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June 27, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10582-15 page 201

A HYSTERESIS EFFECT ON OPTICAL ILLUSION AND

NON-KOLMOGOROVIAN PROBABILITY THEORY

Masanari Asano1, Andrei Khrennikov2, Masanori Ohya3,4and

Yoshiharu Tanaka3,5

1Liberal Arts Division, Tokuyama College of Technology

Gakuendai, Shunan, Yamaguchi, Japan

asano@tokuyama.ac.jp

2International Center for Mathematical Modeling in Physics and Cognitive

Sciences, Linnaeus University

S-35195, V¨axj¨o, Sweden

andrei.khrennikov@lnu.se

3Department of Information Science, Tokyo University of Science

2641, Yamazaki, Noda, Chiba, Japan

4ohya@rs.noda.tus.ac.jp

5tanaka@is.noda.tus.ac.jp

In this study, we discuss a non-Kolmogorovness of the optical illusion

in the human visual perception. We show subjects the ambiguous ﬁgure

of “Schr¨oeder stair”, which has two diﬀerent meanings [1]. We prepare

11 pictures which are inclined by diﬀerent angles. The tendency to ans-

wer “left side is front” depends on the order of showing those pictures.

For a mathematical treatment of such a context dependent phenomena,

we propose a non-Kolmogorovian probabilistic model which is based on

adaptive dynamics.

1. Introduction

In classical probability theory, joint probability of two event systems

P(A, B) must hold usual probability law such as a formulae of total pro-

bability (FTP), i.e. P(A) = PAP(A, B). However, it is known that, in

quantum probability theory, FTP is not always satisﬁed [8, 11, 16]. Also,

we can consider quantum probability is a kind of context dependent pro-

bability. From this point of view, although quantum mechanics is usually

considered as a physics for understanding microscopic phenomena, quan-

tum probability can be applied to non-physical context dependent pheno-

201

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202 M. Asano et al.

mena, e.g. in the ﬁelds of biology, cognitive psychology, decision making

theory [4, 5,14, 15, 17]. In this sense, we can say that quantum probability

theory is one of non-Kolmogorov probability theories.

We have been discussed the violation of FTP on context dependent

phenomena in the various ﬁelds: State change of tongue for sweetness [16],

Bayesian updating biased by Psychological factor [12], Lactose-glucose in-

terference in E. coli growth [10, 13], decision making in two-player game

[6, 7, 9].

In this study, we show an optical illusion on the ambiguous ﬁgure of

Schr¨oder stair [1], and propose its non-Kolmogorov probabilistic model.

First, we introduce the concept of contextual probability, and a deﬁni-

tion of joint probability based on adaptive dynamics. Second, we explain

Schr¨oder stair and our experiment so as to obtain non-Kolmogorov statis-

tical data. In the last part, we construct a model of visual perception on

Shr¨oder stair, based on mathematical frame work of adaptive dynamics.

2. Adaptive Dynamics and Joint Probability

In this section, we reconsider the fundamental concept of probability theory

with a simple example, “taste for sweetness”.

2.1. Taste for sweetness: As an example of adaptive

phenomena

Let us consider sweetness of food. If we taste an orange, then some people

say that “it is sweet” (1), and the other people say that “it is not sweet” (0).

The ratio between 1 and 2 is depend on the orange. Therefore we describe

the property of the orange’s sweetness as the probability distribution of an

random variable which takes 1 or 0.

Here let us consider that one takes other foods, sugar and chocolate.

We denote a random variable for chocolate by C= 1,0, and a random

variable for sugar by S= 1,0. If one takes either sugar or chocolate, then

the probability of answer 1, i.e. P(C= 1) or P(S= 1), is very close to

1. However, when one takes chocolate after taking sugar, the tongue will

become dull. Therefore most of people cannot taste sweet (maybe no taste).

Then P(C= 1|S= 1)P(S= 1) + P(C= 1|S= 0)P(S= 0) is much less

than 1. In this case, we have the following inequality.

P(C= 1) 6=P(C= 1|S= 1)P(S= 1) + P(C= 1|S= 0)P(S= 0).

