Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
A generalized method for proving polynomial calculus degree lower bounds
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.ORCID iD: 0000-0002-2700-4285
2015 (English)In: Leibniz International Proceedings in Informatics, LIPIcs, Dagstuhl Publishing , 2015, Vol. 33, 467-487 p.Conference paper, Published paper (Refereed)
Resource type
Text
Abstract [en]

We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. ’99] also on proof size. [Alekhnovich and Razborov’03] established that if the clause-variable incidence graph of a CNF formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree. We further develop the techniques in [AR03] to show that if one can "cluster" clauses and variables in a way that "respects the structure" of the formula in a certain sense, then it is sufficient that the incidence graph of this clustered version is an expander. As a corollary of this, we prove that the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree when restricted to constant-degree expander graphs. This answers an open question in [Razborov’02], and also implies that the standard CNF encoding of the FPHP formulas require exponential proof size in polynomial calculus resolution. Thus, while Onto-FPHP formulas are easy for polynomial calculus, as shown in [Riis’93], both FPHP and Onto-PHP formulas are hard even when restricted to bounded-degree expanders.

Place, publisher, year, edition, pages
Dagstuhl Publishing , 2015. Vol. 33, 467-487 p.
Keyword [en]
Degree, Functional pigeonhole principle, Lower bound, PCR, Polynomial calculus, Polynomial calculus resolution, Proof complexity, Size
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:kth:diva-187389DOI: 10.4230/LIPIcs.CCC.2015.467Scopus ID: 2-s2.0-84958256296ISBN: 978-393989781-1 (print)OAI: oai:DiVA.org:kth-187389DiVA: diva2:931257
Conference
30th Conference on Computational Complexity, CCC 2015; Portland; United States
Note

QC 20160527

Available from: 2016-05-27 Created: 2016-05-23 Last updated: 2016-11-30Bibliographically approved
In thesis
1. On Complexity Measures in Polynomial Calculus
Open this publication in new window or tab >>On Complexity Measures in Polynomial Calculus
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Proof complexity is the study of different resources that a proof needs in different proof systems for propositional logic. This line of inquiry relates to the fundamental questions in theoretical computer science, as lower bounds on proof size for an arbitrary proof system would separate P from NP.

We study two simple proof systems: resolution and polynomial calculus. In resolution we reason using clauses, while in polynomial calculus we use polynomials. We study three measures of complexity of proofs: size, space, and width/degree. Size is the number of clauses or monomials that appear in a resolution or polynomial calculus proof, respectively. Space is the maximum number of clauses/monomials we need to keep at each time step of the proof. Width/degree is the size of the largest clause/monomial in a proof.

Width is a lower bound for space in resolution. The original proof of this claim used finite model theory. In this thesis we give a different, more direct proof of the space-width relation. We can ask whether a similar relation holds between space and degree in polynomial calculus. We make some progress on this front by showing that when a formula F requires resolution width w then the XORified version of F requires polynomial calculus space Ω(w). We also show that space lower bounds do not imply degree lower bounds in polynomial calculus.

Width/degree and size are also related, as strong lower bounds for width/degree imply strong lower bounds for size. Currently, proving width lower bounds has a well-developed machinery behind it. However, the degree measure is much less well-understood. We provide a unified framework for almost all previous degree lower bounds. We also prove some new degree and size lower bounds. In addition, we explore the relation between theory and practice by running experiments on some current state-of-the-art SAT solvers.

Place, publisher, year, edition, pages
Stockholm, Sweden: KTH Royal Institute of Technology, 2016. 180 p.
Series
TRITA-CSC-A, ISSN 1653-5723 ; 2017:02
National Category
Computer Science
Research subject
Computer Science
Identifiers
urn:nbn:se:kth:diva-197278 (URN)978-91-7729-226-5 (ISBN)
Public defence
2017-01-20, D2, Lindstedtsvägen 5, Stockholm, 14:00 (English)
Opponent
Supervisors
Projects
Understanding the Hardness of Theorem Proving
Funder
EU, FP7, Seventh Framework Programme, 279611
Note

QC 20161206

Available from: 2016-12-06 Created: 2016-11-30 Last updated: 2016-12-26Bibliographically approved

Open Access in DiVA

No full text

Other links

Publisher's full textScopus

Search in DiVA

By author/editor
Mikša, MladenNordström, Jakob
By organisation
Theoretical Computer Science, TCS
Mathematical Analysis

Search outside of DiVA

GoogleGoogle Scholar

Altmetric score

Total: 7 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf