![Figure 6. Rosen’s (relational) model of a Turing machine. The vertex f denotes the hardware that induces a software flow from input (A) to output (B). Based on Figure 9B.3 in [23]. Note that the component f in this figure is different from the observable label f used in Section 2.2.](/figures/figure-6-rosens-relational-model-of-a-turing-machine-the-3br3k0w6.webp)
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Citation for published version
Chu, Dominique and Ho, W. (2006) A Category Theoretical Argument Against the Possibility
of Artificial Life. Artificial Life, 12 (4). pp. 117-135. ISSN 1064-5462.
DOI
https://doi.org/10.1162/106454606775186392
Link to record in KAR
https://kar.kent.ac.uk/14519/
Document Version
UNSPECIFIED
Author proof---not final version!
A Category Theoretical Argument
against the Possibility of Artificial
Life: Robert Rosen’s Central
Proof Revisited
Dominique Chu
Computing Laboratory
University of Kent
Canterbury
United Kingdom
and
School of Computer Science
University of Birmingham
Birmingham B15 2TT
United Kingdom
D.F.Chu@kent.ac.uk
Weng Kin Ho
School of Computer Science
University of Birmingham
Birmingham B15 2TT
United Kingdom
W.K.Ho@cs.bham.ac.uk
Abstract One of Robert Rosen’s main contributions to the
scientific community is summarized in his book Life itself. There
Rosen presents a theoretical framework to define living systems;
given this definition, he goes on to show that living systems are
not realizable in computational universes. Despite being well known
and often cited, Rosen’s central proof has so far not been evaluated by
the scientific community. In this article we review the essence of
Rosen’s ideas leading up to his rejection of the possibility of real
artificial life in silico. We also evaluate his arguments and point out
that some of Rosen’s central notions are ill defined. The conclusion
of this article is that Rosen’s central proof is wrong.
1 Introduction
One of the stated g oals of artificial life is the implementation of living systems in silico and their
simulation over an entire life cycle [5]. So far this undertaking has been at best partially successful.
Several researchers have built computational systems that display lifelike behaviors. Very famous
examples are Ray’s Tier ra [21] and Adami’s Avida [2]; those systems consist of adaptive computer
programs that compete for limited resources, displaying interesting evolutionary effects reminis-
cent of living systems. Other examples of living systems in silico are Ono and Ikegami’s self-
assembling cells [18, 17], McMullin’s SCL model [3, 15, 14], and Rasmussen et al.’s self-assembling
lipid bilayers [20].
Probably the most famous of artificial life creatures is John von Neumann’s self-reproducing
machine [6, 7]. The center piece of this machine is the so-called universal constructor—a machine that,
given enough sup plies and a description of a machine D, can build D. Together with a control unit
that regulates the sequence of events and a description of all parts of the von Neumann machine, the
universal constructor is an essential element allowing self-reproduction of the entire machine (for an
in depth discussion of the intricacies of von Neumann’s design see [13]).
Those and many other systems show interesting lifelike behavior; nevertheless, few researchers
would call any of them ‘‘living.’’ It is widely acknowledged that the problem of research into artificial
life forms is not only of a technical nature (how to design an artificial organisms) but at least equally
of a conceptual nature: We do not know what life is; at least, we do not have a set of criteria that can
be applied to a given system to decide whether or not it is alive.
There have been some attempts to formulate a general theory of life. One of the best-known
approaches in this context goes back more than thirty years in time and has culminated in Maturana
n 2006 Massachusetts Institute of Technolog y Artificial Life 12: 1 – 18 (2006)
Author proof---not final version!
and Varela’s celebrated no tion of autopoiesis [25, 12]. An autopoietic system is characterized by two
main properties:
!
It is separated from its environment by a boundary.
!
It has an inter nal org anization that is capable of dynamically sustaining itself (including
its boundary); this internal sufficiency of the productional processes is often referred to
as closure [4].
While autopoiesis has had considera ble influence on various sciences (notably not including biolog y),
there is no stringent theory of autopoiesis that formulates its contents in an unambiguous way (see
Nomura [16] for an attempt).
A more mathematical approach to the topic, also emphasizing closure prope rties, is due to Robert
Rosen and is comprehensively laid ou t in his book Life itself [23]; also see [24, 22, 7, 8, 26, 9].
