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Kobe University Repository : Kernel

TitleAbstract heterarchy: Time/state-scale re-entrant form

Author(s)Gunji, Yukio-Pegio / Sasai, Kazauto / Wakisaka, Sohei

CitationBiosystems, 91(1): 13-33

Issue date2008-01

Resource TypeJournal Article / 学術雑誌論文

Resource Versionauthor

URL http://www.lib.kobe-u.ac.jp/handle_kernel/90001516

Create Date: 2012-01-05

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Gunji, Y.-P., Sasai, K. and Wakisaka, S. (2008)

Abstract heterarchy: Time/ state-scale re-entrant form

Biosystems,91(1), 13-33

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Abstract Heterarchy:

Time/State-Scale Re-entrant Form

Yukio-Pegio GunjiYukio-Pegio Gunji1,2

, Kazauto Sasai1 1 & Sohei Wakisaka

Abstract Heterarchy:

Time/State-Scale Re-entrant Form

1,2, Kazauto Sasai & Sohei Wakisaka1 1

1. Department of Earth & Planetary Sciences, Faculty of Science, Kobe University 1. Department of Earth & Planetary Sciences, Faculty of Science, Kobe University

2. 2. Graduate School of Science and Technology, Kobe University Graduate School of Science and Technology, Kobe University

yukio@kobe-u.ac.jpyukio@kobe-u.ac.jp

Abstract Abstract

A heterarchy is a dynamical hierarchical system inheriting logical

inconsistencies between levels. Because of these inconsistencies, it is very difficult to

formalize a heterarchy as a dynamical system. Here, the essence of a heterarchy is

proposed as a pair of the property of self-reference and the property of a frame problem

interacting with each other. The coupling of them embodies a one-ity inheriting logical

inconsistency. The property of self-reference and a frame problem are defined in terms

of logical operations, and are replaced by two kinds of dynamical system, temporal

dynamics and state-scale dynamics derived from the same “liar statement”. A modified

tent map serving as the temporal dynamics is twisted and coupled with a tent map

serving as the state-scale dynamics, and this results in a discontinuous self-similar

map as a dynamical system. This reveals that the state-scale and temporal dynamics

attribute to the system, and shows both robust and emergent behaviors.

Key Words: Heterarchy, Chaotic liar, Self-reference, Frame problem, Self-similar map

1. 1. Introduction Introduction

What new ideas have been arising in the field of complex systems? Recently,

the difference between robustness and stability has been addressed with the aim of

accessing a new notion of complexity that is beyond the purview of stability theory. Jen

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(2003) claims, “The concept of stability is regarded as an old one that derives from the

study of the stability of the solar system. Although both stability and robustness are

concepts appropriate to measuring feature persistence, only robustness is relevant for

the interplay between dynamics and organization, the role of future, the anticipation of

insults, along with other questions even more difficult to formulate relating to

creativity, intentionality, and identity”. The key notion is that of a heterarchy

(McCulloch, 1945; Keaher and von Goldammer, 1988; Stark, 1999), which is an

interconnected, overlapping, hierarchical network constituted by individual

components simultaneously belonging to and acting in multiple networks, with the

overall dynamics of the system both emerging and governing interactions of these

components. In human society, individuals act simultaneously as members of familial,

political, and economic groups among others, and this is an example of a heterarchy.

Biological signaling processes (Marder and Calabrese, 1996), evolutionary systems

(Shapiro, 2002; Voigt et al., 2004) and computation in engineering (Cantwell-Smith,

2002; Gunji and Kamiura, 2004) also yield examples of a heterarchy.

In heterarchical systems robustness may exist on the level of individuals, on

an intermediate level, or on the level of the whole system. Conversely, robustness at

one level confers a degree of robustness on any of the other levels. Through the

interplay between dynamics and organization, emergent levels and/or components of

networks can be created. Such interplays cannot be described in advance, and it is

possible to define creativity only when one cannot describe all components playing an

essential role in maintaining a system in advance. Imagine that an observer is

convinced that a hierarchical system is perfectly described by finite numbers of explicit

levels. The system, however, has a hidden level that is not described in advance, and

the hidden level affects the system such as to reorganize it and create a new explicit

component. In this instance, it is possible for an observer to recognize creativity. This

is why an essential property of heterarchy is latency of the environment.

There is an essential difficulty in evaluating the property of a heterarchy,

because of the difficulty in measuring feature persistence. When a system persists as a

unity, the essential cause of the persistence perpetually changes. In other words, even

if an observer can find the essential cause of the persistence at one moment, another

cause will appear at the next moment. This entails that an observer must keep on

describing levels and components constituting the heterarchy. Beyond undertaking

such an enumeration, one has to secure an alternative approach to understanding a

heterarchy.

One of the possible ways to understand a heterarchy is to construct an

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abstract model featuring latency of the environment. Formalization of latency of the

environment, the outside of the formal description, or the reservations of the formal

description is required. On the one hand, an observer has to recognize his own

limitations with respect to descriptions, and on the other hand, he has to refer to the

outside of the description. Although such perspectives have been previously proposed

under the name of internal measurement (Matsuno, 1989; Gunji, 1994, 1995) and/or

endophysics (Rössler, 1988, 1994) some of such models (Gunji et al., 1997, 2001) were

not actually needed in the field of complex systems till the notion of a heterarchy

needed to be explicitly addressed.

How one can construct an abstract model for describing a heterarchy? This

can be illustrated by considering computation in the brain. When computation is

executed in a particular environment, computation by the brain can be considered as

being analogous to computation by a computer. For example, one can see a pen as the

object, “pen”, when neurons in Wernicke and/or Brocca’s area relevant for linguistic

comprehension are firing in the brain. If neural activities of the area are regarded as

constituting the computation, it is possible to see that neural activities in any other

areas of the brain provide a particular environment in which the computation is

executed. Such environmental neural activities are also computation. One can see two

levels of computation, an explicit piece of computation in the Wernicke’s area and the

overall environmental computation. It is important to see that the environmental

computation cannot be limited to a finite region, and that the environment is destined

to be indefinite. That is why computation by the brain yields a typical example of a

heterarchy. If one attempts to describe the entirety of a system consisting of both

explicit and environmental computations, one tends to describe a phenomenal aspect

(e.g., Tye, 1995). In order to progress beyond such phenomenological descriptions, one

has to formalize the indefinite and latent environment by using weakly described

self-reference (Gunji et al., 1995, 2004).

Here, we propose a model for describing the essence of a heterarchy in the

following way. Firstly, we address what a heterarchy is, and reveal that the essence of

a heterarchy is robust behavior against logical collapse. Next, robust behavior against

logical collapse is formalized by self-reference connected by a frame problem. Although

both self-reference and the frame problem reveal a kind of infinite regression, there is

a difference with respect to logical status. A dynamical system based on self-reference

has been developed by Grim et al. (1994). We develop their ideas, and the self-reference

with a latent open environment (or with the frame problem) is expressed as the twisted

coupling of two dynamics inheriting an infinite regression. We finally show how such a

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system reveals robust behavior and discuss the relationship between a heterarchy and

robustness.

2. 2. Heterarchy Heterarchy

A heterarchy is a dynamical hierarchical system in which an action at one level

simultaneously reveals reactions at other levels. The significance of a heterarchy is

manifested with respect to the difference between stability and robustness (Jen, 2003).

First, we start by describing a heterarchy of human beings. The allegory helps us to

understand the essence of a heterarchy.

A man is not only a member of his family, but also a member of the company

in which he is employed. His actions, therefore, affect both his family and company,

simultaneously. Imagine if he goes to work on his day off. Although such an action

would benefit his company, it would be detrimental to his family. Listening to this, you

might think that it satisfies the definition of a heterarchy, i.e., simultaneous

interaction among different levels. You should, however, notice that such simultaneous

interaction results just from a hidden specific operation such that detrimental (or

beneficial) to the man’s family is mapped to beneficial (or detrimental) to the company.

One cannot recognize “simultaneousness” in his actions until one comprehends both

the independence of the two levels (family and company) and their simultaneous

interaction. Because of the independence of the two levels, one must take all possible

operations between two levels into consideration. Moreover, one has to focus on the

process of choosing one operation. Now, let us define a set of values for the family and

the company as S = {0(detrimental), 1(beneficial)}. We call all possible operations from

the family to the company “Interpretations” -0, -1, -2, and -3. These interpretations are

defined as follows.

