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P′′
P′′ (P double prime) is a primitive computer programming language created by Corrado BöhmBöhm, C.: "On a family of Turing machines and the related programming language", ICC Bull. 3, 185-194, July 1964.Böhm, C. and Jacopini, G.: "Flow diagrams, Turing machines and languages with only two formation rules", CACM 9(5), 1966. (Note: This is the most-cited paper on the structured program theorem.) in 1964 to describe a family of Turing machines. Definition \mathcal^ (hereinafter written P′′) is formally defined as a set of words on the four-instruction alphabet \, as follows: Syntax # R and \lambda are words in P′′. # If q_1 and q_2 are words in P′′, then q_1 q_2 is a word in P′′. # If q is a word in P′′, then (q) is a word in P′′. # Only words derivable from the previous three rules are words in P′′. Semantics * \ is the tape-alphabet of a Turing machine with left-infinite tape, \Box being the ''blank'' symbol, equivalent to c_0. * All instruction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corrado Böhm
Corrado Böhm (17 January 1923 – 23 October 2017) was an Italian computer scientist and Professor Emeritus at the Sapienza University of Rome, University of Rome "La Sapienza", known especially for his contributions to the theory of structured programming, constructive mathematics, combinatory logic, lambda calculus, and the semantics and implementation of functional programming languages. Work In his PhD dissertation (in Mathematics, at ETH Zurich, 1951; published in 1954), Böhm describes for the first time a full Böhm's language, meta-circular compiler, that is a translation mechanism of a programming language, written in that same language. His most influential contribution is the so-called structured program theorem, published in 1966 together with Giuseppe Jacopini. Together with Alessandro Berarducci, he demonstrated an isomorphism between the strictly-positive algebraic data types and the polymorphic lambda-terms, otherwise known as Böhm–Berarducci encoding. In the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brainfuck
Brainfuck is an esoteric programming language created in 1993 by Swiss student Urban Müller. Designed to be extremely minimalistic, the language consists of only eight simple commands, a data pointer, and an instruction pointer. Brainfuck is an example of a so-called Turing tarpit: it can be used to write any program, but it is not practical to do so because it provides so little abstraction that the programs get very long or complicated. While Brainfuck is fully Turing-complete, it is not intended for practical use but to challenge and amuse programmers. Brainfuck requires one to break down commands into small and simple instructions. The language takes its name from the slang term '' brainfuck'', which refers to things so complicated or unusual that they exceed the limits of one's understanding, as it was not meant or made for designing actual software but to challenge the boundaries of computer programming. Because the language's name contains profanity, many substitutes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structured Program Theorem
The structured program theorem, also called the Böhm–Jacopini theorem, is a result in programming language theory. It states that a class of control-flow graphs (historically called flowcharts in this context) can compute any computable function if it combines subprograms in only three specific ways ( control structures). These are #Executing one subprogram, and then another subprogram (sequence) #Executing one of two subprograms according to the value of a boolean expression (selection) #Repeatedly executing a subprogram as long as a boolean expression is true (iteration) The structured chart subject to these constraints, particularly the loop constraint implying a single exit (as described later in this article), may however use additional variables in the form of bits (stored in an extra integer variable in the original proof) in order to keep track of information that the original program represents by the program location. The construction was based on Böhm's programmin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijective Numeration
Bijective numeration is any numeral system in which every non-negative integer can be represented in exactly one way using a finite string of digits. The name refers to the bijection (i.e. one-to-one correspondence) that exists in this case between the set of non-negative integers and the set of finite strings using a finite set of symbols (the "digits"). Most ordinary numeral systems, such as the common decimal system, are not bijective because more than one string of digits can represent the same positive integer. In particular, adding leading zeroes does not change the value represented, so "1", "01" and "001" all represent the number one. Even though only the first is usual, the fact that the others are possible means that the decimal system is not bijective. However, the unary numeral system, with only one digit, ''is'' bijective. A bijective base-''k'' numeration is a bijective positional notation. It uses a string of digits from the set (where ''k'' ≥ 1) to encode ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imperative Programming
In computer science, imperative programming is a programming paradigm of software that uses Statement (computer science), statements that change a program's state (computer science), state. In much the same way that the imperative mood in natural languages expresses commands, an imperative program consists of command (computing), commands for the computer to perform. Imperative programming focuses on describing ''how'' a program operates step by step (with general order of the steps being determined in source code by the placement of statements one below the other), rather than on high-level descriptions of its expected results. The term is often used in contrast to declarative programming, which focuses on ''what'' the program should accomplish without specifying all the details of ''how'' the program should achieve the result. Procedural programming Procedural programming is a type of imperative programming in which the program is built from one or more procedures (also termed s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structured Programming
