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Modal μ-calculus
In theoretical computer science, the modal μ-calculus (Lμ, Lμ, sometimes just μ-calculus, although this can have a more general meaning) is an extension of propositional modal logic (with many modalities) by adding the least fixed point operator μ and the greatest fixed point operator ν, thus a fixed-point logic. The (propositional, modal) μ-calculus originates with Dana Scott and Jaco de Bakker, and was further developed by Dexter Kozen into the version most used nowadays. It is used to describe properties of labelled transition systems and for verifying these properties. Many temporal logics can be encoded in the μ-calculus, including CTL* and its widely used fragments—linear temporal logic and computational tree logic. An algebraic view is to see it as an algebra of monotonic functions over a complete lattice, with operators consisting of functional composition plus the least and greatest fixed point operators; from this viewpoint, the modal μ-calculus is o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with A Mathematical Theory of Communication, a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures. For instance, rather than considering groups or rings as the object of studythis is the subject of group theory and ring theory in universal algebra, the object of study is the possible types of algebraic structures and their relationships. Basic idea In universal algebra, an (or algebraic structure) is a set ''A'' together with a collection of operations on ''A''. Arity An ''n''- ary operation on ''A'' is a function that takes ''n'' elements of ''A'' and returns a single element of ''A''. Thus, a 0-ary operation (or ''nullary operation'') can be represented simply as an element of ''A'', or a '' constant'', often denoted by a letter like ''a''. A 1-ary operation (or '' unary operation'') is simply a function from ''A'' to ''A'', often denoted by a symbol placed in front of its argument, like ~'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambda Calculus
In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using variable Name binding, binding and Substitution (algebra), substitution. Untyped lambda calculus, the topic of this article, is a universal machine, a model of computation that can be used to simulate any Turing machine (and vice versa). It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. In 1936, Church found a formulation which was #History, logically consistent, and documented it in 1940. Lambda calculus consists of constructing #Lambda terms, lambda terms and performing #Reduction, reduction operations on them. A term is defined as any valid lambda calculus expression. In the simplest form of lambda calculus, terms are built using only the following rules: # x: A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multimodal Logic
A multimodal logic is a modal logic that has more than one primitive modal operator. They find substantial applications in theoretical computer science. Overview A modal logic with ''n'' primitive unary modal operators \Box_i, i\in \ is called an ''n''-modal logic. Given these operators and negation, one can always add \Diamond_i modal operators defined as \Diamond_i P if and only if \lnot \Box_i \lnot P, to give a classical multimodal logic if it is in addition stable under necessitation (or "possibilization", therefore) of both members of provable equivalences. Perhaps the first substantive example of a two-modal logic is Arthur Prior's tense logic, with two modalities, F and P, corresponding to "sometime in the future" and "sometime in the past". A logic with infinitely many modalities is dynamic logic, introduced by Vaughan Pratt in 1976 and having a separate modal operator for every regular expression. A version of temporal logic introduced in 1977 and intended for program v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Calculus
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contrast it with System F, but it should not be confused with first-order logic. It deals with propositions (which can be Truth value, true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives representing the truth functions of Logical conjunction, conjunction, Logical disjunction, disjunction, Material conditional, implication, Logical biconditional, biconditional, and negation. Some sources include other connectives, as in the table below. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or Quantifier (logic), quantifiers. However, all the machinery of pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Occurrence
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. A ''free variable'' is a notation (symbol) that specifies places in an expression where substitution may take place and is not a parameter of this or any container expression. The idea is related to a ''placeholder'' (a symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol. In computer programming, the term free variable refers to variables used in a function that are neither local variables nor parameters of that function. The term non-local variable is often a synonym in this context. An instance of a variable symbol is ''bound'', in contrast, if the value of that variable symbol has been bound to a specific value or range of valu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity Game
