Simplify a Boolean Algebra expression with don't cares

An arbitrary function with "don't cares" can be defined using either BooleanFunction or BooleanConvert, but in those cases, Mathematica makes no effort to find the minimal representation of such a function.

Instead, use BooleanMinimize with the optional third argument: an assumed condition on the variables. In your case, you can specify that your "don't cares" are 0-9 by making the condition be A ∧ (B ∨ C).

Then the calculation

BooleanMinimize[
  (A ∧ ¬ B ∧ C ∧ ¬ D) ∨ (A ∧ B ∧ ¬ C ∧ ¬ D) ∨ (A ∧ B ∧ C ∧ ¬ D),
  "DNF", 
  A ∧ (B ∨ C)] 

will produce the desired output of !D.

Thank's to @Misha Lavrov (a great thanks !), here is how you can solve the problem :

BooleanMinimize[  
    BooleanFunction[{
        {1,0,1,0}-> 1,
        {1,0,1,1}-> 0,
        {1,1,0,0}-> 1,
        {1,1,0,1}-> 0,
        {1,1,1,0}-> 1,
        {1,1,1,1}-> 0
      },{a,b,c,d}],  
    "SOP" (* = "DNF" *),
    BooleanFunction[{
        {1,1,_,_}-> 1, (* all cases above ... *)
        {1,_,1,_}-> 1, (* ... are "do care" *)
        {_,_,_,_}-> 0 (* other cases are "don't care" *)
      },{a,b,c,d}]]  

! d

Note :

@Misha Lavrov's comment (just below) was written before I added the third argument to BooleanMinimize. At this time there were some "don't care" states in the first BooleanFunction[...]