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Recursively enumerable extensions of R1 by finite functions

  • W. Menzel
  • V. Sperschneider
Section I: Complexity
Part of the Lecture Notes in Computer Science book series (LNCS, volume 171)

Abstract

Let R1 be the total recursive functions from IN to IN, and Fin the set of all partial functions from IN to IN having a finite initial segment of IN as domain. Motivated by earlier studies on "simulation-universal automata" (BUCHBERGER&MENZEL 77/ MAIER, MENZEL&SPERSCHNEIDER 82/ MENZEL &SPERSCHNEIDER 82) we ask what it means that R1UF is recursively enumerable (r.e.), for a subfamily F of Fin. We show that each such family F is, in a certain sense, very rich. A (simple) corollary is that it must be dense in Fin w.r.t. the usual product topology on Fin. As a consequence one obtains simple but useful necessary conditions on F to make R1UF r.e. . We also consider the class Ext(R1) :={F⊑Fin | R1UF r.e.} as a whole. It is also quite rich in structure (e.g. if viewed as an upper semi-lattice w.r.t. union of families of functions). A sufficient criterion for F to be in Ext(R1) provides examples of families "naturally in Ext(R1)", thus demonstrating that richness. On the other hand, there are families which belong to Ext(R1) in a more nonstandard way. Main results:
  • There is an F in Ext(R1) which is itself not r.e..

  • For F in Ext(R1), F is r.e. iff F is "effectively dense in Fin", in some appropriate sense.

We conclude with a first look at Ext*(R1), which is the quotient semilattice of Ext(R1) modulo finite families. We prove that Ext*(R1) does not contain minimal elements.

Keywords

Turing Machine Minimal Element Recursive Function Limit Property Finite Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notation

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,...,x-1}

Pn

the partial recursive functions from Nn to Nn

Rn

the total recursive functions from Nn to Nn

Fin

{δεP1 | dom(δ) is a finite initial segment of N}

lg(δ)=1

for δεFin, iff dom(δ) = {0,...,1-1}, 1 is the length of δ

ϕi(n)

is the function in Pn computed by Turing machine i

ϕi

ϕ i (1)

ϕi,t(x)=y

Turing machine i with input x stops within t steps with output y, and i<t, x<t, y<t

Wi

dom(ϕi)

Wi,t

dom(ϕi,t)

λx, y.<x, y>

some standard pairing function, the corresponding component functions being λz. (z).1 and λz. (z).2

F⊑P1 is r.e.

iff there is some iε IN such that F={ϕk | kεWi}; Wi is then called a basis for F

(Dn)nε N

is the usual canonical numbering (coding) of all finite subsets of N

n)nε n

is some fixed canonical numbering of Fin (obtained for example via our pairing function and the numbering (Dn)nεN)

For F⊑Fin

in addition to being r.e., we say that F is canonically enumerable resp. canonically decidable iff {n | δn εF} is r.e. resp. decidable.

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Literature

  1. BUCHBERGER, B. & MENZEL, W. (1977), Simulation-universal automata, Interner Bericht Nr. 14/77, Institut für Informatik I, Universität KarlsruheGoogle Scholar
  2. MAIER, W., MENZEL, W. & SPERSCHNEIDER, V. (1982), Embedding properties of total recursive functions, Zeitschrift f. Math. Logik und Grundlagen der Mathematik, Band 28Google Scholar
  3. MAL'CEV, A.I. (1970), Algorithms and recursive functions, GroningenGoogle Scholar
  4. MENZEL, W. & SPERSCHNEIDER, V. (1982), Universal automata with uniform bound on simulation time, Information and Control, Vol. 52Google Scholar
  5. POUR-EL, M. & HOWARD, W.A. (1964), A structural criterion for recursive enumeration without repetition, Zeitschrift f. Math. Logik u. Grundl. der Mathematik, Band 10Google Scholar
  6. ROGERS, H. JR. (1967), Theory of recursive functions and effective computability, New YorkGoogle Scholar
  7. SPERSCHNEIDER, V., The Length-Problem, in this lecture notes volumeGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • W. Menzel
    • 1
  • V. Sperschneider
    • 1
  1. 1.Institut für Informatik IUniversität KarlsruheGermany

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