Page 156 - Bebop to The Boolean Boogie An Unconventional Guide to Electronics Fundamentals, Components, and Processes

State Diagrams, State Tables, and State Machines 7 37

But wait, there’s more! A further analysis of the 138 solutions requiring
seven product terms yields the following:

66 solutions requiring 17 literals
24 solutions requiring 18 literals
48 solutions requiring 19 literals

Thus, the chances of a random assignment resulting in an optimal solution
is relatively slight. Fortunately, there are computer programs available to aid
designers in this task.5 One solution resulting in the minimum number of
product terms and literals is shown in Figure 12-6.

Current 5tate %ate
q2 c1? @ Assignments
0 0 0 10-cents
0 0 1 15-cents
0 1 0 -
0 1 1 20-cents
1 0 0 0-cents
1 0 1 5-cents
1 1 0 -
1 1 1 -

d2 dl dO State
State Variable Next 5 ta te Assignments
Reg is-ters

Figure 12-6. Example binary encoded state assignment

A truth table for the controller function can now be derived from the state
table shown in Figure 12-3 by replacing the assignments in the current state
column with the corresponding binary patterns for the state variable outputs
(q2, q~, and q,O), and replacing the assignments in the next state column with
the corresponding binary patterns for the state variable inputs (d2, dl, and do).
The resulting equations can then be derived from the truth table by means of
standard algebraic or Karnaugh map techniques. As an alternative, a computer

5 The author used the program BOQL, which was created by his friend Alon Kfir (a man with a
size-16 brain if ever there was one).