## Abstract

Let N denote the set of natural numbers, N^{N} the set of all total functions from N into N. By functional we mean any function whose domain is N^{N}. 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 ∈ N^{N}, 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 ∈ N^{N} contains a finite ds. F is locally fd (lfd) if there exists a set {F_{i} |i ∈ I} of fd functionals such that F(α) = {F_{i}(α) | i ∈I} and F_{i} (α;) ≠ F_{j} (β) for i ≠ j and α, β ∈ N^{N}. Total continuous operators (from N^{N} to N^{N}) are lfd. Examples for fd F show that F^{*} (α) may contain (even 2^{N}א) 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=1}A_{n} where the A_{n}'s are finite and A_{n} ⊄ A_{m} for n ≠ m, then ∃B ⊂ A such that ∀_{n} (A_{n} ⊄ B) and for an infinite sequence n_{1}, n_{2},..., (A_{ni} - B) ∩ (A_{nj} - 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 language | English |
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Pages (from-to) | 175-190 |

Number of pages | 16 |

Journal | Discrete Mathematics |

Volume | 29 |

Issue number | 2 |

DOIs | |

State | Published - 1980 |

Externally published | Yes |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics