49,575 research outputs found
Computing K-Trivial Sets by Incomplete Random Sets
Every K-trivial set is computable from an incomplete Martin-L\"of random set,
i.e., a Martin-L\"of random set that does not compute 0'
Martin-L\"of reducibility and cost functions
Martin-L\"of (ML)-reducibility compares -trivial sets by examining the
Martin-L\"of random sequences that compute them. We show that every -trivial
set is computable from a c.e.\ set of the same ML-degree. We investigate the
interplay between ML-reducibility and cost functions, which are used to both
measure the number of changes in a computable approximation, and the type of
null sets used to capture ML-random sequences. We show that for every cost
function there is a c.e.\ set ML-above the sets obeying it (called an
ML-complete set for the cost function). We characterise the -trivial sets
computable from a fragment of the left-c.e.\ random real~. This leads
to a new characterisation of strong jump-traceability
Computing from projections of random points: a dense hierarchy of subideals of the -trivial degrees
We study the sets that are computable from both halves of some (Martin-L\"of)
random sequence, which we call \emph{-bases}. We show that the collection
of such sets forms an ideal in the Turing degrees that is generated by its
c.e.\ elements. It is a proper subideal of the -trivial sets. We
characterise -bases as the sets computable from both halves of Chaitin's
, and as the sets that obey the cost function .
Generalising these results yields a dense hierarchy of subideals in the
-trivial degrees: For , let be the collection of sets that
are below any out of columns of some random sequence. As before, this
is an ideal generated by its c.e.\ elements and the random sequence in the
definition can always be taken to be . Furthermore, the corresponding
cost function characterisation reveals that is independent of the
particular representation of the rational , and that is properly
contained in for rational numbers . These results are proved using
a generalisation of the Loomis--Whitney inequality, which bounds the measure of
an open set in terms of the measures of its projections. The generality allows
us to analyse arbitrary families of orthogonal projections. As it turns out,
these do not give us new subideals of the -trivial sets, we can calculate
from the family which it characterises.
We finish by showing that the the union of for is the collection
of sets which are robustly computable from a random, a class previously studied
by Hirschfeldt, Jockusch, Kuyper, and Schupp
Density, forcing, and the covering problem
We present a notion of forcing that can be used, in conjunction with other
results, to show that there is a Martin-L\"of random set X such that X does not
compute 0' and X computes every K-trivial set
Computing strategies for achieving acceptability
We consider a trader who wants to direct his portfolio towards a set of
acceptable wealths given by a convex risk measure. We propose a black-box
algorithm, whose inputs are the joint law of stock prices and the convex risk
measure, and whose outputs are the numerical values of initial capital
requirement and the functional form of a trading strategy to achieve
acceptability. We also prove optimality of the obtained capital.Comment: 17 page
Calculus of Cost Functions
Cost functions provide a framework for constructions of sets Turing below the
halting problem that are close to computable. We carry out a systematic study
of cost functions. We relate their algebraic properties to their expressive
strength. We show that the class of additive cost functions describes the
-trivial sets. We prove a cost function basis theorem, and give a general
construction for building computably enumerable sets that are close to being
Turing complete.
This works dates from 2010 and was submitted in 2013 to the long-delayed
volume "The Incomputable" arising from the 2012 Cambridge Turing year
Aspects of Chaitin's Omega
The halting probability of a Turing machine,also known as Chaitin's Omega, is
an algorithmically random number with many interesting properties. Since
Chaitin's seminal work, many popular expositions have appeared, mainly focusing
on the metamathematical or philosophical significance of Omega (or debating
against it). At the same time, a rich mathematical theory exploring the
properties of Chaitin's Omega has been brewing in various technical papers,
which quietly reveals the significance of this number to many aspects of
contemporary algorithmic information theory. The purpose of this survey is to
expose these developments and tell a story about Omega, which outlines its
multifaceted mathematical properties and roles in algorithmic randomness
Certainty Equivalent and Utility Indifference Pricing for Incomplete Preferences via Convex Vector Optimization
For incomplete preference relations that are represented by multiple priors
and/or multiple -- possibly multivariate -- utility functions, we define a
certainty equivalent as well as the utility buy and sell prices and
indifference price bounds as set-valued functions of the claim. Furthermore, we
motivate and introduce the notion of a weak and a strong certainty equivalent.
We will show that our definitions contain as special cases some definitions
found in the literature so far on complete or special incomplete preferences.
We prove monotonicity and convexity properties of utility buy and sell prices
that hold in total analogy to the properties of the scalar indifference prices
for complete preferences. We show how the (weak and strong) set-valued
certainty equivalent as well as the indifference price bounds can be computed
or approximated by solving convex vector optimization problems. Numerical
examples and their economic interpretations are given for the univariate as
well as for the multivariate case
The Communication Cost of Information Spreading in Dynamic Networks
This paper investigates the message complexity of distributed information
spreading (a.k.a gossip or token dissemination) in adversarial dynamic
networks, where the goal is to spread tokens of information to every node
on an -node network. We consider the amortized (average) message complexity
of spreading a token, assuming that the number of tokens is large. Our focus is
on token-forwarding algorithms, which do not manipulate tokens in any way other
than storing, copying, and forwarding them.
We consider two types of adversaries that arbitrarily rewire the network
while keeping it connected: the adaptive adversary that is aware of the status
of all the nodes and the algorithm (including the current random choices), and
the oblivious adversary that is oblivious to the random choices made by the
algorithm. The central question that motivates our work is whether one can
achieve subquadratic amortized message complexity for information spreading.
We present two sets of results depending on how nodes send messages to their
neighbors: (1) Local broadcast: We show a tight lower bound of on
the number of amortized local broadcasts, which is matched by the naive
flooding algorithm, (2) Unicast: We study the message complexity as a function
of the number of dynamic changes in the network. To facilitate this, we
introduce a natural complexity measure for analyzing dynamic networks called
adversary-competitive message complexity where the adversary pays a unit cost
for every topological change. Under this model, it is shown that if is
sufficiently large, we can obtain an optimal amortized message complexity of
. We also present a randomized algorithm that achieves subquadratic
amortized message complexity when the number of tokens is not large under an
oblivious adversary
Topological Analysis of Syntactic Structures
We use the persistent homology method of topological data analysis and
dimensional analysis techniques to study data of syntactic structures of world
languages. We analyze relations between syntactic parameters in terms of
dimensionality, of hierarchical clustering structures, and of non-trivial
loops. We show there are relations that hold across language families and
additional relations that are family-specific. We then analyze the trees
describing the merging structure of persistent connected components for
languages in different language families and we show that they partly correlate
to historical phylogenetic trees but with significant differences. We also show
the existence of interesting non-trivial persistent first homology groups in
various language families. We give examples where explicit generators for the
persistent first homology can be identified, some of which appear to correspond
to homoplasy phenomena, while others may have an explanation in terms of
historical linguistics, corresponding to known cases of syntactic borrowing
across different language subfamilies.Comment: 83 pages, LaTeX, 44 figure
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