48,030 research outputs found
Information complexity is computable
The information complexity of a function is the minimum amount of
information Alice and Bob need to exchange to compute the function . In this
paper we provide an algorithm for approximating the information complexity of
an arbitrary function to within any additive error , thus
resolving an open question as to whether information complexity is computable.
In the process, we give the first explicit upper bound on the rate of
convergence of the information complexity of when restricted to -bit
protocols to the (unrestricted) information complexity of .Comment: 30 page
Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime
The synthesis of classical Computational Complexity Theory with Recursive
Analysis provides a quantitative foundation to reliable numerics. Here the
operators of maximization, integration, and solving ordinary differential
equations are known to map (even high-order differentiable) polynomial-time
computable functions to instances which are `hard' for classical complexity
classes NP, #P, and CH; but, restricted to analytic functions, map
polynomial-time computable ones to polynomial-time computable ones --
non-uniformly!
We investigate the uniform parameterized complexity of the above operators in
the setting of Weihrauch's TTE and its second-order extension due to
Kawamura&Cook (2010). That is, we explore which (both continuous and discrete,
first and second order) information and parameters on some given f is
sufficient to obtain similar data on Max(f) and int(f); and within what running
time, in terms of these parameters and the guaranteed output precision 2^(-n).
It turns out that Gevrey's hierarchy of functions climbing from analytic to
smooth corresponds to the computational complexity of maximization growing from
polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete)
Computation, Hard Analysis, and Information-Based Complexity
Finite-State Complexity and the Size of Transducers
Finite-state complexity is a variant of algorithmic information theory
obtained by replacing Turing machines with finite transducers. We consider the
state-size of transducers needed for minimal descriptions of arbitrary strings
and, as our main result, we show that the state-size hierarchy with respect to
a standard encoding is infinite. We consider also hierarchies yielded by more
general computable encodings.Comment: In Proceedings DCFS 2010, arXiv:1008.127
The "paradox" of computability and a recursive relative version of the Busy Beaver function
In this article, we will show that uncomputability is a relative property not
only of oracle Turing machines, but also of subrecursive classes. We will
define the concept of a Turing submachine, and a recursive relative version for
the Busy Beaver function which we will call Busy Beaver Plus function.
Therefore, we will prove that the computable Busy Beaver Plus function defined
on any Turing submachine is not computable by any program running on this
submachine. We will thereby demonstrate the existence of a "paradox" of
computability a la Skolem: a function is computable when "seen from the
outside" the subsystem, but uncomputable when "seen from within" the same
subsystem. Finally, we will raise the possibility of defining universal
submachines, and a hierarchy of negative Turing degrees.Comment: 10 pages. 0 figures. Supported by the National Council for Scientific
and Technological Development (CNPq), Brazil. Book chapter published in
Information and Complexity, Mark Burgin and Cristian S. Calude (Editors),
World Scientific Publishing, 2016, ISBN 978-981-3109-02-5, available at
http://www.worldscientific.com/worldscibooks/10.1142/10017. arXiv admin note:
substantial text overlap with arXiv:1612.0522
Effective complexity of stationary process realizations
The concept of effective complexity of an object as the minimal description
length of its regularities has been initiated by Gell-Mann and Lloyd. The
regularities are modeled by means of ensembles, that is probability
distributions on finite binary strings. In our previous paper we propose a
definition of effective complexity in precise terms of algorithmic information
theory. Here we investigate the effective complexity of binary strings
generated by stationary, in general not computable, processes. We show that
under not too strong conditions long typical process realizations are
effectively simple. Our results become most transparent in the context of
coarse effective complexity which is a modification of the original notion of
effective complexity that uses less parameters in its definition. A similar
modification of the related concept of sophistication has been suggested by
Antunes and Fortnow.Comment: 14 pages, no figure
The similarity metric
A new class of distances appropriate for measuring similarity relations
between sequences, say one type of similarity per distance, is studied. We
propose a new ``normalized information distance'', based on the noncomputable
notion of Kolmogorov complexity, and show that it is in this class and it
minorizes every computable distance in the class (that is, it is universal in
that it discovers all computable similarities). We demonstrate that it is a
metric and call it the {\em similarity metric}. This theory forms the
foundation for a new practical tool. To evidence generality and robustness we
give two distinctive applications in widely divergent areas using standard
compression programs like gzip and GenCompress. First, we compare whole
mitochondrial genomes and infer their evolutionary history. This results in a
first completely automatic computed whole mitochondrial phylogeny tree.
Secondly, we fully automatically compute the language tree of 52 different
languages.Comment: 13 pages, LaTex, 5 figures, Part of this work appeared in Proc. 14th
ACM-SIAM Symp. Discrete Algorithms, 2003. This is the final, corrected,
version to appear in IEEE Trans Inform. T
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