Recursively enumerable extensions of R_{1} by finite functions
Abstract

There is an F in Ext(R_{1}) which is itself not r.e..

For F in Ext(R_{1}), F is r.e. iff F is "effectively dense in Fin", in some appropriate sense.
We conclude with a first look at Ext*(R_{1}), which is the quotient semilattice of Ext(R_{1}) modulo finite families. We prove that Ext*(R_{1}) does not contain minimal elements.
Keywords
Turing Machine Minimal Element Recursive Function Limit Property Finite FunctionNotation
 N
the set of natural numbers, including 0
 I
is called an initial segment of N iff I= N or I={x  x<k}, for some kε N
 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)  f(x)⇃}, the range of f
 graph(f)
{(x, f(x))  f(x)⇃}, the graph of f
 f⊑g
graph(f)⊑graph (g), g is an extension of f
 fГA
the restriction of f to the set A
 fГx
fГ{0,...,x1}
 P_{n}
the partial recursive functions from N^{n} to N^{n}
 R_{n}
the total recursive functions from N^{n} to N^{n}
 Fin
{δεP_{1}  dom(δ) is a finite initial segment of N}
 lg(δ)=1
for δεFin, iff dom(δ) = {0,...,11}, 1 is the length of δ
 ϕ_{i}^{(n)}
is the function in P_{n} computed by Turing machine i
 ϕ_{i}
ϕ _{i} ^{(1)}
 ϕ_{i,t}(x)=y
Turing machine i with input x stops within t steps with output y, and i<t, x<t, y<t
 W_{i}
dom(ϕ_{i})
 W_{i,t}
dom(ϕ_{i,t})
 λx, y.<x, y>
some standard pairing function, the corresponding component functions being λz. (z)._{1} and λz. (z)._{2}
 F⊑P_{1} is r.e.
iff there is some iε IN such that F={ϕ_{k}  kεW_{i}}; W_{i} is then called a basis for F
 (D_{n})_{nε N}
is the usual canonical numbering (coding) of all finite subsets of N
 (δ_{n})_{nε n}
is some fixed canonical numbering of Fin (obtained for example via our pairing function and the numbering (D_{n})_{nεN})
 For F⊑Fin
in addition to being r.e., we say that F is canonically enumerable resp. canonically decidable iff {n  δ_{n} εF} is r.e. resp. decidable.
Preview
Unable to display preview. Download preview PDF.
Literature
 BUCHBERGER, B. & MENZEL, W. (1977), Simulationuniversal automata, Interner Bericht Nr. 14/77, Institut für Informatik I, Universität KarlsruheGoogle Scholar
 MAIER, W., MENZEL, W. & SPERSCHNEIDER, V. (1982), Embedding properties of total recursive functions, Zeitschrift f. Math. Logik und Grundlagen der Mathematik, Band 28Google Scholar
 MAL'CEV, A.I. (1970), Algorithms and recursive functions, GroningenGoogle Scholar
 MENZEL, W. & SPERSCHNEIDER, V. (1982), Universal automata with uniform bound on simulation time, Information and Control, Vol. 52Google Scholar
 POUREL, M. & HOWARD, W.A. (1964), A structural criterion for recursive enumeration without repetition, Zeitschrift f. Math. Logik u. Grundl. der Mathematik, Band 10Google Scholar
 ROGERS, H. JR. (1967), Theory of recursive functions and effective computability, New YorkGoogle Scholar
 SPERSCHNEIDER, V., The LengthProblem, in this lecture notes volumeGoogle Scholar