- Email: [email protected]

Life with Autonomously

Emerging

Boundaries

Gunji

Department

of Earth Sciences

Faculty of Science Kobe University Rokkodai 1-l Nada, Kobe 657, Japan and Norio

Kon-no

Department

of General

Education

Muroran Institute of Technology Mizumoto-cho 27-1 Muroran 050, Japan

Transmitted

by John Casti

ABSTRACT The

boundary

perpetual open

conditions

disequilibration

(i.e.

one-to-many

a posteriori. We evolutionary scribed

argue

systems

only

of biological

mapping).

However,

that

an idea

such

and implies

in a forward-time

uncontrollable

boundaries

(a posteriori description).

systems the

unprogrammable description.

can be described With respect

We

quilibrium

systems.

In the last, the local transition

deviates

and its purpose becoming

into constant,

due to the uncontrollable from the Newtonian is not to describe

is uniquely

systems.

But

propose

determined

for biological it cannot

a model

microscopic,

microscopic

paradigm

the physics

(i.e.

boundary the paradigm

of being

but

the

dynamics of boundary

macroscopic,

rule is perpetually

and

be de-

in which

a backward-time

to the extent of the controllability

are classified

paradigm

due to the

for the future are

concept

in introducing

systems

progresses,

future

is a basic

conditions, time

are not controllable,

of force. This means that the possibilities

and dise-

transformed conditions.

as Our

of prediction),

to describe

that

of

(evolution).

APPLIED MATHEMATZCS AND COMPUTATZON 43:271-298 0 Elsevier Science Publishing Co., Inc., 1991 655 Avenue of the Americas, New York, NY 10010

271

(1991)

0096-3003/91/$03.50

272 1.

YUKIO GUNJI AND NOR10 KON-NO INTRODUCTION

We understand that all living things consist of materials which obey physical laws [l]. What is the difference, then, between living and nonliving systems? In this paper we argue that it is possible to define the criteria which distinguish living things from nonliving things from a formal point of view. One important difference between living and nonliving is the velocity of the observation propagation [2, 31, which results in the hierarchical structure in living systems. On the one hand, this velocity in general physics can be approximated by an infinite velocity (strictly speaking, the speed of light). On the other hand, the velocities of particles are much smaller than the velocity of observation propagation. Hence observers can observe all of space at an instant. For example, for classical mechanics, which involves the hidden assumption of infinite propagation velocity in force equilibration, there is no contradiction between the local description (the definition of force) and the nonlocal description (the laws of conservation). In living systems, the velocities of particles are also finite and much smaller than the speed of light. However, the velocity of observation propagation cannot be approximated by an infinite velocity [2, 31. The fact that the velocity of observation propagation is smaller than that of particles entails that the description by an external observer cannot be concordant with the description by an internal observer. The motion of a particle is described in terms of spatial interaction (i.e. the velocity: the description by an external observer). In fact, there are some processes which realize the motion of a particle at a lower level (i.e. the level which constitutes a particle: the description by an internal observer). To describe a unique equation of motion of a particle is possible only when we can accurately deduce the motion of a particle from lower-level dynamics. If the velocity of reaction in lower-level dynamics is large enough and can be approximated by an infinite velocity, we can neglect the time interval required to realize the internal dynamics. This means that the internal observer is not necessary when we adopt the description by an external observer. In this case the descriptions by external and internal observers are equivalent. In nonliving systems any internal dynamics are neglected. However, the internal dynamics in living systems are reactions of enzymes, and the reaction velocities depend on the very small velocity of enzyme propagation. Therefore, we are faced with a situation in which the velocity of observation propagation (lower level/internal) (upper level/external). Although

is smaller than the velocity of particles we adopt the description by an external

observer (i.e. focusing on the upper-level dynamics), we cannot ignore the internal observer. Hence, both descriptions by an external and internal observers are required at the same time, because they are not alternative but

Arti$cial L$e

273

constitutive. In other words, local description is not concordant with nonlocal description. The main problem in the science of living systems is to formalize the contradiction between local (internal) and nonlocal (external) description. Can this problem be solved, in the sense of classical mechanics, in material science? It seems to have been solved, from some points of view, in the study of self-organizing systems [J-7]; however, that can by no means be done in the strict sense. In the fields of dissipative structures, synergetics, and autopoietic systems, no researchers have noticed the logical discordance between local and nonlocal description, which we might call the problem of unprogrammabihty. The logical contradiction originating in the finite velocity of observation propagation is expressed in terms of perpetual disequilibration [2, 31. Such a logical contradiction might be isomorphic to Russell’s paradox, Giidel’s theorem of incompleteness, and the reentrant form [8] of Spencer-Brown’s logic [g-11]. Maturana and Varela [U] emphasized the autonomy of living systems and defined an autopoietic system as a system which consists of its own products. In this sense the autopoietic systems involve the problem of self-reference. However, they did not give serious consideration to the relationship between autonomy and unprogrammability resulting from self-reference. On the other hand, Conrad [13, 141 pointed out that evolvability is closely related to unprogrammability in view of the tradeoff principle, and he found evolvability in the openness to the future, which can be compared to our concept of autonomy. Our approach is, in other words, to formalize Conrad’s concept. First we show the significance of perpetual disequilibration and explain how the logical contradiction (or the unprogrammability) arises from their and process. Secondly, we state the relation between unprogrammability a posterior-i description. We show that unprogrammable systems can be regarded as systems whose behavior we cannot talk about a priori but can describe a posteriori. The Newtonian paradigm [15] concentrated only on a priori descriptions, because prediction is the good in this paradigm. As soon as we introduce both a posteriori and a priori description, prediction loses its sense. We could say that the Newtonian paradigm, which consists of formal, material, and efficient cause, is no longer incompatible with the notion of final cause in our paradigm, because we have to describe dynamics both in forward time from the past to the future (a priori) and in backward time from the future to the past (a posteriori) in order to describe unprogrammable systems [g-11]. Hence we must go beyond the Newtonian paradigm in describing living things. Our main purpose is to give the formalization for perpetual disequilibration, introducing unprogrammability. To express the unprogrammability, we define a posterior-i description in detail, with respect to autonomously

27’4

YUKIO GUNJI AND NOR10 KON-NO

determined

boundary

autonomous

system introducing

2. 2.1.

PERPETUAL

conditions.

This

is the

first

attempt

both a priori and a poster-id

to define

an

description.

DISEQUILIBRATION

Equilibration with Finite Velocity Since information carriers in organisms

obey the laws of physics,

they

have to obey the conservation laws [2, 31. Note that it takes a finite time interval At to realize force equilibration, because the velocity of observation propagation is finite in living systems. Let us consider the effect of finite velocity of observation propagation. Define fij(t) as the internal tensile force from the jth site to the ith site at the time t. Supposing that the ith site interacts with the jth site at the time t, the combined force due to internal tensile force measured by an internal observer sitting on the ith site turns out to be A F(t) = fij(t> + fji(t>. Newton’s third law (force equilibration) is obeyed if A F(t) vanishes. If the velocity of observation propagation is infinite [A F(t) = 01, an observer can measure both a position and its deviation at the same time. Then an observer

sees, at the time t,

(1) Next, let us suppose At is the time interval over which changes in the internal tensile force are propagated over the distance (A 1) between the ith and the jth site. Note that Al/At represents the velocity of observation propagation. In this case, the way the combined force A F due to internal tensile force vanishes depends on how local measurement and the genesis of the internal tensile force proceed. Therefore the force equilibration measured by an internal observer becomes

fij( t + At) + fji( t + At) = 0, and we can rewrite

(2)

this as

[ Lj(t)

+ Afij(

+ [&i(t) + A&i(t)]=O.

(3)

Note that we cannot uniquely determine Afij(t> and AfjJt). These terms depend on the genesis of internal tensile force within the time interval At. Let Afij = - ai A F, ALfji = - bj A F. We finally obtain the form AF=(a,+b,)AF,

(4)

Art$kial

[email protected]

where ai + bj = 1. In this case we can arbitrarily fix the values of Afij and Afji, as long as ai + bj = 1. In other words, we cannot uniquely describe the equation of the transition rule, such as fjj(t + At) = fij(t)+ Afij(t), due to the perpetual disequilibration (PD) [2, 31. We cannot avoid this dilemma. On the one hand, if one describes the conservation realization process proceeding in the time interval At, local description [in this case, fij(t + At) = is impossible. On the other hand, if one uniquely describes a fij(t) + Afij( local rule, he ignores the disequilibration process (nonlocal rule). We now observe objects that are organisms as a double observer: a local (internal) observer and a nonlocal (external) observer. However, the essential concept of perpetual disequilibration is not limited to the domain of classical mechanics. The dilemma of the double observer always appears when we cannot ignore the hierarchical structure and finite velocity of observation propagation of the internal observer. We have to discuss this in more detail. The concept of perpetual disequilibration involves the distinction between a phenomenon (e.g. force equilibration) and the process to realize the phenomenon [e.g. fjj(t + Lt) = fij(t>+ Afij(t In the discussion above, the term finite velocity of observation propagation implies not only the velocity of internal tensile force (vi, the process to realize the phenomenon) but the velocity of force equilibration propagation (ol),, the phenomenon). These velocities correspond to those of internal processes (lower-level dynamics) and external processes (upper-level dynamics), and those of local and nonlocal processes. Note that even if the velocity of observation propagation is finite, if vi > o, there is no conflict between local and nonlocal description. In the context of the relation between ui and o, let us examine the concept of perpetual disequilibrium. Figure l(a) shows a schematic diagram of the relationship between vi and v,. It is assumed that time and space (one-dimensional cell configurations) are discrete, and that the states of cells are also discrete. Black cells contain a particle, and white cells are empty. Upper-level dynamics (i.e. external process) involves the velocity of a particle (v,). In this case v, = 2. Note that if we describe the motion of a particle by the transition rule with the radius of interaction r = 2, the velocity of observation propagation of an internal observer is also 2, because an internal observer observes both the nearest and the next-nearest neighbors. On the other hand, imagine the case in which v, = 2 and vi = 1 (note that vi < v,). Although a particle has to be propagated at a velocity of 2, an internal observer can propagate a particle at a velocity of 1 at most. In order to propagate a particle at velocity ve, the interaction or the transition rule of an internal observer has to be perpetually changed. Intrinsically vi represents the velocity of lower-level reaction. So long as Vi > v,, we can precisely describe the rule for a particle in terms of the

276

YUKIO GUNJI AND NOR10 ICON-NO

A

FIG. 1. propagation internal

(a) is a schematic

observer

is denoted

(black cell)] is represented that the concept

process,

(lower-level

or intracellular

vi and 0,. Here

between

and a nonlocal (external)

by solid line. These are expressed disequilibration

the velocity of observation observer.

process).

the velocity

by the configuration

The velocity of an

observer

[here a particle

by the symbols

oi and o<. Note

implies the distinction between

here the motion of a particle)

oi represents

and v, represents

observer

by broken line, and that of an external

of perpetual

(upper-level

expressed

diagram for the relationship

of a local (internal)

and the process

the phenomenon

to realize the phenomenon

(b) and (c) show various cases of the relationship

the velocity of observation needed

propagation

to realize the conservation

between

of the internal observer,

law. The conservation

law is

of cells: (left cell, right cell) = (black, white) or (white, black).

(b) oi = m, (c) oi = u, = 1, Cd) ui < 1 an d v, = 1. Only when ui < up is the dilemma between internal

(local)

description

and an external

is the central

concept

(nonlocal)

description

of perpetual

found.

disequilibration.

Such

indeterminacy

See text for more

an

in local detailed

discussion.

velocity v,. Th is means that we can neglect the effect of lower-level dynamics. If oi < o,, we are faced with the effect of hierarchical structure. Figure I(b)-(d) show various cases of the relationship between ui and w,. Here the whole space consists of two cells. The state space of cells is {IO}. A black cell represents 1, and a white cell represents 0. The conservation law is expressed by the mass balance of 1 and 0 [i.e., the configuration is (IO) or (0, l)]; ui represents the velocity of observation propagation (i.e. the radii of interaction of an internal observer), and u, represents the process velocity required to realize the conservation law (note that this is not the velocity of a particle; but if you define ne as the velocity of a particle, the essence of this discussion is not different). If vi = ~0 (an internal observer measures its nearest neighbor at an instant; hence ui = l/O), there is always a conservation law at each time step [Figure I(b)]. If ui = u, = 1, we can describe a

Ar#cial

277

Life

local rule with a velocity velocity

oi of 1 to satisfy the conservative

o, of 1, as follows: if (my cell, neighbor)

= (l,O),

law with the

then my cell = 1,

and if (my cell, neighbor) = (0, l), then my cell = 0 [Figure l(c)]. The third case is the circumstance of living systems [Figure l(d)]. If ui < 1 and v, = 1, there are two possible law with v, = 1, namely (left cell, internal observer must not observe interval At, because vi < 1. Hence way to change the transition rule aspect an intrinsic

one-to-many

configurations to satisfy the conservative right cell) = (0,l) or (IO). However, an the state of his neighbor within the time an internal observer cannot decide the of an internal observer. We find in this

mapping in local description.

