Computational Ontogeny

William R. Buckley

California Evolution Institute, San Francisco, California, U.S.A.

Keywords:

Automata, Cellular, Construction, Efﬁcient, Learning, Machine, Replication.

Abstract:

Of interest to the theory of machines that construct is ontogeny, by which process of development the con-

structor is transformed from immature to mature form. Whereas we have already shown that self-replicating

machines generally are able to bootstrap themselves through the construction of sub-machines (such as organs

that rewind a tape, or replicate a tape, or initiate the behavior of a construct), in this paper we present in ab-

stract a constructor that bootstraps its ability to construct, through the construction of sub-constructors. This

is to say, we present a constructor that learns how to construct, and does so by constructing; our constructor is

in truth a proto-constructor. Here, learning occurs by the addition of new machine conﬁguration; each learned

lesson is correlated with speciﬁc additions to machine conﬁguration.

1 INTRODUCTION

The theory of machines that construct began with the

seminal work of von Neumann (Jeffress, 1951; von

Neumann, 1966) via his self-replicating machines.

Such constructing machines are viewed as having uni-

versal competence over construction where the yield

is passive. Passivity

correlates formally with the

absence of signal to be found coursing within and

through conﬁguration, though the practical correlate

is in-animation which is quite contrary to typical ex-

pectations of automata such as clocks. Indeed, von

Neumann ignored many lessons of Nature in devel-

oping his self-replicator model, eliciting observations

such as those of (Shalizi, 2012) who gives cautiously

reserved praise.

Maynard Smith offers his own cau-

tious praise while pointing to a lack of a machine em-

bryology in artiﬁcial life models (Smith, 1986); see

(Buckley, 2008a) for a rudimentary model that ad-

dresses some of Maynard Smith’s concerns.

These constructing machines have been further

examined in the work of McMullin, who mused over

the relationship between construction and evolution,

and how this relationship was the proper question be-

Especially for cellular automata.

The relevant quote is: “CA were not invented, however,

to be realistic models of Nature. They started with John von

Neumann, who wanted to study self-reproduction, and de-

cided that the ﬁrst thing to do was ignore everything biolo-

gists had learned about the way actually existing organisms

reproduce themselves. This is known as hubris, and is es-

pecially galling when it works.” (Our emphasis.)

ing addressed by von Neumann, as opposed to ma-

chine self-replication per se (McMullin, 2000). The

key point of McMullin’s argument is that the kind of

reduction in artifact complexity that is the result of

manufacturing processes could perhaps be countered

by an understanding of how systems of constructors

might be organised to yield ever more complex con-

structs, with the expectation being that some of these

constructs would themselves in fact be constructors.

That is, McMullin addressed notions of knowledge-

able, qualitative leverage over construction processes

as a means of gaining quantitative leverage in the pro-

duction of artifacts of ever increasing complexity.

Missing from McMullin’s model is an example.

Presently, we give one such example.

The traditional view of machine self-replication

holds that the mother machine constructs all of the

daughter machine, and that during construction of the

daughter machine, the daughter machine is passive;

the daughter machine initiates behavior subsequent to

its construction. Such a self-replicating machine is

necessarily composed of many subordinate machines,

with the constructor proper being of special impor-

tance. The portion of the self-replicating machine

that is properly the constructor machine is itself sub-

ject to decomposition, and it happens that proper sub-

sets of resulting subordinate constructor machines are

sufﬁcient as to machine construction even if they are

not sufﬁcient as to self-replication. This is to say,

constructors proper can observe a developmental pro-

cess. In this paper, we present a self-replicating ma-

chine that observes properly the ontogeny of its con-

116

Buckley W..

Computational Ontogeny.

DOI: 10.5220/0004204101160121

In Proceedings of the 4th International Joint Conference on Computational Intelligence (ECTA-2012), pages 116-121

ISBN: 978-989-8565-33-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

←←← ⇓

⇑ ⇐⇐⇐

C Arm

. . . λ

. . .

