30
R. ROSEN
translates into the question whether the system D2 will ever come to
equilibrium. Conversely, any pair of dynamical systems interacting in this
way will give rise to a TURING machine via our correspondence principle.
Hence, the halting problem for TURING machines resolves itself into a
purely dynamical problem concerning interacting systems of a particular
type. All questions regarding computability in digital systems can now be
reformulated entirely in dynamical terms (compare ROSEN [11]).
IV. Continuity and Stability
We have remarked earlier, that because dynamical systems are governed
by local laws, the set of states which may be occupied after a small time
has elapsed, starting from any initial state under the influence of bounded
forces, is itself bounded (like a spreading wave in an optical medium). Thus,
in a dynamical system, two states are close if (roughly) they can be reached
from one another in a small time interval by means of a (small) perturbation.
This observation now allows us to define a measure of closeness in abstract
sequential systems. This measure of closeness allows to define a metric,
and hence a topology, on the state set of any abstract sequential machine in
terms of any of its corresponding dynamical systems. These topologies are
naturally closely related to the quotient topologies induced on the set of
basins of these dynamical systems. The metric thus induced on the state
set of an abstract sequential machine may differ, depending on the particular
dynamical system we choose from the class determined by our correspondence principle, but for dynamical purposes these topologies may be
regarded as equivalent.
Once the metric is determined, it may be specified for the discrete
system without reference to any corresponding dynamical system, purely
in terms of intrinsic properties of the machine. Thus, ideas of continuity,
and with them, of stability, can be re-interpreted direcdy into intrinsic
properties of sequential systems, as required by a suitable correspondence
principle.
V. Some Consequences
Discrete descriptions possess an algorithmic, constructive character
which is not possessed by dynamical systems. If a digital activity can be
specified at all, it can be described by the iteration of a (small) number of
elementary procedures; or what is the same thing, any sequential machine
can be replaced by an equivalent modular net. This explicit constructive
character makes it possible to direcdy characterize the class of all digital
systems which possess any particular functional property which we may
specify.
R. ROSEN
translates into the question whether the system D2 will ever come to
equilibrium. Conversely, any pair of dynamical systems interacting in this
way will give rise to a TURING machine via our correspondence principle.
Hence, the halting problem for TURING machines resolves itself into a
purely dynamical problem concerning interacting systems of a particular
type. All questions regarding computability in digital systems can now be
reformulated entirely in dynamical terms (compare ROSEN [11]).
IV. Continuity and Stability
We have remarked earlier, that because dynamical systems are governed
by local laws, the set of states which may be occupied after a small time
has elapsed, starting from any initial state under the influence of bounded
forces, is itself bounded (like a spreading wave in an optical medium). Thus,
in a dynamical system, two states are close if (roughly) they can be reached
from one another in a small time interval by means of a (small) perturbation.
This observation now allows us to define a measure of closeness in abstract
sequential systems. This measure of closeness allows to define a metric,
and hence a topology, on the state set of any abstract sequential machine in
terms of any of its corresponding dynamical systems. These topologies are
naturally closely related to the quotient topologies induced on the set of
basins of these dynamical systems. The metric thus induced on the state
set of an abstract sequential machine may differ, depending on the particular
dynamical system we choose from the class determined by our correspondence principle, but for dynamical purposes these topologies may be
regarded as equivalent.
Once the metric is determined, it may be specified for the discrete
system without reference to any corresponding dynamical system, purely
in terms of intrinsic properties of the machine. Thus, ideas of continuity,
and with them, of stability, can be re-interpreted direcdy into intrinsic
properties of sequential systems, as required by a suitable correspondence
principle.
V. Some Consequences
Discrete descriptions possess an algorithmic, constructive character
which is not possessed by dynamical systems. If a digital activity can be
specified at all, it can be described by the iteration of a (small) number of
elementary procedures; or what is the same thing, any sequential machine
can be replaced by an equivalent modular net. This explicit constructive
character makes it possible to direcdy characterize the class of all digital
systems which possess any particular functional property which we may
specify.