Boolean Curve Fitting with the Aid of Variable-Entered Karnaugh Maps

Abstract

The Variable-Entered Karnaugh Map is utilized to grant a simpler view and a visual perspective to Boolean curve fitting (Boolean interpolation); a topic whose inherent complexity hinders its potential applications. We derive the function(s) through m points in the Boolean space B^(n+1) together with consistency and uniqueness conditions, where B is a general ‘big’ Boolean algebra of l≥1 generators, L atoms (2^(l-1)<L≤2^l) and 2^L elements. We highlight prominent cases in which the consistency condition reduces to the identity (0=0) with a unique solution or with multiple solutions. We conjecture that consistent (albeit not necessarily unique) curve fitting is possible if, and only if, m=2^n. This conjecture is a generalization of the fact that a Boolean function of n variables is fully and uniquely determined by its values in the {0,1}^n subdomain of its B^n domain. A few illustrative examples are used to clarify the pertinent concepts and techniques.

On the Relation between Boolean Curve Fitting and the Inverse Problem of Boolean Equations

“…In this section, we reproduce from Rudeanu [7] and Rushdi and Balamesh [10] the main results known on Boolean curve fitting. The problem at hand requires the determination of a Boolean curve whose graph passes through given points…”

“…where, with a little twist of notation, we are using 0 , 1 , 2 and 3 to stand for ∈ = {0,1} 2 = {00,01,10,11} or for 00 , 01 , 10 and 11 . The corresponding IPBE is to find the equation ( ) = ( 1 , 2 ) = 0, where : 16 2 → 16 = FB{ , } such that it has a consistency condition {0 = 0} and a set of particular solutions {( ̅ , ̅ ), ( , ), ( ̅ , ̅), ( ∨ ̅ , ̅ ∨ )} (18) Figure 3 displays the VEKM representation for IPBE ( ) expressed via (13).…”

“…A sequel paper by Melter and Rudeanu [8] in 1984 specialized the results in [7] for Boolean functions that are linear in the sense of Löwenheim [9] . Boolean curve fitting witnessed a recent revival [10] , as it finally found a useful engineering application in the area of cryptography [11,12] .…”

“…So, exposing the subtle differences between them is of interest. Our comparison is a part of an ongoing effort [10,13] to transfer these two problems from the domain of pure mathematics to the reach of engineers and problem solvers. The solutions of the two problems are being converted from declarative specifications (mathematical approach) to constructive procedures (engineering approach).…”

The Eigenvectors of the Transition Matrix as Predictors of the Dynamics of a Synchronous Boolean Network

“…Unfortunately, the scalar-equation technique suffers from several shortcomings and limitations [17,19,[45][46][47]. Other candidate methods that might be employed in the analysis of synchronous Boolean networks include solution of Diophantine equations [17,[46][47][48][49][50][51], solution of the Boolean satisfiability problem (SAT) [22,[52][53][54][55][56][57], solution of Boolean equations [58][59][60][61][62][63][64][65][66][67][68][69][70][71], and integer linear programming [22].…”