# Generalized probabilistic independence and utility functions

Didier Dubois

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Didier Dubois. Generalized probabilistic independence and utility functions. Bulletin pour les sous-ensembles flous et leurs applications, 1985, 23, pp.83–98. hal-04584339

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# Generalized probabilistic independence and utility functions

Didier Dubois

► To cite this version:

Didier Dubois. Generalized probabilistic independence and utility functions. Bulletin pour les sous-ensembles flous et leurs applications, 1985, 23, pp.83–98. hal-04584339

HAL Id: hal-04584339

https://hal.science/hal-04584339v1

Submitted on 23 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

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Distributed under a Creative Commons Attribution 4.0 International License
GENERALIZED PROBABILISTIC INDEPENDENCE
AND UTILITY FUNCTIONS

Didier Dubois

Laboratoire Langages et Systèmes Informatiques

Université Paul Sabatier

31062 TOULOUSE, FRANCE

**Key-words:** probability, independence, triangular norm, utility

**Abstract** : The concept of probabilistic independence is generalized into a decomposability criterion by means of a two-place operation on the unit interval. The set of candidate operations is characterized via natural axioms owing to a recent result in the theory of functional equations. This result is applied to the construction of utility functions, by re-laxing Von Neumann and Morgenstern's axiom system.

I - INTRODUCTION

For many authors, the uncertainty pervading decisions can be repre-
sented by probability measures on the set of possible consequences of
these decisions. Justifications of the probabilistic setting are based on
the betting behavior interpretation or qualitative probability relations
[5]. Under the subjectivist point of view the concept of probabilistic
independence is sometimes difficult to justify. This is because indepen-
dence is understood, in the setting of statistics, between experiments.
But in the subjectivist setting only independence between events can be
defined, and the usual definition :

$$
P(A \cap B) = P(A) \cdot P(B) \tag{1}
$$

where A, B are subsets of Ω, and P is a probability measure, may appear
too restrictive outside a frequentist framework.

A qualitative view of independence is developped in Fine [2] which
leads to a decomposability criterion. Namely, A and B are separable if and
only if there is an operation * on [0,1] such that

$$
P(A \cap B) = P(A) * P(B) \tag{2}
$$

i.e. P(A n B) can be computed from the knowledge of P(A) and P(B). In the
following, we study the possible candidates for operation *, and come up
with a parametered family of semi-groups on [0,1]. On such a basis the
question of constructing utility functions is reconsidered.

2 - SEPARABILITY OF JOINT EVENTS

The following requirements are natural for operation * :

i) commutativity

ii) associativity

iii) $\forall a \in [0,1], a*1 = a, 0*0 = 0$

iv) $\forall a, b, c, d \in [0,1] \ a \ge b, c \ge d \Rightarrow a * c \ge b * d$

v) continuity

vi) $\forall a \in (0,1) \ a*a < a$

Commutativity stems from A n B = B n A. Associativity appears if one
considers separability of n events A₁...Aₙ : ∀m ≤ n, ∀k₁...kₘ ≤ n,

$$
P(A_{k_1} \cap A_{k_2} \cap \dots \cap A_{k_m}) = P(A_{k_1}) * P(A_{k_2}) * \dots * P(A_{k_m}) \quad (3)
$$

Associativity of * is due to that of set-intersection. Axiom iii) is due
to A Ω = A 0 Ω = 0, and the fact that A and Ω are independent. iv) is
natural if we interpret a = P(A), b = P(B), c = P(C), d = P(D) and A ⊆ B,
C ⊆ D, where A and B, C and D are separable, respectively. Continuity is
also a natural technical requirement.

Under i)-iv), operation * cannot be choosen but among triangular norms
[6], which are semi-group operations on the unit interval, monotonically
increasing in both places, with identity 1. Triangular norms are all such
that
GENERALIZED PROBABILISTIC INDEPENDENCE
AND UTILITY FUNCTIONS

Didier Dubois

Laboratoire Langages et Systèmes Informatiques

Université Paul Sabatier

31062 TOULOUSE, FRANCE

**Key-words:** probability, independence, triangular norm, utility

**Abstract** : The concept of probabilistic independence is generalized into a decomposability criterion by means of a two-place operation on the unit interval. The set of candidate operations is characterized via natural axioms owing to a recent result in the theory of functional equations. This result is applied to the construction of utility functions, by relaxing Von Neumann and Morgenstern's axiom system.

