The length-problem

Abstract

In MENZEL & SPERSCHNEIDER "R.e. extensions of R₁ by finite functions" families of partial recursive functions of the sort R₁ UF were investigated, where R₁ 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 R₁∪Fin(L) is r.e.. If R₁∪Fin(L) is r.e., then L is in the level Σ₂ of the arithmetical hierarchy. Conversely, for a set L in Σ₁, (LP) is trivially solved in that R₁∪Fin(L) is r.e. iff L is infinite. Sets L in Π₁ are more difficult to treat. Let L be r.e. with infinite complement. If L is not hyper-simple then R₁∪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∈R₁ 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 R₁ and R₁∪Fin(L^C). Since for arbitrary r.e. set L we show that Fin(L^C) is r.e. it follows that R₁∪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 R₁υFin(M^C) is r.e..

A subfamily F of P_n is called recursively enumerable (r.e.) iff there is a r.e. subset B of N such that F={ϕ ⁽ⁿ⁾_b |∈B}; B is called a basis for F.

For a subfamily F of Fin we additionally call F canonically enumerable (or canonically decidable) iff {n|δ_n∈F} is r.e. (or decidable).