Definition: LOOP-Computable Functions

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.


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 \(L O O P\).

  1. Lemma: LOOP-Computable Functions are Total

Lemmas: 1
Proofs: 2 3
Theorems: 4

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  1. Erk, Katrin; Priese, Lutz: "Theoretische Informatik", Springer Verlag, 2000, 2nd Edition