Stable states are marked with circles on a Karnaugh map (or flow table, with columns as input and rows as states).
Example with SR Latch:
| $q/SR$ | 00 | 01 | 11 | 10 |
|---|---|---|---|---|
| 0 | (0) | (0) | X | 1 |
| 1 | (1) | 0 | X | (1) |
Only one input can change at a time (setup/hold times for an FF).
Primitive flow table: One stable state each row
Example:
Control delay: Input $x$ coming infrequently synchronized with clock $c$. Output $z$ geos 1 one clock cycle after $x$ also synced with $c$.
| $q/xc$ | 00 | 01 | 11 | 10 | 00 (z) | 01 (z) | 11 (z) | 10 (z) |
|---|---|---|---|---|---|---|---|---|
| 1 | (1) | 2 | 0 | |||||
| 2 | 1 | (2) | 3 | 0 | ||||
| 3 | (3) | 4 | 0 | |||||
| 4 | 5 | (4) | 0 | |||||
| 5 | (5) | 6 | 0 | |||||
| 6 | 1 | (6) | 1 |
Row 1: no clock, no input; when clock, go to state 2
Row 2: no clock, no input, is next clock tick, go to state 1; still clock, no input
Minimization: (1, 2) (3, 4, 5) (6)
| $q/xc$ | 00 | 01 | 11 | 10 | 00 (z) | 01 (z) | 11 (z) | 10 (z) |
|---|---|---|---|---|---|---|---|---|
| a | (a) | (a) | b | - | 0 | 0 | 0 | - |
| b | (b) | c | (b) | (b) | 0 | - | 0 | 0 |
| c | a | (c) | - | - | - | 1 | - | - |
| $y_2y_1/xc$ | 00 | 01 | 11 | 10 | 00 (z) | 01 (z) | 11 (z) | 10 (z) |
|---|---|---|---|---|---|---|---|---|
| 00 | 00 | 00 | 01 | X | 0 | 0 | 0 | X |
| 01 | 01 | 11 | 01 | 01 | 0 | X | 0 | 0 |
| 11 | 00 | 11 | X | X | X | 1 | X | X |
| 10 | X | X | X | X | X | X | X | X |
Two or more state variables changing.
We can fix this by assigning adjacent states or adding spare states.