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

2005
*
Lecture Notes in Computer Science
*

In this paper, we formally investigate the notions

doi:10.1007/11533719_98
fatcat:ewojc2i2z5gbllf6ynfulngu2a
*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. ...##
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On W[1]-Hardness as Evidence for Intractability

2018
*
International Symposium on Mathematical Foundations of Computer Science
*

This

doi:10.4230/lipics.mfcs.2018.73
dblp:conf/mfcs/Bottesch18
fatcat:snza5gqn6vaahii3brbm5wu2z4
*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 [ ...##
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The polynomial hierarchy and fragments of bounded arithmetic

1985
*
Proceedings of the seventeenth annual ACM symposium on Theory of computing - STOC '85
*

Finally we state some strong

doi:10.1145/22145.22177
dblp:conf/stoc/Buss85
fatcat:rsfbnqqgbfe7pb7jjukrczqix4
*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. ...##
###
Interpolation in Valiant's theory
[article]

2007
*
arXiv
*
pre-print

We investigate the following question: if

arXiv:0710.0360v1
fatcat:yazhjh5cejduhfujxeocvqnt6m
*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). ...##
###
Space-bounded hierarchies and probabilistic computations

1984
*
Journal of computer and system sciences (Print)
*

Third, we give

doi:10.1016/0022-0000(84)90066-7
fatcat:bvcg33bozrbc7beb5fusesbngu
*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. ...##
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Towards a definition of the Quantum Ergodic Hierarchy: Kolmogorov and Bernoulli systems

2014
*
Physica A: Statistical Mechanics and its Applications
*

In this paper we translate the two higher levels

doi:10.1016/j.physa.2013.08.070
fatcat:6pb4tbuqsrdv5p32d7ctrakbui
*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 ...##
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Uniform normal form for general time-bounded complexity classes

1986
*
Journal of computer and system sciences (Print)
*

This permits arithmetization

doi:10.1016/0022-0000(86)90034-6
fatcat:cnnhapneqvgddmrqvrdsaka4se
*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. ...##
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Complexity of counting the optimal solutions

2009
*
Theoretical Computer Science
*

However, for

doi:10.1016/j.tcs.2009.05.025
fatcat:64qcmnafzjdt7pgq2vf4umtgsu
*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. ...##
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On W[1]-Hardness as Evidence for Intractability
[article]

2018
*
arXiv
*
pre-print

This

arXiv:1712.05766v3
fatcat:4atehv3dgvcehcuqivkbt6d5vm
*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. ...##
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On the Tape-Number Problem for Deterministic Time Classes
[article]

2013
*
arXiv
*
pre-print

For any

arXiv:1309.7457v1
fatcat:gcqr4fgdwnecth2iksp7v2vmha
*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 . ...##
###
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

2021
*
Bulletin of Symbolic Logic
*

Using tensed language as an heuristic, the following bare-bones story introduces the idea

doi:10.1017/bsl.2021.14
fatcat:pjl6xoj6zzdm7ewor7vtnqaehi
*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. ...##
###
On Parameterized Intractability: Hardness and Completeness

2007
*
Computer journal
*

We study the theory and techniques developed in the research

doi:10.1093/comjnl/bxm036
fatcat:wjrw57b5nvbfnfib3uii5tviiq
*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. ...##
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Towards a complexity theory for the congested clique
[article]

2018
*
arXiv
*
pre-print

In particular, we prove

arXiv:1705.03284v3
fatcat:veod7eqfgfgndij3ocm7rxmr7y
*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. ...##
###
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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