4,475 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
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
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
Cinnamons: A Computation Model Underlying Control Network Programming
We give the easily recognizable name "cinnamon" and "cinnamon programming" to
a new computation model intended to form a theoretical foundation for Control
Network Programming (CNP). CNP has established itself as a programming paradigm
combining declarative and imperative features, built-in search engine, powerful
tools for search control that allow easy, intuitive, visual development of
heuristic, nondeterministic, and randomized solutions. We define rigorously the
syntax and semantics of the new model of computation, at the same time trying
to keep clear the intuition behind and to include enough examples. The
purposely simplified theoretical model is then compared to both WHILE-programs
(thus demonstrating its Turing-completeness), and the "real" CNP. Finally,
future research possibilities are mentioned that would eventually extend the
cinnamon programming into the directions of nondeterminism, randomness, and
fuzziness.Comment: 7th Intl Conf. on Computer Science, Engineering & Applications
(ICCSEA 2017) September 23~24, 2017, Copenhagen, Denmar
Kolmogorov Complexity of Categories
Kolmogorov complexity theory is used to tell what the algorithmic
informational content of a string is. It is defined as the length of the
shortest program that describes the string. We present a programming language
that can be used to describe categories, functors, and natural transformations.
With this in hand, we define the informational content of these categorical
structures as the shortest program that describes such structures. Some basic
consequences of our definition are presented including the fact that equivalent
categories have equal Kolmogorov complexity. We also prove different theorems
about what can and cannot be described by our programming language.Comment: 16 page
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