Finally, perpetual disequilibration should be interpreted as a dilemma between local and nonlocal observers, not only in classical mechanics, but also in all processes in which we cannot neglect the velocity of internal processes. Generally, researchers on cellular automata take the finite velocity of observation propagation into consideration; however, they deal only with the special case vi > v, (which will be discussed in the next section). The problem of the one-to-many mapping resulting from PD is not solved in physics. It is isomorphic to the problem of unprogrammable systems (consistent but incomplete formal languages) [q-II]. One-to-many mapping implies indeterminacy in a priori decision, but that one of the possible paths is chosen

a posteriori.

This is a central problem

of theoretical

biology.

2.2.

Local and Nonlocal Observer Perpetual disequilibration was ignored in the Newtonian paradigm, even in the paradigm for disequilibrium systems. Classical mechanics consists of three major laws: (1) the hypothesis of inertia, (2) the equation of motion, and (3) the force equilibration or conservation law. In this paradigm, by the

assumption of inertia, we can define which means that we can control the the distribution of the state [16]. We operator and can introduce the first through the hidden assumption of the gation.

the constant state at any time slice, boundary condition corresponding to can distinguish the operand from the and the second law independently infinite velocity of observation propa-

Once the infinite velocity of observation propagation is admitted, there is no contradiction between the local and nonlocal observers. The local observer is indicated by the description of the rule of the system at local sites, and is defined as the force at a site (Newton’s second law) in classical mechanics. On the other hand, we find nonlocal observers who describe global or nonlocal rules of the system in Newton’s third law. Supposing an infinite velocity of observation propagation, we can simultaneously observe all local points in the whole space. Hence the local rule can be defined by a one-to-one mapping. The third law can be obtained only by the integration of

278

YUKIO GUNJI AND NOR10 KON-NO

the local rule over the whole space, and the second law can be obtained

by

differentiation of a nonlocal function (e.g. a potential function). This is the reason why there is no contradiction between local and nonlocal observers in classical mechanics [9, 111. As examples of nonreversible systems, consider the model for finite cellular automata. These systems involve spatial interactions, supposing that time, space, and state values are all discrete. They include the finite velocity of the observation propagation in the form Ar/At, where Ar represents the radius of the interaction and At represents the time interval between time steps. However, they are constructed under a condition where no conservation law and no indication of force equilibration can be found. The local rules of nonreversible systems are formalized, neglecting the realization process of the force equilibration (which is just PD); hence contradiction between local and nonlocal observers cannot happen. In order to describe physical systems in terms cellular automata, invertible (reversible) cellular automata have been proposed (e.g. the Fredkin construction [17], the Margolus neighborhood [IS]). In these systems, local rules themselves are invertible; then force equilibration is always satisfied. Hence there are some conserved quantities. It is remarked that this equilibration process does not reflect perpetual disequilibration. Both in the Margolus neighborhood and in the Fredkin construction, the velocity of observation propagation is the same as the velocity of a particle (ui = u,); however, it is finite. Hence it allows only arbitrary punctuated equilibration, not perpetual disequilibration. This will become clearer in later sections, in connection with our definition of microscopic reversibility. In the process of PD there is only a change in the local force to realize the equilibration; however, we cannot find the virtual force equilibration in any time slice. As a result the deviation from the equilibration increases with the passage of time. The degree of disequilibration also increases, paradoxically, in spite of the process for realizing force equilibration. That is why the process is called perpetual disequilibration. Gunji [19, 201 introduced asynchronous updating for finite cellular automata, motivated by the concept of PD. Due to logical contradiction between local and nonlocal observers, the internal states of cells are different, even when all the external states observed by external observers are the same. Then the internal clock controlling the updating order in the whole space was constructed, and cells were programmed to be operated asynchronously according to the internal clock. However, we have to program the internal clock in order to simulate the time evolution. Hence even in asynchronous automata we cannot express the unprogrammability deduced from PD, because in those models evolution of the whole system (upper-level dynamics) can be described perfectly by lower-level dynamics. The funda-

279

Arti$cial Life mental feature of the unprogrammability

is the significance

of the one-to-many

mapping. It can be expressed only when we require both external and internal observers at the same time. No mechanics is free from the problem arising from PD, insofar as operands are separated from operators. PD is immediately related to evolution, which means decrease of the internal degrees of freedom (or increase of the constraints) in systems [3]. Evolution originates from both the existence of one-to-many mappings (a priori indeterminacy) and endogeneous degeneration [2, 31 (a poster-hi choice), which appear with the introduction of PD. 2.3.

Description a priori and a posteriori What is unprogrammability? Insofar as we accept unprogrammability, do we have to give up description itself? This is not necessary when we accept a reverse-time description. The reason why we cannot describe any local rules to describe unprogrammable systems results from the attributes of the Newtonian paradigm: we have to describe only a priori because our purpose is prediction. Uncontrollability appears only when we confine ourselves to forward time. The state in the future can be described a posteriori, because one-to-many possible paths are chosen as a result under the constraint of force equilibration. The uncontrollability

can now be replaced

by description

a posteriori.

For

we deal with the functions f :A + B where Vu E A, 3b E B such that f(a) = b, uncontrollable local rules are rewritten by uncontrollable microscopic boundary conditions (Figure 2). Consider a one-to-many mapping

A + A, where Vu E A, 3b,

c, or d E A. Unpredictability with respect to the various results b, c, or d can be attributed to the different microscopic boundary conditions. Then we reexpress the time evolution as (a, Ei> + b,

(a,t2) + c, (a, (J -+ d, where (i, (s, and [a represent the boundary conditions. We cannot in general describe these boundary conditions a priori, but can a posteriori (Figure 2). If we can describe them a priori, we do not obtain a one-to-many mapping and do not have to introduce the backward time. The formalization for the selection of boundary conditions a posteriori will be defined in detail in later sections. Since we find some indications of accumulations of the uncontrollable microscopic boundary conditions in the macroscopic boundary condition, we can no longer control the macroscopic boundary a priori. On the contrary, we can classify the dynamics with respect to the extent of controllability of boundary conditions. Here the force equilibration process attributed to the realization of the reversibility can be replaced by the extent of controlling the boundary. If the distribution of state values or the configuration is constant through time, the specific equilibration is realized at each time step. For this

