A function \(f : \mathbb N^k \to \mathbb N\) is LOOP-computable, if there exists a unit-cost Random Access Machine \(M\) with a finite \(L O O P\) 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 \(L O O P\) 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 LOOP-computable functions is denoted by \(L O O P\).
Lemmas: 1
Proofs: 2 3
Theorems: 4