I - INTRODUCTION

For many authors, the uncertainty pervading decisions can be repre-
sented by probability measures on the set of possible consequences of
these decisions. Justifications of the probabilistic setting are based on
the betting behavior interpretation or qualitative probability relations
[5]. Under the subjectivist point of view the concept of probabilistic
independence is sometimes difficult to justify. This is because indepen-
dence is understood, in the setting of statistics, between experiments.
But in the subjectivist setting only independence between events can be
defined, and the usual definition :

$$
P(A \cap B) = P(A) \cdot P(B) \tag{1}
$$

where A, B are subsets of Ω, and P is a probability measure, may appear
too restrictive outside a frequentist framework.

A qualitative view of independence is developped in Fine [2] which
leads to a decomposability criterion. Namely, A and B are separable if and
only if there is an operation * on [0,1] such that

$$
P(A \cap B) = P(A) \ast P(B) \tag{2}
$$

i.e. P(A n B) can be computed from the knowledge of P(A) and P(B). In the
following, we study the possible candidates for operation *, and come up
with a parametered family of semi-groups on [0,1]. On such a basis the
question of constructing utility functions is reconsidered.

2 - SEPARABILITY OF JOINT EVENTS

The following requirements are natural for operation * :

i) commutativity

ii) associativity

iii) $\forall a \in [0,1], a*1 = a, 0*0 = 0$

iv) $\forall a, b, c, d \in [0,1] \ a \ge b, c \ge d \Rightarrow a * c \ge b * d$

v) continuity

vi) $\forall a \in (0,1) \ a*a < a$

Commutativity stems from A n B = B n A. Associativity appears if one
considers separability of n events A₁...Aₙ : ∀m ≤ n, ∀k₁...kₘ ≤ n,

$$
P(A_{k_1} \cap A_{k_2} \cap \dots \cap A_{k_m}) = P(A_{k_1}) * P(A_{k_2}) * \dots * P(A_{k_m}) \quad (3)
$$

Associativity of * is due to that of set-intersection. Axiom iii) is due
to A ∩ Ω = A ∅ ∩ ∅ = ∅, and the fact that A and Ω are independent. iv) is
natural if we interpret a = P(A), b = P(B), c = P(C), d = P(D) and A ⊆ B,
C ⊆ D, where A and B, C and D are separable, respectively. Continuity is
also a natural technical requirement.

Under i)-iv), operation * cannot be choosen but among triangular norms
[6], which are semi-group operations on the unit interval, monotonically
increasing in both places, with identity 1. Triangular norms are all such
that
$$T_W(a,b) \le a \le b \le \min(a,b) \quad (4)$$

$$\text{where } T_W(a,b) = 0 \text{ if } a, b \in [0,1), T_W(a,1) = T_W(1,a) = a.$$

The main continuous triangular norms, apart from $\min$ and $T_W$ are the product and $a \ast b = \max(0,a+b-1)$, also denoted $T_m$.

Axiom vi) is to account for the situation when $P(A) = P(B) = a$ and $P(A-B) \ne 0$, $P(B-A) \ne 0$. Then clearly $P(A \cap B) \le a$. The separability of A and B leads to admit axiom vi). A triangular norm satisfying $\gamma$-vi) is called Archimedean and is pseudo-additive in the sense that for there is a continuous, strictly decreasing mapping $f : [0,1] \to [0, +\infty)$ such that $f(1) = 0$, and (see [6])

$$f(a \ast b) = f(a) + f(b) \quad (5)$$

Operation $\ast$ cannot be arbitrarily chosen among triangular norms. First note that $\forall A, B$

$$\max(0, P(A) + P(B) - 1) \le P(A \cap B) \quad (6)$$

The equality ($T_m$-separability) is valid when A and B simultaneously occur as seldom as possible. With a subjectivist point of view, the equality expresses a pessimistic opinion on the simultaneous occurrence of A and B knowing $P(A)$ and $P(B)$.

On the other hand, $P(A \cap B)$ is bounded from above by $\min(P(A), P(B))$. The minimum operation violates axiom vi). Indeed $P(A \cap B) = \min(P(A), P(B))$ expresses a strong dependence between A and B : if $P(A) \ge P(B)$, it means that A occurs at least wherever B occurs; it expresses an optimistic opinion about the joint occurrence of A and B since it amounts to claiming that if the least probable event occurs then the other event should occur too. Hence $\ast$-separability ranges from maximal exclusiveness ($T_m$-separability) to strong dependence (min-separability). The usual independence ($\ast = \text{product}$) expresses a moderate opinion between these extreme cases.