YUKIO GUNJI AND NOR10 KON-NO

280

A u 1

I

c

b FIG 2. mapping boundary

Schematic

diagram showing the uncontrollable

(a) is translated condition

d

ti

into the function

cannot

be decided

a

priori;

however

a U 5 = g(b) we can decide ti (I posteriori(b). The criterion founded on the concept

of perpetual

boundary

with an uncontrollable

condition. A one-to-many boundary

from both

to construct

condition.

b = f(a

The

U 5) and

the reverse function is

disequilibration.

dynamics class, the local rule f can be described as the identity function, which means that the controllable microscopic boundary condition and the macroscopic boundary condition can be chosen arbitrarily. We call this class of dynamics constant. The second class is called microscopically reversible. The local rule f is described by a one-to-one mapping, but it is no longer the identity function. Whichever time-step pair (t, t + 1) is arbitrarily chosen, both micro- and macroscopic reversibility can be realized, because the local inverse function exists. In other words, reversibility is independent of the macroscopic f-’ boundary condition. Classical mechanics and the reversible automata [17, 18, 211 belong to this class. Because the boundary condition is controllable, we do not have to distinguish description a priori and a posterior-i. In the third class reversibility is dependent on the macroscopic boundary

condition. Now the local rule f is not invertible. Consider the nonlocal map from one configuration to another or between whole spatial distributions from step to step, and denote it by f. If f is surjective, we can describe the reverse function g, adding a suitable boundary condition a posterior-i. Reversibility can be realized only when an adjustable macroscopic boundary

Arti$cial Lafe

281

condition is chosen a posterior+. In other words, uncontrollable microscopic boundary conditions can be perfectly well replaced by the uncontrollable macroscopic boundary condition. We understand that such control of boundary conditions a posteriori can be another interpretation for fluctuations in equilibrium systems, which are ordinarily interpreted as random. We of course recognize that there exists such fluctuation, which can be observed as the Brownian motion. However, the statistical fluctuation is not real, but just an interpretation for a real fluctuation. Then we call the class of dynamics in which we have to introduce only macroscopic boundary conditions equilibrium or macroscopically rez;ersible. The fourth class is the collection of dynamics which we can no longer describe by macroscopic uncontrollable boundary conditions alone. We call them disequilibrium systems in that, even if reverse functions g and macroscopic boundary conditions are selected a posterior-i, the system is nonreversible. In order to realize reversibility we newly control the microscopic boundary condition a posteriori, by the perpetual transformation of the local rule f a posteriori with the passage of time. Controlling macroscopic or microscopic boundary conditions a posterior-i is equivalent to an interpretation of perpetual disequilibration. We call these interpretations macroscopic and microscopic PD respectively.

3.

AUTONOMOUS

BOUNDARY

In this section we give the formal definition for perpetual disequilibration. First, in order to give the criteria by which reversibility is estimated, we introduce the metric and the distance between state values. When we deal with only finite elementary cellular automata 1231 here, the distance is defined by the extent of similarity between cell configurations. Secondly, we formalize microscopic and macroscopic PD (perpetual disequilibration). By this definition, systems can be classified with respect to the controllability of boundary conditions. 3.1. Basic Dejinition Here we concentrate concept can be extended

on elementary cellular automata. easily to arbitrary cellular automata.

DEFINITION3.1 (Class of rules).

Let

However,

our

282

YUKIO GUNJI AND NOR10 KON-NO

Thus fRl equals the class of legal symmetric rules defined by Wolfram [22]. The transition rule f is described in general as CZ:” = f(u:_ i, u:, a:+‘).

DEFINITION3.2 (Distance d(f,g)

=

between

c

vf,

g E 8 (5)

d,(f(~,b,c>,g(~~b~c))

a,b,cs

where

rules).

Cl

dH(~,y)=~~C=l]~i-yi]

t/x,yEfln,

~=(ri,...,r,,),

y=(Yi,...,Y,,),

and R = (0, l}. This definition

for the metric

is that for Hamming

distance.

As well as

distance between rules, we can clearly define distance between configurations. Then we can give the criterion for the identification of configurations.

DEFINITION3.3 (Equivalence). equivalent

For

7,~’

E R,

= {0, l}N and d,,

77 is

to 7’ if dH(77,77’) = 0.

(6)

3.2.

The Formalization for Perpetual Disequilibration In this section we define the formalization of perpetual disequilibration, which is divided into two criteria with respect to the scale of the measurement. Let W =(j:flNU8RN

+ a,}

and 8;’

= {g : fi,

U da,

+ firv U ail,}.

DEFINITION3.4 (Macroscopic ration 77 E Q,, follows: Select

PD). For a rule f E % and each configumacroscopic PD (with respect to f, 77, and d,) is defined as an inverse function gf* E ‘8 gf = {g)“, . . . , gjmCf))} and some

boundary conditions [* E aR, = {0, l}“, p E aKIN= {0,1)’ such that t*> and l* give the minimum of the following collection of distances: (dH( &Y&77 where

u 5) u 6% 77U6):lBk

Z>&~fl&

gf*,

(7)

J‘ E 8

fiN u a%,

is the operation by the local rule f over the configuration and gfck) E %-’ denotes an inverse function of f.

The detailed construction of gjk’ E !Rg will Define %71,= UfE881g,. appear in Section 4. The boundary condition 5 is that at the tth step, and [ is that at the (t + l)th step. Due to the macroscopic PD, we can decide the pair that best

Artijkial

tife

Macros. Frc. process

3.

Diagram

of equilibration

Micros.

PD

for macroscopic is interpreted

the evolution of configurations.

satisfies the condition

and microscopic as the process

perpetual

PD

disequilibrntion

to realize the reversibility

(PD).

The

with respect

to

See text for further discussions.

