Complete problems in the first-order predicate calculus

Abstract

Some problems concerning the satisfiability of first-order predicate calculus formulae in Schonfinkel-Bernays form provide a natural hierarchy of complete problems for various complexity classes. Also, problems concerning the existence of resolution proofs from sets of clauses not necessarily in Schonfinkel-Bernays form provide another such hierarchy. In this way we obtain problems complete for P, NP, PSPACE, deterministic and nondeterministic exponential, deterministic and nondeterministic double exponential time, and exponential space. The results concerning resolution proofs may have practical implications for the design of resolution theorem proving programs. Also, these results enable us to make precise statements about the relative difficulty of various resolution strategies. Some connections with temporal logic and alternating Turing machines are discussed.