7 research outputs found
Comparing hierarchies of total functionals
In this paper we consider two hierarchies of hereditarily total and
continuous functionals over the reals based on one extensional and one
intensional representation of real numbers, and we discuss under which
asumptions these hierarchies coincide. This coincidense problem is equivalent
to a statement about the topology of the Kleene-Kreisel continuous functionals.
As a tool of independent interest, we show that the Kleene-Kreisel functionals
may be embedded into both these hierarchies.Comment: 28 page
Differential calculus with imprecise input and its logical framework
We develop a domain-theoretic Differential Calculus for locally Lipschitz functions on finite dimensional real spaces with imprecise input/output. The inputs to these functions are hyper-rectangles and the outputs are compact real intervals. This extends the domain of application of Interval Analysis and exact arithmetic to the derivative. A new notion of a tie for these functions is introduced, which in one dimension represents a modification of the notion previously used in the one-dimensional framework. A Scott continuous sub-differential for these functions is then constructed, which satisfies a weaker form of calculus compared to that of the Clarke sub-gradient. We then adopt a Program Logic viewpoint using the equivalence of the category of stably locally compact spaces with that of semi-strong proximity lattices. We show that given a localic approximable mapping representing a locally Lipschitz map with imprecise input/output, a localic approximable mapping for its sub-differential can be constructed, which provides a logical formulation of the sub-differential operator
Abstract Datatypes for Real Numbers in Type Theory
Abstract. We propose an abstract datatype for a closed interval of real numbers to type theory, providing a representation-independent approach to programming with real numbers. The abstract datatype requires only function types and a natural numbers type for its formulation, and so can be added to any type theory that extends G枚del鈥檚 System datatype is equivalent in power to programming intensionally with representations of real numbers. We also consider representing arbitrary real numbers using a mantissa-exponent representation in which the mantissa is taken from the abstract interval.
Recursive Solution of Initial Value Problems with Temporal Discretization
We construct a continuous domain for temporal discretization of differential
equations. By using this domain, and the domain of Lipschitz maps, we formulate
a generalization of the Euler operator, which exhibits second-order
convergence. We prove computability of the operator within the framework of
effectively given domains. The operator only requires the vector field of the
differential equation to be Lipschitz continuous, in contrast to the related
operators in the literature which require the vector field to be at least
continuously differentiable. Within the same framework, we also analyze
temporal discretization and computability of another variant of the Euler
operator formulated according to Runge-Kutta theory. We prove that, compared
with this variant, the second-order operator that we formulate directly, not
only imposes weaker assumptions on the vector field, but also exhibits superior
convergence rate. We implement the first-order, second-order, and Runge-Kutta
Euler operators using arbitrary-precision interval arithmetic, and report on
some experiments. The experiments confirm our theoretical results. In
particular, we observe the superior convergence rate of our second-order
operator compared with the Runge-Kutta Euler and the common (first-order) Euler
operators.Comment: 50 pages, 6 figure
Real Number Computability and Domain Theory
We present the different constructive definitions of real number that can be found in the literature. Using domain theory we analyse the notion of computability that is substantiated by these definitions and we give a definition of computability for real numbers and for functions acting on them. This definition of computability turns out to be equivalent to other definitions given in the literature using different methods. Domain theory is a useful tool to study higher order computability on real numbers. An interesting connection between Scott-topology and the standard topologies on the real line and on the space of continuous functions on reals is stated. The main original result in this paper is the proof that every computable functional on real numbers is continuous w.r.t. the compact open topology on the function space. 1 Introduction Turing in 1937 was the first to introduce the notion of computable real number [24]. Since then a great number of different approaches have been us..