31,123 research outputs found
A Note on Universal Measures for Weak Implicit Computational Complexity
Abstract. This note is a case study for finding universal measures for weak implicit computational complexity. We will instantiate âuniver-sal measures â by âdynamic ordinalsâ, and âweak implicit computational complexity â by âbounded arithmeticâ. Concretely, we will describe the connection between dynamic ordinals and witness oracle Turing ma-chines for bounded arithmetic theories
On the necessity of complexity
Wolfram's Principle of Computational Equivalence (PCE) implies that universal
complexity abounds in nature. This paper comprises three sections. In the first
section we consider the question why there are so many universal phenomena
around. So, in a sense, we week a driving force behind the PCE if any. We
postulate a principle GNS that we call the Generalized Natural Selection
Principle that together with the Church-Turing Thesis is seen to be equivalent
to a weak version of PCE. In the second section we ask the question why we do
not observe any phenomena that are complex but not-universal. We choose a
cognitive setting to embark on this question and make some analogies with
formal logic. In the third and final section we report on a case study where we
see rich structures arise everywhere.Comment: 17 pages, 3 figure
Asymptotic Moments for Interference Mitigation in Correlated Fading Channels
We consider a certain class of large random matrices, composed of independent
column vectors with zero mean and different covariance matrices, and derive
asymptotically tight deterministic approximations of their moments. This random
matrix model arises in several wireless communication systems of recent
interest, such as distributed antenna systems or large antenna arrays.
Computing the linear minimum mean square error (LMMSE) detector in such systems
requires the inversion of a large covariance matrix which becomes prohibitively
complex as the number of antennas and users grows. We apply the derived moment
results to the design of a low-complexity polynomial expansion detector which
approximates the matrix inverse by a matrix polynomial and study its asymptotic
performance. Simulation results corroborate the analysis and evaluate the
performance for finite system dimensions.Comment: 7 pages, 2 figures, to be presented at IEEE International Symposium
on Information Theory (ISIT), Saint Petersburg, Russia, July 31 - August 5,
201
Immunity and Simplicity for Exact Counting and Other Counting Classes
Ko [RAIRO 24, 1990] and Bruschi [TCS 102, 1992] showed that in some
relativized world, PSPACE (in fact, ParityP) contains a set that is immune to
the polynomial hierarchy (PH). In this paper, we study and settle the question
of (relativized) separations with immunity for PH and the counting classes PP,
C_{=}P, and ParityP in all possible pairwise combinations. Our main result is
that there is an oracle A relative to which C_{=}P contains a set that is
immune to BPP^{ParityP}. In particular, this C_{=}P^A set is immune to PH^{A}
and ParityP^{A}. Strengthening results of Tor\'{a}n [J.ACM 38, 1991] and Green
[IPL 37, 1991], we also show that, in suitable relativizations, NP contains a
C_{=}P-immune set, and ParityP contains a PP^{PH}-immune set. This implies the
existence of a C_{=}P^{B}-simple set for some oracle B, which extends results
of Balc\'{a}zar et al. [SIAM J.Comp. 14, 1985; RAIRO 22, 1988] and provides the
first example of a simple set in a class not known to be contained in PH. Our
proof technique requires a circuit lower bound for ``exact counting'' that is
derived from Razborov's [Mat. Zametki 41, 1987] lower bound for majority.Comment: 20 page
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