DG/L is a programming language developed by Data General Corp for the Nova , Eclipse , and Eclipse/MV families of minicomputers in the 1970s and early 1980s.
51-411: There were two separate versions: The language itself was an extended version of Algol 60 . It supported Integers , Single and Double precision floating point and complex numbers, and both fixed and arbitrary precision strings . It also supported full arbitrary precision binary-coded decimal (BCD) arithmetic on strings. It had many convenient program control flow features, but being designed in
102-405: A circular definition or self-reference , in which the putative recursive step does not get closer to a base case, but instead leads to an infinite regress . It is not unusual for such books to include a joke entry in their glossary along the lines of: A variation is found on page 269 in the index of some editions of Brian Kernighan and Dennis Ritchie 's book The C Programming Language ;
153-431: A is an element of X . It can be proved by mathematical induction that F ( n ) = G ( n ) for all natural numbers n : By induction, F ( n ) = G ( n ) for all n ∈ N {\displaystyle n\in \mathbb {N} } . A common method of simplification is to divide a problem into subproblems of the same type. As a computer programming technique, this is called divide and conquer and
204-489: A class of objects or methods exhibits recursive behavior when it can be defined by two properties: For example, the following is a recursive definition of a person's ancestor . One's ancestor is either: The Fibonacci sequence is another classic example of recursion: Many mathematical axioms are based upon recursive rules. For example, the formal definition of the natural numbers by the Peano axioms can be described as: "Zero
255-475: A collection of polygons labelled by finitely many labels, and then each polygon is subdivided into smaller labelled polygons in a way that depends only on the labels of the original polygon. This process can be iterated. The standard `middle thirds' technique for creating the Cantor set is a subdivision rule, as is barycentric subdivision . A function may be recursively defined in terms of itself. A familiar example
306-404: A crucial role not only in syntax, but also in natural language semantics . The word and , for example, can be construed as a function that can apply to sentence meanings to create new sentences, and likewise for noun phrase meanings, verb phrase meanings, and others. It can also apply to intransitive verbs, transitive verbs, or ditransitive verbs. In order to provide a single denotation for it that
357-705: A format specification as used in FORTRAN, e.g. A simpler program using an inline format: An even simpler program using the Display statement: An alternative example, using Elliott Algol I/O is as follows. Elliott Algol used different characters for "open-string-quote" and "close-string-quote", represented here by ‘ and ’ . Here's a version for the Elliott 803 Algol (A104) The standard Elliott 803 used 5-hole paper tape and thus only had upper case. The code lacked any quote characters so £ (pound sign)
408-447: A key advance in the rise of structured programming . ALGOL 60 was one of the first languages implementing function definitions (that could be invoked recursively). ALGOL 60 function definitions could be nested within one another (which was first introduced by any programming language), with lexical scope . It gave rise to many other languages, including CPL , PL/I , Simula , BCPL , B , Pascal , and C . Practically every computer of
459-567: A non-recursive definition (e.g., a closed-form expression ). Use of recursion in an algorithm has both advantages and disadvantages. The main advantage is usually the simplicity of instructions. The main disadvantage is that the memory usage of recursive algorithms may grow very quickly, rendering them impractical for larger instances. Shapes that seem to have been created by recursive processes sometimes appear in plants and animals, such as in branching structures in which one large part branches out into two or more similar smaller parts. One example
510-477: A random function passed as actual argument. Call-by-name is known by many compiler designers for the interesting " thunks " that are used to implement it. Donald Knuth devised the " man or boy test " to separate compilers that correctly implemented " recursion and non-local references." This test contains an example of call-by-name. There are 35 such reserved words in the standard Burroughs Large Systems sub-language: There are 71 such restricted identifiers in
561-404: A set X , an element a of X and a function f : X → X , the theorem states that there is a unique function F : N → X {\displaystyle F:\mathbb {N} \to X} (where N {\displaystyle \mathbb {N} } denotes the set of natural numbers including zero) such that for any natural number n . Dedekind was the first to pose
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#1733085441536612-427: A structure in which what follows the verb is another sentence: Dorothy thinks witches are dangerous , in which the sentence witches are dangerous occurs in the larger one. So a sentence can be defined recursively (very roughly) as something with a structure that includes a noun phrase, a verb, and optionally another sentence. This is really just a special case of the mathematical definition of recursion. This provides
663-546: A triple denote the association that an Attribute of an Object has a specific Value. LEAP was created by Jerome Feldman (University of California Berkeley) and Paul Rovner (MIT Lincoln Lab) in 1967. LEAP was also implemented in SAIL. Recursion Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic . The most common application of recursion
714-461: A way of understanding the creativity of language—the unbounded number of grammatical sentences—because it immediately predicts that sentences can be of arbitrary length: Dorothy thinks that Toto suspects that Tin Man said that... . There are many structures apart from sentences that can be defined recursively, and therefore many ways in which a sentence can embed instances of one category inside another. Over
765-435: Is Romanesco broccoli . Authors use the concept of recursivity to foreground the situation in which specifically social scientists find themselves when producing knowledge about the world they are always already part of. According to Audrey Alejandro, “as social scientists, the recursivity of our condition deals with the fact that we are both subjects (as discourses are the medium through which we analyse) and objects of
816-452: Is a natural number, and each natural number has a successor, which is also a natural number." By this base case and recursive rule, one can generate the set of all natural numbers. Other recursively defined mathematical objects include factorials , functions (e.g., recurrence relations ), sets (e.g., Cantor ternary set ), and fractals . There are various more tongue-in-cheek definitions of recursion; see recursive humor . Recursion
867-468: Is an approach to optimization that restates a multiperiod or multistep optimization problem in recursive form. The key result in dynamic programming is the Bellman equation , which writes the value of the optimization problem at an earlier time (or earlier step) in terms of its value at a later time (or later step). In set theory , this is a theorem guaranteeing that recursively defined functions exist. Given
918-457: Is in mathematics and computer science , where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references can occur. A process that exhibits recursion is recursive . Video feedback displays recursive images, as does an infinity mirror . In mathematics and computer science,
969-413: Is key to the design of many important algorithms. Divide and conquer serves as a top-down approach to problem solving, where problems are solved by solving smaller and smaller instances. A contrary approach is dynamic programming . This approach serves as a bottom-up approach, where problems are solved by solving larger and larger instances, until the desired size is reached. A classic example of recursion
1020-534: Is made, the site suggests "Did you mean: recursion ." An alternative form is the following, from Andrew Plotkin : "If you already know what recursion is, just remember the answer. Otherwise, find someone who is standing closer to Douglas Hofstadter than you are; then ask him or her what recursion is." Recursive acronyms are other examples of recursive humor. PHP , for example, stands for "PHP Hypertext Preprocessor", WINE stands for "WINE Is Not an Emulator", GNU stands for "GNU's not Unix", and SPARQL denotes
1071-500: Is no portable hello world program in ALGOL. The following program could (and still will) compile and run on an ALGOL implementation for a Unisys A-Series mainframe, and is a straightforward simplification of code taken from The Language Guide at the University of Michigan -Dearborn Computer and Information Science Department Hello world! ALGOL Example Program page. Where * etc. represented
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#17330854415361122-430: Is properly defined, a recursive procedure is not easy for humans to perform, as it requires distinguishing the new from the old, partially executed invocation of the procedure; this requires some administration as to how far various simultaneous instances of the procedures have progressed. For this reason, recursive definitions are very rare in everyday situations. Linguist Noam Chomsky , among many others, has argued that
1173-544: Is substantially different from ALGOL 60 and was criticised partially for being so, so that in general "ALGOL" refers to dialects of ALGOL 60. ALGOL 60 – with COBOL – were the first languages to seek standardization. ALGOL 60 was used mostly by research computer scientists in the United States and in Europe. Its use in commercial applications was hindered by the absence of standard input/output facilities in its description and
1224-504: Is suitably flexible, and is typically defined so that it can take any of these different types of meanings as arguments. This can be done by defining it for a simple case in which it combines sentences, and then defining the other cases recursively in terms of the simple one. A recursive grammar is a formal grammar that contains recursive production rules . Recursion is sometimes used humorously in computer science, programming, philosophy, or mathematics textbooks, generally by giving
1275-569: Is the Fibonacci number sequence: F ( n ) = F ( n − 1) + F ( n − 2). For such a definition to be useful, it must be reducible to non-recursively defined values: in this case F (0) = 0 and F (1) = 1. Applying the standard technique of proof by cases to recursively defined sets or functions, as in the preceding sections, yields structural induction — a powerful generalization of mathematical induction widely used to derive proofs in mathematical logic and computer science. Dynamic programming
1326-505: Is the definition of the factorial function, given here in Python code: The function calls itself recursively on a smaller version of the input (n - 1) and multiplies the result of the recursive call by n , until reaching the base case , analogously to the mathematical definition of factorial. Recursion in computer programming is exemplified when a function is defined in terms of simpler, often smaller versions of itself. The solution to