Structured programming is a programming paradigm aimed at improving the clarity, quality, and development time of a computer program by making specific disciplined use of the structured control flow constructs of selection ( if/then/else) and repetition ( while and for), block structures, and subroutines. It emerged in the late 1950s with the appearance of the ALGOL 58 and ALGOL 60 programming languages, with the latter including support for block structures. Contributing factors to its popularity and widespread acceptance, at first in academia and later among practitioners, include the discovery of what is now known as the structured program theorem in 1966, and the publication of the influential " Go To Statement Considered Harmful" open letter in 1968 by Dutch computer scientist Edsger W. Dijkstra, who coined the term "structured programming". Structured programming is most frequently used with deviations that allow for clearer programs in some particular cases, such as whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Models Of Computation
In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes how units of computations, memories, and communications are organized. The computational complexity of an algorithm can be measured given a model of computation. Using a model allows studying the performance of algorithms independently of the variations that are specific to particular implementations and specific technology. Categories Models of computation can be classified into three categories: sequential models, functional models, and concurrent models. Sequential models Sequential models include: * Finite-state machines * Post machines ( Post–Turing machines and tag machines). * Pushdown automata * Register machines ** Random-access machines * Turing machines * Decision tree model * External memory model Functional models Functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
99 Bottles Of Beer
"99 Bottles of Beer" or "100 Bottles of Pop on the Wall" is a traditional reverse counting song from the United States and Canada. It is popular to sing on road trips, as it has a very repetitive format which is easy to memorize and can take a long time when sung in full. In particular, the song is often sung by children on long school bus trips, such as class field trips, family road trips, or on Scout or Girl Guide outings. In computer science, printing the lyrics of 99 Bottles of Beer is a commonly used task to demonstrate esoteric programming languages. History Lyrics The song's lyrics are as follows, beginning with n=99: (n) bottles of beer on the wall. (n) bottles of beer. If one of the bottles just happen to fall, (n−1) bottles of beer on the wall. \layout The same verse is repeated, each time with one bottle fewer, until there is none left. Variations on the last verse following the last bottle going down include lines such as: No more bottles of beer o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterated Function
In mathematics, an iterated function is a function that is obtained by composing another function with itself two or several times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial object, the result of applying a given function is fed again into the function as input, and this process is repeated. For example, on the image on the right: : Iterated functions are studied in computer science, fractals, dynamical systems, mathematics and renormalization group physics. Definition The formal definition of an iterated function on a set ''X'' follows. Let be a set and be a function. Defining as the ''n''-th iterate of , where ''n'' is a non-negative integer, by: f^0 ~ \stackrel ~ \operatorname_X and f^ ~ \stackrel ~ f \circ f^, where is the identity function on and denotes function composition. This notation has been traced to and John Frederick William Herschel in 1813. Herschel credited ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computable Function
Computable functions are the basic objects of study in computability theory. Informally, a function is ''computable'' if there is an algorithm that computes the value of the function for every value of its argument. Because of the lack of a precise definition of the concept of algorithm, every formal definition of computability must refer to a specific model of computation. Many such models of computation have been proposed, the major ones being Turing machines, register machines, lambda calculus and general recursive functions. Although these four are of a very different nature, they provide exactly the same class of computable functions, and, for every model of computation that has ever been proposed, the computable functions for such a model are computable for the above four models of computation. The Church–Turing thesis is the unprovable assertion that every notion of computability that can be imagined can compute only functions that are computable in the above sense. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing-complete
In computability theory, a system of data-manipulation rules (such as a model of computation, a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any Turing machine (devised by English mathematician and computer scientist Alan Turing). This means that this system is able to recognize or decode other data-manipulation rule sets. Turing completeness is used as a way to express the power of such a data-manipulation rule set. Virtually all programming languages today are Turing-complete. A related concept is that of Turing equivalence two computers P and Q are called equivalent if P can simulate Q and Q can simulate P. The Church–Turing thesis conjectures that any function whose values can be computed by an algorithm can be computed by a Turing machine, and therefore that if any real-world computer can simulate a Turing machine, it is Turing equivalent to a Turing ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Composition
In mathematics, the composition operator \circ takes two function (mathematics), functions, f and g, and returns a new function h(x) := (g \circ f) (x) = g(f(x)). Thus, the function is function application, 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 #Properties, associativity. Examples * Composition of functions on a finite set (mathematics), 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
