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W-Hardness Under Linear FPT-Reductions: Structural Properties and Further Applications [chapter]

Jianer Chen, Xiuzhen Huang, Iyad A. Kanj, Ge Xia
2005 Lecture Notes in Computer Science  
In this paper, we formally investigate the notions of W [t]-hardness and W [t]-completeness under the linear fpt-reduction, and study structural properties of the corresponding complexity classes.  ...  The notion of linear fpt-reductions has been recently used to derive strong computational lower bounds for well-known NP-hard problems.  ...  The parameterized versions of these problems are just the corresponding parameterized problems in Theorem 5. The theorem follows immediately from Theorem 5 and Theorem 6.  ... 
doi:10.1007/11533719_98 fatcat:ewojc2i2z5gbllf6ynfulngu2a

On W[1]-Hardness as Evidence for Intractability

Ralph Christian Bottesch, Michael Wagner
2018 International Symposium on Mathematical Foundations of Computer Science  
This theorem also has consequences for the A-Hierarchy (a parameterized version of the Polynomial Hierarchy), namely that unless W[P] is a subset of some level A[t], there are structural differences between  ...  We also prove that under the unlikely assumption that W[P] collapses to W[1] in a specific way, the collapse of any two consecutive levels of the A-Hierarchy implies the collapse of the entire hierarchy  ...  For example, a parameterized version of Cook's Theorem connects Boolean circuit satisfiability to W [1] -completeness (see [9] ), a parameterized version of Ladner's Theorem states that if FPT = W [  ... 
doi:10.4230/lipics.mfcs.2018.73 dblp:conf/mfcs/Bottesch18 fatcat:snza5gqn6vaahii3brbm5wu2z4

The polynomial hierarchy and fragments of bounded arithmetic

S R Buss
1985 Proceedings of the seventeenth annual ACM symposium on Theory of computing - STOC '85  
Finally we state some strong versions of the G a e l incompleteness theorems and we give a proof-theoretic principle which is equivalent to NP=co-NP.  ...  Thus, we present a theory S: which defines precisely the class of functions in P, another theory S; which defines precisely the class of functions which are polynomial time relative to NP, a theory U:  ...  Lipton, A. Wilkie, P. Pudlak, S. Cook and M. Dowd for their encouragement and suggestions.  ... 
doi:10.1145/22145.22177 dblp:conf/stoc/Buss85 fatcat:rsfbnqqgbfe7pb7jjukrczqix4

Interpolation in Valiant's theory [article]

Pascal Koiran
2007 arXiv   pre-print
We investigate the following question: if a polynomial can be evaluated at rational points by a polynomial-time boolean algorithm, does it have a polynomial-size arithmetic circuit?  ...  Answering this question positively would imply a transfer theorem from boolean to algebraic complexity.  ...  More precisely, let (f n ) be a family of multivariate polynomials which can be evaluated in polynomial time at integer points (in the sense of Remark 3).  ... 
arXiv:0710.0360v1 fatcat:yazhjh5cejduhfujxeocvqnt6m

Space-bounded hierarchies and probabilistic computations

Walter L. Ruzzo, Janos Simon, Martin Tompa
1984 Journal of computer and system sciences (Print)  
Third, we give a new definition of space-bounded oracle machines, and use it to define a space-bounded "oracle hierarchy" analogous to the original definition of the polynomial time hierarchy.  ...  Unlike its polynomial time analogue, the entire log n space "alternation hierarchy" is contained in the second level of the log n space "oracle hierarchy."  ...  Second, by a relativized version of Theorem 1, assume that N@) has expected time r(n) = 2*0"ogn) on all inputs.  ... 
doi:10.1016/0022-0000(84)90066-7 fatcat:bvcg33bozrbc7beb5fusesbngu

Towards a definition of the Quantum Ergodic Hierarchy: Kolmogorov and Bernoulli systems

Ignacio Gomez, Mario Castagnino
2014 Physica A: Statistical Mechanics and its Applications  
In this paper we translate the two higher levels of the Ergodic Hierarchy [1], the Kolmogorov level and the Bernoulli level, to quantum language.  ...  We illustrate the physical relevance of the Quantum Ergodic Hierarchy with two emblematic examples of the literature: the Casati-Prosen model [4], [5] and the kicked rotator [6], [7], [8].  ...  Concepts like "observables" ❅ ❅ ❅ ❅ ❅ ❅ Lyapunov Exponents Ergodic Hierarchy Complexity Brudno's Theorem Pesin's Theorem Figure 1 : The "chaos pyramid" is a diagram of the relationships beetween the  ... 
doi:10.1016/j.physa.2013.08.070 fatcat:6pb4tbuqsrdv5p32d7ctrakbui

Uniform normal form for general time-bounded complexity classes

Bernard R. Hodgson, Clement F. Kent
1986 Journal of computer and system sciences (Print)  
This permits arithmetization of Turing machine complexity classes determined by quite general time bounds. Applications are made to complexity hierarchies and to obtain a single, uniform, normal form.  ...  Refining techniques of previous works, we obtain a normal form arithmetical representation for non-deterministic computability, in which the polynomial matrix does not involve the time-bounding function  ...  Subsequently Matijasevic has sent us [3] , which contains a closely related version of Theorem 1 obtained independently by Jukna. We comment on the relation of the two results below.  ... 
doi:10.1016/0022-0000(86)90034-6 fatcat:cnnhapneqvgddmrqvrdsaka4se