### 3 - CHARACTERIZATION OF EVENTS SEPARABILITY

So far, we have not used the additivity law of probability measures. We use it now in the following:

**Lemma 1** : if A and B are $\ast$-separable then there is an operation O such that $\bar{A}$ and $\bar{B}$ are O-separable.

$$ \begin{array}{l} \textbf{Proof :} \quad P(\bar{A} \cap \bar{B}) = 1 - P(A) - P(B) + P(A) \ast P(B) \\ \phantom{\textbf{Proof :} \quad P(\bar{A} \cap \bar{B}) =} = P(\bar{A}) + P(\bar{B}) - 1 + (1 - P(\bar{A})) \ast (1 - P(\bar{B})) \\ \phantom{\textbf{Proof :} \quad P(\bar{A} \cap \bar{B}) =} \triangleq P(\bar{A}) \circ P(\bar{B}). \qquad \text{Q.E.D.} \end{array} $$

As a consequence, operation O also belongs to Archimedean triangular norms, and is thus associative. Associativity of operation O implies associativity of operation $a \ast b = a \ast b$, given that $\ast$ is associative. This property is not self-evident and severely restricts the choice of $\ast$ among Archimedean triangular-norms as shown by Frank [3]. Indeed, he has shown that the only Archimedean continuous triangular norms such that $a \ast b$ and $a \ast b = a \ast b$ are simultaneously associative are defined by

$$a \ast b = a F_s b = \log_s \left[ 1 + \frac{(s^a - 1)(s^b - 1)}{s - 1} \right] \quad s > 0 \quad (7)$$

For $s = 0, 1, +\infty$ we respectively recover $a \ast b = \max(0, a+b-1)$, $a, b$, and $\min(a, b)$. Note that $\ast$ is strictly monotonic and satisfies (5) with $f_s(a) = \log_s \frac{s-1}{s^a-1}$, for $s \ne 0, 1, +\infty$. Now it is easy to check that if A and B are $\ast$-separable for $\ast = F_s$ then

* $\bar{A}$ and $\bar{B}$ are $F_s$-separable, i.e. $P(\bar{A} \cap \bar{B}) = P(\bar{A}) F_s P(\bar{B})$

* A and $\bar{B}$, as well as $\bar{A}$ and B, are $F_s$-separable with $s' = 1/s$

$$ \begin{align*} \text{i.e. } P(A \cap B) &= P(A) - P(A \cap B) = P(A) - P(A) F_s P(B) \\ &= P(A) F_s P(B) \end{align*} $$

$$\text{For instance } P(A \cap B) = \min(P(A), P(B)) \Rightarrow P(A \cap \bar{B}) = \max(0, P(A) + P(\bar{B}) - 1)$$

* A and B as well as A and $\bar{B}$ are both $\ast$-separable if and only if they are both independent ($\ne$ product).

* The quantity $a \ast b = a F_s b$ is equal to $1 - (1-a)F_s(1-b)$, and is called a triangular conorm.
$$T_W(a,b) \le a \le b \le \min(a,b) \quad (4)$$

$$\text{where } T_W(a,b) = 0 \text{ if } a, b \in [0,1), T_W(a,1) = T_W(1,a) = a.$$

The main continuous triangular norms, apart from $\min$ and $T_W$, are the product and $a \ast b = \max(0,a+b-1)$, also denoted $T_m$.

Axiom vi) is to account for the situation when $P(A) = P(B) = a$ and $P(A-B) \ne 0$, $P(B-A) \ne 0$. Then clearly $P(A \cap B) \le a$. The separability of A and B leads to admit axiom vi). A triangular norm satisfying $\gamma$-vi) is called Archimedean and is pseudo-additive in the sense that for there is a continuous, strictly decreasing mapping $f : [0,1] \to [0, +\infty)$ such that $f(1) = 0$, and (see [6])

$$f(a \ast b) = f(a) + f(b) \quad (5)$$

Operation $\ast$ cannot be arbitrarily chosen among triangular norms. First note that $\forall A, B$

$$\max(0, P(A) + P(B) - 1) \le P(A \cap B) \quad (6)$$

The equality ($T_m$-separability) is valid when A and B simultaneously occur as seldom as possible. With a subjectivist point of view, the equality expresses a pessimistic opinion on the simultaneous occurrence of A and B knowing $P(A)$ and $P(B)$.

