The Intractability of Resolution

We prove that, for infinitely many disjunctive normal form propositional calculus tautologies ~:, the length of the shortest resolution proof of ~: cannot be bounded by any polynomial of the length of ~=. The tautologies we use were introduced by Cook and Reckhow (1979) and encode the pigeonhole principle. Extended resolution can furnish polynomial length proofs of these formulas. 1. Definitions and background 1.1. Resolution theorem proving Theorem proving using resolution was introduced by Robinson [8]. The method is applicable to first-order predicate calculus or to propositional calculus. If the formula to be proved is a consequence of axioms, resolution is used to prove the disjunction of the original formula with the negations of those axioms it depends on. Predicate calculus formulas are reduced to propositional calculus formulas by using quantifier elimination techniques. Furthermore, the formula to be proved is put into disjunctive normal fo rm (DNF). The theorem proving task is then reduced to proving that a given D N F propositional calculus formula is a tautology. For the purpose of showing nonpolynomial complexity, we consider only D N F propositional calculus formulas. We use the notation '+ ' for logical 'or', juxtaposition for 'and' , and ' " for negation (x' is 'not x'). An example of a D N F propositional tautology is: abc' + ab'd + ab' c 'd ' + a 'd + a' c'd' + c. To define resolution we let s r be a D N F propositional calculus tautology , and we describe how resolution produces a proof of s r. The conjunctions of which ~ is a disjunction are called clauses and ~: is considered to be a set of clauses. The variables and the negated variables of which a clause is a conjunction are called literals. A clause is considered a set of literals~ A clause covers a truth assignment to the variables in ~: if the truth assignment makes the clause true. The resolution procedure shows ~ to be a tautology by demonstrating that every truth assignment is covered by some clause in s r. The procedure starts with the original set of clauses in s r and repeatedly generates new clauses from existing ones. Each new clause is derived from two previously existing clauses and the new one covers only truth assignments 0304-3975/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)