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Electronic Colloquium on Computational Complexity

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All reports by Author John Hitchcock:

TR18-018 | 22nd January 2018
John Hitchcock, Adewale Sekoni, Hadi Shafei

Polynomial-Time Random Oracles and Separating Complexity Classes

Bennett and Gill (1981) showed that P^A != NP^A != coNP^A for a random
oracle A, with probability 1. We investigate whether this result
extends to individual polynomial-time random oracles. We consider two
notions of random oracles: p-random oracles in the sense of
martingales and resource-bounded measure (Lutz, 1992; Ambos-Spies ... more >>>

TR18-013 | 18th January 2018
John Hitchcock, Adewale Sekoni

Nondeterminisic Sublinear Time Has Measure 0 in P

The measure hypothesis is a quantitative strengthening of the P $\neq$ NP conjecture which asserts that NP is a nonnegligible subset of EXP. Cai, Sivakumar, and Strauss (1997) showed that the analogue of this hypothesis in P is false. In particular, they showed that NTIME[$n^{1/11}$] has measure 0 in P. ... more >>>

TR18-011 | 18th January 2018
John Hitchcock, Hadi Shafei

Nonuniform Reductions and NP-Completeness

Nonuniformity is a central concept in computational complexity with powerful connections to circuit complexity and randomness. Nonuniform reductions have been used to study the isomorphism conjecture for NP and completeness for larger complexity classes. We study the power of nonuniform reductions for NP-completeness, obtaining both separations and upper bounds for ... more >>>

TR16-012 | 21st January 2016
John Hitchcock, Hadi Shafei

Autoreducibility of NP-Complete Sets

We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under strong hypotheses for NP. Assuming there is a p-generic set in NP, we show the following:

- For every $k \geq 2$, there is a $k$-T-complete set for NP that is $k$-T autoreducible, but is not $k$-tt autoreducible ... more >>>

TR10-174 | 12th November 2010
Scott Aaronson, Baris Aydinlioglu, Harry Buhrman, John Hitchcock, Dieter van Melkebeek

A note on exponential circuit lower bounds from derandomizing Arthur-Merlin games

We present an alternate proof of the recent result by Gutfreund and Kawachi that derandomizing Arthur-Merlin games into $P^{NP}$ implies linear-exponential circuit lower bounds for $E^{NP}$. Our proof is simpler and yields stronger results. In particular, consider the promise-$AM$ problem of distinguishing between the case where a given Boolean circuit ... more >>>

TR09-071 | 1st September 2009
John Hitchcock, A. Pavan, Vinodchandran Variyam

Kolmogorov Complexity in Randomness Extraction

We clarify the role of Kolmogorov complexity in the area of randomness extraction. We show that a computable function is an almost randomness extractor if and only if it is a Kolmogorov complexity
extractor, thus establishing a fundamental equivalence between two forms of extraction studied in the literature: Kolmogorov extraction
more >>>

TR08-022 | 9th January 2008
Harry Buhrman, John Hitchcock

NP-Hard Sets are Exponentially Dense Unless NP is contained in coNP/poly

We show that hard sets S for NP must have exponential density, i.e. |S<sub>=n</sub>| &#8805; 2<sup>n<sup>&#949;</sup></sup> for some &#949; > 0 and infinitely many n, unless coNP &#8838; NP\poly and the polynomial-time hierarchy collapses. This result holds for Turing reductions that make n<sup>1-&#949;</sup> queries.

In addition we study the instance ... more >>>

TR06-071 | 25th April 2006
John Hitchcock, A. Pavan

Hardness Hypotheses, Derandomization, and Circuit Complexity

We consider hypotheses about nondeterministic computation that
have been studied in different contexts and shown to have interesting

1. The measure hypothesis: NP does not have p-measure 0.

2. The pseudo-NP hypothesis: there is an NP language that can be
distinguished from any DTIME(2^n^epsilon) language by an ... more >>>

TR06-039 | 28th February 2006
John Hitchcock, A. Pavan

Comparing Reductions to NP-Complete Sets

Under the assumption that NP does not have p-measure 0, we
investigate reductions to NP-complete sets and prove the following:

- Adaptive reductions are more powerful than nonadaptive
reductions: there is a problem that is Turing-complete for NP but
not truth-table-complete.

- Strong nondeterministic reductions are more powerful ... more >>>

TR05-161 | 13th December 2005
John Hitchcock

Online Learning and Resource-Bounded Dimension: Winnow Yields New Lower Bounds for Hard Sets

We establish a relationship between the online mistake-bound model of learning and resource-bounded dimension. This connection is combined with the Winnow algorithm to obtain new results about the density of hard sets under adaptive reductions. This improves previous work of Fu (1995) and Lutz and Zhao (2000), and solves one ... more >>>

TR05-105 | 24th September 2005
Lance Fortnow, John Hitchcock, A. Pavan, Vinodchandran Variyam, Fengming Wang

Extracting Kolmogorov Complexity with Applications to Dimension Zero-One Laws

We apply recent results on extracting randomness from independent
sources to ``extract'' Kolmogorov complexity. For any $\alpha,
\epsilon > 0$, given a string $x$ with $K(x) > \alpha|x|$, we show
how to use a constant number of advice bits to efficiently
compute another string $y$, $|y|=\Omega(|x|)$, with $K(y) >
(1-\epsilon)|y|$. ... more >>>

TR04-079 | 30th August 2004
John Hitchcock, Jack H. Lutz, Sebastiaan Terwijn

The Arithmetical Complexity of Dimension and Randomness

Constructive dimension and constructive strong dimension are effectivizations of the Hausdorff and packing dimensions, respectively. Each infinite binary sequence A is assigned a dimension dim(A) in [0,1] and a strong dimension Dim(A) in [0,1].

Let DIM^alpha and DIMstr^alpha be the classes of all sequences of dimension alpha and of strong ... more >>>

TR04-072 | 19th August 2004
John Hitchcock

Hausdorff Dimension and Oracle Constructions

Bennett and Gill (1981) proved that P^A != NP^A relative to a
random oracle A, or in other words, that the set
O_[P=NP] = { A | P^A = NP^A }
has Lebesgue measure 0. In contrast, we show that O_[P=NP] has
Hausdorff dimension 1.

... more >>>

TR04-029 | 7th April 2004
John Hitchcock, Maria Lopez-Valdes, Elvira Mayordomo

Scaled dimension and the Kolmogorov complexity of Turing hard sets

Scaled dimension has been introduced by Hitchcock et al (2003) in order to quantitatively distinguish among classes such as SIZE(2^{a n}) and SIZE(2^{n^{a}}) that have trivial dimension and measure in ESPACE.

more >>>

TR04-025 | 24th January 2004
John Hitchcock, A. Pavan, Vinodchandran Variyam

Partial Bi-Immunity and NP-Completeness

The Turing and many-one completeness notions for $\NP$ have been
previously separated under {\em measure}, {\em genericity}, and {\em
bi-immunity} hypotheses on NP. The proofs of all these results rely
on the existence of a language in NP with almost everywhere hardness.

In this paper we separate the same NP-completeness ... more >>>

TR03-063 | 2nd July 2003
John Hitchcock

The Size of SPP

Derandomization techniques are used to show that at least one of the
following holds regarding the size of the counting complexity class
1. SPP has p-measure 0.
2. PH is contained in SPP.
In other words, SPP is small by being a negligible subset of
exponential time or large ... more >>>

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