3,140 research outputs found
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
Computability, Noncomputability, and Hyperbolic Systems
In this paper we study the computability of the stable and unstable manifolds
of a hyperbolic equilibrium point. These manifolds are the essential feature
which characterizes a hyperbolic system. We show that (i) locally these
manifolds can be computed, but (ii) globally they cannot (though we prove they
are semi-computable). We also show that Smale's horseshoe, the first example of
a hyperbolic invariant set which is neither an equilibrium point nor a periodic
orbit, is computable
On the relevance of the neurobiological analogue of the finite-state architecture
We present two simple arguments for the potential relevance of a neurobiological analogue of the finite-state architecture. The first assumes the classical cognitive framework, is well-known, and is based on the assumption that the brain is finite with respect to its memory organization. The second is formulated within a general dynamical systems framework and is based on the assumption that the brain sustains some level of noise and/or does not utilize infinite precision processing. We briefly review the classical cognitive framework based on Church-Turing computability and non-classical approaches based on analog processing in dynamical systems. We conclude that the dynamical neurobiological analogue of the finite-state architecture appears to be relevant, at least at an implementational level, for cognitive brain systems
A Primer on the Tools and Concepts of Computable Economics
Computability theory came into being as a result of Hilbert's attempts to meet Brouwer's challenges, from an intuitionistc and constructive standpoint, to formalism as a foundation for mathematical practice. Viewed this way, constructive mathematics should be one vision of computability theory. However, there are fundamental differences between computability theory and constructive mathematics: the Church-Turing thesis is a disciplining criterion in the former and not in the latter; and classical logic - particularly, the law of the excluded middle - is not accepted in the latter but freely invoked in the former, especially in proving universal negative propositions. In Computable Economic an eclectic approach is adopted where the main criterion is numerical content for economic entities. In this sense both the computable and the constructive traditions are freely and indiscriminately invoked and utilised in the formalization of economic entities. Some of the mathematical methods and concepts of computable economics are surveyed in a pedagogical mode. The context is that of a digital economy embedded in an information society
Elements of a Theory of Simulation
Unlike computation or the numerical analysis of differential equations,
simulation does not have a well established conceptual and mathematical
foundation. Simulation is an arguable unique union of modeling and computation.
However, simulation also qualifies as a separate species of system
representation with its own motivations, characteristics, and implications.
This work outlines how simulation can be rooted in mathematics and shows which
properties some of the elements of such a mathematical framework has. The
properties of simulation are described and analyzed in terms of properties of
dynamical systems. It is shown how and why a simulation produces emergent
behavior and why the analysis of the dynamics of the system being simulated
always is an analysis of emergent phenomena. A notion of a universal simulator
and the definition of simulatability is proposed. This allows a description of
conditions under which simulations can distribute update functions over system
components, thereby determining simulatability. The connection between the
notion of simulatability and the notion of computability is defined and the
concepts are distinguished. The basis of practical detection methods for
determining effectively non-simulatable systems in practice is presented. The
conceptual framework is illustrated through examples from molecular
self-assembly end engineering.Comment: Also available via http://studguppy.tsasa.lanl.gov/research_team/
Keywords: simulatability, computability, dynamics, emergence, system
representation, universal simulato
Computational Complexity of Iterated Maps on the Interval (Extended Abstract)
The exact computation of orbits of discrete dynamical systems on the interval
is considered. Therefore, a multiple-precision floating point approach based on
error analysis is chosen and a general algorithm is presented. The correctness
of the algorithm is shown and the computational complexity is analyzed. As a
main result, the computational complexity measure considered here is related to
the Ljapunow exponent of the dynamical system under consideration
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