The main objective of this paper is to establish results for PDS with general maxterm (resp., minterm) functions as evolution operators in the sense of the fixed point theorem by Banach.
Section 2 gives some preliminaries concerning PDS on maxterm and minterm Boolean functions.
Particular cases of Boolean functions are maxterms and minterms. Recall that a maxterm (resp., minterm) of n variables [x.sub.1], [x.sub.2], ..., [x.sub.n] is a Boolean function F such as
where [mathematical expression not reproducible] is the coefficient matrix that holds the values of all of the minterms of the logic function, while the [mathematical expression not reproducible] vector holds all of the minterms.
Equation (5) shows that the Kronecker product between two logic variables represents all of the minterms; similarly, the following can also be shown:
When we calculated the minterms, we used variable [x.sub.4] as MSB and [u.sub.1] as LSB.
(b) Identifying a suitable implicant (see Definition 4) that covers the selected minterm,
(d) Repeating steps (a) to (c) until every minterm is covered at least once.
Similarly if p has k' ones then no minterm of F can have less than k' ones.
Let [H.sub.ij] be the subset of G that sends the ith bit of the minterm to the bit position j.
(Dependence Function of Three variables) The Boolean function of variables A, B, and C, whose minterms correspond to the values <[v.sub.A], [v.sub.B], [v.sub.C]> for which P[(A = [v.sub.A] [conjunction] B = [v.sub.B] [conjunction] C = [v.sub.C]) [is greater than] P(A - [v.sub.A] [conjunction] B = [v.sub.B] [conjunction] C = [v.sub.C]).sub.MI] is called the dependence function of variables A, B, and C.
The Boolean function of variables [I.sub.1], [I.sub.2], ..., [I.sub.k], whose minterms correspond to the values <[v.sub.1], [v.sub.2], ..., [v.sub.k]> for which