⇒⇒ ↑

→→ ↑

R/W Arm

Figure 1: High-level architectural depiction of a von Neu-

mann self-replicator. Each of the four organs shown in the

ﬁgure is composed entirely of stages, arranged with input at

ﬁgure top, and output at ﬁgure bottom, except for the R/W

Arm, where stages are arranged from left to right. The ar-

rows show major signal paths within the conﬁguration, and

the capital C symbol represents a conﬂuent state, and so a

point of signal duplication. Additional stages are added to

the left edge of the TL, TL

and C Arm organs; the construc-

tion arm has no access to the R/W Arm organ. No sense of

scale should be inferred from this drawing.

structor, and this in addition to the otherwise general

ontogeny of the self-replicating machine as a whole

(Buckley, 2008a). In accomplishing this ontogeny,

our machine acquires new conﬁguration, thereby in-

creasing the set of conﬁgurations that the machine can

construct; our machine learns to construct, and does

so by constructing and incorporating into its own con-

ﬁguration that newly constructed conﬁguration which

represents those lessons. Further, the set of instruc-

tions acceptable from tape correspondingly increases.

Our example self-replicating machine is imple-

mented in von Neumann 29-state cellular automata

(von Neumann, 1966), and has physical characteris-

tics common to such machines. In particular our ma-

chine reads its tape twice, has distinct construction

and tape replication phases of its behavior and suffers

the consequences of destructive reading of its tape.

For a thorough and yet succinct review of the charac-

ters of state types and their interactions within von

Neumann cellular automata, see the opening three

paragraphs of (Buckley and Mukherjee, 2005). For

discussion of types of signal, see (Burks, 1970) or

(Buckley, 2008b).

2 ABSTRACT

SELF-REPLICATOR DESIGN

We should like to make as concrete as possible the de-

sign of this conﬁguration, so as to strengthen reader

understanding of the details of our model of con-

General Recogniser

Pulser

Figure 2: Abstract structure of the stage. A stage is com-

posed of a general recogniser followed by a (large) pulser.

Stages generally emit a series of one to ﬁve signals for each

accepted signal.

structor ontogeny. To assist with this goal, we give

a self-replicating conﬁguration that has a rather reg-

ular structure, this serving to simplify reader under-

standing of conﬁguration behavior. This conﬁgura-

tion is composed of exactly four different types of

sub-conﬁguration: multiple copies of one kind of sig-

nal processor (emits a set of signals in response to ac-

ceptance of a speciﬁc, recognised signal; this organ is

called a stage), a single read/write arm (allows read-

ing and writing of the tape), a single construction arm

(allows for the alteration of state class of target cells,

and for the articulation of space traversal for con-

struction signal), and simple signal paths (constructed

largely of cells set to one of the four ordinary trans-

mission states, with the occasional conﬂuent cell for

signal duplication). The sequential acceptance of sig-

nals by stages yields a further abstraction, in that the

conﬁguration behaves according to a microprogram.

This microprogram is expressed as the signaling that

is generated by the various stages comprising the con-

ﬁguration. A block diagram of this self-replicator is

given in ﬁgure 1.

A stage is a combination of a signal pulser and a

signal recogniser; see (Thatcher, 1970) for a descrip-

tion of each of these two organs. An understanding of

the detailed operation of stages is not critical to under-

standing of our thesis regarding machine ontogeny;

such understanding is amply served by knowledge of

the relationship between signal acceptance and signal

generation, that generation follows acceptance. In-

stead, the point critical to our thesis is the set of states

used to build these two organs, a point we explore

later in this paper. We see the stage abstractly di-

agrammed in ﬁgure 2. Stages serve the purpose of

translation, bringing about a sequence of operations

in response to a speciﬁc signal, and in so doing yield

the conversion of instruction found on tape into pur-

poseful construction and articulation signal.

The various pairings of signal recogniser and

pulser, the stages of the self-replicator, are organised

into four groupings, these correlating to all portions of

the conﬁguration save the construction and read/write

arms, proper, and the lowly signal paths. Two of the

groupings yield signals that direct articulation of the

two arms; the C Arm organ and the R/W Arm organ.

A few of the signals accepted by the stages of the

C Arm organ serve not articulation but construction.

ComputationalOntogeny

117

. . .

Figure 3: Tree structure of the TL and TL

languages. Edges

of the graph are labeled with the corresponding value of

input read from tape. For each node ν, the successor nodes

are numbered 2ν + 1 (for inputs of value zero) and 2ν + 2

(for inputs of value 1).

