Self-reproduction: principles of structure and operation
Pavel O. Luksha
Higher School of Economics
Moscow, Russia
phone no. +7 902 158 2057
fax no. +7 095 156 9323
email pluksha@yahoo.com
Abstract
The paper concentrates around the core phenomenon of a majority of complex systems – their
ability to self-reproduce, in the attempt to formulate principles common to various selfreproducers. Four aspects of self-reproduction are analyzed. First, the universality of selfreproduction property for complex systems is shown. Second, two kinds of formal definition
and classification of self-reproducers are provided in terms of “ancestor-progeny” and
“system-environment” relationship, based on the definitions suggested by McMullin and
Lofgren. Third, the universal model of self-reproduction is given through consideration of
interactions of a self-reproducer with its material environment. Finally, the model of a “true
self-reproducer” as suggested in the classic work of von Neumann is analyzed in order to
outline possible designs for such a self-reproducer, as well as its dynamic representation in
form of a cognitological map.
1
1.1
Self-reproductive systems in a class of complex systems
Introduction
Since the early days of cybernetics, interest in self-reproductive systems emerged and was
constantly growing. One may only speculate regarding the motivation that drew the scholars.
It probably has the same roots as middle-age alchemist desire to build a homunculus – an
ever-thirst of knowledge and at the same time a wish to become a life-creator. Whatsoever,
the studies were confined either to biological reproduction applications or to artificial life
design.
A phenomenon of self-reproduction is broader than what is traditionally thought, and it has an
ultimate character, at least for our part of the Universe. It can be demonstrate by pointing out
that self-reproduction is one of the core properties to most, if not all, objects of biological and
social sciences, such as:
(1) viruses and similar entities. Reproduction in DNA-RNA-protein system is but single
example in a wide class of auto- and hetero-catalytic reactions, still argued to be considered a
life or a pre-life;
(2) cells – a minimal natural self-reproducer with internal environment, which can be
represented as a system of DNA-RNA-protein reproduction plus membrane self-reproduction;
(3) organisms –multi-cellular systems which reproduce themselves as single unities, and
human being one of them;
(4) populations of biological organisms (prides, flocks, herds, but not yet structured social
systems) – are more or less unified systems which reproduce themselves through reproduction
of their members; these systems tend to maintain a stable number of their habitants.
Obviously, species with sexual divergence can only be reproduced as a population, since
copulation and variation are two requirements of their existence. While some species may
have only a minimal level of unity, others – e.g. ant colonies – reproduce almost as a whole
(Miramontes, 1995).
(5) social groups / collectives / organizations. Michael Rozov argues (Rozov, 1997) that
social collectives and organizations maintain themselves as self-reproducers: while there
individual members may leave a given group or an organization, it may reconstruct itself
through involvement of new individuals1.
(6) societies. It has been shown that a society as a whole can be considered a large natural
self-reproducer (see (Luksha, 2002), (Plekhanov et al., 2001)). It can also be shown that social
skills and social knowledge, including sciences or natural language, are sub-types of a social
memory which is reproduced as a part of self-reproduced cultural system2.
Overall, the importance of self-reproducing systems is still somewhat underestimated by the
scientific community, and the need for their study is undervalued. After the first major
breakthrough accomplished by von Neumann and his short followers in a revolutionary series
of works (von Neumann, Burks, 1966), very little has been done to develop towards better
elaborated conceptual models. A kinematic beast, proposed by von Neumann, seems to be
one of the few models for a designable self-reproducing system. Several years latter, analysis
moved to a sphere of artificial life, a-life-as-it-could-be3. Nowadays, models often simply
assume there is a self-reproductive system, or a population of self reproductive systems (e.g.
CA models, or computer simulated artificial life). Accordingly, the accent has been shifted
from understanding of self-reproduction to understanding of simulated evolution (Taylor,
1999).
The fundamental principles in self-reproduction theory still require certain further shaping. A
particular need in them is obvious: “Principles or laws play the role of expressing the most
basic ideas in a science, establishing a framework or methodology for problem solving. The
domain of General Systems and Cybernetics is in particular need of such principles, since it
purports to guide thought in general, not just in a specific discipline.”(Heylighen, 1992) And,
I would remark, one of the systems and cybernetic research branches of outstanding
importance, theory of self-reproduction, has a particular starvation for principles.
The target of this paper is quite ambitious, as I attempt to summarize some of the existing
knowledge in the field, which I believe can further be applied to more specific areas (such as
applications in biological or social sciences). Therefore, the topics which I will try to cover
are:

to estimate the outstanding importance of self-reproducing systems in the observed
physical Universe (and, through that, to emphasize the importance of a well-developed
theory of self-reproduction);

to give and further clarify the definition of self-reproduction through ancestor-progeny
and system-environment relationship, and to provide a typology of self-reproducers based
on that;

to represent two major types of self-reproduction mechanism and, accordingly, three
general logical models of self-reproduction;

