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Mutual Recursion
In mathematics and computer science, mutual recursion is a form of recursion where two mathematical or computational objects, such as functions or datatypes, are defined in terms of each other. Mutual recursion is very common in functional programming and in some problem domains, such as recursive descent parsers, where the datatypes are naturally mutually recursive. Examples Datatypes The most important basic example of a datatype that can be defined by mutual recursion is a tree, which can be defined mutually recursively in terms of a forest (a list of trees). Symbolically: f: [1 ..., t[k t: v f A forest ''f'' consists of a list of trees, while a tree ''t'' consists of a pair of a value ''v'' and a forest ''f'' (its children). This definition is elegant and easy to work with abstractly (such as when proving theorems about properties of trees), as it expresses a tree in simple terms: a list of one type, and a pair of two types. Further, it matches many algorithms on trees, whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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NFCC
The PFF National Challenge Cup is an annual single-elimination tournament, knockout association football, football competition in men's domestic Football in Pakistan, Pakistani football within the Pakistan football league system. It is organized by and named after the Pakistan Football Federation. Initially named as ''Inter Departmental Championship'', it was introduced in 1979 as a football tournament for departmental selections and Pakistan Armed Forces, armed forces teams excluded from the National Football Championship (Pakistan), National Football Championship of Pakistan''.'' Khan Research Laboratories F.C., Khan Research Laboratories have won the most titles (six). WAPDA F.C., WAPDA are the current champions, winning the 2023–24 PFF National Challenge Cup, 2023–24 edition courtesy of a 1–0 win against SA Gardens FC, SA Gardens in the final. Background Introduced in 1979 to offer nationwide competition to departmental selections and Pakistan Armed Forces, armed force ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Scheme (programming Language)
Scheme is a dialect of the Lisp family of programming languages. Scheme was created during the 1970s at the MIT Computer Science and Artificial Intelligence Laboratory (MIT CSAIL) and released by its developers, Guy L. Steele and Gerald Jay Sussman, via a series of memos now known as the Lambda Papers. It was the first dialect of Lisp to choose lexical scope and the first to require implementations to perform tail-call optimization, giving stronger support for functional programming and associated techniques such as recursive algorithms. It was also one of the first programming languages to support first-class continuations. It had a significant influence on the effort that led to the development of Common Lisp.Common LISP: The Language, 2nd Ed., Guy L. Steele Jr. Digital Press; 1981. . "Common Lisp is a new dialect of Lisp, a successor to MacLisp, influenced strongly by ZetaLisp and to some extent by Scheme and InterLisp." The Scheme language is standardized in the offic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Peter Norvig
Peter Norvig (born 14 December 1956) is an American computer scientist and Distinguished Education Fellow at the Stanford Institute for Human-Centered AI. He previously served as a director of research and search quality at Google. Norvig is the co-author with Stuart J. Russell of the most popular textbook in the field of AI: '' Artificial Intelligence: A Modern Approach'' used in more than 1,500 universities in 135 countries. Early life and education Norvig grew up in an academic family. His father was Danish and came to the United States after World War II to study math at the University of Minnesota. Norvig received a Bachelor of Science in applied mathematics from Brown University and a Ph.D. in computer science from the University of California, Berkeley. Career and research Norvig is a councilor of the Association for the Advancement of Artificial Intelligence and co-author, with Stuart J. Russell, of '' Artificial Intelligence: A Modern Approach'', now the leading col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Prolog
Prolog is a logic programming language that has its origins in artificial intelligence, automated theorem proving, and computational linguistics. Prolog has its roots in first-order logic, a formal logic. Unlike many other programming languages, Prolog is intended primarily as a declarative programming language: the program is a set of facts and Horn clause, rules, which define Finitary relation, relations. A computation is initiated by running a ''query'' over the program. Prolog was one of the first logic programming languages and remains the most popular such language today, with several free and commercial implementations available. The language has been used for automated theorem proving, theorem proving, expert systems, term rewriting, type systems, and automated planning, as well as its original intended field of use, natural language processing. See also Watson (computer). Prolog is a Turing-complete, general-purpose programming language, which is well-suited for inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Meta-circular Evaluator
In computing, a meta-circular evaluator (MCE) or meta-circular interpreter (MCI) is an interpreter which defines each feature of the interpreted language using a similar facility of the interpreter's host language. For example, interpreting a lambda application may be implemented using function application. Meta-circular evaluation is most prominent in the context of Lisp. A self-interpreter is a meta-circular interpreter where the interpreted language is nearly identical to the host language; the two terms are often used synonymously. History The dissertation of Corrado Böhm describes the design of a self-hosting compiler. Due to the difficulty of compiling higher-order functions, many languages were instead defined via interpreters, most prominently Lisp. The term itself was coined by John C. Reynolds, and popularized through its use in the book '' Structure and Interpretation of Computer Programs''. Self-interpreters A self-interpreter is a meta-circular interpreter whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Programming Language
