Post-Karnaugh map simplification

Bipolarity

So I have to implement a 4 input 1 output circuit. I am given the Karnaugh map (obviously a 4by4) and have to build the circuit.

I have already determined the essential prime implicants for my map and three possible permutations of nonessential prime implicants.

So let's say I pick a permutation. I will obtain a boolean expression, but how could I simplify this expression? Most textbooks I have explain only how to draw the K-map itself and not how further simplification can be done after the boolean expression from the K-map has been generated. Is there an algorithm or procedure that allows one to further simplify the expression?

Also, are there heuristics in determining which set of nonessential prime implicants will produce simplified expressions using the fewest number of gates?

Also, are there heuristics in determining whether one should build using 0's logic or 1's logic?

Thanks for all the help.

BiP

NascentOxygen

Just guessing, but I would think that after simplifying the Karnaugh map, there would be little/no further simplification possible.