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Each time an object of class X is created, the constructor of counter<X> is called, incrementing both the created and alive count. Each time an object of class X is destroyed, the alive count is decremented. It is important to note that counter<X> and counter<Y> are two separate classes and this is why they will keep separate counts of Xs and Ys.
The 91 function was chosen for being nested-recursive (contrasted with single recursion, such as defining () by means of ()). The example was popularized by Manna's book, Mathematical Theory of Computation (1974). As the field of Formal Methods advanced, this example appeared repeatedly in the research literature.
where a represents the number of recursive calls at each level of recursion, b represents by what factor smaller the input is for the next level of recursion (i.e. the number of pieces you divide the problem into), and f(n) represents the work that the function does independently of any recursion (e.g. partitioning, recombining) at each level ...
Mathematically, a set of mutually recursive functions are primitive recursive, which can be proven by course-of-values recursion, building a single function F that lists the values of the individual recursive function in order: = (), (), (), (), …, and rewriting the mutual recursion as a primitive recursion.
"A function φ is primitive recursive in ψ 1, ..., ψ k (briefly Ψ), if there is a finite sequence φ 1, ..., φ k of (occurrences of) functions ... such that each function of the sequence is either one of the functions Ψ (the assumed functions), or an initial function, or an immediate dependent of preceding functions, and the last function ...
If the target of a tail is the same subroutine, the subroutine is said to be tail recursive, which is a special case of direct recursion. Tail recursion (or tail-end recursion) is particularly useful, and is often easy to optimize in implementations. Tail calls can be implemented without adding a new stack frame to the call stack.
On the other hand, we can "enumerate" any recursively enumerable set (see also its complexity class RE) by a primitive-recursive function in the following sense: given an input (M, k), where M is a Turing machine and k is an integer, if M halts within k steps then output M; otherwise output nothing.
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 [ 2 ] of his function (which had three non-negative integer arguments), many authors modified it to suit various purposes, so that today "the Ackermann ...