Expander CNFs have Exponential DNNF Size

A negation normal form (NNF) is a circuit on unbounded fanin conjunction (AND) and disjunction (OR) gates, with fanin one negation (NOT) gates applied to input gates only. A NNF is decomposable (DNNF) if, for each AND node, the subcircuits rooted at the children of the node are variable disjoint. We prove an unconditional exponential lower bound on the DNNF size of CNF formulas; thus far, only a superpolynomial lower bound was known, under the condition that the polynomial hierarchy does not collapse. As corollaries, we derive that the negation of a DNNF has in general size exponential in the original DNNF (which was known to hold if P 6= NP), and that the language of prime implicates (PI) is exponentially more succinct than DNNFs (which was unknown). These results settle three open problems in the area of knowledge compilation [DM02].