On the other hand, $P(A \cap B)$ is bounded from above by $\min(P(A), P(B))$. The minimum operation violates axiom vi). Indeed $P(A \cap B) = \min(P(A), P(B))$ expresses a strong dependence between A and B : if $P(A) \ge P(B)$, it means that A occurs at least wherever B occurs; it expresses an optimistic opinion about the joint occurrence of A and B since it amounts to claiming that if the least probable event occurs then the other event should occur too. Hence $\ast$-separability ranges from maximal exclusiveness ($T_m$-separability) to strong dependence (min-separability). The usual independence ($\ast = \text{product}$) expresses a moderate opinion between these extreme cases.

### 3 - CHARACTERIZATION OF EVENTS SEPARABILITY

So far, we have not used the additivity law of probability measures. We use it now in the following:

**Lemma 1** : if A and B are $\ast$-separable then there is an operation $\odot$ such that $\bar{A}$ and $\bar{B}$ are $\odot$-separable.

$$ \begin{array}{l} \textbf{Proof :} \quad P(\bar{A} \cap \bar{B}) = 1 - P(A) - P(B) + P(A) \ast P(B) \\ \phantom{\textbf{Proof :} \quad P(\bar{A} \cap \bar{B}) =} = P(\bar{A}) + P(\bar{B}) - 1 + (1-P(\bar{A})) \ast (1-P(\bar{B})) \\ \phantom{\textbf{Proof :} \quad P(\bar{A} \cap \bar{B}) =} \triangleq P(\bar{A}) \circ P(\bar{B}). \qquad \text{Q.E.D.} \end{array} $$

As a consequence, operation $\odot$ also belongs to Archimedean triangular norms, and is thus associative. Associativity of operation $\odot$ implies associativity of operation $a \ast b = a \ast b$, given that $\ast$ is associative. This property is not self-evident and severely restricts the choice of $\ast$ among Archimedean triangular-norms as shown by Frank [3]. Indeed, he has shown that the only Archimedean continuous triangular norms such that $a \ast b$ and $a \ast b = a \ast b$ are simultaneously associative are defined by

$$a \ast b = a \circ f_s b = \log_s \left[ 1 + \frac{(s^a - 1)(s^b - 1)}{s - 1} \right] \quad s > 0 \quad (7)$$

For $s = 0, 1, +\infty$ we respectively recover $a \ast b = \max(0, a+b-1)$, $a \circ b$, and $\min(a,b)$. Note that $\ast$ is strictly monotonic and satisfies (5) with $f_s(a) = \log_s \frac{s-1}{s^a-1}$, for $s \ne 0, 1, +\infty$. Now it is easy to check that if A and B are $\ast$-separable for $\ast = f_s$ then

* $\bar{A}$ and $\bar{B}$ are $f_s$-separable, i.e. $P(\bar{A} \cap \bar{B}) = P(\bar{A}) f_s P(\bar{B})$

* A and $\bar{B}$, as well as $\bar{A}$ and $\bar{B}$, are $f_s$-separable with $s' = 1/s$

$$ \begin{align*} \text{i.e. } P(A \cap B) &= P(A) - P(A \cap B) = P(A) - P(A) f_s P(B) \\ &= P(A) f_s P(B) \end{align*} $$

$$\text{For instance } P(A \cap B) = \min(P(A), P(B)) \Rightarrow P(A \cap \bar{B}) = \max(0, P(A) + P(\bar{B}) - 1)$$

* A and B as well as A and $\bar{B}$ are both $\ast$-separable if and only if they are both independent ($\ne$ product).

* The quantity $a \ast b = a \circ b$ is equal to $1 - (1-a)f_s(1-b)$, and is called a triangular conorm.
## 4 - EXTENSION OF VON NEUMANN-MORGENSTERN AXIOMS OF UTILITY

Let $X$ be a set containing the possible consequences of potential decisions. A set $\bar{X}$ of uncertain consequences is built over $X$, and is such that

$$X \subseteq \bar{X}; x, y \in X \Rightarrow (x, y) \in \bar{X}$$

where $(x, p, y)$ is short for the uncertain consequence where $x$ occurs with probability $p$ and $y$ with probability $1-p$. The choice among potential decisions is guided by a preference ordering $\geq$ over uncertain consequences. $\geq$ is supposed to be a weak order [4], i.e. is transitive, reflexive, and $\forall x, y \in \bar{X}, x \geq y$ or $y \geq x$. $>$ and $\sim$ respectively denote the strict ordering and equivalence constructed from $\geq$. Notice that :

$$(x, 1, y) \sim x; (x, p, y) \sim (y, 1-p, x).$$

The fundamental axioms of utility theory are then

$$U1) \quad x \sim y \Rightarrow \forall p \in [0,1], \forall z \in X (x, p, z) \sim (y, p, z).$$

