CNF Encodings of Parity

Authors Gregory Emdin, Alexander S. Kulikov , Ivan Mihajlin, Nikita Slezkin

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Author Details

Gregory Emdin
  • St. Petersburg State University, Russia
Alexander S. Kulikov
  • Steklov Mathematical Institute at St. Petersburg, Russian Academy of Sciences, Russia
  • St. Petersburg State University, Russia
Ivan Mihajlin
  • Steklov Mathematical Institute at St. Petersburg, Russian Academy of Sciences, Russia
Nikita Slezkin
  • Steklov Mathematical Institute at St. Petersburg, Russian Academy of Sciences, Russia


Research is partially supported by Huawei (grant TC20211214628).

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Gregory Emdin, Alexander S. Kulikov, Ivan Mihajlin, and Nikita Slezkin. CNF Encodings of Parity. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 47:1-47:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


The minimum number of clauses in a CNF representation of the parity function x₁ ⊕ x₂ ⊕ … ⊕ 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, 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Circuit complexity
  • encoding
  • parity
  • lower bounds
  • circuits
  • CNF


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