A "maximally constrained system" is thus
one in which there is a sufficient number of non-holonomic constraints imposed
to create this condition. So, it is "maximal" in the sense that this certain
number of constraints allows a system to eliminate the maximum number (namely,
all) of the independent velocity variables. On the other hand, this certain
number of constraints is "minimal" in the sense that it only takes this
number of constraints to achieve this condition - adding more such constraints
are not required to eliminate the already-eliminated independent velocity
variables.
So, whether it is "maximal" or "minimal" seems to me to be a
"glass half-full or half-empty" question. Rosen appear to me to have chosen the
term "maximal" to indicate the condition of maximally eliminating the
independent velocity varables. In my view, such a number of constraints
could also be regarded as an optimal number insofar as it achieves this
condition without adding excess constraints.
As for "totally constrained
system", Howard quotes:
>
Rosen (p. 114): "As an example of a totally constrained system,
>
of the type we have discussed in the previous section [the
> section
from which I quoted above], is a neural network, or more
> generally, a
special purpose digital computer, considered as
hardware."
The quote (in
section IV), refers to "previous
section", which would be section III "Constraints and Controls" rather than (as
Howard indicates) section II "Constraints in Mechanical Systems". Section III
discusses the notion of how a maximally constrained system could be controlled
by parameters, restoring degrees-of-freedom from inside the system. The summary
on of section III on p. 113 [ital.
orig.]:
"We can think of the specification of
the "forcing function" Beta(x,t) [i.e., parameters that are a function of
configuration and time - TG] as a program, superimposed on what is a
maximally constrained mechanical system. Thus, these systems are
programmable, even though maximally constrained. It is this factor
which renders such systems "plastic", and which will be of crucial import to
us in our subsequent discussion of machines, cells, and
brains.
We should note explicitly that these "forcings"
have little to do with the mechanical impressed forces of physics, which
reflect themselves in accelerations. These "forcings" instead
manifest themselves in velocities, with impressed forces held fixed.
Thus, the "plasticity" of maximally constrained systems, which we have
described, has a quite different epistemological basis than the behaviors of
conventional mechanical systems."
So, it could be
that by "totally constrained system", Rosen meant: a maximally
constrained system which also has degrees-of-freedom restored to it by virtue of
such parameters. Or, it could simply be that "totally constrained system" is a
synonym for "maximally constrained system". In either case, it was indeed a term he used, and it involves, and
depends centrally on, the definition
of a "maximally constrained system".
Howards main
objection to Rosen's paper:
HP: Both
Morowitz and Rosen have drawn a mistaken conclusion from this example. They
both miss the absolutely essential distinction between rate-independent
(non-dynamic, holonomic) memory constraints and rate-dependent (dynamic,
nonholonomic) control constraints. This is the conceptually equivalent
distinction that von Neumann makes between descriptive (quiescent) constraints
and constructive (nonholonomic, dynamic) constraints, that ironically, Rosen
also misinterpreted.
As I see it,
Rosen's paper does not miss that distinction. As noted at the top of this post,
Rosen precedes his discussion of maximally constrained systems by laying out the
differences between holonomic and non-holonomic constraints in some detail,
including their respective mathematical ramifications for a system's analytical
description.
I don't think
Rosen dwells on the holonomic constraints because they are not central to what
is required in order to achieve a maximally constrained system. That is,
holonomic constraints do not contribute to the property of making all the
independent velocity variables become dependent solely upon configuration
variables.
However, I see
nothing in his argument which precludes that such a system could
also possess holonomic constraints in addition to the required
non-holonomic ones. It seems to me that non-holonomic constraints
could be imposed on holonomically constrained portions (e.g., rigid 'chunks of
particles') of the system just as readily as they could be imposed on
individual 'particles', and could still result in a maximally constrained
system. The resulting system would indeed inherit properties due to those
holonomic constraints along with properties inherited from its non-holonomic
constraints. I just don't see Rosen's math or verbal arguments as
precluding this possibility; I just think that it is not central to his focus in
that paper.
Howard also
states:
Unlike
program-controlled computers, neither folding nor enzyme catalysis is a
"maximally constrained" process as Rosen states. Innumerable rate-dependent,
parallel impressed natural physical forces play an essential role. In fact,
quite the contrary, the one-dimensional sequence constraints can be viewed as
the MINIMUM number of constraints necessary to achieve the enzyme?s specific
rate-controlling function.
This is very unclear to me. As I read this, if folding
depended upon "impressed physical natural forces", then for a protein to fold
reliably into the same shape every time would mean that the outside
world must always be imposing the exact same forces each and every time,
which is not reasonable at all to expect. Additionally, such a situation would
negate the proposal that "one-dimensional sequence constraints can be viewed as
the MINIMUM number of constraints...". Instead, the outside world (via impressed
forces) would determine the resultant conformation. Conversely, if the
"one-dimensional sequence constraints" were "the MINIMUM number of
constraints..." then impressed forces could not "play an essential role" in
determing the final folded shape.
