1,272 research outputs found
Strong Turing Degrees for Additive BSS RAM's
For the additive real BSS machines using only constants 0 and 1 and order
tests we consider the corresponding Turing reducibility and characterize some
semi-decidable decision problems over the reals. In order to refine,
step-by-step, a linear hierarchy of Turing degrees with respect to this model,
we define several halting problems for classes of additive machines with
different abilities and construct further suitable decision problems. In the
construction we use methods of the classical recursion theory as well as
techniques for proving bounds resulting from algebraic properties. In this way
we extend a known hierarchy of problems below the halting problem for the
additive machines using only equality tests and we present a further
subhierarchy of semi-decidable problems between the halting problems for the
additive machines using only equality tests and using order tests,
respectively
Coding-theorem Like Behaviour and Emergence of the Universal Distribution from Resource-bounded Algorithmic Probability
Previously referred to as `miraculous' in the scientific literature because
of its powerful properties and its wide application as optimal solution to the
problem of induction/inference, (approximations to) Algorithmic Probability
(AP) and the associated Universal Distribution are (or should be) of the
greatest importance in science. Here we investigate the emergence, the rates of
emergence and convergence, and the Coding-theorem like behaviour of AP in
Turing-subuniversal models of computation. We investigate empirical
distributions of computing models in the Chomsky hierarchy. We introduce
measures of algorithmic probability and algorithmic complexity based upon
resource-bounded computation, in contrast to previously thoroughly investigated
distributions produced from the output distribution of Turing machines. This
approach allows for numerical approximations to algorithmic
(Kolmogorov-Chaitin) complexity-based estimations at each of the levels of a
computational hierarchy. We demonstrate that all these estimations are
correlated in rank and that they converge both in rank and values as a function
of computational power, despite fundamental differences between computational
models. In the context of natural processes that operate below the Turing
universal level because of finite resources and physical degradation, the
investigation of natural biases stemming from algorithmic rules may shed light
on the distribution of outcomes. We show that up to 60\% of the
simplicity/complexity bias in distributions produced even by the weakest of the
computational models can be accounted for by Algorithmic Probability in its
approximation to the Universal Distribution.Comment: 27 pages main text, 39 pages including supplement. Online complexity
calculator: http://complexitycalculator.com
Complexity of Two-Dimensional Patterns
In dynamical systems such as cellular automata and iterated maps, it is often
useful to look at a language or set of symbol sequences produced by the system.
There are well-established classification schemes, such as the Chomsky
hierarchy, with which we can measure the complexity of these sets of sequences,
and thus the complexity of the systems which produce them.
In this paper, we look at the first few levels of a hierarchy of complexity
for two-or-more-dimensional patterns. We show that several definitions of
``regular language'' or ``local rule'' that are equivalent in d=1 lead to
distinct classes in d >= 2. We explore the closure properties and computational
complexity of these classes, including undecidability and L-, NL- and
NP-completeness results.
We apply these classes to cellular automata, in particular to their sets of
fixed and periodic points, finite-time images, and limit sets. We show that it
is undecidable whether a CA in d >= 2 has a periodic point of a given period,
and that certain ``local lattice languages'' are not finite-time images or
limit sets of any CA. We also show that the entropy of a d-dimensional CA's
finite-time image cannot decrease faster than t^{-d} unless it maps every
initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy
Kolmogorov's Structure Functions and Model Selection
In 1974 Kolmogorov proposed a non-probabilistic approach to statistics and
model selection. Let data be finite binary strings and models be finite sets of
binary strings. Consider model classes consisting of models of given maximal
(Kolmogorov) complexity. The ``structure function'' of the given data expresses
the relation between the complexity level constraint on a model class and the
least log-cardinality of a model in the class containing the data. We show that
the structure function determines all stochastic properties of the data: for
every constrained model class it determines the individual best-fitting model
in the class irrespective of whether the ``true'' model is in the model class
considered or not. In this setting, this happens {\em with certainty}, rather
than with high probability as is in the classical case. We precisely quantify
the goodness-of-fit of an individual model with respect to individual data. We
show that--within the obvious constraints--every graph is realized by the
structure function of some data. We determine the (un)computability properties
of the various functions contemplated and of the ``algorithmic minimal
sufficient statistic.''Comment: 25 pages LaTeX, 5 figures. In part in Proc 47th IEEE FOCS; this final
version (more explanations, cosmetic modifications) to appear in IEEE Trans
Inform T
Complexity Hierarchies Beyond Elementary
We introduce a hierarchy of fast-growing complexity classes and show its
suitability for completeness statements of many non elementary problems. This
hierarchy allows the classification of many decision problems with a
non-elementary complexity, which occur naturally in logic, combinatorics,
formal languages, verification, etc., with complexities ranging from simple
towers of exponentials to Ackermannian and beyond.Comment: Version 3 is the published version in TOCT 8(1:3), 2016. I will keep
updating the catalogue of problems from Section 6 in future revision
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