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. Several important complexity questions could be resolved if we could make substantial improvements in the input size required for lifting with the Index function, 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 shown by Lovett, Meka, Mertz, Pitassi and Zhang using a recent breakthrough improvement on the Sunflower Lemma to show that a certain graph associated with the Index function of near-linear 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;
1) The conjecture of Lovett et al. is false when the size of the Index gadget is logN−\omega(1).
2) Also, the Inner-Product function, which satisfies the disperser property at size O(logN), does not have this property when its size is log N−\omega(1).
3) Nonetheless, using Index gadgets of size at least 4, we prove a lifting theorem for a restricted class of communication protocols in which one of the players is limited to sending parities of its inputs.
4) Using the ideas from this lifting theorem, we derive a strong lifting theorem from decision tree size to parity decision tree size. We use this to derive a general lifting theorem in proof complexity from tree-resolution size to tree-like Res(\oplus) refutation size, which yields many new exponential lower bounds on such proofs.
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A ZPP NP[1] Lifting Theorem
The complexity class ZPP NP[1] (corresponding to zero-error randomized algorithms with access to one NP oracle query) is known to have a number of curious properties. We further explore this class in the settings of time complexity, query complexity, and communication complexity. • For starters, we provide a new characterization: ZPP NP[1] equals the restriction of BPP NP[1] where the algorithm is only allowed to err when it forgoes the opportunity to make an NP oracle query. • Using the above characterization, we prove a query-to-communication lifting theorem , which translates any ZPP NP[1] decision tree lower bound for a function f into a ZPP NP[1] communication lower bound for a two-party version of f . • As an application, we use the above lifting theorem to prove that the ZPP NP[1] communication lower bound technique introduced by Göös, Pitassi, and Watson (ICALP 2016) is not tight. We also provide a “primal” characterization of this lower bound technique as a complexity class.
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- Award ID(s):
- 1942742
- PAR ID:
- 10275802
- Date Published:
- Journal Name:
- ACM Transactions on Computation Theory
- Volume:
- 12
- Issue:
- 4
- ISSN:
- 1942-3454
- Page Range / eLocation ID:
- 1 to 20
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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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 [20] 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.more » « less
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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 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.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1145/3428673