This inequality is called violation of formulae of total probability (FTP),

and makes very important discussion for the non-Kolmogorov feature of

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A Hysteresis Eﬀect 203

adaptive (context-dependent) phenomena. The most important thing in

the above inequality is that the experimental context of LHS is diﬀerent

from that of RHS. That is, P(C= 1) of LHS is given by the experimental

results without taking sugar, but RHS is obtained as the result of both

sugar and chocolate. From the point of Kolmogorov’s probability theory,

the violation is obvious since probability space or measure must be deﬁned

within the single experimental context, e.g. context of “taking sugar” or

“not taking sugar”. However, if we discuss the diﬀerence of these contexts,

the violation of FTP is meaningful. For instance, in order to discuss the

change of tongue state, we need to combine these two diﬀerent contexts. It

is necessary to construct such a theory in which the context dependency of

the probabilistic phenomena can be discussed.

2.2. What is adaptive dynamics?

Adaptive dynamics is a detailed theory to describe the existence (subsis-

tent) by taking various eﬀects surrounding it, and the concept of adaptive

dynamics was proposed by Ohya. In the book [11], adaptive dynamics is

explained as follows.

“We ﬁnd a mathematics to describe a sort of subjective aspects of

physical phenomena, for instance, observation, eﬀects of surroun-

dings”

From mathematical point of view, adaptive dynamics is considered that

the dynamics of a state or an observable after an instant (say the time

t0) attached to a system of interest is aﬀected by the existence of some

other observable and state at that instant. Such dynamics was deﬁned by

lifting [3] as follows.

Let ρ∈ S (H) and A∈ O (H) be a state and an observable before t0,

and let σ∈ S (H ⊗ K) and Q∈ O (H ⊗ K) be a state and an observable to

give an eﬀect to the state ρand the observable A. This eﬀect is described

by a lifting E∗

σQ, so that the state ρbecomes E∗

σQρﬁrst, then it will be

trKE∗

σQρ≡ρσ Q. The adaptive dynamics here is the whole process such as

Adaptive Dynamics :ρ⇒ E∗

σQρ⇒ρσ Q = trKE∗

σQρ.

That is, what we need is how to construct the lifting for each problem to

be studied, that is, we properly construct the lifting E∗

σQ by choosing σand

Qproperly. The expectation value of another observable B∈ O (H) in the

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204 M. Asano et al.

adaptive state ρσQ is

trρσQB= trHtrKB⊗IE∗

σQρ.

Now suppose that there are two quantum event systems

A={ak∈R, Fk∈ P (H)},

B={bj∈R, Ej∈ P (K)},

where we do not assume Fk,Ejare projections, but they satisfy the condi-

tions PkFk=I,PjEj=Ias POVM (positive operator valued measure)

corresponding to the partition of a probability space in classical system.

Then the “joint-like” probability obtaining akand bjmight be given by the

formula

P(ak, bj) = tr EjFkE∗

σQρ,(2.1)

where is a certain operation (relation) between Aand B, more generally,

one can take a certain operator function f(Ej, Fk) instead of EjFk. If σ,

Qare independent from any Fk,Ejand the operation is the usual tensor

product ⊗so that Aand Bcan be considered in two independent systems

or to be commutative, then the above “joint-like” probability becomes the

joint probability. However, if this is not the case, e.g.,Qis related to A

and B, the situation will be more subtle. Therefore, the problem is how

to set the operation and how to construct the lifting E∗

σQ in order to

describe the particular problems associated to systems of interest. Recently

we have been discuss this problem in the context dependent systems like

bio-systems and psycho-systems mentioned in Introduction.

2.3. Non-Kolmogorovian probability in adaptive dynamics

for sweetness

Let |e1iand |e2ibe the orthogonal vectors describing sweet and non-sweet

states, respectively. We set the |e1iand |e2ias 1

0and 0

1in the Hil-

bert space C2.The initial (neutral) state of a tongue is given by a density

operator

ρ≡ |x0i hx0|,

with x0=1

√2(|e1i+|e2i). Here the neutral pure state ρdescribes the state

of tongue before the experiment, and we start from this state ρ.

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A Hysteresis Eﬀect 205

When one tastes sugar, the state of tongue will be changed. Such change

can be written as the operator on the Hilbert space. The operator corre-

sponding to tasting sugar is represented as

S=λ10

0λ2,

where λiare complex numbers satisfying |λ1|2+|λ2|2= 1. This operator

can be regarded as the square root of the sugar state σS:

σS=|λ1|2E1+|λ2|2E2, E1=|e1i he1|, E2=|e2i he2|.

Taking sugar, he will taste that it is sweet with the probability |λ1|2and

non-sweet with the probability |λ2|2, so |λ1|2should be much higher than

|λ2|2for usual sugar. This comes from the following change of the neutral

initial state of a tongue:

ρ→ρS= Λ∗

S(ρ)≡S∗ρS

tr |S|2ρ.(2.2)

This is the state of a tongue after tasting sugar.

The subtle point of the present problem is that, just after tasting sugar,

the state of a tongue is neither ρSnor ρ. However, for a while the tongue

becomes dull to sweetness (and this is the crucial point of our approach

for this example), so the tongue state can be written by means of a certain

“exchanging” operator X=0 1

1 0 such that

ρSX =XρSX.

Similarly, when one tastes chocolate, the state will be given by

ρSX C = Λ∗

C(ρSX )≡C∗ρS X C

tr |C|2ρSX

,

where the operator Chas the form

C=µ10

0µ2

with |µ1|2+|µ2|2= 1. Common experience tells us that |λ1|2≥ |µ1|2≥

|µ2|2≥ |λ2|2and the ﬁrst two quantities are much larger than the last two

quantities.

As can be seen from the preceding consideration, in this example the

“adaptive set” {σ, Q}is the set {S, X, C}. Now we introduce the following

(nonlinear) lifting:

E∗

σQ(ρ) = ρS⊗ρSX C = Λ∗

S(ρ)⊗Λ∗

C(XΛ∗

S(ρ)X).

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206 M. Asano et al.

The corresponding joint probabilities are given by

P(S=j, C =k) = tr Ej⊗EkE∗

σQ(ρ).

The probability that one tastes sweetness of the chocolate immediately after

tasting sugar is

P(S= 1, C = 1) + P(S= 2, C = 1) = |λ2|2|µ1|2

|λ2|2|µ1|2+|λ1|2|µ2|2,

which is P(C= 1). Note that this probability is much less than

P(C= 1) = tr E1Λ∗

C(ρ) = |µ1|2,

which is the probability of sweetness tasted by the neutral tongue ρ. This

means that the usual formula of total probability should be replaced by the

adaptive (context dependent) probability law.

3. Optical Illusions in Human Perception Process

In this section, we shortly introduced the phenomena of “optical illusion”,

and we show the experiment and its results on optical illusion.

A human being does not recognize what comes into our eyes. Our re-

cognition is unconsciously biased. How to bias depends on how it exists in

certain surroundings. See the Figure 1. These letters are of the same bright-

ness of color in reality. However we feel the diﬀerent brightness which is

biased by background colors. This is a famous example of optical illusion.

Psychologists explain that this is caused by “perceptual constancy”.

The perceptual constancy plays a very important role in human perception.

When we see a pyramid, the shape of pyramid is diﬀerent for the angles

Fig. 1. Peceptual constancy: Left letter ‘A’ looks a little brighter than right letter ‘A’.

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A Hysteresis Eﬀect 207

at which we see it. From the side, we can see a triangle, while, from the

sky, we can see a square. However, we can recognize this triangle and that

square means the same object (a pyramid). It is considered that such stable

recognitions of humans are assisted by the perceptual constancya.

3.1. Ambiguous ﬁgure

Ambiguous ﬁgure is a ﬁgure which has two or more meanings. Rubin’s vase

is one of most famous ambiguous ﬁgures (see Figure 2). The intensity of

illusion depends on how it exists in the surroundings [2]. For example, if we

change the size of ﬁgure, how can we see in the same ambiguous ﬁgure? For

the tiny ﬁgure of Rubin’s vase, one can conﬁrm that we have the tendency

of seeing “vase” more than “faces”.

