# The length-problem

## Abstract

In MENZEL & SPERSCHNEIDER "R.e. extensions of R_{1} by finite functions" families of partial recursive functions of the sort R_{1} UF were investigated, where R_{1} is the set of all total recursive functions from N to N and F is a subfamily of Fin which is the set of all partial functions from N to N having a finite initial segment of N as domain. Here we treat families F which are defined by some "length-set" in the following sense: for a function δ∈Fin define lg(δ to be the cardinality of the domain of δ (read "lg(δ)" as "the length of δ"), and for L⊑*P* define Fin(L):={δ∈Fin|1g(δ)∈L}. The length-problem is then stated as the following problem:

(LP) Characterize the sets L⊑*P* such that R_{1}∪Fin(L) is r.e.. If R_{1}∪Fin(L) is r.e., then L is in the level Σ_{2} of the arithmetical hierarchy. Conversely, for a set L in Σ_{1}, (LP) is trivially solved in that R_{1}∪Fin(L) is r.e. iff L is infinite. Sets L in Π_{1} are more difficult to treat. Let L be r.e. with infinite complement. If L is not hyper-simple then R_{1}∪Fin(L^{C}) is r.e.. This is shown by providing for each partial recursive function ϕ_{e} a "follower" ψ_{e} which is equal to ϕ_{e} in case ϕ_{e}∈R_{1} and is an element of Fin(L^{C}) otherwise. Providing such a "follower" ψ_{e} for ϕ_{e} is constructive (we need only one follower for each ϕ_{e} which is never "given up") and extensional (ψ_{e} depends only on ϕ_{e} and not on e). Enumeration of ψ_{e} is uniform in e, hence {ψ_{e}|e∈N} is a r.e. family between R_{1} and R_{1}∪Fin(L^{C}). Since for arbitrary r.e. set L we show that Fin(L^{C}) is r.e. it follows that R_{1}∪Fin(L^{C}) is r.e. as the union of the two r.e. families {ψ_{e}|e∈N} and Fin(L^{C}).

We prove that such a constructive and extensional way of providing a follower for each partial recursive function under the condition that the resulting finite functions have a length in L^{C} can be applied only to non-hyper-simple sets L. By more sophisticated methods we show that there is even a maximal set M such that R_{1}υFin(M^{C}) is r.e..

## Notation

- N
set of all natural numbers, including 0

- f(x)↓
f is defined at x

- f(x)↑
f is undefined at x

- dom(f)
{x|f(x)↓}, the domain of f

- rg(f)
{f(x)|x∈dom(f)}, the range of f

- f⊑g
g is an extension of f

- fГA
the restriction of f to A

- P
_{n} the partial recursive functions from N

^{N}to N- R
_{n} the total recursive functions from N

^{N}to N- Fin
{δ∈P

_{1}|dom(δ) is a finite initial segment of N}- lg(δ)=1
iff dom(δ)={0,...,l−1} (the length of δ)

- ϕ
_{e}^{(n)} the function in P

_{n}computed by Turing-machine e- ϕ
_{e} ϕ

_{e}^{(1)}- ϕ
_{e,s}(x)=y iff Turing-machine e with input x stops within s steps with output y and e<s, x<s, y<s.

- W
_{e} dom(ϕ

_{e})- W
_{e,s} dom(ϕ

_{e,s})- (D
_{n})_{n∈N} is the usual canonical numbering (coding) of all finite subsets of N

- (δ
_{n})_{n∈N} is some fixed canonical numbering of Fin

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## Literature

- MENZEL, W. & SPERSCHNEIDER, V., Recursively enumerable extensions of R
_{1}by finite functions, in this lecture notes volumeGoogle Scholar - ROGERS, H. Jr., Theory of recursive functions and effective computability, McGraw Hill, New YorkGoogle Scholar