5,879 research outputs found
Naming the largest number: Exploring the boundary between mathematics and the philosophy of mathematics
What is the largest number accessible to the human imagination? The question
is neither entirely mathematical nor entirely philosophical. Mathematical
formulations of the problem fall into two classes: those that fail to fully
capture the spirit of the problem, and those that turn it back into a
philosophical problem
Coding-theorem Like Behaviour and Emergence of the Universal Distribution from Resource-bounded Algorithmic Probability
Previously referred to as `miraculous' in the scientific literature because
of its powerful properties and its wide application as optimal solution to the
problem of induction/inference, (approximations to) Algorithmic Probability
(AP) and the associated Universal Distribution are (or should be) of the
greatest importance in science. Here we investigate the emergence, the rates of
emergence and convergence, and the Coding-theorem like behaviour of AP in
Turing-subuniversal models of computation. We investigate empirical
distributions of computing models in the Chomsky hierarchy. We introduce
measures of algorithmic probability and algorithmic complexity based upon
resource-bounded computation, in contrast to previously thoroughly investigated
distributions produced from the output distribution of Turing machines. This
approach allows for numerical approximations to algorithmic
(Kolmogorov-Chaitin) complexity-based estimations at each of the levels of a
computational hierarchy. We demonstrate that all these estimations are
correlated in rank and that they converge both in rank and values as a function
of computational power, despite fundamental differences between computational
models. In the context of natural processes that operate below the Turing
universal level because of finite resources and physical degradation, the
investigation of natural biases stemming from algorithmic rules may shed light
on the distribution of outcomes. We show that up to 60\% of the
simplicity/complexity bias in distributions produced even by the weakest of the
computational models can be accounted for by Algorithmic Probability in its
approximation to the Universal Distribution.Comment: 27 pages main text, 39 pages including supplement. Online complexity
calculator: http://complexitycalculator.com
Empirical Encounters with Computational Irreducibility and Unpredictability
There are several forms of irreducibility in computing systems, ranging from
undecidability to intractability to nonlinearity. This paper is an exploration
of the conceptual issues that have arisen in the course of investigating
speed-up and slowdown phenomena in small Turing machines. We present the
results of a test that may spur experimental approaches to the notion of
computational irreducibility. The test involves a systematic attempt to outrun
the computation of a large number of small Turing machines (all 3 and 4 state,
2 symbol) by means of integer sequence prediction using a specialized function
finder program. This massive experiment prompts an investigation into rates of
convergence of decision procedures and the decidability of sets in addition to
a discussion of the (un)predictability of deterministic computing systems in
practice. We think this investigation constitutes a novel approach to the
discussion of an epistemological question in the context of a computer
simulation, and thus represents an interesting exploration at the boundary
between philosophical concerns and computational experiments.Comment: 18 pages, 4 figure
Towards Autopoietic Computing
A key challenge in modern computing is to develop systems that address
complex, dynamic problems in a scalable and efficient way, because the
increasing complexity of software makes designing and maintaining efficient and
flexible systems increasingly difficult. Biological systems are thought to
possess robust, scalable processing paradigms that can automatically manage
complex, dynamic problem spaces, possessing several properties that may be
useful in computer systems. The biological properties of self-organisation,
self-replication, self-management, and scalability are addressed in an
interesting way by autopoiesis, a descriptive theory of the cell founded on the
concept of a system's circular organisation to define its boundary with its
environment. In this paper, therefore, we review the main concepts of
autopoiesis and then discuss how they could be related to fundamental concepts
and theories of computation. The paper is conceptual in nature and the emphasis
is on the review of other people's work in this area as part of a longer-term
strategy to develop a formal theory of autopoietic computing.Comment: 10 Pages, 3 figure
Bulletin of the Center for Children's Books 15 (10) 1962
published or submitted for publicatio
On the boundaries of solvability and unsolvability in tag systems. Theoretical and Experimental Results
Several older and more recent results on the boundaries of solvability and
unsolvability in tag systems are surveyed. Emphasis will be put on the
significance of computer experiments in research on very small tag systems
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