265 research outputs found
On the algebraic structure of Weihrauch degrees
We introduce two new operations (compositional products and implication) on Weihrauch degrees, and investigate the overall algebraic structure. The validity of the various distributivity laws is studied and forms the basis for a comparison with similar structures such as residuated lattices and concurrent Kleene algebras. Introducing the notion of an ideal with respect to the compositional product, we can consider suitable quotients of the Weihrauch degrees. We also prove some specific characterizations using the implication. In order to introduce and study compositional products and implications, we introduce and study a function space of multi-valued continuous functions. This space turns out to be particularly well-behaved for effectively traceable spaces that are closely related to admissibly represented spaces
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
The Bolzano-Weierstrass Theorem is the Jump of Weak K\"onig's Lemma
We classify the computational content of the Bolzano-Weierstrass Theorem and
variants thereof in the Weihrauch lattice. For this purpose we first introduce
the concept of a derivative or jump in this lattice and we show that it has
some properties similar to the Turing jump. Using this concept we prove that
the derivative of closed choice of a computable metric space is the cluster
point problem of that space. By specialization to sequences with a relatively
compact range we obtain a characterization of the Bolzano-Weierstrass Theorem
as the derivative of compact choice. In particular, this shows that the
Bolzano-Weierstrass Theorem on real numbers is the jump of Weak K\"onig's
Lemma. Likewise, the Bolzano-Weierstrass Theorem on the binary space is the
jump of the lesser limited principle of omniscience LLPO and the
Bolzano-Weierstrass Theorem on natural numbers can be characterized as the jump
of the idempotent closure of LLPO. We also introduce the compositional product
of two Weihrauch degrees f and g as the supremum of the composition of any two
functions below f and g, respectively. We can express the main result such that
the Bolzano-Weierstrass Theorem is the compositional product of Weak K\"onig's
Lemma and the Monotone Convergence Theorem. We also study the class of weakly
limit computable functions, which are functions that can be obtained by
composition of weakly computable functions with limit computable functions. We
prove that the Bolzano-Weierstrass Theorem on real numbers is complete for this
class. Likewise, the unique cluster point problem on real numbers is complete
for the class of functions that are limit computable with finitely many mind
changes. We also prove that the Bolzano-Weierstrass Theorem on real numbers
and, more generally, the unbounded cluster point problem on real numbers is
uniformly low limit computable. Finally, we also discuss separation techniques.Comment: This version includes an addendum by Andrea Cettolo, Matthias
Schr\"oder, and the authors of the original paper. The addendum closes a gap
in the proof of Theorem 11.2, which characterizes the computational content
of the Bolzano-Weierstra\ss{} Theorem for arbitrary computable metric space
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
The descriptive theory of represented spaces
This is a survey on the ongoing development of a descriptive theory of
represented spaces, which is intended as an extension of both classical and
effective descriptive set theory to deal with both sets and functions between
represented spaces. Most material is from work-in-progress, and thus there may
be a stronger focus on projects involving the author than an objective survey
would merit.Comment: survey of work-in-progres
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