$$U2) \quad x > y \Rightarrow \forall p \in (0,1), x > (x, p, y) > y.$$

$$U3) \quad x > y > z \Rightarrow \exists p \in [0,1] \ C \sim (x, p, z)$$

$$U4) \quad ((x, p, y), q, y) \sim (x, p, q, y).$$

Given these axioms, it is proved in [7] that a utility function $u: \bar{X} \to \mathbb{R}$ exists such that $x > y$ iff $u(x) > u(y)$, $x \sim y$ iff $u(x) = u(y)$, and

$$u(x, p, y) = p \ u(x) + (1-p)u(y) \quad (8)$$

Moreover, $u$ is an interval scale [4].

Here we shall focus on axiom U4, which the authors of [7] wished to modify, but, to them, "the mathematical difficulties seem to be considerable" (p. 632). The uncertain consequence $((x, p, y), q, y)$ can be viewed as the result of two successive decisions; the first one enables the second one with probability $p$, and the second one leads to consequence $x$ with probability $p$. On the whole, in this 2-stage decision process, $x$ is got with probability $p$ and $y$ with probability $1-p$, as long as the events following the decisions are independent. Whether this assumption is verified is sometimes up to the decider's judgment.

He may consider on the contrary, that a favorable issue to both decisions (i.e. yielding $x$) is very unlikely, and then only grants a probability

$\max(p+q-1, 0)$ to $x$. Or he is very optimistic and admits that if the most risky decision reaches its goal, then so will the other ones, and so he grants probability $\min(p, q)$ to $x$. More generally, he takes for granted that $P(x) = p \ast q$ where $\ast$ is a Frank triangular-norm. Hence we relax U4 into

$$U4') \quad ((x, p, y), q, y) \sim (x, p \ast q, y)$$

## 5 - CONSTRUCTION OF GENERALIZED UTILITY FUNCTIONS

Following the same reasoning process as in [7], we can prove the following results

**Lemma 2:** $\forall p, q \in (0,1), \forall x, y \in \bar{X}, x > y, p > q \Rightarrow (x, p, y) > (x, q, y)$.

**Proof:** Applying (U2): $x > y \Rightarrow (x, p, y) > y \Rightarrow \forall r \in (0,1), (x, p, y) > ((x, p, y)r, y) = (x, p \ast r, q)$ due to (U4'). It is now enough to solve equation $p \ast r = q$. Using (5), $r$ is unique and is equal to $f_s^{-1}(f_s(q) - f_s(p))$ for $s \neq +\infty, 1, 0$; $r = q$ if $\ast = \min$; $r = q/p$ if $\ast = \text{product}$; $r = 1-p+q$ if $\ast = T_m$. $\forall q \in \mathbb{Q}, \forall p \in \mathbb{P}$.

**Lemma 3:** $\forall p, q \in (0,1) \forall x \neq y \in \bar{X}, x > y$ and $(x, p, y) \sim (x, q, y)$ imply $p = q$.

**Proof:** Obvious with Lemma 2.

**Lemma 4:** $\forall q, \text{ if } p > r \text{ and } x > y \text{ then there is } \alpha \text{ such that} ((x, p, y), q, (x, r, y)) \sim (x, \alpha, y), \alpha \in (0,1]$.

**Proof:** $x > (x, p, y) > (x, r, y) > y$, and there is some $t$ such that $(x, p, y)$
$(x, t(x, r, y))$ using lemma 3.

Now $(x, t, (x, r, y)) \sim (x, t+r-t \ast r, y)$, with $t+r-t \ast r = 1-(1-r) \ast (1-t)$
since $\ast$ is a Frank t-norm. Hence we must find $t$ such that

$$(1-r) \ast (1-t) = 1-p$$

which yields $t = 1 - f_s^{-1}(f_s(1-p) - f_s(1-r)) = \frac{\Delta}{p} \theta r$. This is a kind of subtraction since $t = 0$ for $r = p$. Now it is easy to check that
## 4 - EXTENSION OF VON NEUMANN-MORGENSTERN AXIOMS OF UTILITY

