A perspective on certain polynomial-time solvable classes of satisfiability

The scope of certain well-studied polynomial-time solvable classes of Satisfiability is investigated relative to a polynomial-time solvable class consisting of what we call matched formulas. The class of matched formulas has not been studied in the literature, probably because it seems not to contain many challenging formulas. Yet, we find that, in some sense, the matched formulas are more numerous than Horn, extended Horn, renamable Horn, qHorn, CC-balanced, or Single Lookahead Unit Resolution (SLUR) formulas. In addition, we find that relatively few unsatisfiable formulas are in any of the above classes. However, there are many unsatisfiable formulas that can be solved in polynomial time by restricting resolution so as not to generate resolvents of size greater than the number of literals in a maximum length clause. We use the well-studied random k-SAT model M(n;m;k) for generating CNF formulas with m clauses, each with k distinct literals, from n variables. We show, for all m=n > 2=k(k 1), the probability that a random formula is SLUR, q-Horn, extended Horn, CC-balanced, or renamable Horn tends to 0 as n goes to infinity. We show, for all m=n < :64, that random formulas are matched formulas with probability tending to 1 as n goes to infinity. The propositional connection graph is introduced to represent clause structure for formulas with general-width clauses. The probabilistic analysis highlights the vulnerability of previously studied polynomial-time solvable classes to certain cyclic substructures in this graph. The matched formulas are not vulnerable to such structures. Thus, we believe that part of the significance of this work lies in guiding the future development of polynomial-time solvable classes of Satisfiability. This paper has been submitted to Discrete Applied Mathematics.