Ackermann function

In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive. In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive. After Ackermann's publication of his function (which had three nonnegative integer arguments), many authors modified it to suit various purposes, so that today 'the Ackermann function' may refer to any of numerous variants of the original function. One common version, the two-argument Ackermann–Péter function, is defined as follows for nonnegative integers m and n: Its value grows rapidly, even for small inputs. For example, A(4, 2) is an integer of 19,729 decimal digits (equivalent to 265536−3, or 22222−3). In the late 1920s, the mathematicians Gabriel Sudan and Wilhelm Ackermann, students of David Hilbert, were studying the foundations of computation. Both Sudan and Ackermann are credited with discovering total computable functions (termed simply 'recursive' in some references) that are not primitive recursive. Sudan published the lesser-known Sudan function, then shortly afterwards and independently, in 1928, Ackermann published his function φ {displaystyle varphi } (the Greek letter phi). Ackermann's three-argument function, φ ( m , n , p ) {displaystyle varphi (m,n,p)} , is defined such that for p = 0, 1, 2, it reproduces the basic operations of addition, multiplication, and exponentiation as and for p > 2 it extends these basic operations in a way that can be compared to the hyperoperations: (Aside from its historic role as a total-computable-but-not-primitive-recursive function, Ackermann's original function is seen to extend the basic arithmetic operations beyond exponentiation, although not as seamlessly as do variants of Ackermann's function that are specifically designed for that purpose—such as Goodstein's hyperoperation sequence.) In On the Infinite, David Hilbert hypothesized that the Ackermann function was not primitive recursive, but it was Ackermann, Hilbert's personal secretary and former student, who actually proved the hypothesis in his paper On Hilbert's Construction of the Real Numbers. Rózsa Péter and Raphael Robinson later developed a two-variable version of the Ackermann function that became preferred by many authors. Ackermann's original three-argument function φ ( m , n , p ) {displaystyle varphi (m,n,p)} is defined recursively as follows for nonnegative integers m, n, and p: