24 research outputs found
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
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Continuity, Discontinuity and Dynamics in Mathematics & Economics - Reconsidering Rosser's Visions
Barkley Rosser has been a pioneer in arguing the case for the mathematics of discontinuity, broadly conceived, to be placed at the foundations of modelling economic dynamics. In this paper we reconsider this vision from the broad perspective of a variety of different kinds of mathematics and suggest a broadening of Rosser’s methodology to the study of economic dynamicsContinuity, Discontinuity, Economic Dynamics, Relaxation Oscillations
The degree structure of Weihrauch-reducibility
We answer a question by Vasco Brattka and Guido Gherardi by proving that the
Weihrauch-lattice is not a Brouwer algebra. The computable Weihrauch-lattice is
also not a Heyting algebra, but the continuous Weihrauch-lattice is. We further
investigate the existence of infinite infima and suprema, as well as embeddings
of the Medvedev-degrees into the Weihrauch-degrees
Sequential discontinuity and first-order problems
We explore the low levels of the structure of the continuous Weihrauch
degrees of first-order problems. In particular, we show that there exists a
minimal discontinuous first-order degree, namely that of \accn, without any
determinacy assumptions. The same degree is also revealed as the least
sequentially discontinuous one, i.e. the least degree with a representative
whose restriction to some sequence converging to a limit point is still
discontinuous.
The study of games related to continuous Weihrauch reducibility constitutes
an important ingredient in the proof of the main theorem. We present some
initial additional results about the degrees of first-order problems that can
be obtained using this approach
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
Computability in basic quantum mechanics
The basic notions of quantum mechanics are formulated in terms of separable infinite dimensional Hilbert space H. In terms of the Hilbert lattice L of closed linear subspaces of H the notions of state and observable can be formulated as kinds of measures as in [21]. The aim of this paper is to show that there is a good notion of computability for these data structures in the sense of Weihrauch’s Type Two Effectivity (TTE) [26]. Instead of explicitly exhibiting admissible representations for the data types under consideration we show that they do live within the category QCB0 which is equivalent to the category AdmRep of admissible representations and continuously realizable maps between them. For this purpose in case of observables we have to replace measures by valuations which allows us to prove an effective version of von Neumann’s Spectral Theorem