My reasoning is as follows.
- Every boolean formula can be expressed as a sum-of-products.
- Every sum-of-produts is a list of minterms.
- For each minterm, there is 1 combination of inputs that satisfy the formula.
So if I have the sum-of-products representation of a boolean formula I can tell if it satisfiable.
But that is the SAT problem, which is hard.
So the hard part of the 3 step procedure above must be transforming a boolean formula into a sum of products.
Is this reasoning right?