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                 __                   __ __ __
4.3)  f=(x1 x2 + x1 x3)(x1(x4 + x3) + x1 x3 x4 x5)

a) does the circuit have any static hazards?
                            __ 
yes, f(x1, 0, 1, 1, 0) = x1*x1

b) does the circuit have any dynamic hazards?
                               __
yes, f(x1, 1, 1, 0, 0) = (x1 + x1) * x1

c) find a hazard free SOP realization for the following function:

          x1
         ____
    ---------
    |1   1 1|
    |1 1 1  ||x3
    ---------
       ---
       x2
    __ __   __ __   __                         __
f = x2 x3 + x1 x2 + x1 x3 + x2 x3 + x1 x2 + x1 x3

4.4) For the following state table determine if possible a unicode state assignment in which each transition requires only a single state variable change, under the single input variable change assumption.

    00 01 11 10
   +----------+
 1 |1  2  1  1|
 2 |3  2  2  1|
 3 |3  4  2  3|
 4 |5  4  4  3|
 5 |5  2  4  5|
   +----------+

first, the list of required adjacencies:

1 | 2
2 | 1, 3, 5
3 | 2, 4
4 | 3, 5
5 | 2, 4

the following assignment satisfies all the required adjacencies:
          y1
         ____
    ---------
    |1 2 3  |
    |  5 4  ||y3
    ---------
       ---
       y2



4.7) Find a minimal variable unicode SST assignment for the previous flow table.

The answer to 4.4 IS a minimal variable unicode SST assignment.

4.8) For the following combinational functions derive hazard free SOP ralizations. (we didn't do POS)

a) F(x1, x2, x3, x4) = SUM( m0, m1, m7, m15)

          x1
         ____
    +--------+
    |1       |
 x4||1       | 
   ||  1 1   ||x3
    |        ||
    +--------+
       ---
       x2

In this case, the minimal realization is hazard free:
    __ __ __
f = x1 x2 x3 + x2 x3 x4


4.9) For the combinational function,

f(x1, x2, x3, x4) = SUM(m0, m2, m3, m4, m5, m6, m7, m12, m13, m15)

does the SOP realization 
      __ __   __                 __
  f = x1 x4 + x1 x3 + x2 x4 + x2 x3

have any hazards for single input changes?  What does this imply about the necessity of a hazard free realization containing ALL prime implicants?

          x1
         ____
    ---------
    |1 1 1  |
 x4||  1 1  | 
   ||1 1 1  ||x3
    |1 1    ||
    ---------
       ---
       x2
No, there are no hazards.

                __
Prime implicant x1 x3 is missing.

This implies For a 2-level AND-OR Circuit it is sufficient but not necessary to have the
sum of all prime implicants.