5,528 research outputs found
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
Quantum Hypercomputation - Hype or Computation?
A recent attempt to compute a (recursion--theoretic) non--computable function using the quantum adiabatic algorithm is criticized and found wanting. Quantum algorithms may outperform classical algorithms in some cases, but so far they retain the classical (recursion--theoretic) notion of computability. A speculation is then offered as to where the putative power of quantum computers may come from
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
Nonanonymity and sensitivity of computable simple games
This paper investigates algorithmic computability of simple games (voting games). It shows that (i) games with a finite carrier are computable, (ii) computable games have both finite winning coalitions and cofinite losing coalitions, and (iii) computable games violate any conceivable notion of anonymity, including finite anonymity and measurebased anonymity. The paper argues that computable games are excluded from the intuitive class of gniceh infinite games, employing the notion of ginsensitivityh\-equal treatment of any two coalitions that differ only on a finite set.Voting games, infinitely many players, ultrafilters, recursion theory, Turing computability, finite carriers, finite winning coalitions, algorithms
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
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