10 research outputs found
Representations of measurable sets in computable measure theory
This article is a fundamental study in computable measure theory. We use the
framework of TTE, the representation approach, where computability on an
abstract set X is defined by representing its elements with concrete "names",
possibly countably infinite, over some alphabet {\Sigma}. As a basic
computability structure we consider a computable measure on a computable
-algebra. We introduce and compare w.r.t. reducibility several natural
representations of measurable sets. They are admissible and generally form four
different equivalence classes. We then compare our representations with those
introduced by Y. Wu and D. Ding in 2005 and 2006 and claim that one of our
representations is the most useful one for studying computability on measurable
functions
The Craig Interpolation Property in First-order G\"odel Logic
In this article, a model-theoretic approach is proposed to prove that the
first-order G\"odel logic, , as well as its extension
associated with first-order relational languages enjoy the
Craig interpolation property. These results partially provide an affirmative
answer to a question posed in [Aguilera, Baaz, 2017, Ten problems in G\"odel
logic]
Turing machines on represented sets, a model of computation for Analysis
We introduce a new type of generalized Turing machines (GTMs), which are
intended as a tool for the mathematician who studies computability in Analysis.
In a single tape cell a GTM can store a symbol, a real number, a continuous
real function or a probability measure, for example. The model is based on TTE,
the representation approach for computable analysis. As a main result we prove
that the functions that are computable via given representations are closed
under GTM programming. This generalizes the well known fact that these
functions are closed under composition. The theorem allows to speak about
objects themselves instead of names in algorithms and proofs. By using GTMs for
specifying algorithms, many proofs become more rigorous and also simpler and
more transparent since the GTM model is very simple and allows to apply
well-known techniques from Turing machine theory. We also show how finite or
infinite sequences as names can be replaced by sets (generalized
representations) on which computability is already defined via representations.
This allows further simplification of proofs. All of this is done for
multi-functions, which are essential in Computable Analysis, and
multi-representations, which often allow more elegant formulations. As a
byproduct we show that the computable functions on finite and infinite
sequences of symbols are closed under programming with GTMs. We conclude with
examples of application
Turing machines on represented sets, a model of computation for Analysis
We introduce a new type of generalized Turing machines (GTMs), which areintended as a tool for the mathematician who studies computability in Analysis.In a single tape cell a GTM can store a symbol, a real number, a continuousreal function or a probability measure, for example. The model is based on TTE,the representation approach for computable analysis. As a main result we provethat the functions that are computable via given representations are closedunder GTM programming. This generalizes the well known fact that thesefunctions are closed under composition. The theorem allows to speak aboutobjects themselves instead of names in algorithms and proofs. By using GTMs forspecifying algorithms, many proofs become more rigorous and also simpler andmore transparent since the GTM model is very simple and allows to applywell-known techniques from Turing machine theory. We also show how finite orinfinite sequences as names can be replaced by sets (generalizedrepresentations) on which computability is already defined via representations.This allows further simplification of proofs. All of this is done formulti-functions, which are essential in Computable Analysis, andmulti-representations, which often allow more elegant formulations. As abyproduct we show that the computable functions on finite and infinitesequences of symbols are closed under programming with GTMs. We conclude withexamples of application
Representations of measurable sets in computable measure theory
This article is a fundamental study in computable measure theory. We use the
framework of TTE, the representation approach, where computability on an
abstract set X is defined by representing its elements with concrete "names",
possibly countably infinite, over some alphabet {\Sigma}. As a basic
computability structure we consider a computable measure on a computable
-algebra. We introduce and compare w.r.t. reducibility several natural
representations of measurable sets. They are admissible and generally form four
different equivalence classes. We then compare our representations with those
introduced by Y. Wu and D. Ding in 2005 and 2006 and claim that one of our
representations is the most useful one for studying computability on measurable
functions