Minimal determining sets of locally finitely-determined functionals

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Let N denote the set of natural numbers, NN the set of all total functions from N into N. By functional we mean any function whose domain is NN. if F is a functional and A is a partial function from N into N, we say that A is a determining segment (ds) of F if F has the same value on any two total extensions of A. A ds is called minimal (mds) if it does not properly contain another ds. For α ∈ NN, denote by F* (α) the set of all mds's of F which are subsets of α. A functional F is called finitely-determined (fd) if every α ∈ NN contains a finite ds. F is locally fd (lfd) if there exists a set {Fi |i ∈ I} of fd functionals such that F(α) = {Fi(α) | i ∈I} and Fi (α;) ≠ Fj (β) for i ≠ j and α, β ∈ NN. Total continuous operators (from NN to NN) are lfd. Examples for fd F show that F* (α) may contain (even 2Nא) infinite mds's. The two main results for lfd functionals are that every ds contains a mds and that if F*(α) consists only of finite sets then F* (α) is itself finite. This follows from a combinatorial Theorem. If A = {n-ary union}n=1An where the An's are finite and An ⊄ Am for n ≠ m, then ∃B ⊂ A such that ∀n (An ⊄ B) and for an infinite sequence n1, n2,..., (Ani - B) ∩ (Anj - B) = ∅ when ≠ j. A partial recursive functional F, if undefined on α, behaves differently when fd or non-fd on α. From any oracle-machine for F we can effectively construct another which makes finitely many queries about α when F is undefined and fd on α.

Original languageEnglish
Pages (from-to)175-190
Number of pages16
JournalDiscrete Mathematics
Issue number2
StatePublished - 1980
Externally publishedYes

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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