127,974 research outputs found
Stationary Algorithmic Probability
Kolmogorov complexity and algorithmic probability are defined only up to an
additive resp. multiplicative constant, since their actual values depend on the
choice of the universal reference computer. In this paper, we analyze a natural
approach to eliminate this machine-dependence.
Our method is to assign algorithmic probabilities to the different computers
themselves, based on the idea that "unnatural" computers should be hard to
emulate. Therefore, we study the Markov process of universal computers randomly
emulating each other. The corresponding stationary distribution, if it existed,
would give a natural and machine-independent probability measure on the
computers, and also on the binary strings.
Unfortunately, we show that no stationary distribution exists on the set of
all computers; thus, this method cannot eliminate machine-dependence. Moreover,
we show that the reason for failure has a clear and interesting physical
interpretation, suggesting that every other conceivable attempt to get rid of
those additive constants must fail in principle, too.
However, we show that restricting to some subclass of computers might help to
get rid of some amount of machine-dependence in some situations, and the
resulting stationary computer and string probabilities have beautiful
properties.Comment: 13 pages, 5 figures. Added an example of a positive recurrent
computer se
A generalized characterization of algorithmic probability
An a priori semimeasure (also known as "algorithmic probability" or "the
Solomonoff prior" in the context of inductive inference) is defined as the
transformation, by a given universal monotone Turing machine, of the uniform
measure on the infinite strings. It is shown in this paper that the class of a
priori semimeasures can equivalently be defined as the class of
transformations, by all compatible universal monotone Turing machines, of any
continuous computable measure in place of the uniform measure. Some
consideration is given to possible implications for the prevalent association
of algorithmic probability with certain foundational statistical principles
Algorithmic Information Theory and Foundations of Probability
The use of algorithmic information theory (Kolmogorov complexity theory) to
explain the relation between mathematical probability theory and `real world'
is discussed
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
The Thermodynamics of Network Coding, and an Algorithmic Refinement of the Principle of Maximum Entropy
The principle of maximum entropy (Maxent) is often used to obtain prior
probability distributions as a method to obtain a Gibbs measure under some
restriction giving the probability that a system will be in a certain state
compared to the rest of the elements in the distribution. Because classical
entropy-based Maxent collapses cases confounding all distinct degrees of
randomness and pseudo-randomness, here we take into consideration the
generative mechanism of the systems considered in the ensemble to separate
objects that may comply with the principle under some restriction and whose
entropy is maximal but may be generated recursively from those that are
actually algorithmically random offering a refinement to classical Maxent. We
take advantage of a causal algorithmic calculus to derive a thermodynamic-like
result based on how difficult it is to reprogram a computer code. Using the
distinction between computable and algorithmic randomness we quantify the cost
in information loss associated with reprogramming. To illustrate this we apply
the algorithmic refinement to Maxent on graphs and introduce a Maximal
Algorithmic Randomness Preferential Attachment (MARPA) Algorithm, a
generalisation over previous approaches. We discuss practical implications of
evaluation of network randomness. Our analysis provides insight in that the
reprogrammability asymmetry appears to originate from a non-monotonic
relationship to algorithmic probability. Our analysis motivates further
analysis of the origin and consequences of the aforementioned asymmetries,
reprogrammability, and computation.Comment: 30 page
Quantum Kolmogorov Complexity and Quantum Key Distribution
We discuss the Bennett-Brassard 1984 (BB84) quantum key distribution protocol
in the light of quantum algorithmic information. While Shannon's information
theory needs a probability to define a notion of information, algorithmic
information theory does not need it and can assign a notion of information to
an individual object. The program length necessary to describe an object,
Kolmogorov complexity, plays the most fundamental role in the theory. In the
context of algorithmic information theory, we formulate a security criterion
for the quantum key distribution by using the quantum Kolmogorov complexity
that was recently defined by Vit\'anyi. We show that a simple BB84 protocol
indeed distribute a binary sequence between Alice and Bob that looks almost
random for Eve with a probability exponentially close to 1.Comment: typos correcte
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