7,103 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
Arrow's Theorem, countably many agents, and more visible invisible dictators
For infinite societies, Fishburn (1970), Kirman and Sondermann (1972), and Armstrong (1980) gave a nonconstructive proof of the existence of a social welfare function satisfying Arrowfs conditions (Unanimity, Independence, and Nondictatorship). This paper improves on their results by (i) giving a concrete example of such a function, and (ii) showing how to compute, from a description of a profile on a pair of alternatives, which alternative is socially preferred under the function. The introduction of a certain goracleh resolves Miharafs impossibility result (1997) about computability of social welfare functions.Arrow impossibility theorem, Turing computability, recursion theory, oracle algorithms, free ultrafilters
A Swiss Pocket Knife for Computability
This research is about operational- and complexity-oriented aspects of
classical foundations of computability theory. The approach is to re-examine
some classical theorems and constructions, but with new criteria for success
that are natural from a programming language perspective.
Three cornerstones of computability theory are the S-m-ntheorem; Turing's
"universal machine"; and Kleene's second recursion theorem. In today's
programming language parlance these are respectively partial evaluation,
self-interpretation, and reflection. In retrospect it is fascinating that
Kleene's 1938 proof is constructive; and in essence builds a self-reproducing
program.
Computability theory originated in the 1930s, long before the invention of
computers and programs. Its emphasis was on delimiting the boundaries of
computability. Some milestones include 1936 (Turing), 1938 (Kleene), 1967
(isomorphism of programming languages), 1985 (partial evaluation), 1989 (theory
implementation), 1993 (efficient self-interpretation) and 2006 (term register
machines).
The "Swiss pocket knife" of the title is a programming language that allows
efficient computer implementation of all three computability cornerstones,
emphasising the third: Kleene's second recursion theorem. We describe
experiments with a tree-based computational model aiming for both fast program
generation and fast execution of the generated programs.Comment: In Proceedings Festschrift for Dave Schmidt, arXiv:1309.455
Constructive Algebraic Topology
The classical ``computation'' methods in Algebraic Topology most often work
by means of highly infinite objects and in fact +are_not+ constructive. Typical
examples are shown to describe the nature of the problem. The Rubio-Sergeraert
solution for Constructive Algebraic Topology is recalled. This is not only a
theoretical solution: the concrete computer program +Kenzo+ has been written
down which precisely follows this method. This program has been used in various
cases, opening new research subjects and producing in several cases significant
results unreachable by hand. In particular the Kenzo program can compute the
first homotopy groups of a simply connected +arbitrary+ simplicial set.Comment: 24 pages, background paper for a plenary talk at the EACA Congress of
Tenerife, September 199
- …