to outline a universal functional and structural model of a self-reproducing system.
Rozov uses a metaphor of ‘Theseus ship’: an artifact ship in which any broken component must be
repaired. After some centuries this ship is completely a different one, every detail in it being replaced.
This metaphor perfectly describes process of social institute self-reproduction (Stepin et al., 1996).
2
Which, in its turn, implies certain importance of self-reproduction even for relatively ‘spiritual’
sciences, such as linguistics, culturology, and others. Dawkins suggests that there exists a unit of selfreproducing social memory, meme, resembling a unit of self-reproducing genetic memory, gene
(Dawkins, 1989).
3
John von Neumann himself contributed substantially to this shift. Tim Taylor describes the process as
following: “The original implementation envisaged by von Neumann was a constructive system, which
Burks has referred to both as the `robot model' and as the `kinematic model'. However, von Neumann
decided that the system was too complicated to capture in a set of rules that were both simple and
enlightening, so he turned his attention to developing the cellular automata (CA) framework with
Stanislaw Ulam.” While the utmost interesting branch of CA is well suited to model a process of
evolution, it is obviously some deviation from implementable self-reproducers.
11
The presented paper is in line with the Artificial Life research program as formulated in
(Langton, 1989), concentrating on the crucial aspect of artificial life, self-reproduction. It is
assumed that there exist to a invariant (that is, substrate-independent universal) principles of
organization and operation of self-reproducers of various kinds. Yet, in order to avoid the trap
of “abstract phenomenon” as described by Emmeche (Emmeche, 1992), I will rely on two
fundamental properties of self-reproducers: their materiality and their complexity of
organization. There may also exist principles that are specific to a certain class of selfreproducers, depending on some particular properties of their substrate and the matter-energy
mechanism of their self-reproduction; and some of these I will also discuss in more details.
1.2
Genesis of self-reproducing systems
A self-reproducing system must be capable to produce a system with similar structure and
functions. Thus, any growing system (capable to produce homogenous structural elements of
itself) has certain properties of a self-reproducer – such as e.g. crystals. This has been realized
by von Neumann, who has pointed out that “...one of the difficulties in defining what one
means by self-reproduction is that certain organizations, such as growing crystals, are selfreproductive by any naive definition of self-reproduction, yet nobody is willing to award them
the distinction of being self-reproductive.” (von Neumann, 1966)
The clear way to get around these obstacle (in my opinion, a path that von Neumann himself
takes is subject to Ockham’s razor4) is to hint that there are additional properties that turn a
system into a self-reproducer. Many systems are only capable to self-reproducing through
growth-and-separation cycles (membrane separation in cell reproduction, birth act ending
with a separation of mother and child organisms, social separation is observed as a form of
society reproduction etc.). A diffusion (observed in auto-and-hetero-catalytic reactions, and
also in reproduction of large social systems) is a less explicit, though very similar, mechanism
to distribute systems in space and time.
Thus, if a system is capable to produce its structure and functions in a separate locus (in fact,
in a separate system), that is, in a space and time interval which is not identical to its own,
then this system can be called self-reproductive5. Note that this definition suggests that
systems capable to self-repair / self-restoration are not necessarily self-reproductive, because
self-restoration process occurs inside the same system through a replacement of its older
structure. Also, a growth of a crystal, as long as this crystal remains a single and
comparatively homogenous fragment of matter, is not its self-reproduction. However, should
a crystal have mechanisms for its periodic self-separation (like in a case of periodic division
of a cell membrane which is but a liquid crystal), it could have been called self-reproductive.
A complexity of behavior of such systems implies they must be complex per se.
It is obvious that self-reproducing systems emerged as a result of development of complex
self-organized systems capable to grow and then to distribute themselves in space and time.
Process of their evolution from emergent systems can be visualized as a looping sequence of
selection through external disturbance and internal variation (Figure 1), e.g. as described in
(Kaufmann, 1993).
Von Neumann tries to resolve the problem as following: “… a way around this difficulty is to say that
self-reproduction includes the ability to undergo inheritable mutations as well as the ability to make
another organism like the original.” (von Neumann, 1966). This suggestion is, in my opinion, a
substantial complication of a problem; moreover, von Neumann provides no logical explanation why a
self-reproducing system without mutations cannot be called a self-reproducing one.
5
In fact, a similar definition is found in (Krippendorff, 1986): “The production of a unity with an
organization similar to that which produced it, each occupying different physical spaces, e.g., the
production of biological offsprings”. However, I believe that a wording ‘different physical space’ does
not allow to distinct between complete separation and diffusion, two distinct cases of self-reproduction
in terms of space and time.
4
Figure 1: Development of emergent systems towards self-reproducers
emergent systems
disturbance
quasistationary
systems
pre-evolution
intense
disturbance
evolution
elimination
of non-stable
elimination
of non-reproducing
stable structures
capable to selfreproduce
variation
and selection
1.3
Evolutionary domination of self-reproducing systems
Domination of self-reproducing systems in a class of complex self-organizing systems is a
logical consequence of their ability to self-reproduce. This can be shown as following.
Complex material systems observable at any point in time, may have following one of the
following origins:
(1) spontaneously emerged self-organized systems which have complex structure and exhibit
stable and yet diverse functioning.
(2) complex systems created by other systems, i.e. systems emerged due to goal-directed
operation of some other complex systems. These systems break into two classes:
(2a) self-reproducing systems, which are able to reproduce their structure and functions;
(2b) non-reproducing systems, i.e. systems which cannot reproduce themselves but which are
reproduced by some other systems.
(3) eternal systems (to which a concept of origin is not applicable). Our up-to-date knowledge
does not allow for such systems because they violate the Second Law of Thermodynamics:
since any material system degrades, it cannot exist for substantial periods of time unless some
work is performed to maintain its internal structure.
A basic model can demonstrate that competition of spontaneously emerging complex systems
versus self-reproducing ones leads to the domination of the latter.
Let x=x(t) and y=y(t) designate a quantity of self-reproducing and spontaneously emerging
systems in a period t in time, accordingly.
Let each self-reproducing system reproduce, on average, 1 other systems. At the same time,
2 systems are degraded and destroyed, so the net ‘surplus’ of self-reproduction is =1-2
(obviously, 0, otherwise dynamics of a population leads to its natural extinction).
Let spontaneously emerging systems appear in each period in average quantity  (this
quantity does not depend on a quantity of systems of this class which already exist). At the
same time, systems degrade and disappear at rate .
Then, a dynamic model of two competing classes can be described as
 x    x

