A function \(f : \mathbb N^k \to \mathbb N\) is WHILE-computable, if there exists a unit-cost Random Access Machine \(M\) with a finite \(W H I L E\) program \(P\), such that the machine \(M\) computes the output \(m\in \mathbb N\) for every input of \(k\) natural numbers \((n_1,\ldots,n_k)\in\mathbb N^k\), i.e.
\[f(n_1,\ldots,n_k)=m\]
in the following way: Given the initial state of registers of \(M\) \(r_i:=n_i\) for \(1\le i\le k\) and \(r_i:=0\) for \(i > k\), \(M\) starts the \(W H I L E\) program \(\mathtt {P}\) and terminates at a new state such that * \(r_i=n_i\) for \(1\le i\le k\) (i.e. the initial register states remain unchanged), * \(r_{k+1}=m\) (i.e. the next register contains the output), and * \(r_i=0\) for \(i > k + 1\).
The set of all WHILE-computable functions is denoted by \(W H I L E\).
The function \(f : \mathbb N^k \to \mathbb N\) is partially WHILE-computable, if there is a subset \(S^k\subseteq\mathbb N^k\), for which the restriction \({f|}_{S^k} : \mathbb N^k \to \mathbb N\) is WHILE-computable and for every input \((n_1,\ldots,n_k)\in\mathbb N^k\setminus S^k\) the machine \(M\) behaves as follows: Given the initial state of registers of \(M\) \(r_i:=n_i\) for \(1\le i\le k\) and \(r_i:=0\) for \(i > k\), \(M\) starts the \(W H I L E\) program \(\mathtt {P}\) and never terminates. In this case, we set \[f(n_1,\ldots,n_k)=\text{undefined}.\]
The set of all partially WHILE-computable functions is denoted by \(W H I L E^{part}\).
Corollaries: 1
Corollaries: 1
Proofs: 2 3
Theorems: 4 5
