4,777 research outputs found
From Nonstandard Analysis to various flavours of Computability Theory
As suggested by the title, it has recently become clear that theorems of
Nonstandard Analysis (NSA) give rise to theorems in computability theory (no
longer involving NSA). Now, the aforementioned discipline divides into
classical and higher-order computability theory, where the former (resp. the
latter) sub-discipline deals with objects of type zero and one (resp. of all
types). The aforementioned results regarding NSA deal exclusively with the
higher-order case; we show in this paper that theorems of NSA also give rise to
theorems in classical computability theory by considering so-called textbook
proofs.Comment: To appear in the proceedings of TAMC2017 (http://tamc2017.unibe.ch/
Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length
We provide an implicit characterization of polynomial time computation in
terms of ordinary differential equations: we characterize the class
of languages computable in polynomial time in terms of
differential equations with polynomial right-hand side.
This result gives a purely continuous (time and space) elegant and simple
characterization of . This is the first time such classes
are characterized using only ordinary differential equations. Our
characterization extends to functions computable in polynomial time over the
reals in the sense of computable analysis. This extends to deterministic
complexity classes above polynomial time.
This may provide a new perspective on classical complexity, by giving a way
to define complexity classes, like , in a very simple
way, without any reference to a notion of (discrete) machine. This may also
provide ways to state classical questions about computational complexity via
ordinary differential equations, i.e.~by using the framework of analysis
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
Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?
Classical interpretations of Goedel's formal reasoning imply that the truth
of some arithmetical propositions of any formal mathematical language, under
any interpretation, is essentially unverifiable. However, a language of
general, scientific, discourse cannot allow its mathematical propositions to be
interpreted ambiguously. Such a language must, therefore, define mathematical
truth verifiably. We consider a constructive interpretation of classical,
Tarskian, truth, and of Goedel's reasoning, under which any formal system of
Peano Arithmetic is verifiably complete. We show how some paradoxical concepts
of Quantum mechanics can be expressed, and interpreted, naturally under a
constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version
is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht
A Universal Ordinary Differential Equation
An astonishing fact was established by Lee A. Rubel (1981): there exists a
fixed non-trivial fourth-order polynomial differential algebraic equation (DAE)
such that for any positive continuous function on the reals, and for
any positive continuous function , it has a
solution with for all . Lee A. Rubel
provided an explicit example of such a polynomial DAE. Other examples of
universal DAE have later been proposed by other authors. However, Rubel's DAE
\emph{never} has a unique solution, even with a finite number of conditions of
the form .
The question whether one can require the solution that approximates
to be the unique solution for a given initial data is a well known open problem
[Rubel 1981, page 2], [Boshernitzan 1986, Conjecture 6.2]. In this article, we
solve it and show that Rubel's statement holds for polynomial ordinary
differential equations (ODEs), and since polynomial ODEs have a unique solution
given an initial data, this positively answers Rubel's open problem. More
precisely, we show that there exists a \textbf{fixed} polynomial ODE such that
for any and there exists some initial condition that
yields a solution that is -close to at all times.
In particular, the solution to the ODE is necessarily analytic, and we show
that the initial condition is computable from the target function and error
function
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