Efficient handling of binary data

Abstract

using a method such as that of Quine [1], where the set P of basic cells is listed but the set R of all the vertices is never found. In this case we decompose P into a-components anc~, as shown by Natapoff, we may operate on each a-component separately to find a simplest normal form of the given function. Natapoff [2] defines an equivalence relation p over the set R of all the vertices of an arbitrary truth function, and shows that p has the following property: If R is partitioned into components R1, ... , R= by p, then every minimal covering m of R can be expressed uniquely as m = mllJ m2 U • .-(J mn where each mi is a minimal covering of R~ ; conversely, every such union is a minimal covering of R. The purpose of this note is to show that the method proposed by iVatapoff for partitioning R into p-components can be improved. Let 4~ be a Boolean truth function; we will follow Natapoff in stating our results in the language of sets of points on an N-dimensional Euclidean cube. An arbitrary cell of such a cube is represented as an N-tuple (C1, ... , CN), each Ci = 0, 1 or 2, where 2 indicates a coordinate to which reference has been suppressed. Let P designate the set of all the basic cells (prime im-plicants) of¢; let A = (A~, ..-, AN) and B = (B~, • • • , BN) be arbitrary basic cells. We will say that A and B are compatible if for each i (i = 1, ... , N): (a) A~ = Bi, or (b) A~ = 2, or (c) Bi = 2. It is obvious that A and B are compatible if and only if they contain a vertex V in common. If A and B are compatible, we will abbreviate this by writing A cm B. We now define an equivalence relation a over P as follows: if A and B are basic cells, then Ao-B holds if and only if there exist basic cells D~, ..., D, such that A = D1, D1 cm D2, D2 cm D3, ..-, D,_i cm D~ and D~ = B. It is easy to see that if Sx, • • • : S, are the a-components of P, and if R~ designates the set of all the …

DOI: 10.1145/362814.362832

Cite this paper

@article{Raduchel1970EfficientHO, title={Efficient handling of binary data}, author={William J. Raduchel}, journal={Commun. ACM}, year={1970}, volume={13}, pages={758-759} }