Rosen’s work defines a living system as closed with respect to ‘‘efficient causation.’’ This is often
understood to be roughly equivalent to the closure property of autopoietic systems [11]. However,
the two concepts are not identical. Maturana and Varela themselves illustrated the concept of
autopoiesis by means of a computational model [25]; this model has since been re-implemented by
McMullin and Varela [15, 14, 3]. Rosen’s central message, on the other hand, is precisely that closed
systems are fundamentally different from computing machines. Rosen takes one further step by
claiming that living systems cannot even be implemented in computationa l systems. If he were right,
this would mean that attempts to construct life in silico are futile. Altogether, Rosen’s central
conclusion is that life in silico is impossible.
A detailed presentation of the argument supporting his conclusion is laid out in Rosen’s book Life
itself; a central part of this book is a semi-formal proof for this assertion. Life itself was published in
1991 and has since attracted quite considerable interest, reflected in numerous citings of it. This
interest is not surprising, given that if the proof (and the conclusion) were correct, then this would
have deep implications for the field of artificial life and even our view and understanding of life.
Unfortunately, a clean assessment of the argument is hampered by the inaccessibility of Rosen’s
writing. Its free mixing of mathematical allusions, formal arguments, and comments on the history
and philosophy of science, in addition to poor editing, makes it very hard for the reader to distill the
essential argument from the text, let alone critically assess it. Rosen’s idiosyncratic style is not helpful
in this effort.
In this article we aim at two objectives. Firstly, we wish to clarify a number of concepts that are
essential for Rosen’s proof but not clearly presented by Rosen himself. In order to achieve this we
will (in Section 2) review the notions of synthetic and analytic models. Those concepts are central to
Rosen’s idea of machine versus living system, and thus essential for his central proof. An important
part of this section will also be to clarify the relations between those two models and certain concepts
from category theory. This relation is only hinted at in Rosen’s original work. Having clarified those
basic concepts of synthetic and analytic models, we will then go on to define ‘‘mechanisms’’ (and
‘‘machines’’) in Section 3. The notion of mechanism is of particular importance in that it is
coextensive with the set of systems that can be simulated in computers. In Section 4 we will recap
Rosen’s central proof stating that living systems are not mechanisms, hence not simulable.
The second main goal of this article is to give a critical assessment of Rosen’s central proof. We
will do this in Section 5, where we also show that Rosen’s own concept of machine is flawed. We will
provide a brief discussion and conclusion in Section 6.
2 Models
In this section we will introduce Rosen’s notion of synthetic and analytic model.
Rosen focused his discussion entirely on models; he largely avoided writing about systems. The
reason for this is that as observers we do not have direct access to real systems in the physical world
Q1
Artificial Life Volume 12, Number 12
A Category Theoretical Argument ag ainst the Possibility of Artificial LifeD. Chu and W. Kin Ho
Author proof---not final version!
(the Welt an sich). Knowledge about the real world is thus necessarily incomplete; at the very least,
whatever we know about real system s, we can never be sure what the status of this knowledge
actually is.
This diffi culty disappears when we restrict the discussion to formal systems instead; in con-
trast to real systems, formal systems are accessible. Throughout the remainder of this article, the
word ‘‘system’’ will always refer to formal systems unless explicitly stated otherwise. For all practical
purposes, thus, the notion of system is interchangeable with the notion of model, only that we tend
to use the word ‘‘system’’ to denote given formal systems and their very accurate models.
Rosen’s argument fundamentally rests on the distinction between two types of models, namely
synthetic and analytic models [23, 22]. The main purpose of this section is to introduce them and to
clarify their relation to certain concepts from category theory. Since we do not assume that the
reader is familiar with this branch of mathematics, we will give a brief introduction to some con-
cepts of category theory. An exhaustive discussion of this topic would of course go beyond the
scope of this article; the reader who wishes to explore category theory in any depth is referred to
textbooks [19, 10].
2.1 Categories
We start by defining the notion of category.
A category is a construct that consists of the following:
!
Objects (sets, or anything else; in our case: models).
!
Morphisms between objects; not every object needs to be the doma in or codomain of
a morphism.
!
For every object X, an identity morphism id
X
.
!
For each pair of morphisms f : A ! B, g : B ! C a composite morphism fg : A ! C.
!
Furthermore, we require associativity [( fg)h = f( gh)] and that morphisms can be
combined with the identity map (given a morphism f : A ! B, we have f id
A
= f and
id
B
f = f ).
In the simplest case, the objects of a category are sets and the morphism s are mappings between the
sets. In the more general cas e the objects might be anything, including categories themselves.
Categories are often graphically represented as points (objects) of which some are conn ected by
directed gra phs (the morphisms), usually indicated as arrows. The strength of category theory is that
it allows one to formulate relations between the objects and also relations between categories.
