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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 S n . 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 φ ndoi:10.1093/jigpal/8.6.787 fatcat:hm3bldzb7ngwvly6ogmg56c7p4