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The Ackermann function, due to its definition in terms of extremely deep recursion, can be used as a benchmark of a compiler's ability to optimize recursion. The first published use of Ackermann's function in this way was in 1970 by Dragoș Vaida [27] and, almost simultaneously, in 1971, by Yngve Sundblad. [14]
Here, the function () is the inverse Ackermann function. The inverse Ackermann function grows extraordinarily slowly, so this factor is 4 or less for any n that can actually be written in the physical universe. This makes disjoint-set operations practically amortized constant time.
The Ackermann function A(m,n) is a well-known example of a total recursive function (in fact, provable total), that is not primitive recursive. There is a characterization of the primitive recursive functions as a subset of the total recursive functions using the Ackermann function.
BlooP and FlooP (Bounded loop and Free loop) are simple programming languages designed by Douglas Hofstadter to illustrate a point in his book Gödel, Escher, Bach. [1] BlooP is a Turing-incomplete programming language whose main control flow structure is a bounded loop (i.e. recursion is not permitted [citation needed]).
In 1928, Wilhelm Ackermann defined a 3-argument function (,,) which gradually evolved into a 2-argument function known as the Ackermann function. The original Ackermann function ϕ {\displaystyle \phi } was less similar to modern hyperoperations, because his initial conditions start with ϕ ( a , 0 , n ) = a {\displaystyle \phi (a,0,n)=a} for ...
This article has a mistake somewhere in the definitions. I wrote a C program based on what is here, and it simply goes in an endless loop for m = 4 (A(4,n)). It does not terminate, as the definition states. --Gesslein 16:58, 4 October 2006 (UTC) Nevermind. My computer is just really, really slow. It works now and I verified all the numbers below.
LOOP is a simple register language that precisely captures the primitive recursive functions. [1] The language is derived from the counter-machine model.Like the counter machines the LOOP language comprises a set of one or more unbounded registers, each of which can hold a single non-negative integer.
The primitive recursive functions are a subset of the total recursive functions, which are a subset of the partial recursive functions. For example, the Ackermann function can be proven to be total recursive, and to be non-primitive. Primitive or "basic" functions: Constant functions C k n: For each natural number n and every k