Lattice Functions and Equations

The Inverse Problem for Boolean Equations

“…These include: (a) Subsumptive general solutions, in which each of the variables is expressed as an interval by deriving successive conjunctive or disjunctive eliminants of the original function, (b) Parametric general solutions, in which each of the variables is expressed via arbitrary parameters which are freely chosen elements of the underlying Boolean algebra and (c) Particular solutions, each of which is an assignment from the underlying Boolean algebra to every pertinent variable that makes the Boolean equation an identity (Brown, 2003). The reconstructed function f (X) in every case is set in a canonical form, such as the complete-sum form (the Blake Canonical from) (Brown, 2003;Rudeanu, 1974;2001;Blake, 1938;Tison, 1967;Reusch, 1975;Cutler et al, 1979;Muroga, 1979;Gregg, 1998;Rushdi and Al-Yahya, 2001;Rushdi, 2001b), to facilitate proving its equivalence to the original function. The methods presented herein are a mixture of purely-algebraic methods and map methods that utilize the variableentered Karnaugh map (Rushdi, 1983;1985;1986;1987;1997;2001a;2004;Rushdi and Amashah, 2011;Rushdi and Al-Yahya, 2000a;2000b;2001).…”

“…The reconstructed function f (X) in every case is set in a canonical form, such as the complete-sum form (the Blake Canonical from) (Brown, 2003;Rudeanu, 1974;2001;Blake, 1938;Tison, 1967;Reusch, 1975;Cutler et al, 1979;Muroga, 1979;Gregg, 1998;Rushdi and Al-Yahya, 2001;Rushdi, 2001b), to facilitate proving its equivalence to the original function. The methods presented herein are a mixture of purely-algebraic methods and map methods that utilize the variableentered Karnaugh map (Rushdi, 1983;1985;1986;1987;1997;2001a;2004;Rushdi and Amashah, 2011;Rushdi and Al-Yahya, 2000a;2000b;2001). These methods are demonstrated with carefullychosen illustrative examples over big Boolean algebras of various sizes.…”

“…Conditions (17) and (1) can be disjuncted (ORed) together to form a single equation (Equation 18) whose unique solution is solution u j (Rudeanu, 2001):…”

“…There are m conditions of the form (18) since 1 ≤ j ≤ m. The conjunction (ANDing) of equations (18) produces an equation which has m solutions u j (1 ≤ j≤ n) (Rudeanu, 2001). Hence, the function f(X) can be reconstructed by the formula of Equation 19: Fig.…”

“…By contrast, the Inverse Problem (IV) of Boolean equations aims at reconstructing the equation f (X) = 0 given the set of solutions and hence verifying the correctness of this set. Naturally, the Forward Problem of Boolean Science Publications JCS equations has been extensively treated in the literature (see, for example, (Brown, 2003;Rudeanu, 1974;2001;2010;Tucker and Tapia, 1992;1995;Trabado et al, 1993;Unger, 1994;Jung, 1995;Woods and Casinovi, 1996;Brusentsov and Vladimirova, 1998;Levchenkov, 2000a;2000b;Rushdi, 2001a;2004;Baneres et al, 2009;Rushdi and Amashah, 2011), while the inverse problem seems to have received no or little attention.…”