 y    y
(1.1)
The explicit form of the model (1.1) as time-dependent functions is
 x(t )  e  t  C1  e  t  ln(C1 )

 y (t )  e  t  C 2  e  t  ln(C2 )
(1.2)
where C1 and C2 are constants, determined by initial values x(0), y(0), and parameters , ,
and . Then, a dynamics of proportion of spontaneously emerging systems to selfreproducing (as derived from (1.2)) is
lim
t 

C 
y(t )
 lim (exp  (   )  t  ln( 2 ))  0
x(t ) t 
C1 

(1.3)
Accordingly, a fraction of spontaneously emerging systems in a set of complex systems
reduces to the minimum over time. In other words, considering a material nature of any
complex system (which implies that any given system of any type is bound to disappear over
time), it is obvious that major share of complex systems over time will be self-reproducers.
This implies that self-reproducing systems are a universal and dominating type of complex
systems in the Universe. However, the model outlined is just a basic proof of this intuitively
evidential fact because it considers no competition in and between the system classes, and
thus self-reproducing systems take over by simple outnumbering. The model also does not
consider production/ generation of complex non-reproducing systems by self-reproducers
(case 2b in a classification above).
2 Formal definition of self-reproducing systems
Formalisation of what is a self-reproducing system is required for our further analysis. Some
authors point out that there is no generalized definition of self-reproduction6 (Zegzhda et al.,
1994). However, two main branches found in literature can be attributed as “progenyascendant relationship” and “system-environment relationship” definitions.
2.1
Progeny-ascendant relationship
A formal definition of a self-reproducing system7, proposed by Barry McMullin (McMullin,
2000). Let s be a system8 s in system class , and O(s) is a set of systems that system
s is capable to construct (O as an offspring). System is capable to produce any system, if
O(s). Then, if sO(s), s is self-reproducing.
Network of systems, e.g., systems J1 and J2, can be considered, in which system J1 constructs
system J2, and the latter in its turn constructs system J 1. Each of systems Ji is not a selfreproducer in a sense it is constructed by the other system. However, a network (J1+ J2) is a
6
However, this claim refers mainly to entities such as computer viruses (similar statements regarding
memes are also found). These entities in fact should be called quasi-self-reproducers, as it is shown
further on.
7
In this section, a word “system” means any copies of some system, and not a single exemplar of this
system.
8
While McMullin talks of machines, and his main issue is to find a definition for a designable artificial
life, I believe his definition is good enough to be generalized to a class of systems capable to produce
other systems.
self-reproducing system9 (a similar contra-argument was suggested elsewhere by Bruce
Edmonds who has pointed out to a case of Eigen’s hypercycles).
Thus a following amendment to a definition of self-reproduction could be made.

Let s be a system s in system class , O(s) be a set of systems s is capable to produce.

Let O(A) be a set of systems that a can be produced by all systems in set A.
On(s)=O(O(…(n times)O(s))) can be described as n-th “generation of progenies” to a
system s. Accordingly, O0(s)=s. An inverse function, O-1(s), can then be described as an
“ancestor” of m (this function holds a property that sO(O-1(s)).

Then, if for a system s0, there exists k such that s0Ok(s0), then there exists a self
reproducing system S, which can be defined as a set S={s0, s1, … sk-1}, that
si-1O-1(si)  i=1…k-1, and s0O(sk-1).
The scheme (Figure 2) is a graph in the set , where nodes represent particular si and
branches represent relations of production (sisj = “si produces sj”). This graph is an example
of progeny-ascendant relation network.
Figure 2: Two types of self-reproducing systems
S2
Sj2
Sk2
S1
Si2
Si1
Si3
Sj3
Then, the definition by McMullan is only valid for the system S1, which has a relation si1si1.
Accordingly, the definition suggested here above, is also valid for the system S2, formed by
three particular systems, each of them capable to produce another: si2sj2sk2 si2.
However, should these systems be considered as sub-components of a single S2, then S2
conforms to the McMullan’s definition as well.
System S2 has similarity with cases such as RNA-DNA-protein hetero-catalytic reaction.
None of these chemical agents is reproduced by itself; proteins are reproduced with proteins
and RNA, RNA and DNA reproduced with proteins and DNA. Yet, a unity of these three
agents constitutes a self-reproducing system.
The suggested definitions in general “systems” form have one notable flaw, because, in order
to determine whether a given system is a self-reproducer, one must define class  for which
this is determined. In fact, if  is defined as any material object, the definition describes any
repeated process: e.g. an oscillation in wave-like processes, even in mechanic waves, shall be
“self-reproduction”. To get around this, there are two possible ways. Either we restrict 
phenomenologically: e.g. only objects of engineering (machines), biology (living organisms)
9
Until cloning has been recently developed, any single organism belonging to a species with sexual
divergence (including humans) could not be considered truly self-reproductive, because it was only a
couple of two different sexes that constituted a minimal self-reproducer.
and social sciences (societies and institutions) can be self-reproductive10. Alternatively, we
restrict the minimal level of complexity of objects in  (but then it is possible that some
comparatively simple self-reproducers such as computer viruses will be excluded).
It is evident that not only production of system copies is important for self-reproduction, but
also a way in which these copies are produced. Real self-reproducing systems can only
produce copies similar, but not identical to, their originals (that is, imitations). Therefore, for
a more realistic description, some measure of qualitative difference, d(li,lj), can be introduced,
so that:

