Example: Divide-and-Conquer Programs

•  deriv(U+V,X,DU+DV) :- deriv(U,X,DU), deriv(V,X,DV).
   deriv(X,X,1)       :- atom(X).
   deriv(C,X,0)       :- atom(C), C \= X.

• 
  (size measure: number of functors).

• New Approach:

  

• For each particular computation, we obtain a succession of values for (which depends on the structure of the input).

• In this case, any succession yields the same value.

• Thus, (taking ) equivalent to (the difference equation):

  

• Solution: .

• General Approach:

  

• Solution: .