Theory of Semi-Feasible Algorithms pp 79-103 | Cite as

# Closures

## Abstract

In this chapter we will study several closure properties of the P-selective sets. That is, for various functions *h* we will ask whether *h*(*A* _{1},..., *A* _{ k }) is P-selective whenever all the *A* _{ i }’s are P-selective. Theorem 1.8 states that for complementation (that is, *k* = 1 and *h*(*A* _{1}) = Ā_{1}) the answer is yes; the complement of a P-selective set is always P-selective. In this chapter we will be in part concerned with closure under boolean operations, i.e., with functions of the form *χh(A* _{1}..., *A* _{ k }) = *f*(χ*A* _{1}, ... ,χ*A* _{ k }) for some boolean function ƒ. In particular, we will study so-called boolean *connectives*. A boolean connective *I*(ƒ) is defined using a boolean operator ƒ; *I*(ƒ) works on a collection of sets and returns a single set using, in the fashion just described, the boolean operator ƒ with which it is defined. For instance, let ƒ _{or} denote the boolean function such that ƒ _{or} (*a, b*) = *a* ⋁ *b*. The connective *I*(ƒ_{or}) works on two sets *A* and *B* and, since it applies an element-by-element “or,” returns *A* ∪ *B* (i.e., *I*(*f* _{or}) (*A, B*) = *A* ∪ *B*). Indeed, a boolean connective *I*(ƒ) can be defined for any boolean function ƒ. Since there are \( {2^{{2^k}}} \) boolean functions on *k* variables, it follows that there are \( {2^{{2^k}}} \) potential boolean closures to be investigated for the P-selective sets. Section 5.2 will give a complete account of this investigation. In particular, for each *k* we will determine under exactly how many—and which—*k*-ary boolean connectives the P-selective sets are closed.

### Keywords

Sorting Prefix Hema TanOl## Preview

Unable to display preview. Download preview PDF.