68 research outputs found
Completion of Choice
We systematically study the completion of choice problems in the Weihrauch
lattice. Choice problems play a pivotal role in Weihrauch complexity. For one,
they can be used as landmarks that characterize important equivalences classes
in the Weihrauch lattice. On the other hand, choice problems also characterize
several natural classes of computable problems, such as finite mind change
computable problems, non-deterministically computable problems, Las Vegas
computable problems and effectively Borel measurable functions. The closure
operator of completion generates the concept of total Weihrauch reducibility,
which is a variant of Weihrauch reducibility with total realizers. Logically
speaking, the completion of a problem is a version of the problem that is
independent of its premise. Hence, studying the completion of choice problems
allows us to study simultaneously choice problems in the total Weihrauch
lattice, as well as the question which choice problems can be made independent
of their premises in the usual Weihrauch lattice. The outcome shows that many
important choice problems that are related to compact spaces are complete,
whereas choice problems for unbounded spaces or closed sets of positive measure
are typically not complete.Comment: 30 page
Weihrauch goes Brouwerian
We prove that the Weihrauch lattice can be transformed into a Brouwer algebra
by the consecutive application of two closure operators in the appropriate
order: first completion and then parallelization. The closure operator of
completion is a new closure operator that we introduce. It transforms any
problem into a total problem on the completion of the respective types, where
we allow any value outside of the original domain of the problem. This closure
operator is of interest by itself, as it generates a total version of Weihrauch
reducibility that is defined like the usual version of Weihrauch reducibility,
but in terms of total realizers. From a logical perspective completion can be
seen as a way to make problems independent of their premises. Alongside with
the completion operator and total Weihrauch reducibility we need to study
precomplete representations that are required to describe these concepts. In
order to show that the parallelized total Weihrauch lattice forms a Brouwer
algebra, we introduce a new multiplicative version of an implication. While the
parallelized total Weihrauch lattice forms a Brouwer algebra with this
implication, the total Weihrauch lattice fails to be a model of intuitionistic
linear logic in two different ways. In order to pinpoint the algebraic reasons
for this failure, we introduce the concept of a Weihrauch algebra that allows
us to formulate the failure in precise and neat terms. Finally, we show that
the Medvedev Brouwer algebra can be embedded into our Brouwer algebra, which
also implies that the theory of our Brouwer algebra is Jankov logic.Comment: 36 page
Probabilistic Computability and Choice
We study the computational power of randomized computations on infinite
objects, such as real numbers. In particular, we introduce the concept of a Las
Vegas computable multi-valued function, which is a function that can be
computed on a probabilistic Turing machine that receives a random binary
sequence as auxiliary input. The machine can take advantage of this random
sequence, but it always has to produce a correct result or to stop the
computation after finite time if the random advice is not successful. With
positive probability the random advice has to be successful. We characterize
the class of Las Vegas computable functions in the Weihrauch lattice with the
help of probabilistic choice principles and Weak Weak K\H{o}nig's Lemma. Among
other things we prove an Independent Choice Theorem that implies that Las Vegas
computable functions are closed under composition. In a case study we show that
Nash equilibria are Las Vegas computable, while zeros of continuous functions
with sign changes cannot be computed on Las Vegas machines. However, we show
that the latter problem admits randomized algorithms with weaker failure
recognition mechanisms. The last mentioned results can be interpreted such that
the Intermediate Value Theorem is reducible to the jump of Weak Weak
K\H{o}nig's Lemma, but not to Weak Weak K\H{o}nig's Lemma itself. These
examples also demonstrate that Las Vegas computable functions form a proper
superclass of the class of computable functions and a proper subclass of the
class of non-deterministically computable functions. We also study the impact
of specific lower bounds on the success probabilities, which leads to a strict
hierarchy of classes. In particular, the classical technique of probability
amplification fails for computations on infinite objects. We also investigate
the dependency on the underlying probability space.Comment: Information and Computation (accepted for publication
Devil's staircase phase diagram of the fractional quantum Hall effect in the thin-torus limit
After more than three decades the fractional quantum Hall effect still poses
challenges to contemporary physics. Recent experiments point toward a fractal
scenario for the Hall resistivity as a function of the magnetic field. Here, we
consider the so-called thin-torus limit of the Hamiltonian describing
interacting electrons in a strong magnetic field, restricted to the lowest
Landau level, and we show that it can be mapped onto a one-dimensional lattice
gas with repulsive interactions, with the magnetic field playing the role of a
chemical potential. The statistical mechanics of such models leads to interpret
the sequence of Hall plateaux as a fractal phase diagram, whose landscape shows
a qualitative agreement with experiments.Comment: 5 pages main text, 11 pages supplementary, 2 figure
Effective Choice and Boundedness Principles in Computable Analysis
In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles on closed sets which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. Well-known omniscience principles from constructive mathematics such as and can naturally be considered as Weihrauch degrees and they play an important role in our classification. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example
Interpolation in extensions of first-order logic
We prove a generalization of Maehara's lemma to show that the extensions of
classical and intuitionistic first-order logic with a special type of geometric
axioms, called singular geometric axioms, have Craig's interpolation property.
As a corollary, we obtain a direct proof of interpolation for (classical and
intuitionistic) first-order logic with identity, as well as interpolation for
several mathematical theories, including the theory of equivalence relations,
(strict) partial and linear orders, and various intuitionistic order theories
such as apartness and positive partial and linear orders.Comment: In this up-dated version of the paper a more general notion of
singular geometric theory is provided allowing the extension of our
interpolation results to further fundamental mathematical theorie
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