold such as p ( Y = y j | X = x i ) ≥ α . {\displaystyle p(Y=y_{j}|X=x_{i})\geq \alpha .} In the special case where α = 1 , {\displaystyle \alpha =1,} this recognition system is essentially detecting logical implication of the form X = x i → Y = y j {\displaystyle X=x_{i}\rightarrow Y=y_{j}} or, in words, "if X = x i , {\displaystyle X=x_{i},} then Y = y j {\displaystyle Y=y_{j}} ". The system's ability to recognize such implications (or, in general, conditional probabilities exceeding threshold) is partially contingent on the resolution with which the system analyzes the variables. As an example of this last point, consider the feature space shown to the right. The variables may each be regarded at two different resolutions. Variable X {\displaystyle X} may be regarded at a high (quaternary) resolution wherein it takes on the four values { x 1 , x 2 , x 3 , x 4 } {\displaystyle \{x_{1},x_{2},x_{3},x_{4}\}} or at a lower (binary) resolution wherein it takes on the two values { X 1 , X 2 } . {\displaystyle \{X_{1},X_{2}\}.} Similarly, variable Y {\displaystyle Y} may be regarded at a high (quaternary) resolution or at a lower (binary) resolution, where it takes on the values { y 1 , y 2 , y 3 , y 4 } {\displaystyle \{y_{1},y_{2},y_{3},y_{4}\}} or { Y 1 , Y 2 } , {\displaystyle \{Y_{1},Y_{2}\},} respectively. At the high resolution, there are no detectable implications of the form X = x i → Y = y j , {\displaystyle X=x_{i}\rightarrow Y=y_{j},} since every x i {\displaystyle x_{i}} is associated with more than one y j , {\displaystyle y_{j},} and thus, for all x i , {\displaystyle x_{i},} p ( Y = y j | X = x i ) < 1. {\displaystyle p(Y=y_{j}|X=x_{i})<1.} However, at the low (binary) variable resolution, two bilateral implications become detectable: X = X 1 ↔ Y = Y 1 {\displaystyle X=X_{1}\leftrightarrow Y=Y_{1}} and X = X 2 ↔ Y = Y 2 {\displaystyle X=X_{2}\leftrightarrow Y=Y_{2}} , since every X 1 {\displaystyle X_{1}} occurs iff Y 1 {\displaystyle Y_{1}} and X 2 {\displaystyle X_{2}} occurs iff Y 2 . {\displaystyle Y_{2}.} Thus, a pattern recognition system scanning for implications of this kind would find them at the binary variable resolution, but would fail to find them at the higher quaternary variable resolution. ==== Issues and methods ==== It is not feasible to exhaustively test all possible discretization resolutions on all variables in order to see which combination of resolutions yields interesting or significant results. Instead, the feature space must be preprocessed (often by an entropy analysis of some kind) so that some guidance can be given as to how the discretization process should proceed. Moreover, one cannot generally achieve good results by naively analyzing and discretizing each variable independently, since this may obliterate the very interactions that we had hoped to discover. A sample of papers that address the problem of variable discretization in general, and multiple-variable discretization in particular, is as follows: Chiu, Wong & Cheung (1991), Bay (2001), Liu et al. (2002), Wang & Liu (1998), Zighed, Rabaséda & Rakotomalala (1998), Catlett (1991), Dougherty, Kohavi & Sahami (1995), Monti & Cooper (1999), Fayyad & Irani (1993), Chiu, Cheung & Wong (1990), Nguyen & Nguyen (1998), Grzymala-Busse & Stefanowski (2001), Ting (1994), Ludl & Widmer (2000), Pfahringer (1995), An & Cercone (1999), Chiu & Cheung (1989), Chmielewski & Grzymala-Busse (1996), Lee & Shin (1994), Liu & Wellman (2002), Liu & Wellman (2004). === Variable granulation (clustering/aggregation/transformation) === Variable granulation is a term that could describe a variety of techniques, most of which are aimed at reducing dimensionality, redundancy, and storage requirements. We briefly describe some of the ideas here, and present pointers to the literature. ==== Variable transformation ==== A number of classical methods, such as principal component analysis, multidimensional scaling, factor analysis, and structural equation modeling, and their relatives, fall under the genus of "variable transformation." Also in this category are more modern areas of study such as dimensionality reduction, projection pursuit, and independent component analysis. The common goal of these methods in general is to find a representation of the data in terms of new variables, which are a linear or nonlinear transformation of the original variables, and in which important stati

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