Even if
we view the primary sequence as a constraint, it is insufficient by itself to describe how the folding will
occur. It only says to me that: the primary
sequence is the minimum number of constraints given the forcing (and other)
properties that inhere in and among the constituents of this sequence. To
me, omitting this last italicized portion simply hides the necessary non-holonomic constraint
relationships between aspects of the sequence which do determine how a
given sequence will fold. Informationally, having to specify only a primary
sequence does seem evolutionarily efficient, but such information
implicitly relies upon the aforementioned non-holonomic constraints for it to be
meaningful in any way. Rosen on p. 123 [ital orig]:
"These questions are part of a
larger one; namely, that most system descriptions do not contain sufficient
"information" to allow realization at all. For instance, from a specific
knowledge of kinetic parameters (e.g., rate constants, turnovers, numbers,
specificities, and the like) we cannot pass to the primary (or even tertiary)
structure of an enzyme which manifests these parameters. It appears that only
very special kinds of descriptions, which we may call blueprints,
allow themselves to be realized. The primary structures of
an enzyme, or a sequence of DNA or RNA, exemplify such blueprints, but these
in turn are very special descriptions; they do not generally allow us to pass
to other modes of description. Thus, if I am given the primary sequence of a
protein, I can in general deduce nothing about its functional or kinetic
properties."
Regards,
Tim> -----Original Message-----
> From: ROSEN Forum
[mailto:***]On Behalf Of Howard
> Pattee
> Sent: Sunday,
October 31, 2004 8:30 AM
> To: ***
> Subject:
Re: Maximally constrained
>
>
> Judith wrote:
> I
believe Howard Pattee has misinterpreted what Robert Rosen
> wrote about
maximally constrained systems. In order to understand
> why this is so, we
have to discuss how my father defined and used
> the terms "maximally" and
"constrained".
>
> HP: I was commenting only how Rosen actually uses
the phrase
> "maximal constraint" in the 1986 paper in question. (I am
aware
> of his later usages, as in Life Itself, where he lumps
local
> constraints with natural laws) This is a well-written paper
so
> there is not much room for misinterpretation. In the 1986
paper
> he explicitly uses Hertz's definitions of constraints, the
ones
> accepted by all physicists. Also, by "maximally constrained"
I
> see no way he could mean "optimally constrained" as Judith
wants
> to interpret him. I simply quote the relevant
paragraph:
>
> Rosen (p. 112): "If we impose the maximal number of
nonholonomic
> constraints, then the velocity vector is uniquely
determined by
> the configuration. The result is an autonomous dynamical
system,
> or vector field, on the configuration space. At this point,
the
> impressed forces of conventional analytical dynamics
disappear
> completely; their only role is to get the system
moving."
>
> HP: This is a fairly technical statement for which very
little
> ambiguity is possible. If the forces of analytic dynamics
(i.e.,
> forces resulting from the laws of nature) disappear except
to
> provide the energy to get the system moving then the motion
must
> be entirely the result of the constraints. It is in this
sense
> that the system is maximally constrained. This completely
defines
> the system. There is no room for optimality to play a role
here.
>
> Judith: A "maximally constrained system" is not a
"totally
> constrained system" or a system with the maximum number
of
> constraints. Instead, it is a system with such constraints as
>
maximize its potential.
>
> HP: Again, I simply quote Rosen. He
equates "maximally
> constrained system" with "totally constrained
system." He argues
> that any "potential" must arise from programmability
that is a
> special type of constraint. In any case, no amount of
word
> processing by Judith can interpret Rosen's concept of
"maximal
> constraint" to mean the same as my interpretation of "minimal
constraint."
>
> Rosen (p. 114): "As an example of a totally
constrained system,
> of the type we have discussed in the previous
section [the
> section from which I quoted above], is a neural network, or
more
> generally, a special purpose digital computer, considered as
hardware."
>
> You will also see that this 1986 paper is entirely
consistent
> with Rosen's discussion of nonholonomic constraints in
the
> Appendix of Anticipatory Systems (p. 414 and note 1, p. 427).
In
> fact, he wrote the paper as an elaboration of this too brief
Appendix.
>
> If you actually read the entire paper you will see
that Rosen is
> using the same concept of nonholonomic constraint that I
use to
> distinguish the laws (forces of analytical dynamics) from
the
> informational constraints of genetic memory. We had discussed
the
> concept of nonholonomic constraint at great length, so I
know
> there was no intent to disagree with me. In fact, he was using
my
> views to introduce his discussion. That is why I said he only
>
misinterpreted what I had in mind. I fully understand how I could
> easily
be misinterpreted, because in 1986 things were not as
> clear to me as
they are today. I had not yet clearly stated the
> evolutionary principle
of "minimum constraint" in writing. I had
> proposed only a vague
"principle of optimal constraint." With
> more thought about natural
selection it only made sense that life
> uses the forces of nature as much
as possible and the
> informational constraints only where necessary for
survival. The
> universal life processes of folding and self-assembly are
prime
> examples of the creative potential of using the fewest
>
constraints to control the largest universe of lawful dynamics.
>
>
Howard