Abstract
We consider the question of certifying unsatisfiability of random 3-CNF formulas. At which densities can we hope for a simple sufficient condition for unsatisfiability that holds almost surely? We study this question from the point of view of definability theory. The main result is that first-order logic cannot express any sufficient condition that holds almost surely on random 3-CNF formulas with n/sup 2-/spl alpha// clauses, for any irrational positive number /spl alpha/. In contrast, it can when the number of clauses is n/sup 2+/spl alpha//, for any positive /spl alpha/. As an intermediate step, our proof exploits the planted distribution for 3-CNF formulas in a new technical way. Moreover, the proof requires us to extend the methods of Shelah and Spencer for proving the zero-one law for sparse random graphs to arbitrary relational languages.
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