d(li,li)=0 (a function has its minimum for an exact copy of a self-reproducer li).

d(li,lj)D if system is considered an imitation of a given one, where D is a level of
acceptable variation (see (Eigen et al., 1981) for this kind of measures used in pre-life
models).
Accordingly, a real self-reproductive system be described as a self-reproducing in respect to a
set of a sample and its variations11 (Figure 3).
Figure 3: Natural self-reproducers: set of variations
acceptable variation
d(Sj2, S’j2)<D
sample system
(or its exact
copy)
Sj2
S’j2
imitation
(variation)
of a system
Introduction of acceptable qualitative difference allows to distinct between possible types of
new systems production, analogue to Sipper distinction between self-replication and selfreproduction (Sipper et al, 1997). Three possible types of reproduction (replication, imitation,
and random-walk mutations) are presented in Table 1 (where st be a system s produced in t-th
generation). The important case of random-walk mutations describes a process in which new
species may possibly be formed.
10
but this basically means that the issue is delegated to engineers, biologists and humanity scientists to
decide whether a given system shall be considered as potentially self-reproducing – which adds a
degree of subjectivity to a subject.
11
An issue of a particular interest to evolutionary biology is – when accumulated mutations make an
organism so distinct from its ancestors that this mutant can be named a new species
Table 1: Typology of reproduction
type of new system
production
criterion
description
replication
d(s0, st)0
(or d(st, st+1)0)
difference between each new copy and the
original system must be minimal
imitation
d(s0, st)<D
each new copy imitates the original system,
with possible reversible mutations (Ladenko,
1986)
random-walk mutations
d(st, st+1)<D
a new copy must have resemblance with its
parent, but not necessary with all its ancestors
(and thus this is a process of non-reversible
random-walk mutations); this case has been
sought by von Neumann (Aspray, Burks,
1987)
I further will not particularly distinct between various types of reproduction, denoting all such
processes as self-reproduction. A self-reproducing system, accordingly, is a system capable to
produce its copies or imitations (which is, other self-reproducing systems with the equivalent,
or similar, structure and functions), and it is a system created by another self-reproducing
system with the equivalent, or similar, structure and functions12.
2.2
System-environment relationship
All natural self-reproducers are purely material structures. Therefore, they must have matter
and energy interactions with external environment, and they can only be reproduced through
such interactions.
The importance of system-environment relationship has been to a substantial degree neglected
by von Neumann and his first followers. At a later stage of discussion, when attempts of
practical implementation commenced, the issue came back into a view (Taylor, 1999).
Following closely an approach proposed by Löfgren (Löfgren, 1972), a refined definition can
be suggested to describe a system reproduced in a given environment. A producing system S’
urges its environment F to produce another system S’’, by applying some “effort” (or targeted
action) E to it:
E
( S  
F )  S 
(2.1)
If S’’ is such that S’ and S’’ have a substantial degree of similarity, and S’’ is capable to
produce systems similar to S’ and S’’, then E is a process of self-reproduction13. Since S’ and
S’’ are similar, they can both be denoted as S, and the definition can be written as
E
( S 
F)  S
(2.2)
The effort E, which means transformation of matter and energy, will transform raw material
of environment F into a target system S, also generating some by-product / waste W (which is
a consequence of the Second Law of Thermodynamics, assuming inevitable inefficiency of
12
Note there is a particular case of self-reproducer giving origin to other self-reproducers but not
having been itself produced by another self-reproducing system. The Biblical example is Adam, who
was the first human. A first self-reproducing automaton created by human would be a similar case.
13
Following amendment can be suggested (based on A. Plekhanov’s oral report, with amendments):
if S’ is represented as finite-state automaton with state set {S’i}, and it interacts with a set of
environment elements {Fj} to produce a set of states {S’’k} in this environment, and both sets are
equivalent: {S’i}~{S’’k}, then S’ is self-reproductive.
any transformation). Then, it is possible to represent a process of self-reproduction in a from
of an auto-catalytic reaction:
F + S  2S + W
(2.3)
S is self-reproducing in the environment of F, gradually “consuming” F in this process14.
W denotes degraded matter and energy produced in the reaction which is not usable for
further interactions with S.
Clearly, this conforms to the “progeny-ascendant” definition given above, with an important
amendment: a process of self-reproduction occurs in a given environment. Accordingly,
availability of required elements in the environment is a pre-condition of self-reproduction.
This availability may shrink with the time, since required elements are transformed into a new
system and thereby consumed.
“System and environment” is a complex meta-system which breaks into two sub-components:
the one renewable / multipliable and non-renewable / expiring. It obvious that various types
of systems self-reproductive in their given environment have a completely different physical
structure and also a different complexity of organization and functioning (compare e.g. a
computer virus to a reproduction of multi-cellar organism); also a complexity of their
environment can be different.
It is possible to distinct between types of natural reproducers depending on a degree of
complexity of self-reproducer S in relation to its environment F (Table 2). Although a
measure of complexity is by itself very vague (Edmonds, 1999), it can still be introduced to
indicate some objective characteristics of system complexity. The table should be considered
but a sketch of typology.
Table 2: Types of natural self-reproducers
complexity of F
compared to S
type of selfreproduction
description
examples
higher (more
complex)
quasi-selfreproducers
strictly dependent in their
reproduction on a system of
higher complexity which is not
produced as a part of
reproduction process of which
they are a part
 viruses and genes;
 memes;
 computer viruses and
computer “artificial life”
(e.g. Tierra (Ray, 2001))
comparable
(more or less
equal
complexity)
dependent
selfreproducers
true complex systems
requiring another complex
system (produced or not
produced as a part of their
reproduction process) to selfreproduce
lower (simpler)
true selfreproducers
complex autonomous systems
capable to self-reproduce in an
environment of basic elements