The other two groupings serve the tape read and in-

struction translation processes, proper; they deﬁne a

language by which the tape is read, and instructions

discerned. It is these latter two groupings of stages

that are of prime concern here in this paper, and de-

ﬁne the TL and TL

organs. For our self-replicator, the

reading of the tape occurs by a sequential alternating

pattern of TL generation, followed by TL

recogni-

tion. The TL organ generates signal that is then used

to read the tape, and so by consequence of von Neu-

mann destructive reading yields generation of TL

sig-

nal, that is then recognised by the TL

organ. Upon

such signal recognition it is trivially possible to dis-

criminate between the reading of a one bit and the

reading of a zero bit, and as we have already said that

any one stage accepts only one signal, it is clear that

for each TL

signal, there is one and only one stage

that accepts the signal; no two TL

stages point to any

one TL stage, and no two TL stages point to any one

stage. This implies that the stages of TL are ar-

ranged in a binary tree structure; the ﬁrst three levels

of the tree are shown in ﬁgure 3. Traversal of the tree

always begins at the root, with signal TL

. Traversal

of the tree terminates at level six, where an instruction

from tape is accepted. Terminal nodes in the tree are

numbered 31 through 62, inclusive. Figure 4 shows

the organisation of stages in the TL and TL

organs,

Figure 4: Organisation of stages in the TL

organ. Implied

is that signal IC

triggers generation of TL

by the TL or-

gan. Notice that only for TL

signal is there call to generate

instruction that brings the writing of a zero (W

) upon the

tape; this accounts for the destructive read of zero.

and the microprogram for a few stages of the TL

or-

gan.

We should mention that the TL

organ has exactly

twice as many stages as has the TL organ. Also, the

signal emitted by a TL

stage that acts as a trigger to

bring subsequent emittance of TL signal is itself not

a member of the TL/TL

set of signal. Instead, this

trigger signal is of an internal code (IC) which is quite

different from TL and TL

signals.

We may see how the sequence of TL → TL

→ TL

... yields reading of an instruction from tape

and translation of that instruction into either construc-

tion arm articulation signal or cell state construction

signal. We see that the instruction <011001> is read

with the sequence [TL

: TL

]. This is to say that in reading the instruc-

tion <011001>, the TL organ is directed to generate

the foregoing sequence of signals. The origin of this

direction is the TL

organ. For each TL

signal recog-

nised, a corresponding set of signals is issued. These

issued signals direct the extension and retraction of

the read/write arm and the return signal path, any nec-

essary repair to tape (owing to destructive read), any

signal corresponding to construction arm articulation

and construction, and the next TL signal to issue.

The mechanism of destructive read is simple

enough. In our implementation (which differs

slightly from that of von Neumann), a special signal

<100011> is used to actually read the bit as repre-

sented upon the tape. If the tape at the location of

reading holds a representation of the value one, then

the signal is returned unchanged. If however the tape

at the location of reading holds a representation of the

value zero, then a cell is constructed that represents

the value of one, and the signal is changed, return-

ing as <1>. It is this alteration of read signal that is

responsible for the conversion of TL signal to TL

sig-

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118

nal. So, for the previously given TL sequence, there

will be generated in the read process the TL

sequence

[TL

: TL

]. We see that

in reading bits valued at one, the corresponding TL

signal is unaltered, and that for zero valued bits, the

TL signal is altered to TL

signal; the TL

organ ac-

cepts both TL and TL

signal.

This is the location in the text at which the key

point of our thesis comes into view. It is clearly the

case that only those instructions on tape that are rep-

resented within the read sequence of the set of stages

comprising the TL

organ will be accepted by the con-

ﬁguration. Further, if it so happens that the set of

states of which any stage is composed is itself a proper

subset of the states available for construction, then the

constructor need be able only to construct that proper

subset of states in order to construct a stage, and hence

for the machine to observe ontogeny. Von Neumann

conﬁgurations are composed generally of nine pas-

sive states, yet it happens that for the TL, TL

and C

Arm organs of our self-replicator, the stages and their

interconnections are composed of only six of these

states. It is therefore sufﬁcient for the observation of

ontogeny that our self-replicator be initially able to

construct only these six states.

