47,007 research outputs found
Memory effects can make the transmission capability of a communication channel uncomputable
Most communication channels are subjected to noise. One of the goals of
Information Theory is to add redundancy in the transmission of information so
that the information is transmitted reliably and the amount of information
transmitted through the channel is as large as possible. The maximum rate at
which reliable transmission is possible is called the capacity. If the channel
does not keep memory of its past, the capacity is given by a simple
optimization problem and can be efficiently computed. The situation of channels
with memory is less clear. Here we show that for channels with memory the
capacity cannot be computed to within precision 1/5. Our result holds even if
we consider one of the simplest families of such channels -information-stable
finite state machine channels-, restrict the input and output of the channel to
4 and 1 bit respectively and allow 6 bits of memory.Comment: Improved presentation and clarified claim
Computation with narrow CTCs
We examine some variants of computation with closed timelike curves (CTCs),
where various restrictions are imposed on the memory of the computer, and the
information carrying capacity and range of the CTC. We give full
characterizations of the classes of languages recognized by polynomial time
probabilistic and quantum computers that can send a single classical bit to
their own past. Such narrow CTCs are demonstrated to add the power of limited
nondeterminism to deterministic computers, and lead to exponential speedup in
constant-space probabilistic and quantum computation. We show that, given a
time machine with constant negative delay, one can implement CTC-based
computations without the need to know about the runtime beforehand.Comment: 16 pages. A few typo was correcte
Lurching Toward Chernobyl: Dysfunctions of Real-Time Computation
Cognitive biological structures, social organizations, and computing machines operating in real time are subject to Rate Distortion Theorem constraints driven by the homology between information source uncertainty and free energy density. This exposes the unitary structure/environment system to a relentless entropic torrent compounded by sudden large deviations causing increased distortion between intent and impact, particularly as demands escalate. The phase transitions characteristic of information phenomena suggest that, rather than graceful decay under increasing load, these structures will undergo punctuated degradation akin to spontaneous symmetry breaking in physical systems. Rate distortion problems, that also affect internal structural dynamics, can become synergistic with limitations equivalent to the inattentional blindness of natural cognitive process. These mechanisms, and their interactions, are unlikely to scale well, so that, depending on architecture, enlarging the structure or its duties may lead to a crossover point at which added resources must be almost entirely devoted to ensuring system stability -- a form of allometric scaling familiar from biological examples. This suggests a critical need to tune architecture to problem type and system demand. A real-time computational structure and its environment are a unitary phenomenon, and environments are usually idiosyncratic. Thus the resulting path dependence in the development of pathology could often require an individualized approach to remediation more akin to an arduous psychiatric intervention than to the traditional engineering or medical quick fix. Failure to recognize the depth of these problems seems likely to produce a relentless chain of the Chernobyl-like failures that are necessary, bot often insufficient, for remediation under our system
Gene Expression and its Discontents: Developmental disorders as dysfunctions of epigenetic cognition
Systems biology presently suffers the same mereological and sufficiency fallacies that haunt neural network models of high order cognition. Shifting perspective from the massively parallel space of gene matrix interactions to the grammar/syntax of the time series of expressed phenotypes using a cognitive paradigm permits import of techniques from statistical physics via the homology between information source uncertainty and free energy density. This produces a broad spectrum of possible statistical models of development and its pathologies in which epigenetic regulation and the effects of embedding environment are analogous to a tunable enzyme catalyst. A cognitive paradigm naturally incorporates memory, leading directly to models of epigenetic inheritance, as affected by environmental exposures, in the largest sense. Understanding gene expression, development, and their dysfunctions will require data analysis tools considerably more sophisticated than the present crop of simplistic models abducted from neural network studies or stochastic chemical reaction theory
A Proof of Entropy Minimization for Outputs in Deletion Channels via Hidden Word Statistics
From the output produced by a memoryless deletion channel from a uniformly
random input of known length , one obtains a posterior distribution on the
channel input. The difference between the Shannon entropy of this distribution
and that of the uniform prior measures the amount of information about the
channel input which is conveyed by the output of length , and it is natural
to ask for which outputs this is extremized. This question was posed in a
previous work, where it was conjectured on the basis of experimental data that
the entropy of the posterior is minimized and maximized by the constant strings
and and the alternating strings
and respectively. In the present
work we confirm the minimization conjecture in the asymptotic limit using
results from hidden word statistics. We show how the analytic-combinatorial
methods of Flajolet, Szpankowski and Vall\'ee for dealing with the hidden
pattern matching problem can be applied to resolve the case of fixed output
length and , by obtaining estimates for the entropy in
terms of the moments of the posterior distribution and establishing its
minimization via a measure of autocorrelation.Comment: 11 pages, 2 figure
Skip-Sliding Window Codes
Constrained coding is used widely in digital communication and storage
systems. In this paper, we study a generalized sliding window constraint called
the skip-sliding window. A skip-sliding window (SSW) code is defined in terms
of the length of a sliding window, skip length , and cost constraint
in each sliding window. Each valid codeword of length is determined by
windows of length where window starts at th symbol for
all non-negative integers such that ; and the cost constraint
in each window must be satisfied. In this work, two methods are given to
enumerate the size of SSW codes and further refinements are made to reduce the
enumeration complexity. Using the proposed enumeration methods, the noiseless
capacity of binary SSW codes is determined and observations such as greater
capacity than other classes of codes are made. Moreover, some noisy capacity
bounds are given. SSW coding constraints arise in various applications
including simultaneous energy and information transfer.Comment: 28 pages, 11 figure
Algorithmic complexity of quantum capacity
Recently the theory of communication developed by Shannon has been extended
to the quantum realm by exploiting the rules of quantum theory. This latter
stems on complex vector spaces. However complex (as well as real) numbers are
just idealizations and they are not available in practice where we can only
deal with rational numbers. This fact naturally leads to the question of
whether the developed notions of capacities for quantum channels truly catch
their ability to transmit information. Here we answer this question for the
quantum capacity. To this end we resort to the notion of semi-computability in
order to approximately (by rational numbers) describe quantum states and
quantum channel maps. Then we introduce algorithmic entropies (like algorithmic
quantum coherent information) and derive relevant properties for them. Finally
we define algorithmic quantum capacity and prove that it equals the standard
one
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