Physics and Computation

2009

September 7-11th 2009

Ponta-Delgada, Azores, Portugal

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!! Talk from Mathieu Hoyrup

!!! Title

Dynamical systems: unpredictability vs incomputability.

!!! Abstract

The long-term forecasting of chaotic dynamical systems is generally

not possible, because of the so-called "sensitivity to initial

conditions": if you do not know exactly the initial state of a system,

you cannot predict its long-range evolution. For this reason, the

mathematical theory of dynamical systems is more focused on the

understanding of the global, asymptotic behaviour of dynamical systems

than the prediction of individual trajectories. In particular, several

notions of entropies of a system are defined in order to quantify its

unpredictability.

Computability theory provides another perspective on the problem of

unpredictability: a system is unpredictable if its evolution cannot be

computed by a machine. Unpredictability is then understood as

incomputability, or difficulty to compute.

I will present some works that connect these two approaches: the

classical one from the mathematical theory of dynamical systems, the

recursion-theoretic one using concepts from computability theory,

especially Kolmogorov complexity.

!!! Title

Dynamical systems: unpredictability vs incomputability.

!!! Abstract

The long-term forecasting of chaotic dynamical systems is generally

not possible, because of the so-called "sensitivity to initial

conditions": if you do not know exactly the initial state of a system,

you cannot predict its long-range evolution. For this reason, the

mathematical theory of dynamical systems is more focused on the

understanding of the global, asymptotic behaviour of dynamical systems

than the prediction of individual trajectories. In particular, several

notions of entropies of a system are defined in order to quantify its

unpredictability.

Computability theory provides another perspective on the problem of

unpredictability: a system is unpredictable if its evolution cannot be

computed by a machine. Unpredictability is then understood as

incomputability, or difficulty to compute.

I will present some works that connect these two approaches: the

classical one from the mathematical theory of dynamical systems, the

recursion-theoretic one using concepts from computability theory,

especially Kolmogorov complexity.