3,053 research outputs found
On the Complexity and Behaviour of Cryptocurrencies Compared to Other Markets
We show that the behaviour of Bitcoin has interesting similarities to stock
and precious metal markets, such as gold and silver. We report that whilst
Litecoin, the second largest cryptocurrency, closely follows Bitcoin's
behaviour, it does not show all the reported properties of Bitcoin. Agreements
between apparently disparate complexity measures have been found, and it is
shown that statistical, information-theoretic, algorithmic and fractal measures
have different but interesting capabilities of clustering families of markets
by type. The report is particularly interesting because of the range and novel
use of some measures of complexity to characterize price behaviour, because of
the IRS designation of Bitcoin as an investment property and not a currency,
and the announcement of the Canadian government's own electronic currency
MintChip.Comment: 16 pages, 11 figures, 4 table
Sub-computable Boundedness Randomness
This paper defines a new notion of bounded computable randomness for certain
classes of sub-computable functions which lack a universal machine. In
particular, we define such versions of randomness for primitive recursive
functions and for PSPACE functions. These new notions are robust in that there
are equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorov
complexity, and (3) martingales. We show these notions can be equivalently
defined with prefix-free Kolmogorov complexity. We prove that one direction of
van Lambalgen's theorem holds for relative computability, but the other
direction fails. We discuss statistical properties of these notions of
randomness
On Resource-bounded versions of the van Lambalgen theorem
The van Lambalgen theorem is a surprising result in algorithmic information
theory concerning the symmetry of relative randomness. It establishes that for
any pair of infinite sequences and , is Martin-L\"of random and
is Martin-L\"of random relative to if and only if the interleaved sequence
is Martin-L\"of random. This implies that is relative random
to if and only if is random relative to \cite{vanLambalgen},
\cite{Nies09}, \cite{HirschfeldtBook}. This paper studies the validity of this
phenomenon for different notions of time-bounded relative randomness.
We prove the classical van Lambalgen theorem using martingales and Kolmogorov
compressibility. We establish the failure of relative randomness in these
settings, for both time-bounded martingales and time-bounded Kolmogorov
complexity. We adapt our classical proofs when applicable to the time-bounded
setting, and construct counterexamples when they fail. The mode of failure of
the theorem may depend on the notion of time-bounded randomness
Compressibility, laws of nature, initial conditions and complexity
We critically analyse the point of view for which laws of nature are just a
mean to compress data. Discussing some basic notions of dynamical systems and
information theory, we show that the idea that the analysis of large amount of
data by means of an algorithm of compression is equivalent to the knowledge one
can have from scientific laws, is rather naive. In particular we discuss the
subtle conceptual topic of the initial conditions of phenomena which are
generally incompressible. Starting from this point, we argue that laws of
nature represent more than a pure compression of data, and that the
availability of large amount of data, in general, is not particularly useful to
understand the behaviour of complex phenomena.Comment: 19 Pages, No figures, published on Foundation of Physic
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