Let $X$ be a set containing the possible consequences of potential decisions. A set $\bar{X}$ of uncertain consequences is built over $X$, and is such that

$$X \subseteq \bar{X}; x, y \in X \Rightarrow (x, y) \in \bar{X}$$

where $(x, p, y)$ is short for the uncertain consequence where $x$ occurs with probability $p$ and $y$ with probability $1-p$. The choice among potential decisions is guided by a preference ordering $\geq$ over uncertain consequences. $\geq$ is supposed to be a weak order [4], i.e. is transitive, reflexive, and $\forall x, y \in \bar{X}, x \geq y$ or $y \geq x$. $>$ and $\sim$ respectively denote the strict ordering and equivalence constructed from $\geq$. Notice that :

$$(x, 1, y) \sim x; (x, p, y) \sim (y, 1-p, x).$$

The fundamental axioms of utility theory are then

$$U1) \quad x \sim y \Rightarrow \forall p \in [0,1], \forall z \in X (x, p, z) \sim (y, p, z).$$

$$U2) \quad x > y \Rightarrow \forall p \in (0,1], x > (x, p, y) > y.$$

$$U3) \quad x > y > z \Rightarrow \exists p \in [0,1] \ C \sim (x, p, z)$$

$$U4) \quad ((x, p, y), q, y) \sim (x, p, q, y).$$

Given these axioms, it is proved in [7] that a utility function $u: \bar{X} \to \mathbb{R}$ exists such that $x > y$ iff $u(x) > u(y)$, $x \sim y$ iff $u(x) = u(y)$, and

$$u(x, p, y) = p \ u(x) + (1-p)u(y) \quad (8)$$

Moreover, $u$ is an interval scale [4].

Here we shall focus on axiom U4, which the authors of [7] wished to modify, but, to them, "the mathematical difficulties seem to be considerable" (p. 632). The uncertain consequence $((x, p, y), q, y)$ can be viewed as the result of two successive decisions; the first one enables the second one with probability $p$, and the second one leads to consequence $x$ with probability $p$. On the whole, in this 2-stage decision process, $x$ is got with probability $p$ and $y$ with probability $1-p$, as long as the events following the decisions are independent. Whether this assumption is verified is sometimes up to the decider's judgment.

He may consider on the contrary, that a favorable issue to both decisions (i.e. yielding $x$) is very unlikely, and then only grants a probability

$\max(p+q-1, 0)$ to $x$. Or he is very optimistic and admits that if the most risky decision reaches its goal, then so will the other ones, and so he grants probability $\min(p, q)$ to $x$. More generally, he takes for granted that $P(x) = p \ast q$ where $\ast$ is a Frank triangular-norm. Hence we relax U4 into

$$U4') \quad ((x, p, y), q, y) \sim (x, p \ast q, y)$$

## 5 - CONSTRUCTION OF GENERALIZED UTILITY FUNCTIONS

Following the same reasoning process as in [7], we can prove the following results

**Lemma 2:** $\forall p, q \in (0,1), \forall x, y \in \bar{X}, x > y, p > q \Rightarrow (x, p, y) > (x, q, y)$.

**Proof:** Applying (U2): $x > y \Rightarrow (x, p, y) > y \Rightarrow \forall r \in (0,1), (x, p, y) > ((x, p, y)r, y) = (x, p \ast r, q)$ due to (U4'). It is now enough to solve equation $p \ast r = q$. Using (5), $r$ is unique and is equal to $f_s^{-1}(f_s(q) - f_s(p))$ for $s \neq +\infty, 1, 0$; $r = q$ if $\ast = \min$; $r = q/p$ if $\ast = \text{product}$; $r = 1-p+q$ if $\ast = T_m$. $\forall q \in \mathbb{Q}, \forall p \in \mathbb{P}$.

**Lemma 3:** $\forall p, q \in (0,1) \forall x \neq y \in \bar{X}, x > y$ and $(x, p, y) \sim (x, q, y)$ imply $p = q$.

**Proof:** Obvious with Lemma 2.

**Lemma 4:** $\forall q, \text{ if } p > r \text{ and } x > y \text{ then there is } \alpha \text{ such that} ((x, p, y), q, (x, r, y)) \sim (x, \alpha, y), \alpha \in [0, 1]$.

