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The existence and density of generalized complexity cores

Ronald V. Book, Ding-Zhu Du
1987 Journal of the ACM  
If g is a class of sets and A is not in S then an infinite set H is a proper hard core for A with respect to g, if H C A and for every C E Q such that C c A, C G H is finite. It is shown that if E?  ...  generalized complexity cores is studied.  ...  An existence theorem for "partially immune" hard cores is established and the complexity of hard cores is discussed.  ... 
doi:10.1145/28869.28880 fatcat:7sekb2ncjnfzdmwqiwx6bp3f5a

The structure of generalized complexity cores

Ronald V. Book, Ding-Zhu Du
1988 Theoretical Computer Science  
Structural properties of proper hard cores (generalized complexity cores) are studied.  ...  The setting involves a countable class C of sets of strings that is closed under finite union and under finite variation. For any set A let C" denote the collection of all C E C such that C E A.  ...  If A has a maximal proper hard core H, then A-H must be in C For otherwise, there is a proper hard core G for A-H with respect to C. By Theorem 3.5, G is also a proper hard core for A.  ... 
doi:10.1016/0304-3975(88)90119-3 fatcat:rhahfnfwxbhwhcwofywsgyu2tu

The Complexity and Distribution of Hard Problems

David W. Juedes, Jack H. Lutz
1995 SIAM journal on computing (Print)  
Particular attention is given to the complexity measured by the size of complexity cores and distribution abundance in the sense of measure of languages that are P m -hard for E and other complexity classes  ...  First, every P m -hard language for E has a dense P-complexity core. Second, if P 6 = NP, then every P m -hard  ...  For simplicity w e only consider the class E. i Every weakly P m -hard for E has a dense exponential complexity core Theorem 4.9. ii For every language A 2 E, at least one of the spans P m A, P ,1 m A  ... 
doi:10.1137/s0097539792238133 fatcat:twap7ewd2jfbdmdf5imi7kwpvm

The density and complexity of polynomial cores for intractable sets

Pekka Orponen, Uwe Schöning
1986 Information and Control  
Lynch has shown that A then has an infinite recursive polynomial complexity core.  ...  We further study how dense the core sets for A can be, under various assumptions about the structure of A.  ...  Russo for their helpful comments on an earlier version of this paper, and J. Balc~izar (again) and O. Watanabe for discussions in which the present version was greatly improved.  ... 
doi:10.1016/s0019-9958(86)80024-9 fatcat:auqt3cqdzbeshgypklgyvxjltu

Page 6626 of Mathematical Reviews Vol. , Issue 88m [page]

1988 Mathematical Reviews  
The paper under review proposes a different notion of hard core for arbitrary classes, based on a characterization of the polynomial time complexity cores: X is a hard core for A with respect to class  ...  .], and with other properties for other classes, and then give very general sufficient conditions on A for having a core with respect to an arbitrary class of sets C (Theorem 2.7).  ... 

The Quantitative Structure of Exponential Time [chapter]

Jack H. Lutz
1997 Complexity Theory Retrospective II  
The measure structure of these classes is seen to interact in informative w ays with bi-immunity, complexity cores, polynomial-time reductions, completeness, circuit-size complexity, Kolmogorov complexity  ...  , natural proofs, pseudorandom generators, the density of hard languages, randomized complexity, a n d lowness.  ...  I especially thank my colleagues David Juedes and Elvira Mayordomo for working with me in this area, and for allowing me to include so much of our work in this survey.  ... 
doi:10.1007/978-1-4612-1872-2_10 fatcat:ez5zkrti5vacrctw3ubltjubxm

On hard instances

Martin Mundhenk
2000 Theoretical Computer Science  
Hard instances are shown to be stronger than complexity cores (introduced by Lynch (1975) Nevertheless, NP-hard sets must have super-polynomially dense hard instances, unless P = NP. : S 0 3 0 4 -3 9 7  ...  Comparing instance complexity to Kolmogorov complexity, they introduced the notion of p-hard instances, and conjectured that every set not in P has p-hard instances.  ...  Acknowledgements The author would like to thank for the comments of two referees.  ... 
doi:10.1016/s0304-3975(98)00262-x fatcat:mrhp3hfxnjd75ikwdd322c2uxy

Completeness and Weak Completeness under Polynomial-Size Circuits

David W. Juedes, Jack H. Lutz
1996 Information and Computation  
complexity cores and space-bounded Kolmogorov complexity) that are violated by almost every element of ESPACE.  ...  A Small Span Theorem for PÂPoly-Turing reductions in ESPACE is proven and used to show that every PÂPoly-Turing degree including the complete degree has measure 0 in ESPACE.  ...  Both authors thank two anonymous referees for their careful reading, for pointing out an error in the first version of Theorem 3.8, and for several useful suggestions.  ... 
doi:10.1006/inco.1996.0017 fatcat:qn3waymvqzbjxa3mvrxkulbmbm