(this case of self-reproducers
has been described by
 elements of chemical
hetero-catalytic reaction;
 any species with sexual
divergence;
 certain biological systems
with symbiotic / parasitic
reproduction (basic case is
bacteria conjunction)
 prokaryotic / eukaryotic
cell;
 organism with asexual
reproduction (e.g. hydra);
 self-reproducing society;
14
Some self-reproducers, such as computer viruses or memes, can be thought of as reproduced without
F expenditure. However, nothing can be reproduced at no cost, although a cost may be quite low so it
can be neglected (energy required to reproduce each new copy of a computer virus – that is, a series of
electronic signals in a hardware – is insignificant, especially when compared with amounts of energy
required for basic self-maintenance, but it is still inevitable energy expense).
complexity of F
compared to S
type of selfreproduction
description
examples
Maturana & Varela, and prior
to that it has been considered
by von Neumann)
 artificial self-reproducing
plants (e.g. a NASA lunar
factory)
Comparative complexity is not the only issue in self-reproduction. For each of these types of
self-reproductive structures, there obviously exists a lower limit of complexity that would
allow them to operate purposefully and in particular to self-reproduce. There are clear
evidences from cellular biology that such a limit exists for biological self-reproducers, such as
prokaryotic cells. A minimum structure of a cell must have 15%-20% of components of
E.Coli (Watson, 1976).
Bacteria (such as E.Coli) is quite a complex structure capable to self-reproduce in a mixture
of rather basic organic molecules. Yet, a computer virus is a comparatively simple program
which requires quite a complicated hardware and software to get executed (i.e. to selfreproduce). This may imply that there exists a lower limit of complexity for “system and
environment” aggregate structure, allowing a system to self-reproduce in a given
environment.
Obviously, natural systems have developed a number of strategies, and a range of supporting
mechanisms, that shall suit their ultimate purpose to self-reproduce. The classical publication
to describe these strategies and mechanisms was von Neumann’s work. However (most likely
due to specific purposes pursued by von Nemann himself), this work has number of
inadequacies on a way to better comprehension of a subject. Since his model has been derived
from an Turing automate (which has been but a convenient mathematical abstraction of a
calculation process (Turing, 1936)), it lacks consideration of material aspect of selfreproduction, and it also has a degree of non-universality. Efforts made in a scholars
movement ever since were an attempt to elaborate this heritage as well as to some extent
deliberate from it.
3
3.1
Principles of self-reproduction
Logic of self-reproduction
A complex system performing targeted operations may be represented as an information
system (Poletaev, 1971), and so can be a self-reproducer (Korogodin, Korogodina, 2000). It
should be emphasized that a system must have a source of information about itself in order to
produce its copy. Two possible sources of information about a system inside its own structure
can be:
(a) a system itself as a physical organization;
(b) some object which represents a system, or system self-representation, which can be called
an instruction (or memory about a system).
Correspondingly, a system can reproduce in two ways:
(a) through self-inspection, by copying or imitating. The original structure is copied by
finding elements, similar to those inside the existing system, in the external environment.
(b) based on instructions, “executing” them. This implies there is a part of system named
“instruction”, and another part of system named “instruction execution unit” (a physical
implementation of a Turing automate with a instruction tape and environment tape could be
an example of such a unit). Obviously, instructions in this case must be reproduced as a part
of self-reproducing system. Yet, there is a problem of instruction infinite regress: an
instruction cannot hold a description of itself, therefore there should be some “instructionindependent” structure that will reproduce instructions. This problem can typically be
resolved through introduction of a copying unit which will produce a copy of instructions
without any reference to their semantic content.
All self-reproducing systems (artificial or evolutionary emerged) combine two indicated
methods of self-reproduction. A principle universal to all self-reproducing structures
suggested by Meller as it appears in (Korogodin, 1991), claims that self-reproduction process
can either be represented as reproduction of system structure, or it can be represented as
reproduction of a memory about this system (instructions) and devices used to reproduce it.
It gives a raise to three possible basic models of self-reproduction, namely, Laing’s automate,
von Neumann’s automate and a superposition of n mutually-instructing automata. Certainly,
more complicated constructions may also be thought of. To describe these models, let us use
following notions: let A, B etc be functional units of a system, while (A), (B) etc will
denote some source of information about these units, respectively.
(1) Laing’s automate is introduced as following. Let an executing unit be denoted as A, and
instruction denoted as (X), A such that it can create another system X based on the
instructions:
A: A + (X)  X
(3.1.1)
Now let X = A, then this system reproduces unit A:
A + (A)  A
(3.1.2)
Now, if (A)=A, then this is a minimal self-reproducing system reproduced via selfinspection. It is a superposition of two identical poly-functional components A, one of them
acting as a self-reproducing device and another as an instructing device.
This kind of basic self-reproducing systems has been first proposed by Richard Laing: strands
or strings of molecule-like finite state automata which may exist in either passive or active
state, a passive being analogue to Turing machine tape and active being analogue to Turing
machine read-head (Laing, 1977). This means that a self-reproducing system minimally
consists of two self-reproducing automata. An example of such system may not be found in
biology but is found in sociology: maintenance of social memory is performed through
demonstration and imitation (and an act of imitation is by itself a demonstration). A computer
virus is a also a self-executing code which copies itself onto a new media using itself a source
of information.
(2) another case of self-reproducing system is a superposition of n poly-functional instructors.
For sake of simplicity, let n=2. Then let A and B be two different execution units, (A) an
instructing unit for execution unit A to reproduce, and (B) an instructing unit for execution
unit B to reproduce. Then these execution units reproduce as
 A   ( A)  A

