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A generalized method for proving polynomial calculus degree lower boundsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2015 (English)In: Leibniz International Proceedings in Informatics, LIPIcs, Dagstuhl Publishing , 2015, Vol. 33, p. 467-487Conference paper, Published paper (Refereed)
##### Resource type

Text
##### Abstract [en]

##### Place, publisher, year, edition, pages

Dagstuhl Publishing , 2015. Vol. 33, p. 467-487
##### 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
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##### Note

##### In thesis

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.

QC 20160527

Available from: 2016-05-27 Created: 2016-05-23 Last updated: 2016-11-30Bibliographically approved1. On Complexity Measures in Polynomial Calculus$(function(){PrimeFaces.cw("OverlayPanel","overlay1051001",{id:"formSmash:j_idt707:0:j_idt711",widgetVar:"overlay1051001",target:"formSmash:j_idt707:0:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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