1377-416: Is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be 'recursive'. To understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps based on a set of rules, while the running of a procedure involves actually following
1428-429: Is the recursive nature of management hierarchies , ranging from line management to senior management via middle management . It also encompasses the larger issue of capital structure in corporate governance . The Matryoshka doll is a physical artistic example of the recursive concept. Recursion has been used in paintings since Giotto 's Stefaneschi Triptych , made in 1320. Its central panel contains
1479-511: The "SPARQL Protocol and RDF Query Language". The canonical example of a recursively defined set is given by the natural numbers : In mathematical logic, the Peano axioms (or Peano postulates or Dedekind–Peano axioms), are axioms for the natural numbers presented in the 19th century by the German mathematician Richard Dedekind and by the Italian mathematician Giuseppe Peano . The Peano Axioms define
1530-677: The ALGOL Bulletin I was drawn into the international discussions of the language and was selected to be member of the European language design group in November 1959. In this capacity I was the editor of the ALGOL 60 report, produced as the result of the ALGOL 60 meeting in Paris in January 1960." The following people attended the meeting in Paris (from January 11 to 16): Alan Perlis gave a vivid description of
1581-468: The academic discourses we produce (as we are social agents belonging to the world we analyse).” From this basis, she identifies in recursivity a fundamental challenge in the production of emancipatory knowledge which calls for the exercise of reflexive efforts: we are socialised into discourses and dispositions produced by the socio-political order we aim to challenge, a socio-political order that we may, therefore, reproduce unconsciously while aiming to do
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1632-428: The actual parameters that are passed in are an integer variable and an array that is indexed by that same integer variable. Think of passing a pointer to swap(i, A[i]) in to a function. Now that every time swap is referenced, it's reevaluated. Say i := 1 and A[i] := 2, so every time swap is referenced it'll return the other combination of the values ([1,2], [2,1], [1,2] and so on). A similar situation occurs with
1683-770: The committee. ALGOL 60 inspired many languages that followed it. Tony Hoare remarked: "Here is a language so far ahead of its time that it was not only an improvement on its predecessors but also on nearly all its successors." To date there have been at least 70 augmentations, extensions, derivations and sublanguages of ALGOL 60. The Burroughs dialects included special system programming dialects such as ESPOL and NEWP . ALGOL 60 as officially defined had no I/O facilities; implementations defined their own in ways that were rarely compatible with each other. In contrast, ALGOL 68 offered an extensive library of transput (ALGOL 68 parlance for input/output) facilities. ALGOL 60 provided two evaluation strategies for parameter passing:
1734-430: The common call-by-value , and call-by-name . The procedure declaration specified, for each formal parameter, which was to be used: value specified for call-by-value, and omitted for call-by-name. Call-by-name has certain effects in contrast to call-by-reference . For example, without specifying the parameters as value or reference , it is impossible to develop a procedure that will swap the values of two parameters if
1785-482: The contrary. The recursivity of our situation as scholars – and, more precisely, the fact that the dispositional tools we use to produce knowledge about the world are themselves produced by this world – both evinces the vital necessity of implementing reflexivity in practice and poses the main challenge in doing so. Recursion is sometimes referred to in management science as the process of iterating through levels of abstraction in large business entities. A common example
1836-477: The era had a systems programming language based on ALGOL 60 concepts. Niklaus Wirth based his own ALGOL W on ALGOL 60 before moving to develop Pascal . Algol-W was intended to be the next generation ALGOL but the ALGOL 68 committee decided on a design that was more complex and advanced rather than a cleaned simplified ALGOL 60. The official ALGOL versions are named after the year they were first published. ALGOL 68
1887-507: The extensions to the Algol 60 standard introduced in DG/L or carried over from Data General's previous Algol implementation of 1971: Algol 60 ALGOL 60 (short for Algorithmic Language 1960 ) is a member of the ALGOL family of computer programming languages . It followed on from ALGOL 58 which had introduced code blocks and the begin and end pairs for delimiting them, representing
1938-417: The first commercial release, targeting 16-bit Eclipse and Nova, several subsequent updates and major versions were released, approximately one a year. Appendix A of Data General's 1982 revision of its DG/L Language Reference Manual, 093-00229-01 describes DG/L as based on the ALGOL 60 programming language, but gives "data types, operations and statements that ALGOL 60 lacks". Specific differences are: Some of