(7) a posteriori. 6;

Such a pair is expressed

= (‘$&*)

We use the diagram shown in Figure

by

E{%}“. 3(a) for the macroscopic

perpetual

disequilibration. As shown in the legend of Figure 3, 5 is selected in the form of the pair (5, 6). B ecause we have to compute the configuration at the (t + l)th step in order to decide the boundary condition at the tth step, it can be said that the boundaries are decided a posterior-i. It is remarked that can no longer have a definite boundary at any time slice for the system, that we must execute the perpetual disequilibration. Definite states definite boundaries are not at each time slice, but within the time interval. this sense time proceeds not from moment to moment, but from interval interval.

we so or In to

DEFINITION 3.5 (Microscopic PD). We always execute the microscopic perpetual disequilibration after the procedure of macroscopic perpetual disequilibration. Given a subspace %71,of %. For each rule f E % and each configuration 77 E R,, microscopic PD (with respect to f, 77, the distance d,, CRa, and the boundary condition t*), is defined as follows: Select g* E zF1 that gives the minimum of the following collection of distances:

284

YUKIO GUNJI AND NOR10 KON-NO

As well as the one for macroscopic PD, we use the diagram showing the microscopic PD as shown in Figure 3(b). f* is always selected in one unit procedure of PD. First we compute g(f(q

U l)U

s’> starting

from the configuration

77 U 5, and select the bound-

ary pair (t*, e) satisfying the macroscopic PD. Then we determine the configuration g*(j(f‘(rl U [*>U p> as the definite configuration at the t-th step. Secondly, the microscopic PD is executed; however f* is not applied in the time interval between the tth and the (t + 1)th step. This is all for the unit procedure of PD. In the next unit procedure, we start from the configuration J‘(T U e*)U t*, which is the preliminary configuration at the (t + I)th step. After the macroscopic PD the definite configuration for the (t + 0th step is decided as the form of f*(g,-.(f(v U e*)U @)U (‘*), where g,-* represents the reverse fknction corresponding to f *. Note that the distance between configurations cannot be equal to 0 in any system, for any selection of (t*, p), g’*, and compared to the solution for the configuration language.

In some systems

requiring

f *. A

both macro-

distance 0 might be originating in the specific and microscopic

PD, the

contradictions between rules and configurations (here expressed by d, f 0) are by no means resolved, and then these contradictions are continued and transformed step by step. The perpetual prolongation of the resolution self-contradiction would bring forth the perpetual transformation system, which is a basic feature of an evolutionary system. 3.3.

The Classijkation of Boundaries Because

for the of the

of the System with Respect to the Controllability

we now give the definition

for the transformation

of the rule f

through the time evolution, we can no longer define the system as a unique dynamics or rule. The system is defined as a subclass of the rule class by the following.

DEFINITION3.6 (The system). where

8 18,

Systems

and R,

The system is defined as (Rts,fiRN,afiN),

= (0, l}N, an,

are sometimes

= {O,l)‘.

denoted as !R8 for brevity.

DEFINITION3.7 (Constant

system).

8 s is constant

if there exists f E !R s

such that &,(~(~W,$=O for any 77 E 0,

and e E aa,.

(9)

285

[email protected] Lijk A constant

system

always

falls

into

the

constant

state

owing

to the

existence of the rule f satisfying the condition (9), if it is started from any initial rule and any initial configuration. For example, the system including the rule

0 is the constant

010 1

001 0

000

100 0

011 1

system where each transition

101 0 is

110 1

111 1

a:, a:+ i) + af”.

DEFINITION 3.8 (Microscopically reversible system). s2, is microscopically reversible if there exists f E Si, such that Vn E flN and Ve E {aKt)2

where

for f E SS,

g E % = {f: (0, 1}3 + {O, 1)) and

whole configuration (over 77). Similarly, LR, is microscopically such that Vn E R,

reversible

g

operates

over

the

for !JIg if there exists f E ?ll T

and Ve E {a0)2

(11) where for f E ‘RS, 3gck’ E 8,.

COROLLARY 3.9. (%,R,,aR,) microscopically reversible system.

is not only a constant

system

but a

PROOF. We have to prove the existence of a function f satisfying Equation (11) in %. It is sufficient to find f such that Vn E {O, 1)3, 35 E an, and 3F E ZRr E {F : {O, 1j3 -+ {O, 113} such that

F(T) =f(77) u 5, Then we express

F-i

where

F is bijective.

E {F : (0, 1}3 -+ (0, 1}3) so that

VrlEf(77)U5>

3~

such that

77 = F-‘(T).

In order to find such a rule f, we examine all rules in the rule class SF. The transition diagram shown in Figure 4 expresses the deterministic graph

YUKIO GUNJI AND NOR10 KON-NO

286

I

0

bc

FIG. 4. The transition diagram for a rule in the rule class 8, = (F :(O, 1y + (0, 1)3). Connecting two nodes by a directional edge shows the transition represented by the function F (e.g. (O,O, 1) + (a, b, c)). By using the diagram the microscopically reversible system can be found. See text.

for F. Then F is bijective if

a=b=d,

c = e,

dzc.

These relations satisfy the condition for which F is bijective,

Vv-/-{0>1)3> Hence we can constitute (i) a=b=d=l,

if 77 # T’, then

such that

# F(q’).

F(q)

F as the following two cases:

c=e=O:

F, : (O,O,0) + (O,O, O), (0, 1, 1) + (O,l, 0, (1, LO> + (1, 1,01,

(0,0, 1) -+ (0,0, 11,

(0,LO) --f (0, l,O),

(l,O, 0) -+ (LO, 01, (1, 1,l) + (1, 1,1x

(l,O, 1) -+ (LO, 0,

(O,O, 1) + (1, l,O), (LO, 0) + (0, 1,0, (1, 1,l) + (O,O, 0).

(0, LO> +

Cl,& 0,

C&O, 1) +

(0, LO),

(ii) a = b = d = 0, c = e = 1:

F2 : (O,O,0) + (1, 1,11, (0, 1, 1) + (LO, ok (1, 1,O)+ (o,o, 11,

Artificial L,ife

287

For these cases we can define f, and for such f we can constitute (i’)

(ii’)

fi:

000 0

001 0

010 1

011 1

100 0

101 0

110 1

111 1

g,:

000

001

010

011

100

101

110

111

0

0

1

1

0

0

1

1

000

001

010

011

100

101

110

111

1

1

0

0

1

1

0

0

000 1

001 1

010 0

011 0

100 1

101 1

110 0

111 0.

fi:

g,:

It is remarked that fi = gi, i = 1,2. Therefore (%, R,,aR,) cally reversible system. The rule fi is thus the rule example for the constant system.

