Nonclassical Computation: a dynamical systems perspective.

In this chapter we investigate computation from a dynamical systems perspective.

A dynamical system is described in terms of its abstract *state
space*, the system’s current state within its state space, and a
rule that determines its motion through its state space. In a classical
computational system, that rule is given explicitly by the computer
program; in a physical system, that rule is the underlying physical law
governing the behaviour of the system. So a dynamical systems approach
to computation allows us to take a unified view of computation in
classical discrete systems and in systems performing non-classical
computation. In particular, it gives a route to a computational
interpretation of physical embodied systems exploiting the natural
dynamics of their material substrates.

We start with *autonomous* (closed) dynamical systems: those
whose dynamics is not an explicit function of time, in particular, those
with no inputs from an external environment. We begin with
computationally conventional discrete systems, examining their
computational abilities from a dynamical systems perspective. The aim
here is both to introduce the necessary dynamical systems concepts, and
to demonstrate how classical computation can be viewed from this
perspective. We then move on to continuous dynamical systems, such as
those inherent in the complex dynamics of matter, and show how these too
can be interpreted computationally, and see how the material embodiment
can give such computation “for free”, without the need to
explicitly implement the dynamics.

We next broaden the outlook to *open* (non-autonomous) dynamical
systems, where the dynamics is a function of time, in the form of inputs
from an external environment, and which may be in a closely coupled
feedback loop with that environment.

We finally look at *constructive*, or developmental, dynamical
systems, where the structure of the state space is changing during the
computation. This includes various growth processes, again investigated
from a computational dynamical systems perspective.

These later sections are less developed than for the autonomous cases, as the theory is less mature (or even non-existent); however these are the more interesting computational domains, as they move us into the arena of considering biological and other natural systems as computational, open, developmental, dynamical systems.

**Full paper** : PDF 3005K |
doi : 10.1007/978-3-540-92910-9_59

@incollection(SS-NCDynSys, author = "Susan Stepney", title = "Nonclassical Computation: a dynamical systems perspective", chapter = 59, pages = "1979-2025", doi = "10.1007/978-3-540-92910-9_59", crossref = "HBNC" ) @book(HBNC, editor = "Grzegorz Rozenberg and Thomas B{\"a}ck and Joost N. Kok", title = "Handbook of Natural Computing, volume 4", booktitle = "Handbook of Natural Computing, volume 4", publisher = "Springer", year = 2012 )