59 research outputs found
Computability and Complexity
We study the uniform computational content of the Vitali Covering Theorem for
intervals using the tool of Weihrauch reducibility. We show that a more
detailed picture emerges than what a related study by Giusto, Brown, and
Simpson has revealed in the setting of reverse mathematics. In particular,
different formulations of the Vitali Covering Theorem turn out to have
different uniform computational content. These versions are either computable
or closely related to uniform variants of Weak Weak K\H{o}nig's Lemma.Comment: 13 page
Generalized Effective Reducibility
We introduce two notions of effective reducibility for set-theoretical
statements, based on computability with Ordinal Turing Machines (OTMs), one of
which resembles Turing reducibility while the other is modelled after Weihrauch
reducibility. We give sample applications by showing that certain (algebraic)
constructions are not effective in the OTM-sense and considerung the effective
equivalence of various versions of the axiom of choice
Computability of the Radon-Nikodym derivative
We study the computational content of the Radon-Nokodym theorem from measure
theory in the framework of the representation approach to computable analysis.
We define computable measurable spaces and canonical representations of the
measures and the integrable functions on such spaces. For functions f,g on
represented sets, f is W-reducible to g if f can be computed by applying the
function g at most once. Let RN be the Radon-Nikodym operator on the space
under consideration and let EC be the non-computable operator mapping every
enumeration of a set of natural numbers to its characteristic function. We
prove that for every computable measurable space, RN is W-reducible to EC, and
we construct a computable measurable space for which EC is W-reducible to RN
Evolving Computability
We consider the degrees of non-computability (Weihrauch degrees) of finding
winning strategies (or more generally, Nash equilibria) in infinite sequential
games with certain winning sets (or more generally, outcome sets). In
particular, we show that as the complexity of the winning sets increases in the
difference hierarchy, the complexity of constructing winning strategies
increases in the effective Borel hierarchy.Comment: An extended abstract of this work has appeared in the Proceedings of
CiE 201
Reverse mathematics of matroids
Matroids generalize the familiar notion of linear dependence from linear algebra. Following a brief discussion of founding work in computability and matroids, we use the techniques of reverse mathematics to determine the logical strength of some basis theorems for matroids and enumerated matroids. Next, using Weihrauch reducibility, we relate the basis results to combinatorial choice principles and statements about vector spaces. Finally, we formalize some of the Weihrauch reductions to extract related reverse mathematics results. In particular, we show that the existence of bases for vector spaces of bounded dimension is equivalent to the induction scheme for \Sigma^0_2 formulas
Effective local connectivity properties
We investigate, and prove equivalent, effective versions of local
connectivity and uniformly local arcwise connectivity for connected and
computably compact subspaces of Euclidean space. We also prove that Euclidean
continua that are computably compact and effectively locally connected are
computably arcwise connected.Comment: Final versio
A CDCL-style calculus for solving non-linear constraints
In this paper we propose a novel approach for checking satisfiability of
non-linear constraints over the reals, called ksmt. The procedure is based on
conflict resolution in CDCL style calculus, using a composition of symbolical
and numerical methods. To deal with the non-linear components in case of
conflicts we use numerically constructed restricted linearisations. This
approach covers a large number of computable non-linear real functions such as
polynomials, rational or trigonometrical functions and beyond. A prototypical
implementation has been evaluated on several non-linear SMT-LIB examples and
the results have been compared with state-of-the-art SMT solvers.Comment: 17 pages, 3 figures; accepted at FroCoS 2019; software available at
<http://informatik.uni-trier.de/~brausse/ksmt/
Continuous Team Semantics
We study logics with team semantics in computable metric spaces. We show how to define approximate versions of the usual independence/dependence atoms. For restricted classes of formulae, we show that we can assume w.l.o.g.~that teams are closed sets. This then allows us to import techniques from computable analysis to study the complexity of formula satisfaction and model checking
Recent Advances in ÎŁ-definability over Continuous Data Types
The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Sigma-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler's Lemma for Sigma-definability over the reals without the equality test which relates Sigma-definability with definability in the constructive infinitary language L_{omega_1 omega}. Thus, a relation over the real numbers is Sigma-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Sigma-definability and also gives topological characterisation of Sigma-definable relations over the reals without the equality test. We also illustrate how computability over the real numbers can be expressed in the language of Sigma-formulas
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