Fig. 2. Rubins vase: It has two meanings of “two human faces” (white) and “a vase”

(black).

3.2. Depth-inversion for Schr¨oder stair

Another famous example of ambiguous ﬁgure is Schr¨oder stair (see Figure

3).

We can see two diﬀerent types of stair seen in the Figure 3.

1) The surface of A is front, and the surface of B is back.

2) The surface of B is front, and the surface of A is back.

We can change our recognition consciously or unconsciously. Such phe-

nomena is called “depth inversion”. Tendency of recognitions 1) and 2)

aThere is a similar phenomenon that we can feel that snow at night is white (not gray).

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208 M. Asano et al.

Fig. 3. Schr¨oder stair: One can see the steps in the center of box (which is leaning).

depends on various features of picture; e.g. Relative size of the part A for

B, Color (or shadow) in picture, Angle to the horizon, etc. We test the

tendency with respect to leaning angle.

We show the subjects of experiment the 11 diﬀerent pictures.

Case A The value of θis randomly selected from {0, 10, 20, 30, 40, 45, 50,

60, 70, 80, 90}, but each value is selected only once.

Case B The value of θis regularly changed from 0 to 90.

Case C The value of θis regularly changed from 90 to 0.

We divided the 151 subjects into three groups: (A) 55 persons, (B) 48

persons, (C) 48 persons. Computer show a picture to a subject. Recognition

and decision making. Subjects answer 1) or 2) by means of typing corre-

sponding keys. In case (A), computer decide the order of showing pictures

for each subject.

Here we describe the answer for the ﬁgure with angle θas a random

variable Xθsuch that

Xθ=L(L is front)

R(R is front) .

Figure 4 shows the probability distribution P(Xθ=L) with respect to

angle θ.

We can see the diﬀerence among three cases especially at the middle

range of angles. Tendency of human perception is aﬀected by unconscious

cognitions for the dynamical change of the angle. We conﬁrm that the

probability is decreasing with respect to θfor any cases. However, in the

result of case B, the speed of decreasing is faster than that in case A. In

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A Hysteresis Eﬀect 209

Fig. 4. Probability distribution P(Xθ=L) with respect to angle θ.

the case C, we remark that angle θis decreasing from 90 degrees. Therefore

this kind of acceleration eﬀect is similarly seen in case C. These eﬀects may

come from perceptual consistency explained in the previous section.

4. Non-Kolmogorovian Model of Optical Illusion

In this section, we show a non-Kolmogorovian model of optical illusion in

Schr¨oder stair.

Let us consider a subject making his answer. We hypothesize the follo-

wing things:

H1 In the subject’s brain, there are some imaginary “Agents” which are

copies of the subject shown in the picture.

H2 Subject’s decision is given by a majority opinion of all the agents.

Conceptually, this type of decision-making is a kind of “self-dialog”

which is unconsciously proceeding in the perception process. Further, we

add another hypothesis:

H3 Each agent is mathematically expressed as the superposition state of

two alternative answers L or R as follows.

|xi=r1

2|Li+r1

2|Ri.

Here |Li=1

0and |Ri=0

1be the orthogonal vectors describing

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210 M. Asano et al.

the answers (i) and (ii), respectively. This state vector |xidescribes pre-

decision state of the agent, and represents the neutral mind for |Liand

|Ribefore the decision making starts. In the sequel sections, we discuss the

more detailed process of decision making, and propose a model based on

adaptive dynamics theory.

4.1. Model of a majority among Nagents

Let Hbe the two dimensional Hilbert space C2. An initial state of a sub-

ject’s mind is given by

ρ≡ |xi hx|.

When a subject is shown a picture leaning at angle θ, the subject recognizes

the lean of the picture. Such a recognition process is a state-change given

by the operator

M(θ) = cos θ0

0 sin θ.

