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Closures

  • Lane A. Hemaspaandra
  • Leen Torenvliet
Part of the Monographs in Theoretical Computer Science An EATCS Series book series (EATCS)

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) = Ā1) 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 ) = fA 1, ... ,χA 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) = ab. The connective Ior) works on two sets A and B and, since it applies an element-by-element “or,” returns AB (i.e., I(f or) (A, B) = AB). 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

Boolean Function Closure Property Transition Count Selector Function Bibliographic Note 
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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Lane A. Hemaspaandra
    • 1
  • Leen Torenvliet
    • 2
  1. 1.Department of Computer ScienceUniversity of RochesterRochesterUSA
  2. 2.Department of Computer ScienceUniversity of AmsterdamAmsterdamThe Netherlands

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