26
R. ROSEN
descriptions of the same system are possible. To be satisfactory, such a
principle must allow us to express the basic concepts of each description
in terms of the other in an effective manner. The purpose of the present
note is to suggest such a principle and to examine some of its consequences
for the description of a variety of biological systems. The arguments which
follow represent a formalization of ideas implicit in many discussions of
discrete and continuous systems, put into a general system theoretic
perspective. For the purpose of this exposition, we omit mathematical
details; the reader should consult ROSEN [13] for notation and terminology.
II. Continuous and Discrete Descriptions
The continuous description of system activity is invariably formulated
within the framework of dynamical (or control) system theory. In this
theory, the set of states of the system forms a manifold S, and the forces
imposed on the system are related to the kinetic properties of the system
through equations of motion. These equations describe the rate of change
of appropriately chosen state variables in terms of the variables and the
forces. Each possible kinetic activity of the system is represented by a trajectory in this state space. The state space is further contoured into regions
of stability and instability in terms of the equations of motion in a wellknown way.
The discrete description, on the other hand, generally is expressed in
the form of an abstract sequential machine. Here, the set of states forms a
discrete (usually finite) set. According to the homologies we have mentioned,
the forces that can be imposed on the system are represented by abstract
alphabet symbols, and the role of the equations of motion is played by the
next state function. The time variable for sequential machines ranges over
an abstract set of instants, which in general has no relation to any kind of
real time (in sampled-data systems, on the other hand, the set of instants is
directly specified in terms of real time).
Now let us introduce some convenient terminology. Any attribute of
a continuous or discrete description will be denoted by prefixing that
attribute with the letters C-, d- respectively. Thus, e.g. a d-state will refer
to a state occurring in a discrete description, etc.
Suppose that we are given a system 6, and two independent descriptions
of its activity, one of which is continuous and one of which is discrete.
How can we hope to establish a general correspondence between these
descriptions, using nothing more than the fact the descriptions refer to the
same system? The key observation to be made here is the following: the
crucial property of the c-description is the fact that any property whatsoever
of the system 6 must necessarily be specified in terms of the observables
of that description. In other words, if S represents the manifold of c-states
R. ROSEN
descriptions of the same system are possible. To be satisfactory, such a
principle must allow us to express the basic concepts of each description
in terms of the other in an effective manner. The purpose of the present
note is to suggest such a principle and to examine some of its consequences
for the description of a variety of biological systems. The arguments which
follow represent a formalization of ideas implicit in many discussions of
discrete and continuous systems, put into a general system theoretic
perspective. For the purpose of this exposition, we omit mathematical
details; the reader should consult ROSEN [13] for notation and terminology.
II. Continuous and Discrete Descriptions
The continuous description of system activity is invariably formulated
within the framework of dynamical (or control) system theory. In this
theory, the set of states of the system forms a manifold S, and the forces
imposed on the system are related to the kinetic properties of the system
through equations of motion. These equations describe the rate of change
of appropriately chosen state variables in terms of the variables and the
forces. Each possible kinetic activity of the system is represented by a trajectory in this state space. The state space is further contoured into regions
of stability and instability in terms of the equations of motion in a wellknown way.
The discrete description, on the other hand, generally is expressed in
the form of an abstract sequential machine. Here, the set of states forms a
discrete (usually finite) set. According to the homologies we have mentioned,
the forces that can be imposed on the system are represented by abstract
alphabet symbols, and the role of the equations of motion is played by the
next state function. The time variable for sequential machines ranges over
an abstract set of instants, which in general has no relation to any kind of
real time (in sampled-data systems, on the other hand, the set of instants is
directly specified in terms of real time).
Now let us introduce some convenient terminology. Any attribute of
a continuous or discrete description will be denoted by prefixing that
attribute with the letters C-, d- respectively. Thus, e.g. a d-state will refer
to a state occurring in a discrete description, etc.
Suppose that we are given a system 6, and two independent descriptions
of its activity, one of which is continuous and one of which is discrete.
How can we hope to establish a general correspondence between these
descriptions, using nothing more than the fact the descriptions refer to the
same system? The key observation to be made here is the following: the
crucial property of the c-description is the fact that any property whatsoever
of the system 6 must necessarily be specified in terms of the observables
of that description. In other words, if S represents the manifold of c-states