630 research outputs found
Dimension Extractors and Optimal Decompression
A *dimension extractor* is an algorithm designed to increase the effective
dimension -- i.e., the amount of computational randomness -- of an infinite
binary sequence, in order to turn a "partially random" sequence into a "more
random" sequence. Extractors are exhibited for various effective dimensions,
including constructive, computable, space-bounded, time-bounded, and
finite-state dimension. Using similar techniques, the Kucera-Gacs theorem is
examined from the perspective of decompression, by showing that every infinite
sequence S is Turing reducible to a Martin-Loef random sequence R such that the
asymptotic number of bits of R needed to compute n bits of S, divided by n, is
precisely the constructive dimension of S, which is shown to be the optimal
ratio of query bits to computed bits achievable with Turing reductions. The
extractors and decompressors that are developed lead directly to new
characterizations of some effective dimensions in terms of optimal
decompression by Turing reductions.Comment: This report was combined with a different conference paper "Every
Sequence is Decompressible from a Random One" (cs.IT/0511074, at
http://dx.doi.org/10.1007/11780342_17), and both titles were changed, with
the conference paper incorporated as section 5 of this new combined paper.
The combined paper was accepted to the journal Theory of Computing Systems,
as part of a special issue of invited papers from the second conference on
Computability in Europe, 200
Stable Leader Election in Population Protocols Requires Linear Time
A population protocol *stably elects a leader* if, for all , starting from
an initial configuration with agents each in an identical state, with
probability 1 it reaches a configuration that is correct (exactly
one agent is in a special leader state ) and stable (every configuration
reachable from also has a single agent in state ). We show
that any population protocol that stably elects a leader requires
expected "parallel time" --- expected total pairwise interactions
--- to reach such a stable configuration. Our result also informs the
understanding of the time complexity of chemical self-organization by showing
an essential difficulty in generating exact quantities of molecular species
quickly.Comment: accepted to Distributed Computing special issue of invited papers
from DISC 2015; significantly revised proof structure and intuitive
explanation
Design of Geometric Molecular Bonds
An example of a nonspecific molecular bond is the affinity of any positive
charge for any negative charge (like-unlike), or of nonpolar material for
itself when in aqueous solution (like-like). This contrasts specific bonds such
as the affinity of the DNA base A for T, but not for C, G, or another A. Recent
experimental breakthroughs in DNA nanotechnology demonstrate that a particular
nonspecific like-like bond ("blunt-end DNA stacking" that occurs between the
ends of any pair of DNA double-helices) can be used to create specific
"macrobonds" by careful geometric arrangement of many nonspecific blunt ends,
motivating the need for sets of macrobonds that are orthogonal: two macrobonds
not intended to bind should have relatively low binding strength, even when
misaligned.
To address this need, we introduce geometric orthogonal codes that abstractly
model the engineered DNA macrobonds as two-dimensional binary codewords. While
motivated by completely different applications, geometric orthogonal codes
share similar features to the optical orthogonal codes studied by Chung,
Salehi, and Wei. The main technical difference is the importance of 2D geometry
in defining codeword orthogonality.Comment: Accepted to appear in IEEE Transactions on Molecular, Biological, and
Multi-Scale Communication
Finite-State Dimension and Lossy Decompressors
This paper examines information-theoretic questions regarding the difficulty
of compressing data versus the difficulty of decompressing data and the role
that information loss plays in this interaction. Finite-state compression and
decompression are shown to be of equivalent difficulty, even when the
decompressors are allowed to be lossy.
Inspired by Kolmogorov complexity, this paper defines the optimal
*decompression *ratio achievable on an infinite sequence by finite-state
decompressors (that is, finite-state transducers outputting the sequence in
question). It is shown that the optimal compression ratio achievable on a
sequence S by any *information lossless* finite state compressor, known as the
finite-state dimension of S, is equal to the optimal decompression ratio
achievable on S by any finite-state decompressor. This result implies a new
decompression characterization of finite-state dimension in terms of lossy
finite-state transducers.Comment: We found that Theorem 3.11, which was basically the motive for this
paper, was already proven by Sheinwald, Ziv, and Lempel in 1991 and 1995
paper
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