Implicants and Cubes

An examination of Table 6.2 reveals several rows that differ in only one posi-tion. For example, the rows 0010 and 0011 differ only in the rightmost (least signficant) position. Thus, if we allow bits of in to be X (matches either 0 or 1), we can replace the two rows 0010 and 0011 by the single row 001X. This new row 001X corresponds to a product term that includes just three of the four inputs (or their complements):

f_001X= d∧ c ∧ b = (d ∧ c ∧ b ∧ a) ∨ (d ∧ c ∧ b ∧ a). (6.3) The 001X product term subsumes the two minterms corresponding to 0010 and 0011 because it is true when at least one of them is true and nowhere else. Thus, in a logic function we can replace the two minterms for 0010 and 0011 with the simpler product term for 001X without changing the function.

A product term like 001X (d∧ c ∧ b) that is only true when a function is true is called an implicant of the function. This is just a way of saying that the product term implies the function. A minterm may or may not be an implicant of a function. The minterm 0010 (d∧ c ∧ b ∧ a) is an implicant of the prime

000 001

Figure 6.4: A cube visualization of the three-bit prime number function. Each vertex corresponds to a minterm, each edge to a product of two variables, and each face to a single variable. The bold vertices, edges and the shaded face show implicants of the three-bit prime number function.

function because it implies the function — when 0010 is true, the function is true. However, minterm 0100 (d∧ c ∧ b ∧ a) is a minterm, it is a product that includes each input or its complement, but it is not an implicant of the prime function. When 0100 is true, the prime function is false because 4 is not a prime.

If we say that a product is a minterm of a function we are saying that it is both a minterm, and an implicant of the function.

It is often useful to visualize implicants on a cube as shown in Figure 6.4.

This figure shows a bit prime number function mapped onto a three-dimensional cube. Each vertex of the cube represents a minterm. The cube makes it easy to see which minterms and implicants can be combined into larger implicants.¹ Minterms that differ in just one variable (e.g., 001 and 011) are ad-jacent to each other and the edge between two vertices (e.g., 01X) represents the product that includes the two minterms (the OR of the two adjacent minterms).

Edges that differ in just one variable (e.g., 0X1 and 1X1) are adjacent on the cube and the face between the edges represents the product that includes the two edge products (e.g., XX1). In this figure, the three-bit prime number func-tion is shown as five bold vertices (001, 010, 011, 101, and 111). Five bold edges connecting these vertices represent the five two variable implicants of the func-tion (X01, 0X1, 0X1, X11, and 1X1). Finally, the shaded face (XX1) represents the single one variable implicant of the function.

A cube representation of the full four-bit prime number function is shown

1One implicant is larger than another if it contains more minterms. For example, implicant 001 has size 1 because it contains just one minterm. Implicant 01X has size 2 because it contains two minterms (001 and 011) and hence is larger.

0001 0011

1011 0101

0111

0010

1101

Figure 6.5: A cube visualization of the four-bit prime number function.

Number of variables

4 3 2 1

0001 001X 0XX1

0010 00X1 0011 0X01 0101 0X11 0111 01X1 1011 X011 1101 X101

Table 6.3: All implicants of the 4-bit prime number function. Prime implicants are shown in bold.

in Figure 6.5. To avoid clutter only the minterms of the function are labeled.

To represent four variables, we draw a four-dimensional cube as two three-dimensional cubes, one within the other. As before, vertices represent minterms, edges represent products with one X, and faces represent products with two Xs.

In four dimensions, however, we also have eight volumes that represent products with three Xs. For example, the outside cube represents 1XXX — all minterms where the leftmost (most significant) bit d is true. The four-bit prime number function has seven vertices (minterms). Connecting adjacent vertices gives seven edges (implicants with a single X). Finally, connecting adjacent edges gives a single face (implicant with two Xs). All of these implicants of the four-bit prime number function are shown in Table 6.3.

Computer programs that synthesize and optimize logic functions, such as we will be using in this class, use an internal representation of logic functions as a set of implicants where each implicant is represented as a vector with elements 0, 1, or X. To simplify a function, the first step is to generate all of the implicants of the function, such as that shown in Table 6.3. A systematic procedure to do

this is to start with all minterms of the function (the ‘4’ column of Table 6.3).

For each minterm, attempt to insert an X into each variable position. If the result is an implicant of the function, insert it in a list of single X implicants (the ‘3’ column of Table 6.3). Then for each implicant with one X, attempt to insert an X into each of the remaining non X positions and if the result is an implicant, insert it in a list of two X implicants. The process is repeated for two X implicants and so on until no further implicants are generated. Such a procedure will, given a list of minterms, generate a list of implicants.

If an implicant x has the property that replacing any 0 or 1 digit of x with an X results in a product that is not an implicant, then we call x a prime implicant.² A prime implicant is an implicant that cannot be made any larger and still be an implicant. The prime implicants of the prime number function are shown in bold in Table 6.3.

If a prime implicant of a function x is the only prime implicant that contains a particular minterm of the function y, we say that x is an essential prime implicant. x is essential because no other prime implicant includes y. Without x a collection of prime implicants will not include minterm y. All four of the prime implicants of the four-bit prime number function are essential. Implicant 0XX1 is the only prime implicant that includes 0001 and 0111. Minterm 0010 is included only in prime implicant 001X, X101 is the only prime implicant that includes 1101, and 1011 is only included in prime implicant X011.