License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2023.14
URN: urn:nbn:de:0030-drops-175172
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Beame, Paul ; Koroth, Sajin

On Disperser/Lifting Properties of the Index and Inner-Product Functions

LIPIcs-ITCS-2023-14.pdf (0.9 MB)


Query-to-communication lifting theorems, which connect the query complexity of a Boolean function to the communication complexity of an associated "lifted" function obtained by composing the function with many copies of another function known as a gadget, have been instrumental in resolving many open questions in computational complexity. A number of important complexity questions could be resolved if we could make substantial improvements in the input size required for lifting with the Index function, which is a universal gadget for lifting, from its current near-linear size down to polylogarithmic in the number of inputs N of the original function or, ideally, constant. The near-linear size bound was recently shown by Lovett, Meka, Mertz, Pitassi and Zhang [Shachar Lovett et al., 2022] using a recent breakthrough improvement on the Sunflower Lemma to show that a certain graph associated with an Index function of that size is a disperser. They also stated a conjecture about the Index function that is essential for further improvements in the size required for lifting with Index using current techniques. In this paper we prove the following;
- The conjecture of Lovett et al. is false when the size of the Index gadget is less than logarithmic in N.
- The same limitation applies to the Inner-Product function. More precisely, the Inner-Product function, which is known to satisfy the disperser property at size O(log N), also does not have this property when its size is less than log N.
- Notwithstanding the above, we prove a lifting theorem that applies to Index gadgets of any size at least 4 and yields lower bounds for a restricted class of communication protocols in which one of the players is limited to sending parities of its inputs.
- Using a modification of the same idea with improved lifting parameters we derive a strong lifting theorem from decision tree size to parity decision tree size. We use this, in turn, to derive a general lifting theorem in proof complexity from tree-resolution size to tree-like Res(⊕) refutation size, which yields many new exponential lower bounds on such proofs.

BibTeX - Entry

  author =	{Beame, Paul and Koroth, Sajin},
  title =	{{On Disperser/Lifting Properties of the Index and Inner-Product Functions}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-175172},
  doi =		{10.4230/LIPIcs.ITCS.2023.14},
  annote =	{Keywords: Decision trees, communication complexity, lifting theorems, proof complexity}

Keywords: Decision trees, communication complexity, lifting theorems, proof complexity
Collection: 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)
Issue Date: 2023
Date of publication: 01.02.2023

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