After the recognition, the state of mind is changed from initial state ρto

an adaptive state

ρθ= Λ∗

M(ρ)≡M∗ρM

tr |M|2ρ=cos2θcos θsin θ

cos θsin θsin2θ.

The ﬂuctuation between (i) and (ii) is expressed as the above ρθ.

Here, let us recall the hypothesis H1. At the beginning of the process, a

subject imagine and create an imaginary agent in the brain, and this agent

has own mind which is expressed as the adaptive state ρθ. A subject repe-

atedly create the imaginary agents during an experiment for Xθ. We can

consider the adaptive state describing the whole agents as the N-composite

state of ρθ:

σ=ρθ⊗ · · · ⊗ ρθ

| {z }

N

,

where Nis the number of the agents which are created by a subject during

making the answer (i) or (ii). If a subject takes a short time to answer, then

Nmight be small. After creation of agents (or on the way to create them),

the subject can talk with the imaginary agents in the brain, and he/she

knows the answer of each agent. Through this dialogue, the subject can

know the opinions of all the agents. For the Nagents, there are 2Npossible

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A Hysteresis Eﬀect 211

opinions, and one of them is appeared. The subject’s answer is determined

by reference to the opinions of all the agents.

At the ﬁnal step of this dialogue, we additionally assumed that the σis

changed into L⊗Nor R⊗Nsince every subject in this experiment can

not answer anything except (i) or (ii). Along this assumption, we introduce

the observable-adaptive operator Qwhich describes the process of making

a common decision. For example, in the case of N= 2,3 and 4, the operator

Qis

Q(2) =|LLi hLL|+|RRi hRR|,

Q(3) =|LLLi(hLLL|+hLLR|+hLRL|+hRLL|)

+|RRRi(hRRR|+hRRL|+hRLR|+hLRR|),

Q(4) =|LLLLi(hLLLL|+hLLLR|+hLLRL|+hLRLL|+hRLLL|)

+|RRRRi(hRRRR|+hRRRL|+hRRLR|+hRLRR|+hLRRR|).

By applying the operator Qto the state of agents σ, the minority opinions

of the agents are ignored, and changed to the majority ones. In this sense,

we call this decision making as the majority system among the Nagents.

Here, the lifting E∗

σQ :SC2→ S C2Nis deﬁned as

E∗

σQ(ρ) = QσQ∗

tr |Q|2σ=Q{Λ∗

M(ρ)}⊗NQ∗

tr |Q|2{Λ∗

M(ρ)}⊗N∈ S C2N,

and the probabilities for Xθare given with this lifting as

P(Xθ=L)≡tr |Li hL|⊗NE∗

σQ(ρ),

P(Xθ=R)≡tr |Ri hR|⊗NE∗

σQ(ρ).

In N= 2,3 and 4, the probability P(Xθ=L) has the following forms:

P(2) (Xθ=L) = cos4θ

cos4θ+ sin4θ,

P(3) (Xθ=L) = cos3θ+ 3 cos2sin θ2

(cos3θ+ 3 cos2sin θ)2+sin3θ+ 3 sin2cos θ2,

P(4) (Xθ=L) = cos4θ+ 4 cos3θsin θ2

(cos4θ+ 4 cos3θsin θ)2+sin4θ+ 4 sin3cos θ2.

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212 M. Asano et al.

Fig. 5. Comparison of the values of P(Xq =L) in our model with the experimental

data (A).

Fig. 6. Noise of biased phase; (Left) N= 2, (Center) N= 3, (right) N= 4.

We compare these probabilities with experimental data of case (A), see

Figure 5. One can ﬁnd that the probabilities P(Xθ=L) in N= 2,3 and 4

coincide with experimental data of A.

In the situation (B) or (C), we take another adaptive operator M(θ+φ)

which angle is shifted by an unknown psychological bias φ, instead of M(θ).

This value of φcan be also calculated by experimental data. We show it

in the Figure 6. The values of φin (B) are higher than those in (C), and

especially the diﬀerence between φ(B)and φ(C)is clearly seen in middle

range of angle θ. As increasing N, this diﬀerence become smaller.

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