DEFINITION

3.10 (Macroscopically

tally reversible if there 3gF E !Ri,, such that

exists

g:

is a microscopiillustrated by the

reversible

f E Bi,

system). Ri, is macroscopisuch that tlq E R,, 35* E {an}‘,

dH(gf*(J‘(~uV*)U5*),77u5*)=0. COROLLARY

microscopic

3.11. PD (i.e.

By the perpetual

lf f satisjies Equation (121, with only the iteration off ). PD, we describe

there

the configuration

is no process

at t = 0 as

(from Definition Because

f * = f, we can describe

where J‘(“) represents

k iterations

the configuration

of the operation

at t + 1 as

j.

(12)

3.10).

of

YUKIO GUNJI AND NOR10 KON-NO

288

TABLE 1 RELATIONSHIP BETWEENTHE SYSTEMSAND OF BOUNDARY

system

THE

CONTROLLABILITY

CONDITIONS

Macro-BC

Micro-BC

constant microscopic reversible

controllable controllable

macroscopic reversible

uncontrollable; if system is attracted to reversible local rule, controllable a posteriori uncontrollable

controllable uncontrollable; controllable a posteriori

disequilibrium

DEFINITION 3.12 (Disequilibrium librium

system if %,

system).

is not macroscopically

uncontrollable ___________ uncontrollable

The system

8,

is a disequi-

reversible.

We complete the classification of the system with respect to the controllability of the boundary conditions. Table 1 shows the relationship between the systems and the controllability of boundary conditions. Controllability means programmability with respect to a mathematical map. Hence uncontrollability involves indecision of boundary condition a priori, however there are two cases which reversibility is realized a posterior-i (controllable a posteriori) or not. Macroscopic boundary condition (Macro-B(Z) represents boundary condition in the sense of general physics, and microscopic boundary condition (Micro-BC) represents the decision of local rule. Note that macroscopic boundary condition is arisen from the results of disequilibration of local rule which has universality in whole space. See text for further discussion.

DEFINITION 3.13 (Fluctuation by perpetual disequilibration). The process of PD consists of both macroscopic and microscopic PD. Denote the configuration produced by T iterations of the process of PD as q* E R,, and the configuration produced by T iterations of the process of a macroscopitally reversible rule as 17: E a,. FPD (fluctuation by PD) is defined by

In general

it cannot be uniquely

decided.

Hence,

289

Arti$cial Life Under

these definitions,

we discuss

the concrete

St,-construction

in the

next section.

4.

CONSTRUCTION FOR %g (REVERSE AND THE SIMULATION RESULTS

I>

4.1

The Main Flow Diagram We have to construct the rule class !Ilg satisfying at least the microscopic and macroscopic reversibility defined in the above section. For this purpose we define rigorously how to construct the reverse function (in backward time) g, following the procedure proposed by Gunji and Nakamura [9, 111. Then we examine whether this !ltg satisfies Definitions 3.8 and 3.10.

DEFINITION 4.1. [The flow diagram (FD) for f E %I. For f E %, we can describe the transition rule from t + 1 to t as shown in Table 2,where di E (O,l}, i = 1,. . .) 8. For each element (d, -+ cyipiyi), we can construct the spatial transition rule by the following two nodes (box B,) and one directional edge (arrow):

TABLE 2

t+1

t

4 4

ooo 001

4

010

4

011

4

100

4

101

4

110

4

111

t’+1

17s

&fd+1

YUKIO GUNJI AND NOR10 KON-NO

290

In order to examine

the character

of the box B,, we introduce

the kin-pair

(KP) relation such that the pair (d,, d,), k z s, is KP for (ai,Pi> if ok = (Y, = oi and pk = /?, = pi. It is remarked that we are dealing with a symmetric

rule class. Hence

(I) If the KP (dk, d,)

we can classify the boxes B,, as follows:

for (ai, pi)

satisfies

d, = d,, then

X is uniquely

determined. (I)(l)

Similarly, if the KP (dk,, d,,) for (p,,r,) satisfies d,, = d,,, then Y is also uniquely determined. In this case B, has one unique descendant box B s+ 1. We call such a B, a doting mother box (DMB). (I)(Z) Suppose d,. z d,, has two descendent boxes:

\

Here we call B, a nor& (II)

If the

character

KP

(dk, d,)

mother box (NMB). for (czi,pi)

satisfies

d, # d,,

we decide

the

of B, for each case, X = 0 or X = 1.

(III) After completing the procedure for all elements (di + aipiyi. i = 1 , . . . ,8), we sometimes have the same B, in spite of having different di. In this case, B, has more than two descendants. We call such a B, a pro&

mother box (PMB). (IV) Because a box consists of four variables E (0, l}, there are 24 kinds of boxes. Thus we can find B, which has no descendant for some rule f E 8. Such a B, is called a sterile mother box (SMB). Each mother box is either an NMB, a DMB, an SMB, or a PMB. By connecting mother boxes of all types except for SMB, we can construct an FD for f E 8, From now on, following the notation due to Wolfram [22], the rule f E !I? is indicated by its rule number RN defined as RN = C&,ldi2i-1, and is denoted f aN.

Artificial

291

Life

These

FDs have to express the transition graph for g :{O, HNf” -+ IO, 1]N+2, which include both spatial (s * s + 1) and temporal (t + 1 + t) shifts. However, if we find a PMB in the FD, then the FD no longer expresses a function, because it involves a one-to-many mapping. Only FDs which consist of NMBs and DMBs express functions. Therefore we have to modify the FD in this case.

DEFINITION 4.2 (The modification

of the FD).

After completing

the FD

for f by the procedure defined in Definition 4.1, we can modify any PMB into an NMB by selecting just two boxes (B,,,). Then we have to compensate for the reduced number of boxes by transforming a DMB into an NMB, or an SMB into a DMB (the number of boxes after modification is equal to or greater than that before). Owing

to Definition

4.2, we can construct

g in the form of a function

[9, 111. Here we call the path (box and arrow) newly constructed by modification an accretion. Because g has the direction in space given by the spatial shift, once the path created by g becomes an accretion, it cannot be changed back to the original path. This is a statement which should be proven. 4.2.

Macro- and Microscopic Reversibility for FDs First we check whether there exist systems !Xs which are macroscopically reversible for 8, defined here. It is shown that the FDs for faO and fi5a consists only of NMB [g-11]. Here choosing the pair (t*, t*) is replaced by choosing the initial MB. If a possible MB has been tested by checking the condition (I2), then an acceptable pair can be found by various means. Therefore a system LRs which involves f, and/or fi,,, is macroscopically reversible. Due to the macroscopic PD, the macroscopic boundary condition is autonomously (a posteriori) decided. Because at each unit procedure between t and t + 1 we have f(q U (*)U e* = 77U (*, there is no microscopic PD. We can also confirm that a system !Rs including the rule fzo4 is not only constant but also microscopically reversible for ag, because we can construct an FD (shown in Figure 5) for f,,,, due to Definitions 4.1 and 4.2.

