Example
Bits received on line $x$ synchronized w/ clock.
Output 2 is one if 4 consecutive 0s or 2 consecutive 1s are received.
If a 3rd 1 is received for example, the output remains 1.
y2n y1n y0n
x^n ---- D ----- D ------ D ----
| | |
------------------------
CLK |
zn
$y_2^n = x^{n-1}$
$y_1^n = y_2^{n-1} = x^{n-2}$
$y_0^n = y_1^{n-1} = x^{n-3}$
$z^n = \bar x^n \bar y^n_2 \bar y_1^n \bar y_0^n + x^ny_2^n$
| $(y_2y_1y_0)^n$ | $x^n = 0$ | $x^n = 1$ |
|---|---|---|
| 000 | 000,1 | 100,0 |
| 001 | 000,0 | 100,0 |
| 010 | 001,0 | 101,0 |
| 011 | 001,0 | 101,0 |
| 111 | 011,0 | 111,1 |
| 110 | 011,0 | 111,1 |
| 101 | 010,0 | 110,1 |
| 100 | 010,0 | 110,1 |
$(y_2y_1y_0)^{n+1}, z^n$
| $q^n$ | $x^n = 0$ | $x^n = 1$ |
|---|---|---|
| a | a,1 | d,0 |
| b | a,0 | h,0 |
| c | b,0 | g,0 |
| d | b,0 | g,0 |
| e | d,0 | e,1 |
| f | d,0 | e,1 |
| g | c,0 | f,1 |
| h | c,0 | f,1 |
c equals d since they both have the same output and the same next state.
ef equivalent to gh since same output and equal next state.
| $q^n$ | $x^n = 0$ | $x^n = 1$ |
|---|---|---|
| A | A,1 | D,0 |
| B | A,0 | D,0 |
| C | B,0 | D,0 |
| D | C,0 | D,1 |
We went from 8 states to 4! (Only 2 FFs).
We can systematically minimize the state tables.
We have to find equal (or equivalent states).
By implication: what states are implied by a given pair?