 B   ( B)  B
(3.2.1)
Let (A)=B, (B)=A. Then,
(A+B)  (A+B)
(3.2.2)
If A=B, then this case is reduced to Laing’s automate.
A minimal structure of such a reproducer consists of two sub-blocks, A and B, able to polyfunctionally act as instructing and executing units. If a newly constructed system must be
built unified, a copy of one of the sub-blocks (e.g. B) may be constructed on a platform of a
copy of superposition sub-block A (thus ensuring a unity); copy of block B can therefore be a
trigger for block A to commence operation in external environment. Accordingly, a trigger for
B to commence operation is a copy of A in external environment (and thus also a copy of
A+B). A specific feature of such a self-reproducing system is that it is mutation-proof, i.e.
despite any mutations that may occur in A or B it will always retain components of the
original system.
This model can obviously be extended to cases n>2. If it is represented as a graph (in the
same manner as in Section 2.1), this graph must hold a property that each node must have one
incoming branch and one outcoming branch. The hypercycle of Eigen (Eigen, Schuster, 1977)
may be considered an example of such a system.
(3) JVN automate
Von Neumann first supposed, based on the design of Turing automate and further proved that
if there exists a universal Turing automate that can simulate any automate, there also exists a
universal constructor which is capable to build any automate based on its description.
If A is an universal constructor that can built any automate X based on instructions (X), then
A: A + (X)  X
(3.3.1)
If B is a copying unit that can copy (X), then
B: B + (X)  (X)
(3.3.2)
15
A controller C can be introduced to synchronise operations of both units :
C: (A+B+C) + (X)  X + (X)
(3.3.3)
Then, if to assume that
X = A+B+C
(3.3.4)
then X is a minimal self-replicating unit, since
(A+B+C) + (A+B+C)  (A+B+C) + (A+B+C)
(3.3.5)
Moreover, if
X = A+B+C+D
(3.3.6)
where D is any additional structural element of an automaton, this element can be reproduced
inside the given system as well:
(A+B+C+D) + (A+B+C+D)  (A+B+C+D) + (A+B+C+D)
(3.3.7)
This implies that components which are not related to a function of self-reproduction still can
be reproduced as a part of a system. Unlike this, in two previous cases non-reproductive
functions can only be performed by structural units poly-functionally.
Logical design of a JVN automate assumes that a complex structure is self-reproduced in an
environment of comparatively basic elements. Certain types of self-reproducers do not have
need for this complicated structure (quasi- and dependent self-reproducers do not always
conform to this model). Accordingly, quasi-self-reproducers are more likely represented as
Laing’s automata. Dependent self-reproducers, likewise, resemble the case of a superposition
of mutual instructors (assuming that functional units are distributed in a population of m
various types of dependent self-reproducers, m=2 for case of males/females).
3.2
Matter and energy in self-reproduction
One of the criticisms frequently raised against von Neumann’s model of self-reproduction
(and the similar ones, e.g. by Burks) claims that these models do not consider physical
properties of self-reproducing system and its environment. Although there are evidences that
von Neumann thought that some input is required (his kinematic beast “flows” in a pool of
15
Note that this unit should not necessarily exist as a separate structure of an automate, since its
functions can be performed poly-functionally by some other component.
available details from which it constructs new copies of itself), yet the obvious need to
introduce the material side of system-environment interactions was without a due attention.
There are evidences that this may have been done purposefully. Tim Taylor writes: “Von
Neumann concentrated on the logic required for a self-replicating machine to be able to
evolve increased complication. He therefore did not specifically deal with various biological
concerns, most notably concerns of energy. In fact, von Neumann was well aware of these
concerns and warned that by restricting attention to the logic of self-reproduction, ‘one has
thrown half the problem out the window, and it may be the more important half’” (Taylor,
1999).
However, engineers seeking solutions for true self-reproducers that could be implemented had
to face a substantial gap between the theoretical framework and practical problems (Freitas,
Gilbreath, 1980). Neither the von Neumann’s self-reproducer requires material components
for part of its operation cycle such as instruction copying16 (while even a simple copying
machine would need a paper and ink to make a copy), nor this automate requires energy to
operate.
Eventually, a self-reproducing system, which can be physically implemented, must hold the
following properties:
(1) Self-reproducing system transforms its inner and outer physical space when creating its
copy; accordingly, it requires matter and energy which will become a “construction material”
for a new self-reproducing system.
(2) Any targeted transformation (including self-reproduction and self-maintenance17) is
energy-consuming (or, no transformation can be performed at zero energy “cost”). Because
energy and matter degrades as it is transformed, input energy and matter must be high-grade.
Thus, self-reproduction can only be performed if a self-reproducer is an open system, which
has an inflow (and also possibly a renewing deposit) of resources for self-reproduction.
Suppose, it is not so. Let us consider a chain of ascendant-progeny relation (a chain of
generations). Let R be some particular critical resource required for self-reproduction. Let R0
be the initial deposit of this resource in the ‘patriarch’ system (the first system which starts
the chain). Let Ri be a deposit of the resource in i-th generation. Clearly, Ri>0 must hold in
order to allow for reproduction.
Let Ei be an amount of resource R spent to produce generation i. Ei>0 for any i, because this
resource is used to produce a new system. Then, a deposit Ri in i-th generation of selfreproducers is defined as
i
Ri  R0   E j
(3.1)
j o
Since a sequence of “sum of Ej“ has no upper limit (it is growing with every next generation),
there exists T such that RT0, however large (but finite) R0 were. Therefore, a closed system
with a deposit of critical resource will not be efficient as a self-reproducer, because a
sequence of self-reproduction will cease over time.
If a critical resource R is not available (or is in deficit) in the external environment, it can also
be produced in a system’s internal environment with externally available resources, say R’.
Then, in order to operate successfully, a system must maintain a transformation R’R at
Suggested in A. Plekhanov’s oral report on issues of self-reproduction.
As a complex dynamic material structure, a self-reproducer must maintain itself against degradation.
In logical models of self-reproduction (von Neumann’s model in particular) the function of selfmaintenance has never been considered, assuming that JVN automata exist only to produce their
copies, at the expense of their own structural degradation).
16
17
approximately the same rate as R goes into self-reproduction process18. This production, as a
rule, is a transformation of some basic environment into more complex internal components
(by certain manufacturing / productive activities or synthesis reactions). In some cases, they
also have some internal environment in which complex elements, constructed from basic
external environment, are stored.
Figure 4: Matter and energy flows in a self-reproducing system
use of matter and
energy for system
maintenance
self-reproductive
system
inflow of
high-grade
energy
and matter
copy of
a self-reproductive
system
outflow of degraded
energy and matter
Self-reproducing systems must be open systems with an inflow of high grade matter and
energy which they use to maintain themselves and to make a copy of a new system, and the
residual is cast away as a degraded energy and matter (Figure 4); similar representation for
living systems in particular are well-known, e.g. (Kay, 1984). It has also been proven as