A pervasive concept in category theory is the idea of duality. Two concepts in category theory are
dual if one is like the other but with all arrows reversed. For example, if A is a category, then the
opposite category A
op
is A with all arrows reversed. A and A
op
are then dual to each other.
Throughout this article the concept of duality will not be explored in any depth, but it will be used to
indicate relations between definitions and categories; specifically, the concepts of direct sum and
direct product are dual; the reader might compare the definitions in Figures 1 and 3.
The reason the direct product and sum need to be introduced here is that Rosen essentially
identifies them with analytic and synthetic models respectively. The concepts of analytic and synthetic
model in turn are essential to the concept of mechanism —the class of systems that can be simulated
in computers.
2.2 Analytic Models
While the concepts of analytic and synthetic model are crucial to the entire argument, Rosen only
hinted at a mathematical definition. The purpose of this section is to make the notion of analytic
model precise.
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A Category Theoretical Argument against the Possibility of Artificial LifeD. Chu and W. Kin Ho
Author proof---not final version!
Before doing so, however, we will briefly discuss the informal idea behind analytic models.
Rosen’s notion of model might to some appear somewhat counter-intuitive at first. A typical model
in science formulates a dynamic relationship between entities, as for example a differential equation.
Rosen’s idea of model is fully compatible with this conventional view, yet somewhat different in
emphasis. Instead of focusing on the dynamics of components of a system, a model in Rosen’s sense
is a certain way to measure properties of the system. Given a system S, an analytic model of S is a set
of observables that describe the system. Clearly, the quality of the description depends on the
accuracy with which the observables are measured. An observable that can have say, 100 different
values, is better, or at least of higher resolution, than an obser vable that can have only two values.
Increasing the resolution of an observable thus leads to a refined model.
Besides an increase of resolution of an observable, there is a second type of refinement: Not only
might it be that the observables used are inaccurate, measuring the system only crudely; it might also
be tha t the set of observables used is not sufficient to give a full description of the system. In this
situation adding a new observable will also refine the model. One might for example describe a car in
terms of its speed and the number of passeng ers it takes; this description would, in Rosen’s sense, be
a simple analytic model. Another model of a car would describe it in terms of its milage and how
much fuel it carries. Putting those two models together, thus describing the car in terms of speed,
number of passengers, milage, and fuel content, will give a more accurate picture of the state of the
car than either of the models and thus refine both of them.
For every system S there will usually be many different models that are in a partial order
relationship with each other, generated by the refinement relationships between them. Note that, while
models in Rosen’s sense do not explicitly take into account the dynamics of the system, the y never-
theless are complete descriptions of the system. This reflects the assumption that a complete de-
scription of a system at a specific time already determines all future behavior (although possibly only
in a statistical sense). In esse nce, an analytic model is a set of observables with a given resolution.
Formally, an observable is a mapping f from a system S (the system to be modeled) to a set
(usually R
n
). The observable f induces an equivalence relation on S: Two elements s
1
, s
2
2 S are
equivalent (s
1
" s
2
) if f (s
1
) = f(s
2
). The size of the equivalence classes thus indicates how well f
discriminates between states of the system; the smaller the equivalence classes, the better is the
corresponding observable. An analytic model M of S is the partition of S into equivalence classes
generated by some f.
One and the sam e system S will typically have many different analytic models corresponding to all
possible observables f ; those models will be related to each other and can be compared by the way
they partition S into equivalence classes. The relevant concept here is that of refinement: Given a
model M
1
, the model M
2
is a refinement of M
1
if all equivalence relations induced on S by M
2
are
subsets of the equivalence relations induced by M
1
on S. Intuitively, this means that M
2
can distinguish
between at least as many states of S as M
1
. Given two models M
1
, M
2
of S, those models need not be
in a refinement relation to each other, but may be unlinked.
1
This will be the case if the observables
measure unrelated aspects of the system. If there is no specific refinement relation between M
1
Figure 1. Direct product: If S, M
1
, M
2
are objects of the category C, then S is the direct product if we can find an
object X and morphisms k
1
: S ! M
1
, k
2
S ! M
1
, f : X ! M
2
, and c : X ! M
2
such that there is a unique morphism
u : X ! S. The direct product is a generalization of the familiar concept of the Cartesian product of two sets to arbitrary
categories.
1 It is an important technicality that observables and hence models may be partially linked. This is not of fundamental importance, though,
and will be ignored in the present discussion; the reader interested in this aspect is referred to Rosen’s discussion in [22].
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A Category Theoretical Argument ag ainst the Possibility of Artificial LifeD. Chu and W. Kin Ho