Structural Complexity I

Balcázar, José L.; Dı́az, Josep; Gabarró, Joaquim

doi:10.1007/978-3-642-97062-7

Cited by 384 publications

(128 citation statements)

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“…We assume the reader is familiar with basic complexity theory [BDG88] and Kolmogorov complexity [LV08]. We use ≤ p T and P A when referring to polynomial-time Turing reductions, and ≤ p tt and P A tt for polynomial-time truth-table (or non-adaptive) reductions.…”

Section: Preliminariesmentioning

confidence: 99%

Reductions to the Set of Random Strings: The Resource-Bounded Case

Allender

Buhrman

Friedman

et al. 2012

Mathematical Foundations of Computer Science 2012

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Abstract. This paper is motivated by a conjecture [All12, ADF + 13] that BPP can be characterized in terms of polynomial-time nonadaptive reductions to the set of Kolmogorovrandom strings. In this paper we show that an approach laid out in [ADF + 13] to settle this conjecture cannot succeed without significant alteration, but that it does bear fruit if we consider time-bounded Kolmogorov complexity instead.We show that if a set A is reducible in polynomial time to the set of time-t-bounded Kolmogorov-random strings (for all large enough time bounds t), then A is in P/poly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogorov complexity, then A is in PSPACE.

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Section: Preliminariesmentioning

confidence: 99%

Reductions to the Set of Random Strings: The Resource-Bounded Case

Allender

Buhrman

Friedman

et al. 2012

Mathematical Foundations of Computer Science 2012

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“…7 What about computers and their "minimal programs"? Any computation can be done in infinitely many different ways.…”

Section: From Minimal Art To Minimal Programsmentioning

confidence: 99%

“…How can one prove that these methods do not work? We illustrate this approach on the most important problem P =?NP (see, for instance, [64,7,8,116]). …”

Section: Dlog ⊆ Nlog ⊆ P ⊆ Np ⊆ P Sp Ace ⊆ Exp T Imementioning

confidence: 99%

Complexity: A Language-Theoretic Point of View

Calude¹,

Hromkovič²

1997

Handbook of Formal Languages

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“…The textbooks [BDG,BC93,KST93,Pa94,Sch86] can be consulted for the standard notations used in the paper and for basic results in complexity theory. For definitions of probabilistic complexity classes like ZPP see also [Gi77].…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…Furthermore, it is well-known that complexity classes like NP, Σ P k , Π P k , k ≥ 1, PP, C = P, Mod m P, m ≥ 2, PSPACE, and EXP have many-one complete self-reducible sets (see, for example, [BDG,Ba90,OL93]). …”

Section: Preliminaries and Notationmentioning

confidence: 99%

New collapse consequences of NP having small circuits

Köbler

Watanabe

1995

Automata, Languages and Programming

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We show that if a self-reducible set has polynomial-size circuits, then it is low for the probabilistic class ZPP(NP). As a consequence we get a deeper collapse of the polynomial-time hierarchy PH to ZPP(NP) under the assumption that NP has polynomial-size circuits. This improves on the well-known result of Karp, Lipton, and Sipser [KL80] stating a collapse of PH to its second level Σ P 2 under the same assumption.As a further consequence, we derive new collapse consequences under the assumption that complexity classes like UP, FewP, and C = P have polynomialsize circuits.Finally, we investigate the circuit-size complexity of several language classes. In particular, we show that for every fixed polynomial s, there is a set in ZPP(NP) which does not have O(s(n))-size circuits.

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Structural Complexity I

Cited by 384 publications

References 0 publications

Reductions to the Set of Random Strings: The Resource-Bounded Case

Reductions to the Set of Random Strings: The Resource-Bounded Case

Complexity: A Language-Theoretic Point of View

New collapse consequences of NP having small circuits

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