License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.MFCS.2022.47
URN: urn:nbn:de:0030-drops-168455
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Emdin, Gregory ; Kulikov, Alexander S. ; Mihajlin, Ivan ; Slezkin, Nikita

CNF Encodings of Parity

LIPIcs-MFCS-2022-47.pdf (0.7 MB)


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.

BibTeX - Entry

  author =	{Emdin, Gregory and Kulikov, Alexander S. and Mihajlin, Ivan and Slezkin, Nikita},
  title =	{{CNF Encodings of Parity}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{47:1--47:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-168455},
  doi =		{10.4230/LIPIcs.MFCS.2022.47},
  annote =	{Keywords: encoding, parity, lower bounds, circuits, CNF}

Keywords: encoding, parity, lower bounds, circuits, CNF
Collection: 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)
Issue Date: 2022
Date of publication: 22.08.2022

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