Combination switching networks with multiple outputs may be represented by Boolean functions. Report has been prepared which describes derivation and use of extraction algorithm that may be adapted to simplification of such simultaneous Boolean functions

Perlman, M.

English

NASA Technical Reports Server

June 1973	 B73-10165 
NASA TECH BRIEF 
NASA Pasadena Office	
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commercial application. Tech Briefs are available on a subscription basis from the National Technical Information 
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be directed to the Technology Utilization Office, NASA, Code KT, Washington, D.C. 20546. 
Theory and Calculus of Cubical Complexes 
.
Combination switching networks with multiple out-
puts may be represented by Boolean functions. A report 
has been prepared which describes the derivation and 
use of an extraction algorithm that may be adapted to 
the simplification of such simultaneous Boolean func-
tions, a problem that often arises in logic minimization 
and fault diagnosis. 
The algorithms are derived through cubical complexes, 
an n-variable, algebraic and topological system that may 
be used to represent a Boolean function of n-variables. 
The particular problem that motivates the formula-
tion of the extraction algorithm is a "covering" problem, 
i.e.: given a Boolean function, find an equivalent one in 
disjunctive form (the logical sum of logical products) 
that requires a minimum number of Boolean variables. 
Each term of a Boolean function may be represented 
as:
a1	 a2a
X, X2 ... Xnn 
where	 a1 = X, ifa 1, and a Xifa =0 
Each vertex of an n-cube (an n-dimensional hyper. 
cube) can be represented by the coordinates (a 1 , a2 
an) where a = 0, 1. Thus a one-to-one correspondence 
exists between the 2'' possible terms of the Boolean 
function and the 2'' vertices of an n-cube. 
In developing the extraction algorithm, several con-
cepts are defined. A 0-cube is a single vertex, a 1-cube 
consists of two zero cubes that differ in exactly one 
place (e.g., 1101 and 1100 which may be written as
1 lox where x is a free or "don't care" variable). A 
2-cube would be of the form 1 1X, and by induction 
the collection of r-cubes Kr is defined for all r where 
O:5r:5n. 
A cubical complex is defined as K°, K 1 , ... K" and 
two operators, the face and co-face operators. Several 
other operations (a star product, a sharp product, and a 
"less than" operation) are defined for us in the extraction 
algorithm. 
Several types of subcomplexes are defined and, along 
with the various operations, are used in an extraction 
algorithm to find a K-cover of minimum cost for a 
subset of K. This procedure, then, is analogous to 
finding the minimum number of literals and (logical) 
product terms necessary to represent a Boolean functon 
in disjunctive form. 
Note: 
Requests for further information may be directed to: 
Technology Utilization Officer 
NASA Pasadena Office 
4800 Oak Grove Drive 
Pasadena, California 91103 
Reference: B73-10165 
Source: Marvin Perlman of

Caltech/JPL
under contract to
NASA Pasadena Office
(NPO-1149 1) 
. Category 09 
This document was prepared under the sponsorship of the National
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Aeronautics and Space Administration. Neither the United States
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Government nor any person acting on behalf of the United States
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Theory and calculus of cubical complexes

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