**Proof:** $x > (x, p, y) > (x, r, y) > y$, and there is some $t$ such that $(x, p, y)$
$(x, t(x, r, y))$ using lemma 3.

Now $(x, t, (x, r, y)) \sim (x, t+r-t \ast r, y)$, with $t+r-t \ast r = 1-(1-r) \ast (1-t)$
since $\ast$ is a Frank t-norm. Hence we must find $t$ such that

$$(1-r) \ast (1-t) = 1-p$$

which yields $t = 1 - f_s^{-1}(f_s(1-p) - f_s(1-r)) = \frac{\Delta}{p} \theta r$. This is a kind of subtraction since $t = 0$ for $r = p$. Now it is easy to check that
$$((x,p,y),q,(x,r,y)) \sim ((x,p\theta r,(x,r,y)),q(x,r,y)) \\ \sim (x,(p\theta r)*q,(x,r,y)) \quad (U4') \\ \sim (x,(p\theta r)*q+r-(r\theta p)*q*r,y)$$

$$\text{Hence } \alpha = (p\theta r)*r+r-(p\theta r)*q*r \quad (9)$$

$$\text{It is unique due to lemma 3.} \qquad \underline{Q.E.D.}$$

Now if $p < r$, using $(x,p,y) \sim (y,1-p,x)$, we come up with the some expression for $\alpha$, where $q$ is changed into $1-q$, $r$ into $p$ and conversely.

For $s = 1$, i.e. $*$ = product we get $\alpha = pq + (1-p)r \ \forall \ p,r$.

This is the usual assumption of independence, and it suggests the usual form of utility.

For $s = 0$, i.e. $a*b = \max(a+b-1,0)$, we get $\forall p,r$

$$\alpha = \min(\max(r,q+p-1),\max(p,r-q)) \quad (10)$$

this is the pessimistic case.

For $s = +\infty$ i.e. $a*b = \min(a,b)$ we get, when $p > r$, $\alpha = \text{med}(p,q,r)$ where med stands for median, it is noticeable that the median is found here instead of an expectation-like expression (a weighted mean) in the usual case. A median is indeed the qualitative counterpart of a mean. More generally for $* = \min$ we get

$$\alpha = \min(\text{med}(p,q,r),\text{med}(p,1-q,r)) \quad (11)$$

Note that when $q = 0$, $\alpha = r$, and when $q = 1$, $\alpha = p$, so that $\alpha$ moves from $r$ to $p$ when $q$ moves from $0$ to $1$. Actually, for $0 < s < +\infty$, $\alpha$ is strictly increasing with $q$ when $p > r$.

In the following let us assume that $\underline{x}$ contains a maximal element $x_1$ and a minimum element $x_0$. We introduce a function $u_s: \underline{x} \to [0,1]$ with $u_s(x_0) = 0$, $u_s(x_1) = 1$, $u_s(x) = q$ as soon as $x = (x_1,q,x_0)$. Parameter $s$ is that of the separability operation $*$.

The following result is obtained :

**Theorem 1**: $\forall s > 0, s < +\infty$, then

$$\forall x > y, u_s(x) > u_s(y) \quad (12)$$

$$\forall x > y, \forall q \in [0,1]$$

$$u_s(x,q,y) = (u_s(x)\theta u_s(y)) * q + u_s(y) - (u_s(x)\theta u_s(y)) * q * u_s(y) \quad (13)$$

*Proof*: (12) is obvious by using $x = (x_1,p,x_0)$, $y = (x_1,q,x_0)$.

(13) is obvious using the expression of $\alpha$ from Lemma 3.

$$\underline{Q.E.D.}$$

For $s = 1$ we get the usual utility function (8).

Now changing the scale $[0,1]$ into $I = [a_0,a_1]$ comes down to changing operations $\circ = \Theta,*$, etc... into $\circ_I$ in $I$, such that $\forall a,b \in [a_0,a_1]$,

$$a \circ_I b = a_0 + (a_1 - a_0) \left[ \frac{a - a_0}{a_1 - a_0} \circ \frac{b - a_0}{a_1 - a_0} \right]$$

and utility function $u_s(x)$ into $u_s(x,I) = (a_1 - a_0)u_s(x) + a_0 u_s(.,I)$ also satisfies (12) and a generalized from of (13), changing $\Theta, *$ and $q$ into $\Theta_I, *I$ and $a_1q + a_0(1-q)$ respectively. We can also prove a unicity theorem :

**Theorem 2**: if $u_s(.,I)$ and $u_s(.,I') = u'_s$ satisfy the generalized form of (13), and also (12), then $\forall \alpha,\beta : u_s(.,I) = \alpha u_s(.,I') + \beta$.

*Proof*: there is an increasing bijection $f$ between $u$ and $u'$ such that

$$f(a_0) = a'_0, f(a'_1) \text{ and } u' = f \circ u. \text{ Particularly}$$

$$f(u_s(x_1,q,x_0)) = f((a_1-a_0)q+a_0) \\ = u'_s(x_1,q,x_0) = [f(a_1)-f(a_0)]q+f(a_0)$$

$$\text{Hence } f \text{ is of the form } f(a) = \alpha a + \beta \ \forall a \in I. \quad \underline{Q.E.D.}$$

Hence, $u_s(.,I)$ is an interval scale.