A programming language is a system of notation for writing computer programs. Programming languages are described in terms of their Syntax (programming languages), syntax (form) and semantics (computer science), semantics (meaning), usually defined by a formal language. Languages usually provide features such as a type system, Variable (computer science), variables, and mechanisms for Exception handling (programming), error handling. An Programming language implementation, implementation of a programming language is required in order to Execution (computing), execute programs, namely an Interpreter (computing), interpreter or a compiler. An interpreter directly executes the source code, while a compiler produces an executable program. Computer architecture has strongly influenced the design of programming languages, with the most common type (imperative languages—which implement operations in a specified order) developed to perform well on the popular von Neumann architecture. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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ML (programming Language)
ML (Meta Language) is a general-purpose, high-level, functional programming language. It is known for its use of the polymorphic Hindley–Milner type system, which automatically assigns the data types of most expressions without requiring explicit type annotations ( type inference), and ensures type safety; there is a formal proof that a well-typed ML program does not cause runtime type errors. ML provides pattern matching for function arguments, garbage collection, imperative programming, call-by-value and currying. While a general-purpose programming language, ML is used heavily in programming language research and is one of the few languages to be completely specified and verified using formal semantics. Its types and pattern matching make it well-suited and commonly used to operate on other formal languages, such as in compiler writing, automated theorem proving, and formal verification. Overview Features of ML include a call-by-value evaluation strategy, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Lisp (programming Language)
Lisp (historically LISP, an abbreviation of "list processing") is a family of programming languages with a long history and a distinctive, fully parenthesized prefix notation. Originally specified in the late 1950s, it is the second-oldest high-level programming language still in common use, after Fortran. Lisp has changed since its early days, and many dialects have existed over its history. Today, the best-known general-purpose Lisp dialects are Common Lisp, Scheme, Racket, and Clojure. Lisp was originally created as a practical mathematical notation for computer programs, influenced by (though not originally derived from) the notation of Alonzo Church's lambda calculus. It quickly became a favored programming language for artificial intelligence (AI) research. As one of the earliest programming languages, Lisp pioneered many ideas in computer science, including tree data structures, automatic storage management, dynamic typing, conditionals, higher-order function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Sierpiński Curve
Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit n \to \infty completely fill the unit square: thus their limit curve, also called the Sierpiński curve, is an example of a space-filling curve. Because the Sierpiński curve is space-filling, its Hausdorff dimension (in the limit n \to \infty ) is 2 . The Euclidean length of the nth iteration curve S_n is : \beginl_n&= (1+\sqrt 2) 2^n - (2-\sqrt 2) ,\\ &= \frac \sinh(n\log(2)+\mathrm(\frac))\end i.e., it grows ''exponentially'' with n beyond any limit, whereas the limit for n \to \infty of the area enclosed by S_n is 5/12 \, that of the square (in Euclidean metric). Uses of the curve The Sierpiński curve is useful in several practical applications because it is more symmetrical than other commonly studied space-filling curves. For example, it has been used as a basis for the rapid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Hofstadter Female And Male Sequences
In mathematics, a Hofstadter sequence is a member of a family of related integer sequences defined by non-linear recurrence relations. Sequences presented in ''Gödel, Escher, Bach: an Eternal Golden Braid'' The first Hofstadter sequences were described by Douglas Richard Hofstadter in his book ''Gödel, Escher, Bach''. In order of their presentation in chapter III on figures and background (Figure-Figure sequence) and chapter V on recursive structures and processes (remaining sequences), these sequences are: Hofstadter Figure-Figure sequences The Hofstadter Figure-Figure (R and S) sequences are a pair of complementary integer sequences defined as follows: : \begin R(1) &= 1,\ S(1) = 2; \\ R(n) &= R(n - 1) + S(n - 1), \quad n > 1, \end with the sequence S(n) defined as a strictly increasing series of positive integers not present in R(n). The first few terms of these sequences are : ''R'': 1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, 98, 114, 131, 150, 170, 191, 213, 236, 260,&nbs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Minimax
Minimax (sometimes Minmax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, combinatorial game theory, statistics, and philosophy for ''minimizing'' the possible loss function, loss for a Worst-case scenario, worst case (''max''imum loss) scenario. When dealing with gains, it is referred to as "maximin" – to maximize the minimum gain. Originally formulated for several-player zero-sum game theory, covering both the cases where players take alternate moves and those where they make simultaneous moves, it has also been extended to more complex games and to general decision-making in the presence of uncertainty. Game theory In general games The maximin value is the highest value that the player can be sure to get without knowing the actions of the other players; equivalently, it is the lowest value the other players can force the player to receive when they know the player's action. Its formal definition is: :\underline = \max_ \min_ W ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Coroutine
Coroutines are computer program components that allow execution to be suspended and resumed, generalizing subroutines for cooperative multitasking. Coroutines are well-suited for implementing familiar program components such as cooperative tasks, exceptions, event loops, iterators, infinite lists and pipes. They have been described as "functions whose execution you can pause". Melvin Conway coined the term ''coroutine'' in 1958 when he applied it to the construction of an assembly program. The first published explanation of the coroutine appeared later, in 1963. Definition and types There is no single precise definition of coroutine. In 1980 Christopher D. Marlin summarized two widely-acknowledged fundamental characteristics of a coroutine: # the values of data local to a coroutine persist between successive calls; # the execution of a coroutine is suspended as control leaves it, only to carry on where it left off when control re-enters the coroutine at some later s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |