Abstract heterarchy: time/statescale reentrant form.
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 selfreference and the property of a frame problem interacting with each other. The coupling of them embodies a oneity inheriting logical inconsistency. The property of selfreference and a frame problem are defined in terms of logical operations, and are replaced by two kinds of dynamical system, temporal dynamics and statescale 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 statescale dynamics, and this results in a discontinuous selfsimilar map as a dynamical system. This reveals that the statescale and temporal dynamics attribute to the system, and shows both robust and emergent behaviors.

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ABSTRACT: Libet et al. (1983) revealed that brain activity precedes conscious intention. For convenience in this study, we divide brain activity into two parts: a conscious field (CF) and an unconscious field (UF). Most studies have assumed a comparator mechanism or an illusion of CF and discuss the difference of prediction and postdiction. We propose that problems to be discussed here are a twisted sense of agency between CF and UF, and another definitions of prediction and postdiction in a mediation process for the twist. This study specifically examines the definitions throughout an observational heterarchy model based on internal measurement. The nature of agency must be emergence that involves observational heterarchy. Consequently, awareness involves processes having duality in the sense that it is always open to the world (postdiction) and that it also maintains self robustly (prediction).Frontiers in Psychology 01/2013; 4:686. · 2.80 Impact Factor  SourceAvailable from: Koji Sawa[Show abstract] [Hide abstract]
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Kobe University Repository : Kernel
TitleAbstract heterarchy: Time/statescale reentrant form
Author(s)Gunji, YukioPegio / Sasai, Kazauto / Wakisaka, Sohei
CitationBiosystems, 91(1): 1333
Issue date200801
Resource TypeJournal Article / 学術雑誌論文
Resource Versionauthor
URL http://www.lib.kobeu.ac.jp/handle_kernel/90001516
Create Date: 20120105
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Gunji, Y.P., Sasai, K. and Wakisaka, S. (2008)
Abstract heterarchy: Time/ statescale reentrant form
Biosystems,91(1), 1333
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Abstract Heterarchy:
Time/StateScale Reentrant Form
YukioPegio GunjiYukioPegio Gunji1,2
, Kazauto Sasai1 1 & Sohei Wakisaka
Abstract Heterarchy:
Time/StateScale Reentrant 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@kobeu.ac.jpyukio@kobeu.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 selfreference and the property of a frame problem
interacting with each other. The coupling of them embodies a oneity inheriting logical
inconsistency. The property of selfreference and a frame problem are defined in terms
of logical operations, and are replaced by two kinds of dynamical system, temporal
dynamics and statescale 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 statescale dynamics, and this results in a discontinuous selfsimilar
map as a dynamical system. This reveals that the statescale and temporal dynamics
attribute to the system, and shows both robust and emergent behaviors.
Key Words: Heterarchy, Chaotic liar, Selfreference, Frame problem, Selfsimilar 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 (CantwellSmith,
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
selfreference (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 selfreference connected by a frame problem. Although
both selfreference and the frame problem reveal a kind of infinite regression, there is
a difference with respect to logical status. A dynamical system based on selfreference
has been developed by Grim et al. (1994). We develop their ideas, and the selfreference
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.
Interpretation0: 0→0; 1→0, Interpretation1: 0→1; 1→1,
Interpretation2: 0→0; 1→1, Interpretation3: 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 Interpretation3) 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 intralevel dynamics and interlevel dynamics. In the
example of going to work, the intralevel dynamics is just a choice of a value of S (i.e., a
value of a particular level) and the interlevel dynamics is a choice of an interpretation
from Interpretations0~3. 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→Interpretation3, and 1→
Interpretation1. Simultaneous choice requires that each interpretation is assigned a
value of S when the choice is made. For the choice, one can recognize that
Interpretation1 has a value of 1 and Interpretation3 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 Interpretations0 and
2 are not chosen, they also have to have values of S. Assume that Interpretation0 has
a value of 0. If so, the map property collapses because a value of 0 is mapped to both
Interpretations0 and 3. As a result, the map property and the simultaneous making
value have a tradeoff relationship therebetween. If each level is defined by a set, S, a
set of the interlevel 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 codomain
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
onetomany 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
reorganization 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
31. Selfreference and frame problem 31. Selfreference 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 selfreference
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 selfreferential 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 selfreference.
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 decisionmaking 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 encodingism 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 multiagent system in which each agent has no intelligence but has a
particular motivation to move (Sterenlny, 1997). For example, an antrobot is
implemented to pick up and carry an object if it currently is carrying no object and put
it down if the antrobot 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 antrobots 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 antrobot 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 antrobot. The observer enters the
robots’ world, and the frame problem cannot be resolved in that such an internal
observer is not formally described.
Logical selfreference and a frame problem are still the major problems
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preventing decisionmaking from being implemented in a formal way. There are few
investigations being undertaken into the relationship between selfreference 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
selfreference, 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 refound. As a result, the environment surrounding an observer
making a decision is found a posteriori. In this sense, selfreference 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 selfreference 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
decisionmaking using a formal description. Independent of an external observer’s
description, an internal observer’s decisionmaking perpetually proceeds as
materialistic interaction itself (Bickhard and Terveen, 1996). However, if one gives up
describing decisionmaking 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 selfreference 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
selfreference in a statement, the situation is exposed by the frame problem.
The coupling of selfreference and the frame problem allows one to make a
decision in spite of either selfreference 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
selfreference. 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 selfreference 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 selfreference. 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 selfreference is invalidated by the frame problem. In contrast,
the premise of the frame problem is invalidated by selfreference. 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 selfreference, 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 selfreference.
We propose a model featuring selfreference coupled with the frame problem
as the twisted coupling of two dynamics. There has been some research on the
relationship between selfreference and dynamics (Grim et al., 1994; SpncerBrown,
1969). According to SpencerBrown (1969), time (i.e., timeshift) is a particular device
to resolve a contradiction resulting from selfreference, such as x = not(x). If one
recognizes timeshift from the right to the left, and, i.e., xt+1 = not(xt), there is no
contradiction. We disagree with this. However, the timeshift 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 timeshift is introduced, it needs to be
coupled with the frame problem.
How can the frame problem be coupled with timeshift 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 statescale 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 statescale 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 statescale dynamics, and vice versa (Fig. 2).
Our perspective can be applied to a ubiquitous nonlinear 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 selfreference. A problem resulting from selfreference
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 statescale dynamics
is defined, and two kinds of problems are invalidated complementarily. That is an
abstract expression for the essence of a heterarchy inheriting selfreference invalidated
by the coupling with the frame problem.
32. Chaotic liar32. 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 statescale dynamics derived from the same logical expression. If the former
carries the property of selfreference and the latter the property of the frame problem,
it is reasonable to consider that statescale 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 statescale dynamics,
respectively. Therefore, we examine the property of selfreference 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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liarstatement 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 twovalued 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), antisymmetric (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 twovalued 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 selfreferential. 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 timeshift, 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 timeshift. Compared with a chaotic liar, dynamics derived from a classical
liar shows just a simple oscillation. Because a chaotic liar inherits selfunderstanding
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of selfreference, 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 selfreference. A chaotic liar reserves
his own description, and that evokes the property of selfreference. Tsuda and Tadaki
(1997) also enhance the aspect of the liar’s selfunderstanding regarding selfreference.
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 selfunderstanding of selfreference.
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 statescale 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. TimeStateScale ReEntrant FormTimeStateScale ReEntrant Form
41. Temporal dynamics and statescale dynamics derived from chaotic liar
41. Temporal dynamics and statescale dynamics derived from chaotic liar
It is easy to see the property of selfreference 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 powerset, 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.