Discrete and Continuous Representations of Metabolic Models
29
representatives from equivalence classes of forcings). In the next two
sections, we shall show that these constructions allow us to interpret the
basic concepts of both the continuous and the discrete descriptions, each
in terms of the other.
Thus, our constructions satisfy the basic properties which we laid down
in the preceding section, and which should be satisfied by a correspondence
principle between discrete and continuous descriptions. Finally we may
note that, according to these constructions, a dynamical system with a
small number of very deep basins will have very strongly digital characteristics,
while on the other hand, a dynamical system with a very large number of
shallow basins will not. A system whose state space is partly contoured
into a small number of deep basins and partly contoured into a large
number of shallow basins will exhibit a hybrid character.
III. Computability
Let us now sketch how the discrete concept of computability may be
related to dynamical concepts in terms of the correspondence principle we
have outlined above. Computability refers not to ordinary sequential
machines, but to sequential machines with an appropriate feedback (i.e.
memory); a sequential machine with such a feedback is generally called a
TURING machine. More specifically, a TURING machine is a pair of interacting finite sequential machines, one of which serves as a memory of the
previous activity of the other. This memory machine further chooses
inputs to the first machine, to be supplied at subsequent instants as a function of this activity (cf. [13]). In the language of TURING machines, the
first machine corresponds to a tape scanner and printer. The second
machine corresponds to a tape scanner and mover, capable of moving the
tape in either direction. The tape initially is regarded as having a finite
number of input alphabet symbols printed upon it, all the rest of the tape
being blank.
Problems concerning computability can always be translated into a
halting problem for TURING machines, e.g. deciding whether a given
TURING machine with a given input tape will halt its computation after
a finite time. What does this mean dynamically? Using the principle we
have enunciated above, it is possible to pass from the pair of interacting
sequential machines to a pair of interacting dynamical systems whose
stability properties are of the correct type. Call these dynamical systems
D 1 , D 2 , respectively. Let D2 correspond to the tape mover (or "memory"
for D 1 ). Any transition induced in Dl because of external forcings (corresponding to the alphabet symbols on the tape) will in general induce a
similar transition in D 2 , while any transition in D2 will constitute a forcing
of D 1 • The halting problem in a TURING machine context obviously
29
representatives from equivalence classes of forcings). In the next two
sections, we shall show that these constructions allow us to interpret the
basic concepts of both the continuous and the discrete descriptions, each
in terms of the other.
Thus, our constructions satisfy the basic properties which we laid down
in the preceding section, and which should be satisfied by a correspondence
principle between discrete and continuous descriptions. Finally we may
note that, according to these constructions, a dynamical system with a
small number of very deep basins will have very strongly digital characteristics,
while on the other hand, a dynamical system with a very large number of
shallow basins will not. A system whose state space is partly contoured
into a small number of deep basins and partly contoured into a large
number of shallow basins will exhibit a hybrid character.
III. Computability
Let us now sketch how the discrete concept of computability may be
related to dynamical concepts in terms of the correspondence principle we
have outlined above. Computability refers not to ordinary sequential
machines, but to sequential machines with an appropriate feedback (i.e.
memory); a sequential machine with such a feedback is generally called a
TURING machine. More specifically, a TURING machine is a pair of interacting finite sequential machines, one of which serves as a memory of the
previous activity of the other. This memory machine further chooses
inputs to the first machine, to be supplied at subsequent instants as a function of this activity (cf. [13]). In the language of TURING machines, the
first machine corresponds to a tape scanner and printer. The second
machine corresponds to a tape scanner and mover, capable of moving the
tape in either direction. The tape initially is regarded as having a finite
number of input alphabet symbols printed upon it, all the rest of the tape
being blank.
Problems concerning computability can always be translated into a
halting problem for TURING machines, e.g. deciding whether a given
TURING machine with a given input tape will halt its computation after
a finite time. What does this mean dynamically? Using the principle we
have enunciated above, it is possible to pass from the pair of interacting
sequential machines to a pair of interacting dynamical systems whose
stability properties are of the correct type. Call these dynamical systems
D 1 , D 2 , respectively. Let D2 correspond to the tape mover (or "memory"
for D 1 ). Any transition induced in Dl because of external forcings (corresponding to the alphabet symbols on the tape) will in general induce a
similar transition in D 2 , while any transition in D2 will constitute a forcing
of D 1 • The halting problem in a TURING machine context obviously