A parity game is played on a colored directed graph, where each node has been colored by a priority – one of (usually) finitely many natural numbers. Two players, 0 and 1, move a (single, shared) token along the edges of the graph. The owner of the node that the token falls on selects the successor node (does the next move). The players keep moving the token, resulting in a (possibly infinite) path, called a play. The winner of a finite play is the player whose opponent is unable to move. The winner of an infinite play is determined by the priorities appearing in the play. Typically, player 0 wins an infinite play if the largest priority that occurs infinitely often in the play is even. Player 1 wins otherwise. This explains the word "parity" in the title. Parity games lie in the third level of the Borel hierarchy, and are consequently determined. Games related to parity games were implicitly used in Rabin's proof of decidability of the monadic second-order theory of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Information
Perfect information is a concept in game theory and economics that describes a situation where all players in a game or all participants in a market have knowledge of all relevant information in the system. This is different than complete information, which implies Common knowledge (logic), common knowledge of each agent's utility functions, payoffs, strategies and "types". A system with perfect information may or may not have complete information. In economics this is sometimes described as "no hidden information" and is a feature of perfect competition. In a market with perfect information all consumers and producers would have complete and instantaneous knowledge of all market prices, their own utility and cost functions. In game theory, a sequential game has perfect information if each player, when making any decision, is perfectly informed of all the events that have previously occurred, including the "initialization event" of the game (e.g. the starting hands of each player ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-player Game
A two-player game is a multiplayer game that is played by precisely two players. This is distinct from a solitaire game, which is played by only one player. Examples The following are some examples of two-player games. This list is not intended to be exhaustive. * Board games: ** Chess ** Checkers ** Go ** Some wargames, such as '' Hammer of the Scots'' * Card games: ** Cribbage ** Whist ** Rummy ** 66 ** Pinochle ** '' Magic: The Gathering'', a collectible card game in which players duel * Sports: ** Cue sports, a family of games that use cue sticks and billiard balls ** Many athletic games, such as tennis ( singles) * Video games: **''Pong'' ** A Way Out See also * List of types of games * Zero-sum game Zero-sum game is a Mathematical model, mathematical representation in game theory and economic theory of a situation that involves two competition, competing entities, where the result is an advantage for one side and an equivalent loss for the o ... References {{Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Semantics
Game semantics is an approach to Formal semantics (logic), formal semantics that grounds the concepts of truth or Validity (logic), validity on Game theory, game-theoretic concepts, such as the existence of a winning strategy for a player. In this framework, logical formulas are interpreted as defining games between two players. The term encompasses several related but distinct traditions, including dialogical logic (developed by Paul Lorenzen and Kuno Lorenz in Germany starting in the 1950s) and game-theoretical semantics (developed by Jaakko Hintikka in Finland). Game semantics represents a significant departure from traditional Model theory, model-theoretic approaches by emphasizing the dynamic, interactive nature of logical reasoning rather than static truth assignments. It provides intuitive interpretations for various logical systems, including classical logic, intuitionistic logic, linear logic, and modal logic. The approach bears conceptual resemblances to ancient Socratic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Set Algebra
Power may refer to: Common meanings * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power, a type of energy * Power (social and political), the ability to influence people or events Mathematics, science and technology Computing * IBM POWER (software), an IBM operating system enhancement package * IBM POWER architecture, a RISC instruction set architecture * Power ISA, a RISC instruction set architecture derived from PowerPC * IBM Power microprocessors, made by IBM, which implement those RISC architectures * Power.org, a predecessor to the OpenPOWER Foundation Mathematics * Exponentiation, "''x'' to the power of ''y''" * Power function * Power of a point * Statistical power Physics * Magnification, the factor by which an optical system enlarges an image * Optical power, the degree to which a lens converges or diverges light Social sciences and politics * Economic power, encompassing several concepts that economists ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Composition
In mathematics, the composition operator \circ takes two functions, f and g, and returns a new function h(x) := (g \circ f) (x) = g(f(x)). Thus, the function is applied after applying to . (g \circ f) is pronounced "the composition of and ". Reverse composition, sometimes denoted f \mapsto g , applies the operation in the opposite order, applying f first and g second. Intuitively, reverse composition is a chaining process in which the output of function feeds the input of function . The composition of functions is a special case of the composition of relations, sometimes also denoted by \circ. As a result, all properties of composition of relations are true of composition of functions, such as associativity. Examples * Composition of functions on a finite set: If , and , then , as shown in the figure. * Composition of functions on an infinite set: If (where is the set of all real numbers) is given by and is given by , then: * If an airplane's altitude at time&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