4.3. To

Disequilibrium fw FDs illustrate the behavior

of disequilibrium

systems,

we discuss

the

system 8 s = Cfra, faa, f,,]. Of course !Rt, does not include microscopic or macroscopic systems. The system is started from the initial local rule f,,. Even fis sometimes satisfies the condition for macroscopic reversibility;

YUKIO GUNJI AND NOR10 KON-NO

292

FIG. 5. R,

FD for fz04, constructed

including

f204

is a constant

by the procedure and

microscopically

defined in Definitions 4.1 and 4.2. Here reversible

system,

in the

sense

of

Definition 3.8.

however, most configurations cannot be reversible with respect to f,,. Hence the microscopic boundary is so uncontrollable that the local rule f is perpetually transformed. The behavior of this system is strongly dependent on the initial configuration and the choice of g’s (FDs) because these are according to Definitions 4.1 and 4.2, we can choose them arbitrarily). Figure 6(a) shows that the system is attracted to the local rule fs4, and on the other hand the system in many cases is not stabilized with respect to local rules [Figure 6(b)]. Note that any rules which are not reversible are one-to-many mappings due to the reverse function or the construction of the FDs. In particular, by Definition 4.2, in these FDs, DMBs have to be changed to NMBs, and SMBs to one or more NMBs. Therefore local transitions coexist which contradict the original local rule. We find one-to-many mappings in this situation. That is the reason why the local rule can be transformed.

5.

AUTOPOIESIS,

PD, AND UNCONTROLLABILITY

In this section we discuss the relationship between autopoiesis, proposed by Maturana and Varela [7, 121,and perpetual disequilibration. The system of autopoiesis has no concept of unprogrammability, and it emphasizes intrinsic or internal control, in spite of the example of Brownian algebra that is regarded as typical of autopoiesis. Hence tessellation automata [12, can be as exemplifying autopoiesis, in the boundary of system consists of and generated by interaction between components.

Artificial Life

293

a bc

FIG 6. The time evolution of the disequilibrium system !RS including f18, fiz, and fS4. A black square represents the state value 1, and a blank represents 0. These two patterns start from different local rules (A: initial local rule fz2; B: f,,). Zigzag curves accompanying the time evolution show the evolution of the transformation of local rules.

Our concept the paradox of as autopoiesis; his book titled construct the

of autonomy or self-organization [t&11] also originated from the private indication proposed by Spencer-Brown [S] as well however, we find unprogrammability in it. Spencer-Brown, in Laws of Form, started from the stance that it is possible to private language by oneself and completed the calculus of

indication in the form of a primary algebra. However, in the end he pointed out that the algebra involves self-contradiction because the form which contradicts the axiom is itself deduced from the axiom using rules of inference defined in primary algebra. Figure 7 shows the essence of his discussion. In this context, the boundary of the primary algebra (or the primary algebra itself) is not definite and cannot be separated from the outside, which means that one cannot arbitrarily define the boundary without interference with other subjects. In other words, the boundary of the private indication (the specific formal language) must be accompanied by others (contradiction to the axiom). The

YUKIO

infinite

GUNJI

AND

NOR10

KON-NO

operation

finite operation

/I axiom

I

Ii finite

form

value

infinite

,

form

_

CONTRADICTION (APPEARANCE OF

I

PRIMARY

OTHERS)

semantics

syntax

ALGEBRA (SUBJECT) .

FIG. 7. Schematic diagram showing the paradox of the private indication [8]. Although the private indication is described in the form of primary algebra, it cannot be operationally closed independently of the environment (if the infinite form is transformed into the finite form, it can contradict the axiom). Because the whole system includes self-contradiction. it also includes an autonomous boundary (thick line). It is remarked that the semantics of others cannot be positively described by a subject; it is indicated by the question mark. See text.

fact that the semantics of others is different from that of the subject (the owner of the specific language) can be recognized by a subject only when there is contradiction between the subject and others. Hence the boundary of the logic (consistent language) is founded on the environment of the logic, and can be described only when we describe a whole picture which consists of a consistent formal language and contradiction (the existence of others). For the sake of convenience, we now introduce the terms reentrance and autonomous boundary.

DEFINITION5.1 (Reentrance). the foundation of the definition definition is described. It is clear that the foundation

A definition is reentrant (or recursive) if is also in the language by which the of the definition

is generally

outside of the

language. There is no foundation on which one can distinguish the inside from the outside within the language. In a system including primary algebra and a generator for contradiction, we have a reentrant boundary. However,

295

Artificial Life

the whole system itself is not a consistent language, because the meaning (value) of the form (proposition) cannot be uniquely decided (Figure 7). Thus we find that reentrant definition can occur only in an inconsistent language.

DEFINITION 5.2 (Autonomous its definition is reentrant.

boundary).

A boundary

is autonomous

if

From this definition, the boundary between primary algebra and contradiction is autonomous. In contrast, there is no autonomous boundary within the consistent language. When you distinguish all true propositions from false propositions in a consistent language, the foundation for this distinction is outside the language. Now let us consider living things as an autonomous Varela [7], such a system this sense the foundation the autonomous system conclude that it cannot be

system. According

to

is defined as one that generates itself by itself. In of the boundary must be in the system. Therefore has to have an autonomous boundary, and we described in any consistent language. Hence, if we

describe the local rule for element interactions in the system, such a rule has to be a one-to-many mapping, which means that the system is unprogrammable. Inasmuch as the tessellation automata [7, 231 have no unprogrammability, they have no autonomous boundary, and so are incompatible with Varela’s own definition of an autonomous system. We state the following.

