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Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity
KTH, School of Electrical Engineering and Computer Science (EECS), Theoretical Computer Science, TCS.ORCID iD: 0000-0001-8923-1240
KTH, School of Electrical Engineering and Computer Science (EECS), Theoretical Computer Science, TCS.ORCID iD: 0000-0002-2700-4285
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(English)Manuscript (preprint) (Other academic)
Abstract [en]

We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We apply our generalized theorem to solve two open problems:

  • We present the first result that demonstrates a separation in proof power for cutting planes with unbounded versus polynomially bounded coefficients. Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting planes with unbounded coefficients, but for which there are no refutations in subexponential length and subpolynomialline space if coefficients are restricted to be of polynomial magnitude.
  • We give the first explicit separation between monotone Boolean formulas and monotone real formulas. Specifically, we give an explicit family of functions that can be computed with monotone real formulas of nearly linear size but require monotone Boolean formulas of exponential size. Previously only a non-explicit separation was known.

An important technical ingredient, which may be of independent interest, is that we show that the Nullstellensatz degree of refuting the pebbling formula over a DAG G over any field coincides exactly with the reversible pebbling price of G. In particular, this implies that the standard decision tree complexity and the parity decision tree complexity of the corresponding falsified clause search problem are equal.

National Category
Computer Sciences
Research subject
Computer Science
Identifiers
URN: urn:nbn:se:kth:diva-249607OAI: oai:DiVA.org:kth-249607DiVA, id: diva2:1304720
Note

QC 20190529

Available from: 2019-04-12 Created: 2019-04-12 Last updated: 2019-05-29Bibliographically approved
In thesis
1. Lower Bounds and Trade-offs in Proof Complexity
Open this publication in new window or tab >>Lower Bounds and Trade-offs in Proof Complexity
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Propositional proof complexity is a field in theoretical computer science that analyses the resources needed to prove statements. In this thesis, we are concerned about the length of proofs and trade-offs between different resources, such as length and space.

A classical NP-hard problem in computational complexity is that of determining whether a graph has a clique of size k. We show that for all k ≪ n^(1/4) regular resolution requires length n^Ω(k) to establish that an Erdős–Rényi graph with n vertices and appropriately chosen edge density does not contain a k-clique. In particular, this implies an unconditional lower bound on the running time of state-of-the-artalgorithms for finding a maximum clique.

In terms of trading resources, we prove a length-space trade-off for the cutting planes proof system by first establishing a communication-round trade-off for real communication via a round-aware simulation theorem. The technical contri-bution of this result allows us to obtain a separation between monotone-AC^(i-1) and monotone-NC^i.

We also obtain a trade-off separation between cutting planes (CP) with unbounded coefficients and cutting planes where coefficients are at most polynomial in thenumber of variables (CP*). We show that there are formulas that have CP proofs in constant space and quadratic length, but any CP* proof requires either polynomial space or exponential length. This is the first example in the literature showing any type of separation between CP and CP*.

For the Nullstellensatz proof system, we prove a size-degree trade-off via a tight reduction of Nullstellensatz refutations of pebbling formulas to the reversible pebbling game. We show that for any directed acyclic graph G it holds that G can be reversibly pebbled in time t and space s if and only if there is a Nullstellensatzrefutation of the pebbling formula over G in size t + 1 and degree s.

Finally, we introduce the study of cumulative space in proof complexity, a measure that captures the space used throughout the whole proof and not only the peak space usage. We prove cumulative space lower bounds for the resolution proof system, which can be viewed as time-space trade-offs where, when time is bounded, space must be large a significant fraction of the time.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2019. p. 247
Series
TRITA-EECS-AVL ; 2019:47
Keywords
Proof complexity, trade-offs, lower bounds, size, length, space
National Category
Computer Sciences
Research subject
Computer Science
Identifiers
urn:nbn:se:kth:diva-249610 (URN)978-91-7873-191-6 (ISBN)
Public defence
2019-06-14, Kollegiesalen, Brinellvägen 8, Stockholm, 14:00 (English)
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Note

QC 20190527

Available from: 2019-05-27 Created: 2019-05-24 Last updated: 2019-05-27Bibliographically approved

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de Rezende, Susanna F.Nordström, JakobVinyals, Marc

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