evolutionary efficient to accumulated extra resources including energy inside a system in
forms available for further utilization (as a stable flow from outside cannot always be
ensured).
3.3
Functional and structural model of a self-reproducing system
From the fact that a self-reproducer is a complex material structure, it is evident that a selfreproducter must have a following minimal set of functions:
(1) self-reproduction (creation of new copies of a system);
(2) self-maintenance of an existing structure;
(3) production, including:
(3a) extraction of external resources (energy and matter) from external environment;
(3b) production of internal environment (in a case when self-reproducer has higher
complexity than it external environment, it should be capable to convert basic elements of
external environment into complex elements intended for its actions).
Based on this set of functions, it is possible to elaborate a functional and structural model of a
self-reproducer (Figure 5). Obviously, each of the functions shall be supported by some
physical structure (executing unit) in a self-reproducer, and regulated via some particular
additional structure (a controller). The organization may be mono-functional (each structure
supports a single function, as in logical automata described above) or poly-functional (several
functions are supported by a structural element).
18
Because energy for transformation is one of the critical resources required, self-reproducer must be
capable to extract energy from external environment and convert it into inner energy.
Figure 5: Functional and structural model of a self-reproducer
External environment
wastes
Self-reproduction
Production
high grade
energy & matter
Extraction of
external resources
Production of
system internal
environment
Waste
disposal W
(non-usable
elements)
wastes
Reproduction of
memory /
instructions
Production of
working
components
Copy S’ of a
self-reproducing system
Resource
pool F
(usable
elements)
Selfmaintenance
Self-reproducing system S
There are two particular aspects of the scheme.
First, production is not always a necessary component of a self-reproducing system, since a
system may use “building blocks” and energy sources produced by other systems. In fact, this
happens in case of “quasi-self-reproducers” (e.g. computer or biological viruses) which
parasitically use virtually all the resources from their host.
Second, in many cases, there is no straight difference between self-maintenance and selfreproduction of a system. This case has been considered by Maturana and Varela (Maturana,
Varela, 1980) who claimed that self-reproduction is a sub-case of self-production (or, selfmaintenance). Also, in von Neumann’s model no particular consideration of self-maintenance
is made.
It shall be pointed out that all three functions (production, self-maintenance and selfreproduction) are fully implemented and realized in “true self-reproducers” – systems which
reproduce themselves in an environment of comparatively simple structural elements by
producing internal environment of complex elements and further using it to self-reproduce
(e.g. this structural and functional model has been shown valid for large social systems in
(Luksha, 2002) and also partially in (Plekhanov, 1980); it can also be proven valid for social
institutes, multi-cellular organisms, and cells). This kind of systems has been of a particular
interest for artificial life scholars since it has been revealed long ago that self-reproductive
robotics in order to reproduce requires production of internal environment from basic
resources extracted from soil, water, air etc. An example of such a design is NASA lunar
plant project for extra-planetary colonization (Freitas, Gilbreath, 1980) and earlier projects of
automated self-reproductive plants (e.g. flying plants in (Moore, 1956)). Such plants must
include resource extractors (mining facilities, solar batteries etc), units producing internal
environment (e.g. casting facilities or chemical plant), component assembly units (production
of new robotics) and self-maintenance devices (robots for repairing/replacement of broken/
non-functioning units).
3.4
Design for a true self-reproducer
True self-reproducers, as it has been noted, conform to a self-reproduction model suggested
by von Neumann, which has at least the following structural elements: (1) a set of
instructions, (2) a “factory” (execution / assembling unit) to construct a new system based on
instruction, and (3) a copying unit for instruction duplication. It may also be pointed out that
there should also exist (4) a resource extractor which will transport external resources
(construction material and energy) from the external environment into a system. A resource
extractor must be a part of any potential design of a true self-reproducer, and its operation
requires a separate regulator (which should also be taken into consideration).
There are, however, two possible designs of such a system which depends on a role which is
taken by the instruction unit ( Figure 6):
(a) Structure type I: instructions are a passive object which is used by some execution unit
(“active” Turing machine) through an “interpretation process” – or, they have the same
function as memory in contemporary computer systems. A copying device may perform
interpretation by making two copies of an instruction, one of them is sent to a newlyconstructed system, while the other goes into a factory for further execution. In fact, this
means that a copying unit must have two different functions: it must be a copying device for
one part of a system (called instructions), and it must be a regulatory device for another part
of a system (regulating it based on instruction). A copying unit cannot be operated by the
same set of instructions, and it therefore requires a separate controller. Operation for this
design of self-reproducers has been actually discussed before scholar interests shifted into a
cellular automata area, and it is a design that von Neumann originally has suggested. This
structural type, as it has been argued a number of times (e.g. (Burks, 1975)), is an adequate
representation of a prokaryotic cell.
(b) Structure type II: instructions are a component of a regulatory unit – since any targeted
action requires a regulator, it may poly-functionally contain instructions in a way in which
e.g. Watt governor “knows” when a steam machine pressure becomes too high or two low. A
copying unit, with a separate controller, is responsible for the production of a system
regulatory unit in a new system.
Figure 6: Two possible designs of a true self-reproducer
selfreproducer
controller
of CU
instruction
copying unit
instructions
instruction copying unit
controller
of CU
copying unit
(CU)
1
2
resource extractor
copying unit (CU)
3
resource extractor
selfreproducer
1
controller
with instructions
2
production unit
production unit
Structure type I
Structure type II
It has been noted by the designers of one of the best elaborated engineering projects of
artificial life, a lunar factory, that each set of operations requires an independent controller,
and all the controllers in the plant must be subsequently reproduced as a part of the plant
operations. Their design is quite close to what has been called Structure type II. Thus, both
structural types should be kept in consideration when it comes to modeling of selfreproducers, and to engineering solutions. Eventually, the resemblance between these designs
and the self-reproducing social systems is quite vague, pointing out that other potential
designs may also exist.
Finally, a cognitological mapping of a true self-reproducer (or, a revised JVN automate) can
be introduced (Figure 7). Cognitological mapping is a graph allowing to represent a system
dynamics. Its nodes correspond to locations in which self-reproducer components are
generated. Accordingly, branches of a graph represent transport of these components for
utilization in a given location; each branch can be assigned a positive or a negative sign,
depending on whether it is an inflow or an outflow.
Figure 7: Cognitological mapping of a true self-reproducer
+