Of course, the reason for engaging in ontogeny is

that the conﬁguration needs be able to construct all

nine passive von Neumann states in order to engage

in the act of self-replication. The limit on usage of

just six cell states applies only to those stages used

in the construction of the TL, TL

and C Arm organs.

For example, the stages of R/W Arm organ employ a

different subset of six cell states. Thus, our current

demonstration of constructor ontogeny, as opposed

to the more general argument for self-replicator on-

togeny given earlier (Buckley, 2008a).

While this has implications for the tree suggested

in ﬁgure 3, that not all of the stages represented in

the ﬁgure need be constructed at the start of machine

behavior, the issue is more broad, for it is also true

that the C Arm organ need only include stages sufﬁ-

cient to construction of these six states employed in

construction of TL, TL

and C Arm organ stages, and

therefore it too can exhibit ontogeny. Thus, we show

directly and distinctly the ontogeny both of the control

mechanism over construction, and of the mechanism

of construction itself. While we do not take effort in

this paper to show the result, it happens that within

the given model one may even observe ontogenic de-

velopment of the R/W Arm organ, at the cost of using

exactly one perfect signal crossing organ; see (Buck-

ley, 2008b) for a discussion of signal crossing in von

Neumann cellular automata.

000000 RX 010001 LUR

000001 RR 010010 DX

000010 RUX 010011 DR

000011 RUR 010100 DRX

000100 RDX 010101 DRR

000101 RDR 010110 DLX

000110 UX 010111 DLR

000111 UR 011000 TM

001000 ULX 011001 OD

001001 ULR 011010 OL

001010 URX 011011 OR

001011 URR 011100 OU

001100 LX 011101 CN

001101 LR 011110 SD

001110 LDX 011111 SL

001111 LDR 100000 SR

010000 LUX 100001 SU

Figure 5: Code assignments for instruction set of self-

replicator. Codes for construction arm articulation come in

four groups of six signals. The six signals are, for each

group, of a symmetrical nature. For instance, RX is right

extend, and RR is right retract. Similarly, DX is down ex-

tend, and DR is down retract. For rounding corners, we

see that ULX is up-to-left extend, ULR is up-to-left retract,

and LDR is left-to-down retract. TM is the tape mark, and

partitions the code assignment list into two parts, with con-

struction arm articulation codes coming before the codes

for conﬁguration construction. Extension always increases

the length of the construction arm, and retraction always re-

duces the length of the construction arm.

3 ONTOGENY OF A

CONSTRUCTOR

We have now to address the length of instructions

on tape. For the example self-replicator, it happens

that universal articulation (over all quadrants) is nec-

essary to our model of ontogeny. Therefore, it is nec-

essary that the constructor support a total of 24 dif-

ferent motions, thus necessitating at least 24 differ-

ent instruction codes on tape. The need for construct-

ing nine state types increases the required instruction

code count to 33, and the addition of a Tape Mark

symbol stretches the number to 34 different codes re-

quired to express a conﬁguration description on the

tape. We see in ﬁgure 5 the instruction codes assigned

for these 34 different operations; hence the six levels

of the binary tree suggested in ﬁgure 3.

The six von Neumann states necessary to con-

struction of stages employed in the TL, TL

and C

Arm organs are the four ordinary transmission states

{←,↑,→,↓}, the conﬂuent {C} state and the down-

ward pointing special transmission state {⇓}. There-

fore, those stages that correspond to the instruction

ComputationalOntogeny

119

codes for the special transmission states {⇐, ⇑, ⇒}

need not be represented in the TL, TL

and C Arm or-

gans. Indeed, the TL stages corresponding to the sig-

nals {TL

, TL

} need not be con-

structed prior to conﬁguration start, nor need the TL

stages corresponding to these signals be constructed

prior to conﬁguration start. Further, the TL

stage

corresponding to signal {TL

} does not need to be

constructed prior to conﬁguration start. Clearly, cor-

responding stages from the C Arm organ also need

not be constructed prior to conﬁguration start, for a

total of 19 stages that need not be constructed prior to

the start of conﬁguration behavior.

