Abstract

We introduce methods to generate uniform families of hard propositional tautologies. The tautologies are essentially generated from a single propositional formula by a natural action of the symmetric group Sn.The basic idea is that any Second Order Existential sentence Ψ can be systematically translated into a conjunction φ of a finite collection of clauses such that the models of size n of an appropriate Skolemization Ψ are in one-to-one correspondence with the satisfying assignments to φn: the Sn-closure of φ, under a natural action of the symmetric group Sn. Each φn is a CNF and thus has depth at most 2. The size of the φn's bounded by a polynomial in n. Under the assumption NEXPTIME ≠ co-NEXPTIME, for any such sequence φn for which the spectrum S := {n : φn satisfiable] is NEXPTIME-complete, the tautologies ¬φ∉s do not have polynomial length proofs in any propositional proof system.Our translation method shows that most sequences of tautologies being studied in propositional proof complexity can be systematically generated from Second Order Existential sentences and moreover, many natural mathematical statements can be converted into sequences of propositional tautologies in this manner.We also discuss algebraic proof complexity issues for such sequences of tautologies. To this end, we show that any Second Order Existential sentence Ψ can be systematically translated into a finite collection of polynomial equations Q=0 such that the models of size n of an appropriate skolemization Ψ are in one-to-one correspondence with the solutions to Qn=0: the Sn-closure of Q=0, under a natural action of the symmetric group Sn. The degree of Qn is the same as that of Q, and hence is independent of n, and the number of variables is no more than a polynomial in n