1) The 1959 BMB paper "On a Logical Paradox Implicit in the Notion of a
Self-Reproducing Automaton". Rosen begins the paper:
"The purpose of this note is to point out a paradox which arises when one
attempts to provide a precise formulation of the notion of
self-reproducing automaton, as originally introduced by J. von
Neumann (1951) and discussed subsequently by a number of different authors. If
this paradox cannot be resolved, then the conclusion to which we are forced is
that the existence of a self-reproducing automaton is a logical impossibility.
We shall investigate the implications of this result for biological problems,
and we shall also discuss in the light of this result several of the models
for replicating structures which have been proposed."
Rosen then states the argument asserting the paradox, which you yourself
agree in your post of 3/22/04 is "formally correct". Essentially, it consists of
the following: if an arbitrary automaton f can be represented by a set-mapping
f:A->B, where A is the input and B the output, then self-reproduction
consists of f being able to generate a copy of itself as part of the output, B.
But, f cannot be defined until its range, B, is defined and yet B cannot be
defined until f (which is a member of B) is defined. Thus the paradox. Note that
at this point, the argument is about any arbitrary automaton which can
be mapped as f:A->B.
Later in the paper he notes: "It may be instructive at this point to
review von Neumann's construction of a self-reproducing automaton and to observe
the manifestation of the above paradox in his model." He then spends three
paragraphs applying the paradox to an set-mapping model of such an
automaton and concluding, "Hence it appears that the notion of universal
automaton involves the same type of difficulty as does the notion of
self-reproduction."
He then discusses the paradox relative to the
self-replicating structures of Penrose (1958) and Jacobson (1958). Since they
are not truly self-reproducing the paradox does not arise in them. However,
Rosen's own models do not emerge uncriticized. At the conclusion of the
paper, Rosen discusses the issue of the paradox relative to his
own (M,R)-system:
"Finally, it may be useful to examine the relation of
the results enunciated above to the notion of the (M,R)-system, as introduced
by us (Rosen, 1958a, 1958b, 1959). The (M,R)-system was primarily constructed
to provide a model for metabolic activities of a single cell, but as we have
remarked, it is necessary for such a theory to deal with the replication
problem if it is to have any claim to completeness. In one of these works
(1959) we proposed a model of the duplication of the components R-sub-f which
correspond to the genetic material of the cell, based on the notion of induced
mappings which exist formally in the abstract representation of the
(M,R)-system. It will be noticed that these induced mappings play a role very
similar to the auxilary automata which appear to be required in order to
fabricate structures which can be duplicated. It will further be seen
that since the domains and ranges of all the induced mappings are perfectly
well-defined sets, there is no question of encountering the paradox we have
described above. Since these induced mappings will correspond to environmental
objects (at least in the form in which they were enunciated in Rosen
1959), there is no need to consider mechanisms for their replication, so
that we avoid thereby an infinite regress which would destroy the usefulness
of the model.
Nevertheless, it appears that the replicative mappings
induced in abstract (M,R)-systems in the manner described (Rosen 1959) may not
correspond to the actual coarse structure of biological systems. In order to
conform to the discussion set forth above, it seems that the mappings which
are of importance for replicative procedures should be chosen so as to be of
the form of the auxiliary automata occurring in the models of Penrose and
Jacobson. This modification does not appear to be too difficult, and
appropriate investigations are currently in progress."
I do not find this paper to be particularly structured as an
assault on von Neumann or on his ideas. Certainly, as Rosen states up front,
the question of a paradox germinates from the very idea of a
self-reproducing automaton, which in turn was originated by von Neumann.
I think you may have previously argued that Rosen was too narrow in
his interpretation of "self-reproducing automata" as conceived by von Neumann,
thereby causing this paper to be some kind of straw-man argument by Rosen.
But Rosen's characterization seems quite valid. Burk, in the Introduction
of Essays on Cellular Automata remarks:
"In the first essay I also describe von Neumann's kinematic model of
self-reproduction and compare it with his cellular model. This comparison is
of interest because of the vague nature of the general problem of
self-reproduction which von Neumann posed. Once a particular cellular space
(e.g., von Neumann's 29-state system) and a "universal" class of automata
(e.g., initially quiescent automata) are defined, the question of the
existence of a universal constructor is a precisely logical one. But von
Neumann was interested in this question because it is a special case of a more
general question: What kind of logical organization is sufficient for an
automaton to reproduce itself? This question is not precise and admits to
trivial versions as well as interesting ones. Von Neumann had the familiar
natural phenomenon of self-reproduction in mind when he posed it, but he was
not trying to simulate the self-reproduction of a natural system at the level
of genetics and biochemistry. He wished to abstract from the natural
self-reproduction problem its logical form." [p. xv]
So, Burk asserts that the logical form of self-reproduction, divorced
from concerns of chemistry and genetics, was the focus of von Neumann,
which is precisely what Rosen engaged.
Further, the cellular automata, as well as the more complicated
kinematic automata (controlled by a Turing machine), are clearly not beyond the
scope of Turing-computability. Is Burk then mischaracterizing von
Neumann or improperly focusing on these Turing-computable models?
To my knowledge, there are only two rebuttals to this paper. One is Guttman
(1966) and the other is Moore, which I find in Burk's Essays on
Cellular Automata (1970). In neither case do the authors
question Rosen's characterization of von Neumann's automaton as too narrow or
otherwise misrepresenting von Neumann's automata.
