Generating Hard Tautologies Using Predicate Logic and the Symmetric
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 .
The basic idea is that any Second Order Existential sentence can be systematically translated into a conjunction of a finite collection of clauses such that the models of size n of an appropriate Skolemization are in one-to-one correspondence with the satisfying assignments to : the -closure of , under a natural action of the symmetric group . Each is a CNF and thus has depth at most 2. The size of the 's is bounded by a polynomial in n. Under the assumption NEXPTIME co-NEXPTIME, for any such sequence for which the spectrum is NEXPTIME-complete, the tautologies 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 can be systematically translated into a finite collection of polynomial equations such that the models of size n of an appropriate skolemization are in one-to-one correspondence with the solutions to : the -closure of , under a natural action of the symmetric group . The degree of is the same as that of , and hence is independent of n, and the number of variables is no more than a polynomial in n. Furthermore, we briefly describe how, for the corresponding sequences of tautologies , the rich structure of the closed, uniformly generated, algebraic systems has profound consequences on on the algebraic proof complexity of