W1
–
F1 +
+
+
+
R
+
F
F2
B
+
+
W2
+
–
A
+
Here, instructions are denoted as , production unit as A, copying unit as B, resource
extractor as R. The inflow of matter and energy from external environment is represented as F
flow, which further breaks into F1 for instruction copying and F2 for components
reproduction. Both processes shall generate wastes / by-products, denoted as W1 and W2.
Instructions  enter either a copying process (where they are multiplied with help of a
copying unit B), or resource extraction (R), or structure reproduction (A) process. Structure
reproduction yields new resource extractors (R), new manufacturing units (A), and new
instruction copying units (B).
Presented scheme allows to define a dynamic model of a true self-reproducing system (with
or without variations in A, B, R, and ; concentrated or distributed19). It is important to say
for all such cases of true self-reproducers, their basic design is the same.
19
The concentrated dynamic system example is a mechanic implementation of a JVN automate, or a
chemical reverbator, while the distributed model would be an oregonator (Beloussov-Zhabotinsky
chemical reaction is an example). There are evidences that von Neumann planned to extend his work in
order to come up with non-discrete form of a JVN automate which would resemble systems such as
BZh reaction.
This cognitological mapping similar to a dynamic representation of a society in (Luksha,
2002). The latter, however, have only considered the multiplication of instructions indirectly,
while, as one can see, this is an important activity in a self-reproduction process.
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