FACT 5.3. An autonomous boundary can be described only in an inconsistent language. Hence the outside and inside dejined by an autonomous boundary have different logical status in formal description. The meaning of autonomy cannot be separated from that of uncontrollability. Conrad 113, 141argued that unprogrammability is found in biological systems, arising from the interference between the intracellular and intercellular computations. Because of the difference of velocity of computations, intra- and intercellular computations have different logical status. Therefore we find that the unprogrammability proposed by Conrad is as same as ours. We say that the brain as neural networks has an autonomous boundary. Conrad and Matsuno [3, 241 reexpress this problem as the essential problem originating from the fact that the boundary condition is independent of the equation of motion. It has been suggested that the notion of autonomous boundary in an incomplete language may be a breakthrough in describing such biological systems, and the possibility of description a posteriori has been argued [9-111. The system proposed here is one of the possible ways to

YUKIO GUNJI AND NOR10 KON-NO

296

describe uncontrollable evolutionary systems. It has both microscopic and macroscopic autonomous boundaries, because they are unprogrammable. We here repeat that the uncontrollability can be described with the introduction of description a poster-b-i. Finally we discuss the relation between our system and evolutionary systems. In the sense that local rules can be transformed perpetually, disequilibrium systems as proposed here can be called evolutionary. However, not all systems in our paradigm are evolutionary systems. This corresponds to the fact that all natural systems are of course not evolutionary. We find both biological disequilibrium systems and “physical” equilibrium. Whether a system evolves or not is dependent on its environment. In establishing the relationship between our systems and natural systems, !Ri, (the set of local rules) should depend on the environment. However, the dynamics in our systems are not initially obvious or definite, due to PD. It is the property of indeterminacy in description that is expressed and emphasized here. Great events in environments are not described here without the transformation of XIIS itself. However, our system is much more sensitive to environmental change than other systems. Even if environments are gradually and slowly changed, the behavior of the system can be drastically changed due to the microscopic PD. What we emphasize in our system is that such a sensitivity is described only a posterior-i. Therefore it is suggested that an evolutionary system has some essential indeterminacy in description. Our mode1 proposed here, introducing forward and backward dynamics, is the first attempt to express one-to-many biological indeterminacy principle.

6.

mapping positively and to discover a

CONCLUSION

The evolution of a system may be viewed as an increase in the internal constraint of the system. For this the system must be open to the outside. However, we observers have to observe systems from the outside and then have to describe them definitely. In differential equations or cellular automata including the interactions between spatial compartments or particles, the indefiniteness or fuzziness in description is replaced by the uncontrollable boundary conditions. The uncontrollable boundary conditions result in perpetual disequilibration owing to the finite velocity of observation propagation. Hence in the macroscopic and microscopic boundary conditions we find one-to-many mapping. Admitting one-to-many mapping means that we have to describe systems in a bad (inconsistent) language. Bad languages include self-contradiction originating from the misidentification of syntax with semantics, or

Artijicial Life

297

from the mixture of the propositions with their own foundation. One-to-many mappings in forward time are only another expression of autonomous boundaries in bad languages. Such indeterminate aspects can be described with respect to the relationship between the configuration at the tth step and the (t + 1)th step. In this context we have proposed here the autopoietic boundary, which cannot be predicted by an external observer. Since we have to know the configuration at both the tth and the (t + 0th step in order to decide the boundary condition at the tth step, we must not describe any time slices definitely, and we cannot control boundary condi-

tions externally. Therefore we find the equations or local rules themselves as the representations resulting from the stabilizing process, which is the main significance of the model of autopoietic boundaries. We have to reinterpret the models for living things or self-organizing systems in this sense. This paper is the first attempt at such research. We thank Professor M. Conrad very much for deep discussions and careful reading of manuscripts. We also thank Professor K Ito and T. Nakamura for discussions on various topics.

REFERENCES 1 2

3 4 5 6 7 8 9 10

11

12

E. Schrijdinger, What Is Life?, Cambridge U.P., 1945. K. Matsuno. Cell motility: An interplay between local and non-local measurement, Biosystems 22:117-126 (1989). K. Matsuno, Protobiology: Physical Basis of Biology, CRC Press, New York, 1989. H. Haken, Synergetics, 2nd ed., Springer-Verlag, New York, 1978. H. Haken, Information and Self-Organization. Springer-Verlag, New York, 1988. P. Gransdorff and I. Prigogine, Thermodynamic Theo y of Structure, Stability and Fluctuations, Wiley-Interscience, New York, 1971. F. J. Varela, Principles of Biological Autonomy, North-Holland, New York, 1979. G. Spencer-Brown, Laws of Form, George Allen & Unwin, London, 1969. Y. Gunji, Autopoiesis and two kinds of time streams, J. SocioE. Biol. Struct., submitted for publication. Y. Gunji and T. Nakamura, Nondefinite form for autopoiesis, illustrating automata in the forward and backward time, IEEE Trans. EMBS 12(4):1778-1780 (1990). Y. Gunji and T. Nakamura, Time reverse automata pattern generated by Spencer-Brown’s modulator: Invertibility based on the autopoiesis, Biosystems, to appear. U. Maturana and F. J. Varela, Autopoiesis & Cognition-The Realization of the Lioing, Reidel, London, 1973.

YUKIO

298 13

M. Conrad,

Microscopic-macroscopic

ing, Biosystems 14

M. Conrad,

16:345-363

GUNJI

interface

AND NOR10

KON-NO

in biological information

process-

(1984).

The price of programmability,

in The Universal

Turing Machine, A

Half-Century Survey (R. Herken, Ed.), 1988. 15

R. Rosen, Organisms

as causal systems which are not mechanisms:

the nature of complexity,

An essay into

in Theoretical Biology and Complexity (R. Rosen, Ed.),

1985. 16

K.

Matsuno,

79:269-285 17

E.

From

physics

to biology

and

back,

Riu. Biol. (Biol.

Forum)

(1986).

Fredkin

21:219-253

and

T.

Toffoli,

Conservative

logic,

Internat. J. Theoret.

Phys.

(1982).

18

N. Margolus,

19

Y. Gunji,

Physics-like

Molluscan

intrinsic time, illustrating

Physica lOD:Bl-95

models of computation,

pigment

pattern

subjective

generation autonomy,

by a dynamic

(1984).

structure

with

in Dynamical Structure in Biol-

ogy (B. C. Goodwin, A. Sibatani, and G. Webster,

Eds.),

1989.

20

Y. Gunji, Pigment color patterns of mollusks as an autonomous

21

Biosystems 23:317-334 (1990). T. Toffoli and N. Margolus, Cellular Automata Machines, MIT Press, Cambridge, by asynchronous

process generated

automata,

Mass., 1987.

23

S. Wolfram, Statistical mechanics of cellular automata, Reu. Modern Phys. 55(3):601-644 (1983). M. Zeleny, Autopoiesis: A Theory of Living Organization, North-Holland, New

24

York, 1981. M. Conrad

and K. Matsuno,

universality

of differential

22

The boundary

equations,

condition

paradox:

A limit to the

AppZ. Math. Comput. 37:67-74

(1990).

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