CNF Encodings of Parity

03/02/2022

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by   Gregory Emdin, et al.

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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(2^n/(s+1)/n) and k ≥ n/(s+1); 2) the minimum number of clauses is at least 3n. We derive the first two bounds from the Satisfiability Coding Lemma due to Paturi, Pudlak, and Zane.