SAT for pedestrians

The aim of this short note is mainly pedagogical. It summarizes some knowledge about Boolean satisfiability (SAT) and the P=NP? problem in an elementary mathematical language. A convenient scheme to visualize and manipulate CNF formulae is introduced. Also some results like the formulae for the number of unsatisfied clauses and the number of solutions might be unknown. Introduction. Specifically, I will formulate the problem of finding solutions for SAT in fairly elementary terms. Ordinary school calculus will be sufficient to do most of the calculations. One will see how the exponential character of the assignment space undermines any straightforward attempt of finding an efficient algorithm for the problem. Some new necessary conditions (“checks”) for a formula to be satisfiable (or not) will be deduced. Most checks can be calculated in polynomial time(“p.t.”). Their usefulness has not yet been tested for large formulae. The search for “efficient algorithms” (i.e. polynomial time procedures), is important for settling the “P=NP?” problem, one of the outstanding problems of 21 century mathematics. For a definition of complexity classes P and NP a vast amount of literature, both in print and online is available. A simple Google search will do more than necessary. The essence is: problems in class P can be solved on a classical Turing computer, and their running times are bounded by a polynomial in the length of the problem (suitably defined). The definition of NP, on the other hand, does not address the solution of