Interpretation-0: 0→0; 1→0, Interpretation-1: 0→1; 1→1,

Interpretation-2: 0→0; 1→1, Interpretation-3: 0→1; 1→0.

An observer has to describe the man’s action of going to work on his day off as a

simultaneous process of choosing one interpretation. What is a simultaneous process?

A chosen interpretation has a value of S. The situation is, actually, described by the

following.

On his day off, a man decides to go to work and puts on his shoes in the

hallway of his home, where his son and wife, who were expecting to go to the zoo with

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him, are disappointed. The husband hesitates as to whether to go work, and considers

that going to work would be detrimental to his family but beneficial to the company.

This consideration (i.e., choosing Interpretation-3) proceeds within a finite time in the

hallway. Therefore, such a process itself can have the value of S, in a family.

Meanwhile, the man’s wife begins to feel that her husband feels bad about leaving

them to go to work, and thinks that her own attitude is making him feel too guilty. She

thinks that she should send her husband off to work with a smile. Finally, she decides

to hide her disappointment and wishes him farewell with a smile.

The husband hesitates in the hallway. This means that choosing an

interpretation makes sense even in a family, and this triggers the emergence of a new

value, “smile though disappointed” within S. As a result, the value in a family changes

from {0, 1} to {0, 1, 2(smile though disappointed)} since choosing an interpretation that

proceeds within a finite time can be a new kind of interpretation.

Here, we generalize such a process as the following. A heterarchy is defined by

simultaneous interaction among a plurality of levels. This is replaced by the

simultaneous choice between intra-level dynamics and inter-level dynamics. In the

example of going to work, the intra-level dynamics is just a choice of a value of S (i.e., a

value of a particular level) and the inter-level dynamics is a choice of an interpretation

from Interpretations-0~3. The simultaneous choice is defined by two properties; (1) a

map property, and (2) a simultaneous making value. The map property is defined by

the following; for all elements of S, there exists an interpretation. The second property

is defined by the following; each possible chosen interpretation must have a value in a

level (e.g., family). The map property looks natural, however, it requires all possible

correspondences between elements of S and all interpretations. The simultaneous

making value is defined so as to expand such a stance.

Imagine that a map is defined by the following; 0→Interpretation-3, and 1→

Interpretation-1. Simultaneous choice requires that each interpretation is assigned a

value of S when the choice is made. For the choice, one can recognize that

Interpretation-1 has a value of 1 and Interpretation-3 has a value of 0, as soon as

either interpretation is chosen. However, the property of the simultaneous making

value requires a making value for all interpretations. Although Interpretations-0 and

-2 are not chosen, they also have to have values of S. Assume that Interpretation-0 has

a value of 0. If so, the map property collapses because a value of 0 is mapped to both

Interpretations-0 and -3. As a result, the map property and the simultaneous making

value have a trade-off relationship therebetween. If each level is defined by a set, S, a

set of the inter-level operations is defined by Hom(S, S), that is, a set of functions from

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S to S. Simultaneous choice is defined by; f : S→Hom(S, S) is a subjective map. The

map, f, has to cover all elements of Hom(S, S) (i.e., for all elements y of the co-domain

of f, there is an element x in the domain of f such that y=f(x)). Such a requirement is

bound to fail, in principle. The map cannot cover all elements of Hom(S, S) (Gunji,

1992).

However, even though simultaneous choice collapses, the heterarchy proceeds

as a real system. In this situation, one has to focus on the notion of a heterarchy as a

real system persisting against the collapse of the observer’s logical framework.

Remember the above example of a man going to work on his day off. A proceeding

motion against the collapse happens in that example. The appearance of an emergent

state “smile though disappointed” can be explained by an event occurring that proceeds

against the collapse. The situation of which choice of interpretation also makes sense

in a family is expressed as an assumption of a subjective map from S to Hom(S, S) (i.e.,

the map requires simultaneous choice). If one attempts to make a system satisfy

simultaneous choice in spite of the collapse of the assumption, one must find a new

source that is mapped to possible elements of Hom(S, S) out of S. In order to avoid

one-to-many mapping, a new source of an arrow is constructed out of {beneficial,

detrimental}. This is nothing but a new family state, such as “smile though

disappointed”. The collapse of the assumption termed simultaneous choice makes

re-organization of the system possible. This is the essence of a heterarchy;

maintenance of feature persistence against logical collapse.

3. 3. Formal notion of indefinite environments

Formal notion of indefinite environments

3-1. Self-reference and frame problem 3-1. Self-reference and frame problem

The essence of a heterarchy is proceeding motion against logical collapse. How

is it possible to describe such a phenomenon? Our answer is to couple self-reference

with a frame problem.

in the form of a subjective map of f : S→Hom(S, S). According to Lawvere (1969), this is

In the above section, it was shown that a heterarchy inherits logical collapse

called a self-referential property, and is the essence of the diagonal argument. Any

maps g:S→S are expressed as f(x)(x) = g’(x, x) for any x in S, where g’(x, x) = (g(x), g(x)).

Assume that f : S→Hom(S, S) is subjective, and there exists y in S such that f(y)(x) = h

f(x)(x) where h:S→S is an arbitrary map. In substituting y for x, one obtains that f(y)(y)

= h f(y)(y). Because it is a fixed point with respect to an arbitrary map, h, it shows a

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contradiction. The assumption of subjectivity allows the ambiguity of indicating both

an element of S and Hom(S, S). This is the essence of self-reference.

On the other hand, the frame problem is argued in the field of artificial

intelligence (Dreyfuus, 1972, McCarthy and Hayes, 1969). If one attempts to

implement decision-making in a machine, one must describe the situation and/or

context in a formal way to enable a decision to be made. Such an attempt always fails

because it is impossible to distinguish what is necessary and adequate to describe the

situation from among all of the constituents of the whole world. The frame problem

refers to the notion of an indefinite boundary of a situation that can be generalized to

be an indefinite boundary of an entity, matter, and a system.

Recently, in the context of artificial life, it is said that the frame problem has

already been resolved. Researchers who are committed to the notion of situational

subjects think that the frame problem results from encoding-ism by which an

intelligent agent has to connect real entities with their representations in a formal

world (Brooks, 1991). To demonstrate the invalidation of the frame problem, they have

proposed a multi-agent system in which each agent has no intelligence but has a

particular motivation to move (Sterenlny, 1997). For example, an ant-robot is

implemented to pick up and carry an object if it currently is carrying no object and put

it down if the ant-robot encounters another object. They are also implemented to walk

randomly in an arena. In spite of having no particular intelligence, the presence of a

large number of ant-robots results in behavior that appears to be intelligent. Objects

are gathered and become piled up in certain places. Some researchers think that this

can be interpreted as being analogous to the way in which consciousness arises from

the global behavior of many neurons, which are similar to simple machines in many

ways (Brooks, 1991).

The frame problem can never be resolved using the notion of situational

subjects. From that perspective, an observer is separated from an agent, and an

observer makes a decision against the frame problem. An observer prepares the

environment in which an agent can work. Imagine a situation in which half of a stone

is buried in the ground. An ant-robot is unable move it and becomes permanently stuck.

There is no intelligent behavior as a result. Therefore, the frame problem is just

resolved by a superficial solution in which one ignores that the observer enters the

robots’ world and makes a decision instead of an ant-robot. The observer enters the

robots’ world, and the frame problem cannot be resolved in that such an internal

observer is not formally described.

Logical self-reference and a frame problem are still the major problems

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preventing decision-making from being implemented in a formal way. There are few

investigations being undertaken into the relationship between self-reference and the

frame problem. Although these two problems are similar to each other, and both of

them refer to the notion of wholeness and/or of an indefinite boundary of a context,

their expressions of the indefinite wholeness are different from each other. In

self-reference, the notion of indefinite wholeness is expressed as the ambiguity of

indication. In the statement, “This is false”, the term, this, indicates both “This” (a part

of the statement) and “This is false” (whole statement). Although the ambiguity of

indication is used to express the indefinite indication, it is assumed that indicating

wholeness is possible. In contrast, the notion of wholeness is negatively expressed in

the frame problem. Once a particular context is formally confirmed, other necessary

conditions are always re-found. As a result, the environment surrounding an observer

making a decision is found a posteriori. In this sense, self-reference is based on the

positive expression of an indefinite world, and the frame problem is based on the

negative expression of it. They are two sides of the same coin.

We think that both self-reference and the frame problem are problems that do

not require resolving. As discussed in the perspective of artificial life, universal biology

and interactivism, they are only problems if one attempts to implement

decision-making using a formal description. Independent of an external observer’s

description, an internal observer’s decision-making perpetually proceeds as

materialistic interaction itself (Bickhard and Terveen, 1996). However, if one gives up

describing decision-making or interactions, it leads to the erroneous notion that

language can be separated from phenomena. Therefore, we construct an external

expression for an internal observer with invalidation of the external perspective.

We address the relationship between self-reference and the frame problem in

the following. Usually, if one is faced with a particular statement, one believes that it

is trivially possible to determine what the frame is, which surrounds the statement.

Imagine the statement, “This is false” written on a blackboard, where there is also

some graffiti on the same blackboard, such as “NO”. If one thinks that the reference of

“This” is “This is false”, one finds a liar statement. However, if one mixes the graffiti,

“NO” with the above statement, one can read the statement as “This is not false”, and

one does not find a liar statement. There is a problem as to whether one can determine

the wholeness that the term “This” indicates (Fig. 1). Even when one finds

self-reference in a statement, the situation is exposed by the frame problem.

The coupling of self-reference and the frame problem allows one to make a

decision in spite of either self-reference or the frame problem. Imagine that you say, “I

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am a liar”. Such a saying is very ubiquitous in everyday life, although it contains

self-reference. How is it possible to say that? Note that, in everyday life, all statements

are used in indefinite situations, and this entails the creation of the frame problem.

This gives rise to the idea that the frame problem invalidates self-reference and vice

versa. If the term “I” indicates both “I” in the sentence “I am a liar” and the whole

sentence “I am a liar”, the sentence contains self-reference. The frame problem also

exists, however, for the sentence, and then the wholeness or the whole sentence cannot

be indicated. In this sense, the term “I” has no ambiguous meaning due to the frame

problem. The premise of self-reference is invalidated by the frame problem. In contrast,

the premise of the frame problem is invalidated by self-reference. The frame problem

can be expressed as the following; once a particular situation has been explicitly

described, a flaw can be pointed out in the description thereof. The premise of the

frame problem is the presence of a subject who recognizes the situation. Such a subject

is invalidated by self-reference, because such a subject has to be defined by subject = a

subject who recognizes the situation. The term “subject” indicates both part of and the

whole of the expression. That is why the saying “I am a liar” is possible despite the

occurrence of self-reference.

We propose a model featuring self-reference coupled with the frame problem

as the twisted coupling of two dynamics. There has been some research on the

relationship between self-reference and dynamics (Grim et al., 1994; Spncer-Brown,

1969). According to Spencer-Brown (1969), time (i.e., time-shift) is a particular device

to resolve a contradiction resulting from self-reference, such as x = not(x). If one

recognizes time-shift from the right to the left, and, i.e., xt+1 = not(xt), there is no

contradiction. We disagree with this. However, the time-shift is introduced, and there

is no resolution in principle. The question regarding the origin of initial state cannot be

avoided and cannot be answered. Even if the time-shift is introduced, it needs to be

coupled with the frame problem.

How can the frame problem be coupled with time-shift dynamics? Let us call

xt+1 = not(xt) an example of temporal dynamics, and imagine that a state is defined as a

finite binary sequence as the approximation of a real value. A question regarding the

origin of an initial state is replaced by a question regarding how to determine a binary

sequence. Recall the frame problem. As soon as a particular frame (or premise) for

making a decision is stated, a frame including the former frame is found. Although this

is an example of an expression of skepticism, it can also be regarded as a positive way

of generating a frame. It is a recursive algorithm for generating frames, each frame

being generated from the previous frame, one by one. The situation is the same as

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approximating a real value using a finite binary sequence. Once a digit is determined,

a finite sequence of digits is determined, one digit at a time. Such a rule can be

expressed using particular dynamics; given a digit and a boundary value of either 0 or

1, the dynamics generates a finite binary sequence. In contrast with temporal

dynamics, we call this dynamics state-scale dynamics. In this sense, it seems as though

the problem concerning temporal dynamics, the initial value problem, can be settled by

using such an algorithm. Strictly speaking, however, it cannot be solved, because a

boundary value is required to generate a finite binary sequence. As well as temporal

dynamics requiring state-scale dynamics, the problem of a boundary condition has to

be settled using temporal dynamics. We no longer use the term settling or solving, but

instead just the term invalidation. The initial value problem concerning temporal

dynamics is invalidated by state-scale dynamics, and vice versa (Fig. 2).

Our perspective can be applied to a ubiquitous non-linear dynamical system.

Any dynamics can be regarded as an expression for an object that cannot be described

without a dynamic (i.e., temporal) property. Dynamics is, therefore, an expression

resulting from a contradiction or self-reference. A problem resulting from self-reference

also remains as a problem concerning the origin of the initial condition. In computing it

using a digital computer, the origin of boundary values of digits also remains. In our

perspective, instead of dynamics, a pair of temporal dynamics and state-scale dynamics

is defined, and two kinds of problems are invalidated complementarily. That is an

abstract expression for the essence of a heterarchy inheriting self-reference invalidated

by the coupling with the frame problem.

3-2. Chaotic liar3-2. Chaotic liar

We construct an abstract model for describing a heterarchy as follows. Given a

contradictory logical expression, we define a complementary pair of temporal dynamics

and state-scale dynamics derived from the same logical expression. If the former

carries the property of self-reference and the latter the property of the frame problem,

it is reasonable to consider that state-scale dynamics and temporal dynamics

invalidate the problem of the origin of an initial value concerning the temporal

dynamics and the problem of a boundary value concerning the state-scale dynamics,

respectively. Therefore, we examine the property of self-reference and the frame

problem in terms of a contradictory logical expression, and their dynamical expression.

For this purpose, we examine the previous research conducted by Grim et al. (1992).

Grim et al. (1992) describe the detailed relationship between dynamics and a

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liar-statement such as “This is false”. They first introduce Lukasiewictz logic in order

to describe an endomorphism in a real number interval [0.0, 1.0]. Functions in classical

propositional calculus or Boolean algebra are replaced by the following. Truth values,

{True, False} or {0, 1} are replaced by [0.0, 1.0]. The two-valued function, xANDy for x,

y∈{0, 1} is replaced by min(x, y) for x, y∈[0.0, 1.0], and xORy is replaced by max(x, y),

where min(x, y) = x, if x <y; y, otherwise, and max(x, y) = x, if x>y; =y, otherwise. The

negation operator, NOTx, is also replaced by notx : = 1－x. Clearly, min and max

satisfy the definition of infimum and supremum in a lattice (Birkhoff, 1967);

min(x, y)≦x, y (max(x, y)≧x, y) (1)

z≦x, y ⇒ z≦min(x, y), (z≧x, y ⇒ z≧max(x, y)) (2)

x = min(x, y) ⇔ x≦y (x = max(x, y) ⇔ x≧y ). (3)

A lattice is a partial ordered set (L, ≦) closed with respect to infimum (the

greatest lower bound) and supremum (the least upper bound). A binary relation, ≦,

satisfies reflective (x≦x), anti-symmetric (x≦y, y≦x ⇔ x=y) and transitive (x≦y, y≦z

⇒ x≦z) laws. Given X is a subset of L, the lower bound and upper bound of X are

defined by c such that for all x in X c≦x and x≦c, respectively. The infimum and

supremum of X are defined by the greatest lower bound and the least upper bound,

respectively, and are represented by X and X, respectively. In particular, if X

consists of two elements, x and y, infimum and supremum are represented by xy and

xy, respectively. In this sense, condition (1) implies that min(x, y) is the lower bound

of {x, y}, and condition (2) implies that if z is the lower bound, min(x, y) is the greatest

lower bound. Condition (3) is also verified only by the properties of infimum and

supremum. In this sense, in a two-valued function, min and max exactly correspond to

infimum and supremum, respectively. In addition, Boolean algebra is defined by a

complemented distributive lattice. A complemented lattice is defined as a lattice

having a greatest element of 1 and a least element of 0, in which, for any element x,

there exists xc such that xxc = 0 and xxc = 1. A distributive lattice is a lattice in which,

for any elements x, y and z, x(yz) = (xy)(xz). The xc is negation of x, and then the

law of excluded middle holds in Boolean algebra.

because it does not satisfy the excluded middle law; min(x, 1－x) = 0 and max(x, 1－x)

The negation in Lukasiewictz logic is no longer a Boolean negation, however,

= 1. This, therefore, leads to the result that the implication function, x IMPy, does not

coincide with max(1－x, y), while xIMPy = NOT(x)ORy in Boolean algebra. In

focusing on a liar statement, the most fundamental function is an equivalence relation,

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denoted by EQ. In Boolean algebra, xEQy = (x IMPy) AND (yIMPx) = (NOT(x)ORy)

AND (NOT(y)ORx). If one adopts this form as an equivalence relation in Lukasiewictz

logic, x eq y : = max (min(1－y, x), min(1－x, y)). This form is controversial because it

does not hold that x eq x = 1 for all x in [0.0, 1.0]. That is why Grim et al. introduce the

definition;

x eq y : = 1－abs(y－x), (4)

where abs represents a function taking an absolute value.

Grim et al. propose two kinds of liar statement, namely, a classical liar and a

chaotic liar. A classical liar is expressed as

x = notx = 1－x. (5)

A liar statement that is a classical liar is, for example, “This is false”. “This” is

represented by x and “is” is represented by the equivalence relation =.

A chaotic liar is defined as, for example, the statement, “x is as true as x is

false”. This statement implies that a liar knows what he says is self-referential. A

chaotic liar is expressed as

x = x eq (notx) = x eq (1－x) = 1－abs((1－x)－x) = 1－abs(1－2x). (6)

The final expression implies

if 0≦x<0.5, x = 2x; (7)

if 0.5≦x<1.0, x = 2(1－x).

If one regards this expression as a map (i.e., one recognizes time-shift, t→t+1, from

the right term to the left term), one finds a tent map such as

xt+1 = 2xt if 0≦xt <0.5; (8)

= 2(1－xt) otherwise

As well as a chaotic liar, a classical liar can be replaced by dynamics by

introducing time-shift. Compared with a chaotic liar, dynamics derived from a classical

liar shows just a simple oscillation. Because a chaotic liar inherits self-understanding

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of self-reference, the behavior of the dynamics shows chaotic behavior. In other words,

a chaotic liar inherits indefiniteness of his understanding of the world by making his

own description of the world carry the property of self-reference. A chaotic liar reserves

his own description, and that evokes the property of self-reference. Tsuda and Tadaki

(1997) also enhance the aspect of the liar’s self-understanding regarding self-reference.

They use a simultaneous equation for a chaotic liar [20], such as xt+1 = f(xt, yt) and yt+1

= g(xt, yt), and examine the significance for which only xt+1 = f(xt, yt) and yt+1 = g(xt+1, yt)

show chaotic behavior. According to them, the latter has an internal time attribute to

the system in its own right because xt+1 appears as an argument of g(xt+1, yt). They call

such asynchronous updating “internal measurement”, in contrast to synchronous

updating called “external measurement”. This scheme can be regarded as a way of

making a point of self-understanding of self-reference.

The essential point here is that a contradiction resulting from a chaotic liar

cannot be resolved in principle. As mentioned before, remaining problems are

expressed as the origin of the initial value and/or the origin of the boundary value.

Although they cannot be resolved, we propose complementary interaction between

temporal dynamics and state-scale dynamics. Hereinafter, we define a method for

deriving two kinds of dynamics from a chaotic liar, and then propose a description of

the complementary interaction between them.

4. 4. Time-State-Scale Re-Entrant FormTime-State-Scale Re-Entrant Form

4-1. Temporal dynamics and state-scale dynamics derived from chaotic liar

4-1. Temporal dynamics and state-scale dynamics derived from chaotic liar

It is easy to see the property of self-reference and a frame problem as a

contradiction if one investigates Cantor’s diagonal argument (Whitehead and Russell

1925) in the following form. Assume an infinite set, S, can cover its power set. If so, all

elements of S counted as s1, s2, … can correspond to elements of its power-set, and then

elements of the power set are also counted as p1, p2, … Because each element of the

power set is a subset of S, it can be expressed as a binary sequence such as pi = (qi1, qi2,

…), where qij = 1 if sj is included in the subset; otherwise qij = 0. One obtains a table

with infinite columns and rows by arranging pi = (qi1, qi2, …), for i = 1, 2., …. In the

diagonal argument, an infinite diagonal sequence (q11, q22, …) is taken and is modified

by f(qii) = 1－qii. As a result, one obtains x =(f(q11), f(q22), …). Because of the above

assumption, one can add the sequence, x, to the infinite table, and that leads to a

contradiction at the crossing point between x and the diagonal sequence.