SOP 中的布尔表达式

Question

I'm new to boolean expressions.

I've been given the task to simplify F(w,x,y,z) = xy' + x'z' + wxz + wx'y by using K map.

I've done it and the result is wx'+w'y'+xyz .

Now I have to "Write it in a standard SOP form. You need to provide the steps through which you get the standard SOP".

And i have no idea how to do it. I thought result after k map is sop.

Answer 1

Yes, you already have it in SOP form. But the second question is about Standard (aka canonical) SOP form. That's much simpler to find than having to use K-maps (but it's often long), it's just the sum of minterms.

Answer 2

I think your solution does not cover all ones . These Karnaugh maps show the original expression, the simplified version (minimal SOP) and the canonical SOP, where every product contains all literals (all given variables or their negation).

The original expression is

F(w,x,y,z) = x·¬y + ¬x·¬z + w·x·z + w·¬x·y

– there are two fours and two pairs circled in the corresponding (first one) K-map.

The original expression simplified using K-map (shown in the second one):

F(w,x,y,z) = x·¬y + ¬x·¬z + w·y·z

is different than yours, but you can check for example with wolframalpha online tool, that it is the simplified original expression.

It is also the minimal DNF, but not a sum of minterms (where the output is equal to 1), because there are not all variables in every product of the sum.

The third K-map shows ten minterms circled. They form the canonical DNF:

F(w,x,y,z) = m0 + m2 + m4 + m5 + m8 + m10 + m11 + m12 + m13 + m15 =
           = ¬w·¬x·¬y·¬z + ¬w·¬x·y·¬z + ¬w·x·¬y·¬z + ¬w·x·¬y·z + w·¬x·¬y·¬z 
             + w·¬x·y·¬z + w·¬x·y·z + w·x·¬y·¬z + w·x·¬y·z + w·x·y·z

I checked your simplified expression, but there are not all ones covered (even if there were some useful do not care states (marked X)). Maybe you made a typo. Or could there be a typo in the original expression?

Answer 3

We can implement the K-Map algorithm in python for 4 variables, as shown below. The function accepts the Boolean function in SOP (sum of products) form and the names of the variables and returns a simplified reduced representation. Basically you need to create rectangular groups containing total terms in power of two like 8, 4, 2 and try to cover as many elements as you can in one group.

For example, the function can be represented F(w,x,y,z) = xy' + x'z' + wxz + wx'y in SOP form as f(w,x,y,z)=∑(0,2,4,5,8,10,11,12,13,15), as can be seen from the below table:

As can be seen from the output of the next code snippet, the program outputs the simplified form x¬y + ¬x¬z + wyz , where negation of a boolean variable x is represented as ¬x in the code.

from collections import defaultdict
from itertools import permutations, product
    
def kv_map(sop, vars):
    
    sop = set(sop)
    not_covered = sop.copy()
    sop_covered = set([])
    
    mts = [] # minterms
    
    # check for minterms with 1 variable
    all_3 = [''.join(x) for x in product('01', repeat=3)]
    for i in range(4):
        for v_i in [0,1]:
                if len(not_covered) == 0: continue
                mt = ('' if v_i else '¬') + vars[i]
                s = [x[:i]+str(v_i)+x[i:] for x in all_3]
                sop1 = set(map(lambda x: int(x,2), s))
                if len(sop1 & sop) == 8 and len(sop_covered & sop1) < 8: # if not already covered
                    mts.append(mt)
                    sop_covered |= sop1
                    not_covered = not_covered - sop1
        if len(not_covered) == 0:
           return mts
    
    # check for minterms with 2 variables
    all_2 = [''.join(x) for x in product('01', repeat=2)]
    for i in range(4):
        for j in range(i+1, 4):
            for v_i in [0,1]:
                for v_j in [0,1]:
                    if len(not_covered) == 0: continue
                    mt = ('' if v_i else '¬') + vars[i] + ('' if v_j else '¬') + vars[j]
                    s = [x[:i]+str(v_i)+x[i:] for x in all_2]
                    s = [x[:j]+str(v_j)+x[j:] for x in s]
                    sop1 = set(map(lambda x: int(x,2), s))
                    if len(sop1 & sop) == 4 and len(sop_covered & sop1) < 4: # if not already covered
                        mts.append(mt)
                        sop_covered |= sop1
                        not_covered = not_covered - sop1
    if len(not_covered) == 0:
        return mts

    # check for minterms with 3 variables similarly (code omitted)
    # ... ... ...
    
    return mts
    
mts = kv_map([0,2,4,5,8,10,11,12,13,15], ['w', 'x', 'y', 'z'])
mts
# ['x¬y', '¬x¬z', 'wyz']

The following animation shows how the above code (greedily) simplifies the Boolean function given in SOP form (the basic goal is to cover all the 1s with minimum number of power-2 blocks). Since the algorithm is greedy it may get stuck to some local minimum, that we need to be careful about.