Construction of General Subsumptive Solutions of Boolean Equations via Complete-Sum Derivation

Abstract

Boolean-equation solving permeates many diverse areas of modern science. To solve a system of Boolean equations, one usually combines them into an equivalent single Boolean equation {f (X) 0} = whose set of solutions is exactly the same as that of the original system of equations. One of the general classes of solutions for Boolean equations is the subsumptive general solution, in which each variable is expressed as an interval decided by a double inequality in terms of the succeeding variables. The solution validity depends on the satisfaction of a required consistency condition. In this study, we introduce a novel method (henceforth called the CS method) for producing subsumptive Boolean-equation solutions based on deriving the complete sum (CS(f (X)) of the pertinent Boolean function f (X). The complete sum CS(f (X)) is a disjunction of all prime implicants of f (X) and nothing else. It explicitly shows all information about f (X) in the most compact form. We demonstrate the proposed CS solutions in terms of four examples, covering Boolean algebras of different sizes and using two prominent methods for deriving CS(f (X)). Occasionally, the consistency condition results in a collapse of the underlying Boolean algebra into a smaller subalgebra. We also illustrate how an expansion tree (typically reduced to an acyclic graph) can be used to deduce a complete list of all particular solutions from the subsumptive solution. The present CS method yields correct solutions, since it fits into the frame of the most general subsumptive solution. Among competing subsumptive methods, the CS method strikes a reasonable tradeoff between the conflicting requirements of less computational cost and more compact form for the solution obtained. In fact, it is the second best known method from both criteria of efficiency and compactness of solution.

Novel Characterizations of the JK Bistables (Flip Flops)

“…In this section, we employ methods of Booleanequation solving [39][40][41][42][43][44] to obtain all possible solutions for the excitations in terms of the present state and next state. We consider two cases.…”

Satisfiability in Big Boolean Algebras via Boolean-Equation Solving

“…The SAT problem over a big Boolean algebra is handled herein by solving a Boolean equation. There are three main types of Boolean-equation solutions, which can be identified as subsumptive general solutions [22,[25][26][27][28][29][31][32][33][34] , parametric general solutions [22,23,25,26,30,31,[38][39][40][41] and particular solutions. In a subsumptive general solution, each of the variables is expressed as an interval based on successive conjunctive or disjunctive eliminants of the underlying function ( ).…”

“…3. This natural map is typically called a Variable-Entered Karnaugh Map (VEKM) [23,[28][29][30][31][32][33][34][46][47][48][49] with map variable = { , , } and with an entered 'variable' that is not really a variable but is the generator of the underlying Boolean algebra = {0, 1, , }. The map can be read [38,39] to express ( ) in pos (CNF) form as: (5) or in sop (DNF) form as…”

“…The aim of this paper is to handle the SAT problem for the formula ( ) over an arbitrary or big Boolean algebra [22,23] by using a conventional method and a novel one for deriving parametric general solutions of the Boolean equation { ( ) = 1}, and subsequently utilizing expansion trees for generating all particular solutions of the aforementioned Boolean equation. These methods could be cast in pure algebraic form [24][25][26][27] , but become much easier to visualize and comprehend when presented via the natural map of a big Boolean algebra, which (for historical reasons) is called the variable-entered Karnaugh map (VEKM) [23,[28][29][30][31][32][33][34][35][36][37] . In the classical method, the number of parameters used is minimized and compact solutions are obtained [22,23,30,[38][39][40][41] .…”

Switching-Algebraic Analysis of Relational Databases

“…where, CS(g) stands for the complete sum of the function g. We derive CS (g) via any appropriate algorithm such as Tison algorithm (Tison, 1967;Cutler et al, 1979;Brown, 1990;Rushdi and Al-Yahya, 2001; Rushdi and Albarakati, 2014), or the algorithm of VEKM folding (Rushdi and Al-Yahya, 2001). Each of the prime implicants (prime consequents) in CS(g) is interpreted as an equation of the form (1) and hence converted to a propositional implication or equivalently to a functional dependency that is a member of S + .…”