Complexity of counting the optimal solutions

Miki Hermann, Reinhard Pichler
2009 Theoretical Computer Science  
However, for a large variety of counting problems arising from optimization problems, a precise complexity classification turns out to be impossible with these classes.  ...  Moreover, we establish the completeness of several natural counting complexity problems for these new classes. * A preliminary version without proofs appeared in COCOON 2008.  ...  Acknowledgment: We thank Arnaud Durand and Yann Strozecki for their remarks and for the discussion on the proof of Theorem 9.  ... 
doi:10.1016/j.tcs.2009.05.025 fatcat:64qcmnafzjdt7pgq2vf4umtgsu

On W[1]-Hardness as Evidence for Intractability [article]

Ralph C. Bottesch
2018 arXiv   pre-print
This theorem also has consequences for the A-Hierarchy (a parameterized version of the Polynomial Hierarchy), namely that unless W[P] is a subset of some level A[t], there are structural differences between  ...  We also prove that under the unlikely assumption that W[P] collapses to W[1] in a specific way, the collapse of any two consecutive levels of the A-Hierarchy implies the collapse of the entire hierarchy  ...  The counter-example in the proof of Proposition 15 is due to Harry Buhrman. I am especially grateful to Ronald de Wolf and Leen Torenvliet for helpful comments on drafts of the paper.  ... 
arXiv:1712.05766v3 fatcat:4atehv3dgvcehcuqivkbt6d5vm

On the Tape-Number Problem for Deterministic Time Classes [article]

Armin Hemmerling
2013 arXiv   pre-print
For any time bound f, let H(f) denote the hierarchy conjecture which means that the restriction of the numbers of work tapes of deterministic Turing machines to some b generates an infinite hierarchy of  ...  H(f) follows from the gap property, G(f), which says that there is a time-constructible bound f_2 such that f \in o(f_2) and DTIME(f)=DTIME(f_2). G(f) implies further separations.  ...  More precisely, the negation, ¬ G(f ), would just mean a stronger hierarchy result for the complexity classes of time bounds immediately above f .  ... 
arXiv:1309.7457v1 fatcat:gcqr4fgdwnecth2iksp7v2vmha

Page 660 of Mathematical Reviews Vol. , Issue 99a [page]

1991 Mathematical Reviews  
Summary: “The classical ‘impossibility theorem’ is considered under some weaker version of the usual properties.  ...  A hierarchy of collective decision for a given team is obtained by the specification of three elements: first, the number of levels or tiers of the hierar- chy, i.e., the height of the hierarchy, second  ... 

LEVEL THEORY, PART 2: AXIOMATIZING THE BARE IDEA OF A POTENTIAL HIERARCHY

TIM BUTTON
2021 Bulletin of Symbolic Logic  
Using tensed language as an heuristic, the following bare-bones story introduces the idea of a potential hierarchy of sets: 'Always: for any sets that existed, there is a set whose members are exactly  ...  What we need to do is to replace the language of time and activity by the more bloodless language of potentiality and actuality. Parsons [1977: 293]  ...  In §6, I consider a deductive version of this simulation.  ... 
doi:10.1017/bsl.2021.14 fatcat:pjl6xoj6zzdm7ewor7vtnqaehi

On Parameterized Intractability: Hardness and Completeness

J. Chen, J. Meng
2007 Computer journal  
We study the theory and techniques developed in the research of parameterized intractability, emphasizing on parameterized hardness and completeness that imply (stronger) computational lower bounds for  ...  Moreover, the fundamentals of the structural properties in parameterized complexity theory, relationships to classical complexity theory and more recent developments in the area are also introduced.  ...  Then the parameterized version of Q can be solved in time b(jxj, 2k). THEOREM 4 . 4 13 [32]. Let Q be an NP optimization problem.  ... 
doi:10.1093/comjnl/bxm036 fatcat:wjrw57b5nvbfnfib3uii5tviiq

Towards a complexity theory for the congested clique [article]

Janne H. Korhonen, Jukka Suomela
2018 arXiv   pre-print
In particular, we prove a time hierarchy theorem for the congested clique, showing that there are decision problems of essentially all complexities, both in the deterministic and nondeterministic settings  ...  congested clique model (analogous to NP) and show that there is a natural canonical problem family that captures all problems solvable in constant time with nondeterministic algorithms.  ...  This work was supported in part by the Academy of Finland, Grant 285721.  ... 
arXiv:1705.03284v3 fatcat:veod7eqfgfgndij3ocm7rxmr7y

Page 6374 of Mathematical Reviews Vol. , Issue 2001I [page]

2001 Mathematical Reviews  
In the paper under review, the authors prove (Theorem 1) that any global solution in H' which at the initial time is close to a traveling wave and which does not disperse has to be precisely the traveling  ...  As a consequence they obtain an asymptotic stability result for the global solutions in H' of the critical KdV which are close to the traveling wave at time zero (Theorem 2).  ... 
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