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| {{Other uses|LOOP (disambiguation){{!}}LOOP}}
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| '''LOOP''' is a pedagogical programming language designed by [[Uwe Schöning]], along with [[GOTO (programming language)|GOTO]] and [[WHILE (programming language)|WHILE]]. The only operations supported in the language are assignment, addition and looping.
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| The key property of the LOOP language is that the functions it can compute are exactly the [[primitive recursive function]]s.<ref>{{cite book|author=Herbert Enderton|year=2012|title=Computability Theory|publisher=Academic Press}}</ref>
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| == Features ==
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| Each [[primitive recursive function]] is LOOP-computable and vice versa.<ref>{{Cite book | first=Uwe | last=Schöning | title=Theoretische Informatik-kurz gefasst | edition=5 | publisher=Oxford University Press | location=London | year=2008 | page=105 | ISBN=978-3-8274-1824-1 | dnb=986529222}}</ref>
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| In contrast to [[GOTO (programming language)|GOTO]] programs and [[WHILE (programming language)|WHILE]] programs, LOOP programs always [[terminating computation|terminate]].<ref>{{Cite book | first=Uwe | last=Schöning | title=Theoretische Informatik-kurz gefasst | edition=5 | publisher=Oxford University Press | location=London | year=2008 | page=93 | ISBN=978-3-8274-1824-1 | dnb=986529222}}</ref> Therefore, the set of functions computable by LOOP-programs is a proper subset of [[computable function]]s (and thus a subset of the computable by WHILE and GOTO program functions).<ref>{{Cite book | first=Uwe | last=Schöning | edition=4 | title=Theoretische Informatik-kurz gefasst | year=2001 | publisher=Oxford University Press | location=London | page=122 | ISBN=3-8274-1099-1}}</ref>
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| An example of a total computable function that is not LOOP computable is the [[Ackermann function]].<ref>{{Cite book | first=Uwe | last=Schöning | title=Theoretische Informatik-kurz gefasst | edition=5 | publisher=Oxford University Press | location=London | year=2008 | page=112 | ISBN=978-3-8274-1824-1 | dnb=986529222}}</ref>
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| == Formal definition ==
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| === Syntax ===
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| LOOP-programs consist of the symbols <code>LOOP</code>, <code>DO</code>, <code>END</code>, <code>:=</code>, <code>+</code>, <code>-</code> and <code>;</code> as well as any number of variables and constants. LOOP-programs have the following [[syntax]] in modified [[Backus-Naur Form]]:
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| :<math>\begin{array}{lrl}
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| P & := & x_i := x_j + c \\
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| & | & x_i := x_j - c \\
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| & | & P;P \\
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| & | & \mathrm{LOOP} \, x_i \, \mathrm{DO} \, P \, \mathrm{END}
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| \end{array}
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| </math>
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| Here, <math>Var := \{ x_0, x_1, ... \}</math> are variable names and <math>c \in \mathbb{N}</math> are constants.
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| === Semantics ===
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| If '''P''' is a LOOP program, '''P''' is equivalent to a function <math>f: \mathbb{N}^k \rightarrow \mathbb{N}</math>. The variables <math>x_1</math> through <math>x_k</math> in a LOOP program correspond to the arguments of the function <math>f</math>, and are initialized before program execution with the appropriate values. All other variables are given the initial value zero. The variable <math>x_0</math> corresponds to the value that <math>f</math> takes when given the argument values from <math>x_1</math> through <math>x_k</math>.
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| A statement of the form
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| x<sub>0</sub> := x<sub>1</sub> + c
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| means the value of the constant <math>c</math> is added to the value of the variable <math>x_1</math>, and the result is set as the value of the variable <math>x_0</math>. <math>c</math> can have the value zero, which allows the value of one variable to be assigned to another variable:
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| x<sub>0</sub> := x<sub>1</sub> + 0
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| A statement of the form
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| x<sub>0</sub> := x<sub>1</sub> - c
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| means the value of the constant <math>c</math>is subtracted from the value of the variable <math>x_1</math>, and the result is set as the value of the variable <math>x_0</math>. Negative numbers aren't allowed, and are replaced by zeros.
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| Variables are allowed to be simultaneously on the left and right side of an assignment. A statement of the form:
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| x<sub>1</sub>: = x<sub>1</sub> + c
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| for example, adds the value of the constant <math>c</math> to the variable <math>x_1</math>.
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| A statement of the form
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| ''P''<sub>1</sub>; ''P''<sub>2</sub>
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| represents the sequential execution of sub-programs <math>P_1</math> and <math>P_2</math>, in that order.
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| A statement of the form
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| LOOP x DO ''P'' END
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| means the repeated execution of the partial program <math>P</math> a total of <math>x</math> times, where the value that <math>x</math> has at the beginning of the execution of the statement is used. Even if <math>P</math> changes the value of <math>x</math>, it won't affect how many times <math>P</math> is executed in the loop. If <math>x</math> has the value zero, then <math>P</math> is not executed inside the <code>LOOP</code> statement. This allows for [[Conditional (programming)|branches]] in LOOP programs, where the conditional execution of a partial program depends on whether a variable has value zero or one.
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| == Example Programs ==
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| === Addition ===
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| In the following program, the variable <math>x_0</math> is set to the sum of the variables <math>x_1</math> and <math>x_2</math>.
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| x<sub>0</sub> := x<sub>1</sub> + 0;
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| LOOP x<sub>2</sub> DO
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| x<sub>0</sub> := x<sub>0</sub> + 1
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| END
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| #
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| <math>x_0</math> is first assigned the value of <math>x_1</math>. Then, <math>x_0</math> is incremented a total of <math>x_2</math> times by the <code>LOOP</code> statement. This program can be used as a subroutine in other LOOP programs. The LOOP syntax can be extended with the following statement, equivalent to calling the above as a subroutine:
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| x<sub>0</sub> := x<sub>1</sub> + x<sub>2</sub>
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| === Multiplication ===
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| The following LOOP program sets the value of the variable <math>x_0</math> to the product of the variables <math>x_1</math> and <math>x_2</math>.
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| LOOP x<sub>1</sub> DO
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| x<sub>0</sub> := x<sub>0</sub> + x<sub>2</sub>
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| END
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| #
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| This multiplication program uses the syntax introduced by the addition subroutine from the previous example. The multiplication is performed here by adding the value of <math>x_2</math> a total of <math>x_1</math> times, storing results in <math>x_0</math>.
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| == See also ==
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| * [[μ-recursive function]]
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| == Notes and references ==
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| <references />
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| == External links ==
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| * [http://web.archive.org/web/20120311032544/http://loopgotowhile.eugenkiss.com/ Loop, Goto & While]
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| [[Category:Computability theory]]
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Letrado desde hace 20 años
My page; abogado divorcio