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Author Razborov, Alexander
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©2016
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Hardness compression ♦ Resolution ♦ Tradeoff
Abstract We exhibit an unusually strong tradeoff in propositional proof complexity that significantly deviates from the established pattern of almost all results of this kind. Namely, restrictions on one resource (width, in our case) imply an increase in another resource (tree-like size) that is exponential not only with respect to the complexity of the original problem, but also to the whole class of $\textit{all}$ problems of the same bit size. More specifically, we show that for any parameter $\textit{k}$ = $\textit{k}(\textit{n}),$ there are unsatisfiable $\textit{k}-CNFs$ that possess refutations of width $\textit{O}(\textit{k}),$ but such that any tree-like refutation of width $\textit{n}1$ ™ $ε/\textit{k}$ must necessarily have $\textit{doubly}$ exponential size $exp (n^{Ω(k)}).$ This means that there exist contradictions that allow narrow refutations, but in order to keep the size of such a refutation even within a single exponent, it must necessarily use a high degree of parallelism. Our construction and proof methods combine, in a non-trivial way, two previously known techniques: the hardness escalation method based on substitution formulas and expansion. This combination results in a hardness compression approach that strives to preserve hardness of a contradiction while significantly decreasing the number of its variables.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2016-04-07
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 63
Issue Number 2
Page Count 14
Starting Page 1
Ending Page 14

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Source: ACM Digital Library