On ‘Big’ Boolean-Equation Solving and Its Utility in Combinatorial Digital Design

This chapter considers the problem of solving a system of Boolean equations over a finite (atomic) Boolean  algebra  other  than  the  two-valued  one.  A  prominent  “misnomer”  in  mathematical  and engineering  circles  is  the  term  ‘Boolean  algebra’.  This  term  is  widely  used  to  refer  to  switching algebra, which is just one particular case of a ‘Boolean algebra’ that has 0 generators, 1 atom and two elements belonging to B={0,1}.The chapter outlines classical and novel direct methods for deriving the general parametric solution of such a system and for listing all its particular solutions. A detailed example over Bis  used to illustrate these two methods as well as a third method that  starts by deriving  the  subsumptive  solution  first.  The  example  demonstrates  how  the  consistency  condition forces  a  collapse  of  the  underlying  Boolean  algebra  to a  subalgebra,  and  also  how  to list  a  huge number of particular solutions in a very compact space. Subsequently, the chapter proposes some potential  applications  for  the  techniques  of  Boolean-equation  solving.  These  techniques  are  very promising as useful extensions of classical techniques based on two-valued Boolean algebra.

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