As I mentioned in a previous post (3/21/04), Guttman does not
refute Rosen's paradox, but instead provides an alternative mechanism by
which to circumvent the paradox. In doing so, he utilizes the same set-mapping
characterization of automaton as did Rosen. Nor did Rosen fail to "engage his
critics", since Guttman notes that Rosen acknowledged to him that Rosen's
argument was restricted to a specific form of self-reproduction, and did
not include all possible forms of self-reproduction (such as Guttman's
mutation approach).
Edward Moore begins his essay "Machine Models of Self-Reproduction" as
follows:
"The ability of living organisms to reproduce themselves has long been
considered to be one of their most characteristic features. Von Neumann was
the first to treat in any detail the problem of how to make machines reproduce
themselves in a purely mechanistic fashion as a way of throwing light on some
fundamental problems of biology and as a problem (of intrinsic interest aside
from biology) concerning the capabilities and limitations of machines." [p.
187]
Here again, Moore asserts that von Neumann approached self-reproduction
from a "purely mechanistic fashion". Moore's complaint [p. 191] about Rosen's
paper is that he feels that Rosen has confused the distinction between the
tesselation space (which stands in for the "universe" in the tesselation model)
with the configuration of cells, and that no paradox exists. However, this to me
seems confused since Moore acknowledges [p. 188] that the state transition rules
are part of the underlying tesselation structure, and not part of the
configuration. Therefore, any changes in the configuration of the cells -
including any replication of a configuration - relies on the computations
induced by those rules external to the configuration. Replication of a
configuration is therefore not self-reproduction, since the tesselation
structure which drives the process is not also reproduced. As Rosen notes at the
end of chapter 15 "Morphogenesis in Networks" in Essays on Life
Itself:
"Von Neumann's original problem was the following. Suppose that we define
an initial configuration of the tesselation, in which a finite number of the
constituent automata are in active states, and all the other automata are
"off". Can we define state transition rules such that at the end of some
definite time, the configuration of the tesselation will consist of two copies
of the original active configuration, with everything else "off"? This
question has been answered in the affirmative by a number of authors, and a
large literature with strongly morphogenetic overtones has been elaborated
within this framework. However, these problems are entirely network problems,
interpretable as differential birth-and-death by allowing birth to
mean the forcing of a neighboring inactive cell into an active state, and
death to mean the opposite." [p. 245]
This post has become longer than I originally intended. For time and size
considerations, I will break it up into at least two posts.
I also note here in passing that Kampis, in his book
Self-Modifying Systems in Biology and Cognitive
Science, spends many, many pages (primarily in chapter
7 "Self-Reproduction and Computation") on von Neumann's self-reproducing
automata, and comes to many (and perhaps more) of the same kinds of critical
conclusions about von Neumann self-reproducing automata as Rosen. Was
Kampis therefore also "falsely accusing" von Neumann?
Tim
> -----Original Message-----
> From: ROSEN Forum
[
mailto:***]On Behalf Of Howard
> Pattee
> Sent: Tuesday,
March 30, 2004 9:34 PM
> To: ***
> Subject:
Re: Howard's challenge #1
>
>
> Judith,
>
> I
apologize for calling Bob's friends (is that OK) by other
> names. I'm
generally in agreement with most of your long
> responses, but I think you
have lost the issue that I first raised.
>
> Judith: The biggest
problem my father had with von Neumann's
> theory was not the secondary
stuff, which as you mentioned, seems
> to agree quite a bit with
>
"Rosennean" findings-- it's the primary concern that made my father
reject
> the whole shebang. Von Neumann's basic premise was what he
disagreed with
> entirely and wanted to distance himself
from.
>
> HP: I understand all that, but it is not the issue I
raised. All
> I suggested is that it would be best for Bob's reputation if
his
> friends, colleagues, relatives, (what should I call them?),
would
> not continue to defend Bob's specific no-longer defensible
>
argument with von Neumann over self-replication. This has nothing
> to do
with whether von Neumann's had a good philosophy or even a
> good model of
replication. It has nothing to do with the
> technical meanings of
"equivocation" or "confounding" or
> "invalidation" or any other words
that Bob has used to discredit
> von Neumann's model.
>
> The
issue, to put it bluntly, is whether Bob falsely accused von
> Neumann. In
other words, the science of the matter is not the
> issue. It is the
ethics of Bob's refusal to engage his critics on
> the issue and his
unresponsive, uncritical repetition of
> essentially the same charge for
over 40 years.
>
> If this is the case, then I think you would
understand why it is
> not good for Bob's reputation to continue to
justify it by a
> diversion, explaining that the biggest problem Bob had
was that
> he just disagreed entirely with von Neumann's basic
premise
> (whatever that is?) and that led him to reject the whole
shebang.
> That is not considered a good scientific or
philosophical
> argument, although it may be a good psychological
explanation.
>
> I have given brief quotations from von Neumann
showing that Bob's
> assumptions (in his paper, Bull. Math. Biophysics,
1959) were
> mistaken. There are a lot more. I can also find Bob's
>
restatements of this misinterpretation in many papers. I think it
> had
become a habit. In Life Itself, p. 234, he says it
> "parenthetically,"
and with no references, as if it were an
> established fact that von
Neumann confounded simulation with construction.
>
> I would like go
on to discuss why I think some of von Neumann's
> ideas were consistent
with Bob's, specifically his Theory of
> Games. I think that iterative
game theory might fall in the
> category of Bob's impredicative
models.
>
> Howard