CNF Encodings of Parity

Abstract

The minimum number of clauses in a CNF representation of the parity function x 1 ⊕ x 2 ⊕ · · · ⊕ x n is 2 n − 1 . One can obtain a more compact CNF encoding by using non-deterministic variables (also known as guess or auxiliary variables). In this paper, we prove the following lower bounds, that almost match known upper bounds, on the number m of clauses and the maximum width k of clauses: 1) if there are at most s auxiliary variables, then m ≥ Ω (cid:0) 2 n/ ( s +1) /n (cid:1) and k ≥ n/ ( s + 1); 2) the minimum number of clauses is at least 3 n . We derive the first two bounds from the Satisfiability Coding Lemma due to Paturi, Pudlák, and Zane using a tight connection between CNF encodings and depth-3 circuits. In particular, we show that lower bounds on the size of a CNF encoding of a Boolean function imply depth-3 circuit lower bounds for this function. of