1989-516: The first edition of The C Programming Language . The joke is part of the functional programming folklore and was already widespread in the functional programming community before the publication of the aforementioned books. Another joke is that "To understand recursion, you must understand recursion." In the English-language version of the Google web search engine, when a search for "recursion"
2040-646: The index entry recursively references itself ("recursion 86, 139, 141, 182, 202, 269"). Early versions of this joke can be found in Let's talk Lisp by Laurent Siklóssy (published by Prentice Hall PTR on December 1, 1975, with a copyright date of 1976) and in Software Tools by Kernighan and Plauger (published by Addison-Wesley Professional on January 11, 1976). The joke also appears in The UNIX Programming Environment by Kernighan and Pike. It did not appear in
2091-411: The lack of an upper bound on the number of grammatical sentences in a language, and the lack of an upper bound on grammatical sentence length (beyond practical constraints such as the time available to utter one), can be explained as the consequence of recursion in natural language. This can be understood in terms of a recursive definition of a syntactic category, such as a sentence. A sentence can have
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2142-515: The lack of interest in the language by large computer vendors. ALGOL 60 did however become the standard for the publication of algorithms and had a profound effect on future language development. John Backus developed the Backus normal form method of describing programming languages specifically for ALGOL 58. It was revised and expanded by Peter Naur for ALGOL 60, and at Donald Knuth 's suggestion renamed Backus–Naur form . Peter Naur: "As editor of
2193-413: The meeting: "The meetings were exhausting, interminable, and exhilarating. One became aggravated when one's good ideas were discarded along with the bad ones of others. Nevertheless, diligence persisted during the entire period. The chemistry of the 13 was excellent." The language originally did not include recursion . It was inserted into the specification at the last minute, against the wishes of some of
2244-402: The mid 70s, lacked user defined data structures . DG/L had a substantial runtime library for its day, and was used for systems programming both within and outside of Data General. Originally called Algol/5, the product renamed DG/L shortly before the first commercial release in 1978. Officially, the name is meaningless but it was apparently supposed to imply "Data General Language". After
2295-465: The natural numbers referring to a recursive successor function and addition and multiplication as recursive functions. Another interesting example is the set of all "provable" propositions in an axiomatic system that are defined in terms of a proof procedure which is inductively (or recursively) defined as follows: Finite subdivision rules are a geometric form of recursion, which can be used to create fractal-like images. A subdivision rule starts with
2346-508: The problem is then devised by combining the solutions obtained from the simpler versions of the problem. One example application of recursion is in parsers for programming languages. The great advantage of recursion is that an infinite set of possible sentences, designs or other data can be defined, parsed or produced by a finite computer program. Recurrence relations are equations which define one or more sequences recursively. Some specific kinds of recurrence relation can be "solved" to obtain
2397-466: The problem of unique definition of set-theoretical functions on N {\displaystyle \mathbb {N} } by recursion, and gave a sketch of an argument in the 1888 essay "Was sind und was sollen die Zahlen?" Take two functions F : N → X {\displaystyle F:\mathbb {N} \to X} and G : N → X {\displaystyle G:\mathbb {N} \to X} such that: where
2448-427: The rules and performing the steps. Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure. When a procedure is thus defined, this immediately creates the possibility of an endless loop; recursion can only be properly used in a definition if the step in question is skipped in certain cases so that the procedure can complete. Even if it
2499-461: The standard Burroughs Large Systems sub-language: and also the names of all the intrinsic functions. Implementations differ in how the text in bold must be written. The word 'INTEGER', including the quotation marks, must be used in some implementations in place of integer , above, thereby designating it as a special keyword. Following is an example of how to produce a table using Elliott 803 ALGOL: Since ALGOL 60 had no I/O facilities, there
2550-541: The years, languages in general have proved amenable to this kind of analysis. The generally accepted idea that recursion is an essential property of human language has been challenged by Daniel Everett on the basis of his claims about the Pirahã language . Andrew Nevins, David Pesetsky and Cilene Rodrigues are among many who have argued against this. Literary self-reference can in any case be argued to be different in kind from mathematical or logical recursion. Recursion plays
2601-489: Was used for open quote and ? (question mark) for close quote. Special sequences were placed in double quotes (e.g., £L?? produced a new line on the teleprinter). The ICT 1900 series Algol I/O version allowed input from paper tape or punched card. Paper tape 'full' mode allowed lower case. Output was to a line printer. Note use of '(', ')', and %. LEAP is an extension to the ALGOL 60 programming language which provides an associative memory of triples. The three items in
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