research-article

Hardness amplification in proof complexity

Authors:

Toniann PitassiAuthors Info & Claims

STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

Pages 87 - 96

https://doi.org/10.1145/1806689.1806703

Published: 05 June 2010 Publication History

Abstract

We present a general method for converting any family of unsatisfiable CNF formulas that is hard for one of the simplest proof systems -- tree resolution -- into formulas that require large rank in very strong proof systems, including any proof system that manipulates polynomials of degree at most k (known as Th(k) proofs). These include high degree versions of Lovasz-Schrijver and Cutting Planes proofs.

We introduce two very general families of these proof systems, denoted Tcc(k) and Rcc(k). The proof lines of Tcc(k) are arbitrary Boolean functions, each of which can be evaluated by an efficient k-party randomized communication protocol. Tcc(k) proofs include Th(k-1) proofs as a special case. Rcc(k) proofs generalize Tcc(k) proofs and require only that each inference be checkable by a short k-party protocol.

For all k in O(loglog n), our main results are as follows: First, any unsatisfiable CNF formula of high resolution rank can be efficiently transformed into another CNF formula requiring high rank in all Rcc(k) systems, and exponential tree size in all Tcc(k) systems. Secondly, there are strict rank hierarchies for all Rcc(k) systems, and strict tree-size hierarchies for all Tcc(k) systems. Finally, we apply our general method to give optimal integrality gaps for low rank Rcc(2) proofs for MAX-2t-SAT, which imply optimal integrality gaps for low rank Cutting Planes and Th(1) proofs.

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Cited By

Limaye NSrinivasan STavenas S(2024)Superpolynomial Lower Bounds Against Low-Depth Algebraic CircuitsCommunications of the ACM10.1145/361109467:2(101-108)Online publication date: 25-Jan-2024
https://dl.acm.org/doi/10.1145/3611094
Limaye NSrinivasan STavenas S(2022)Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00083(804-814)Online publication date: Mar-2022
https://doi.org/10.1109/FOCS52979.2021.00083
Vinyals MElffers JJohannsen JNordström J(2020)Simplified and Improved Separations Between Regular and General Resolution by LiftingTheory and Applications of Satisfiability Testing – SAT 202010.1007/978-3-030-51825-7_14(182-200)Online publication date: 26-Jun-2020
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Index Terms

Hardness amplification in proof complexity
1. Theory of computation

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cover image ACM Conferences

STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

June 2010

812 pages

ISBN:9781450300506

DOI:10.1145/1806689

General Chair:
Michael Mitzenmacher
Harvard
,
Program Chair:
Leonard J. Schulman
Caltech

Copyright © 2010 ACM.

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Published: 05 June 2010

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Cited By

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Limaye NSrinivasan STavenas S(2022)Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00083(804-814)Online publication date: Mar-2022
https://doi.org/10.1109/FOCS52979.2021.00083
Vinyals MElffers JJohannsen JNordström J(2020)Simplified and Improved Separations Between Regular and General Resolution by LiftingTheory and Applications of Satisfiability Testing – SAT 202010.1007/978-3-030-51825-7_14(182-200)Online publication date: 26-Jun-2020
https://doi.org/10.1007/978-3-030-51825-7_14
Chattopadhyay AKoucký MLoff BMukhopadhyay S(2019)Simulation Theorems via Pseudo-random Propertiescomputational complexity10.1007/s00037-019-00190-7Online publication date: 18-Jul-2019
https://doi.org/10.1007/s00037-019-00190-7
Chattopadhyay AKoucký MLoff BMukhopadhyay SDiakonikolas IKempe DHenzinger M(2018)Simulation beats richness: new data-structure lower boundsProceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3188745.3188874(1013-1020)Online publication date: 20-Jun-2018
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Garg AGöös MKamath PSokolov DDiakonikolas IKempe DHenzinger M(2018)Monotone circuit lower bounds from resolutionProceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3188745.3188838(902-911)Online publication date: 20-Jun-2018
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de Rezende SNordstrom JVinyals M(2016)How Limited Interaction Hinders Real Communication (and What It Means for Proof and Circuit Complexity)2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2016.40(295-304)Online publication date: Oct-2016
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Worah P(2015)Rank bounds for a hierarchy of Lovász and SchrijverJournal of Combinatorial Optimization10.1007/s10878-013-9662-430:3(689-709)Online publication date: 1-Oct-2015
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