Thus, the machine begins its behavior with con-

struction competence restricted to those states of

which TL, TL

and C Arm stages are comprised, and

completes its behavior having acquired unrestricted

construction competence

By placing upon the tape a description of these

stages (that are missing from the initial state of the

conﬁguration), all of these stages can be added to the

conﬁguration post-initiation of behavior. This yields

an increase in the number of instructions acceptable

from the tape. Careful design of the interface of stage

to signal line allows the stage to be fully constructed

before it is linked into the corresponding organ, and

the acceptance of signal in a highly discriminatory

way ensures that no spurious signal is generated dur-

ing ontogeny. The conﬁguration remains well be-

haved throughout any and all ontogeny.

4 DISCUSSION

Simply put, ontogeny is genome-governed develop-

ment.

Development is the acquisition of new features, be

they physical or otherwise. For biological organisms,

ontogeny is very complex, with many sources of in-

formation giving their affect ultimately to biological

metabolism, and this metabolism yielding emergent

features, like hands and eyes and legs and hearts. It is

commonly understood that biology sees the genome

not as a blueprint but as a recipe, and yet we know

that those recipes are sufﬁciently regular that resem-

blances between generations of individuals is strong,

if not uncanny. We suggest that there is within that

recipe a hint of blueprint, yet.

This leads to justiﬁcation of our model. In this

case, the blueprint analogy is strong. Indeed, for typ-

ical von Neumann self-replicators, the description is

exactly a blueprint; the state of every cell is strictly

mapped, and instructions to construct these cells are

placed within a bed of other instructions that direct

space articulation of the construction arm. It becomes

a real challenge to show how such a machine can de-

velop from an immature state into a mature state. The

use of stages to represent the means to control ma-

chine function allows the machine to be partition-able

down to the level of the stage; the proper function

of any one stage is not dependent upon the proper

function of any other stage. Stages are mutually in-

dependent, and yet by combining them, higher-order

functionality is obtainable, all according to the pro-

gramming (accepted and emitted signals) represented

within constructed stages; self-replication becomes

an emergent property of the machine.

In the von Neumann model of machine self-

replication, machine M has a description of itself D

expressed in a language L that is accepted by M, with

acceptance of D by M yielding construction of an-

other M and another D. Further, the (daughter) copies

of M and D are placed adjacent to each other in the

same pose as was assumed by the original (or parent)

M and D. The important point is that D is a complete

description of M; it has not more nor less information

than is needed to describe M in the language L. M and

D represent a distribution of total complexity for the

system M(D).

In our model, we alter that complexity distribu-

tion, by placing more information about M into the

description D, thus reducing the complexity of M and

increasing the complexity of D, and we do so in such

a way that M is able still to construct modiﬁcations

to itself. We suggest that the development of biologi-

cal zygotes is more than analogous with the ontogeny

expressed in our model; the chief differences are per-

haps in complexity of process as opposed to funda-

mental difference of process.

5 CONCLUSIONS

We have presented in abstract a self-replicating ma-

chine that observes ontogeny, demonstrating a direct

link between development and learning within au-

tomata. We have also shown that there are pathways

of construction that facilitate the development of con-

structors from a state of restricted construction com-

petence to a state of unrestricted (general) construc-

tion competence.

The architecture of our example self-replicator

is sufﬁciently ﬂexible that it may provide a useful

framework for the modeling of open-ended evolution

within machines.

One may see also within the architecture of our

example self-replicator the suggestion of an alterna-

tive cellular automata architecture, one based upon

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120

cells that either implement the functionality of a stage,

or of a simple signal line. Such a transition of au-

tomata deﬁnition might well improve computational

performance sufﬁcient to make practicable the use of

such automata in more general study of biological

processes.

ACKNOWLEDGMENTS

The work in this paper responds to the reservations

of Daniel Mange regarding the ability of the model

given in our paper Computational Ontogeny to sup-

port further machine decomposition, and particularly

decomposition of the constructor (Mange, 2005).

Many thanks are extended to Bruce H. Weber and

David Depew for their many helpful suggestions and

comments, particularly regarding details of biological

ontogeny.

To Cosma for his wisdom and his notebooks.

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