On inefficient special cases of NP-complete problems

Ding-Zhu Du, Ronald V. Book
1989 Theoretical Computer Science  
Turning to the intractable class DEXT= lJrzO DTIME@"'), it is shown that every set that is <L-complete for DEXT has an infinite proper polynomial complexity core that is nonsparse and recursive.  ...  Every intractable set A has a polynomial compfexity core, a set H such that for any P-subset S of A or of & S A H is finite. A complexity core H of A is proper if H c A.  ...  If, in addition, H is a subset of A, then H is a proper ha+ core. me version of the general existence theorem for hard cores that is useful here is Theorem 2.10 of 131: If C is a recursively enumerable  ... 
doi:10.1016/0304-3975(89)90015-7 fatcat:qqkaadctgng6zfexhsbadjxtpq

A Finitely presented group whose word problem has sampleable hard instances [article]

Robert H Gilman
2016 arXiv   pre-print
More precisely the problem has a complexity core sampleable in linear time.  ...  In this note we present an example of a natural decision problem, the word problem for a certain finitely presented group, whose hard instances are easy to find.  ...  Such an X is called a complexity core for the decision problem.  ... 
arXiv:1602.02432v1 fatcat:rdtix5ftmzgvxmlmwoz5u5cup4

Complexity of Hard-Core Set Proofs

Chi-Jen Lu, Shi-Chun Tsai, Hsin-Lung Wu
2011 Computational Complexity  
Next, we show that any weakly black-box proof must be inherently non-uniform-to have a hard-core set for a class G of functions, we need to start from the assumption that f is hard against a non-uniform  ...  complexity class with Ω( 1 ε log |G|) bits of advice.  ...  Acknowledgements The authors would like to thank anonymous referees for their useful comments. The research of Chi-Jen Lu was supported in part by the National Science  ... 
doi:10.1007/s00037-011-0003-7 fatcat:zuihqq5lsjhnjhh3dm3jt3uxfq

Kolmogorov Complexity, Complexity Cores, and the Distribution of Hardness [chapter]

David W. Juedes, Jack H. Lutz
1992 Kolmogorov Complexity and Computational Complexity  
Speci cally, e v ery complete problem must have ususually small complexity cores and unusually low space-bounded Kolmogorov complexity.  ...  Problems that are complete for exponential space are provably intractable and known to be exceedingly complex in several technical respects.  ...  Acknowledgment We thank Osamu W atanabe and two a n o n ymous reviewers for suggestions that have improved the exposition of this paper.  ... 
doi:10.1007/978-3-642-77735-6_4 fatcat:xd3j2fefj5c23emuokmbn6u2oq

Boosting and Hard-Core Set Construction

Adam R. Klivans, Rocco A. Servedio
2003 Machine Learning  
Using alternate boosting methods we give an improved bound for hard-core set construction which matches known lower bounds from boosting and thus is optimal within this class of techniques.  ...  This paper connects hard-core set construction, a type of hardness amplification from computational complexity, and boosting, a technique from computational learning theory.  ...  Acknowledgments We thank Jeff Jackson for useful conversations concerning the section on learning DNF formulae. We thank Salil Vadhan for insights on Impagliazzo (1995) .  ... 
doi:10.1023/a:1022949332276 dblp:journals/ml/KlivansS03 fatcat:anbj62oisvd6fmlzidtiqemkdu

Completeness and weak completeness under polynomial-size circuits [chapter]

David W. Juedes, Jack H. Lutz
1995 Lecture Notes in Computer Science  
complexity cores and space-bounded Kolmogorov complexity) that are violated by almost every element o f E S P ACE.  ...  A Small Span Theorem for P/Poly-Turing reductions in ESPACE is proven and used to show t h a t every P/Poly-Turing degree | including the complete degree | has measure 0 in ESPACE.  ...  Both authors thank two anonymous referees for their careful reading, for pointing out an error in the rst version of Theorem 3.8, and for several useful suggestions.  ... 
doi:10.1007/3-540-59042-0_59 fatcat:74qkuxadljbstc6tc4j2jgh35m

A Trichotomy in the Complexity of Counting Answers to Conjunctive Queries [article]

Hubie Chen, Stefan Mengel
2015 arXiv   pre-print
We present a trichotomy theorem, which shows essentially that this problem on a set of conjunctive queries is either tractable, equivalent to the parameterized CLIQUE problem, or as hard as the parameterized  ...  In particular, we study the complexity of this problem relative to sets of conjunctive queries.  ...  Thus we can compute | hom(A ′ , B ′ , S)| in polynomial time by Theorem 11 which completes the proof. Hardness results In this section we will prove the hardness results for Theorem 22.  ... 
arXiv:1408.0890v3 fatcat:bwk3g3yirngkbbgnxh7rt6fdgy
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