## 5 - DISCUSSION

These results indicate that it is possible to build utility functions without the independence assumption, only assuming a separability condition. Note that theorems 1 and 2 are valid only for $0 < s < +\infty$. Indeed for $s = 0$ and $s = +\infty$, (12) does not hold. i.e. we can have $x > y$ and $u(x) = u(y)$.
$$ ((x,p,y),q,(x,r,y)) \sim ((x,p\theta r,(x,r,y)),q(x,r,y)) \\ \sim (x,(p\theta r)*q,(x,r,y)) \quad (U4') \\ \sim (x,(p\theta r)*q+r-(r\theta p)*q*r,y) $$

$$ \text{Hence } \alpha = (p\theta r)*r+r-(p\theta r)*q*r \quad (9) $$

$$ \text{It is unique due to lemma 3.} \qquad \underline{Q.E.D.} $$

Now if $p < r$, using $(x,p,y) \sim (y,1-p,x)$, we come up with the some expression for $\alpha$, where $q$ is changed into $1-q$, $r$ into $p$ and conversely.

$$ \text{For } s = 1, \text{ i.e. } * = \text{product we get } \alpha = pq + (1-p)r \ \forall \ p,r. $$

This is the usual assumption of independence, and it suggests the usual form of utility.

$$ \text{For } s = 0, \text{ i.e. } a*b = \max(a+b-1,0), \text{ we get } \forall p,r $$

$$ \alpha = \min(\max(r,q+p-1),\max(p,r-q)) \quad (10) $$

this is the pessimistic case.

$$ \text{For } s = +\infty, \text{ i.e. } a*b = \min(a,b) \text{ we get, when } p > r, \alpha = \text{med}(p,q,r) $$

where med stands for median, it is noticeable that the median is found here instead of an expectation-like expression (a weighted mean) in the usual case. A median is indeed the qualitative counterpart of a mean. More generally for * = min we get

$$ \alpha = \min(\text{med}(p,q,r),\text{med}(p,1-q,r)) \quad (11) $$

Note that when $q = 0, \alpha = r$, and when $q = 1, \alpha = p$, so that $\alpha$ moves from $r$ to $p$ when $q$ moves from $0$ to $1$. Actually, for $0 < s < +\infty$, $\alpha$ is strictly increasing with $q$ when $p > r$.

In the following let us assume that $\underline{x}$ contains a maximal element $x_1$ and a minimum element $x_0$. We introduce a function $u_s: \underline{x} \to [0,1]$ with $u_s(x_0) = 0$, $u_s(x_1) = 1$, $u_s(x) = q$ as soon as $x = (x_1,q,x_0)$. Parameter $s$ is that of the separability operation *.

The following result is obtained :

**Theorem 1** : $\forall s > 0, s < +\infty$, then
$\forall x > y, u_s(x) > u_s(y)$ (12)

$$ \forall x > y, \forall q \in [0,1] \\ u_s(x,q,y) = (u_s(x)\theta u_s(y))*q+u_s(y)-(u_s(x)\theta u_s(y))*q*u_s(y) \quad (13) $$

**Proof** : (12) is obvious by using $x = (x_1,p,x_0)$, $y = (x_1,q,x_0)$.

(13) is obvious using the expression of $\alpha$ from Lemma 3.

For $s = 1$ we get the usual utility function (8).

Now changing the scale $[0,1]$ into $I = [a_0,a_1]$ comes down to changing operations $O = \Theta,*$, etc... into $O_I$ in $I$, such that $\forall a,b \in [a_0,a_1]$,

$$ a \circ_I b = a_0 + (a_1-a_0) \left[ \frac{a-a_0}{a_1-a_0} \circ \frac{b-a_0}{a_1-a_0} \right] $$

and utility function $u_s(x)$ into $u_s(x,I) = (a_1-a_0)u_s(x)+a_0;u_s(.,I)$ also satisfies (12) and a generalized from of (13), changing $\Theta,*$ and $q$ into $\Theta_I,*I$ and $a_1q+a_0(1-q)$ respectively. We can also prove a unicity theorem :

**Theorem 2** : if $u_s(.,I)$ and $u_s(.,I') = u'_s$ satisfy the generalized form of (13), and also (12), then $\forall \alpha,\beta : u_s(.,I) = \alpha u_s(.,I') + \beta$.

**Proof** : there is an increasing bijection $f$ between $u$ and $u'$ such that

$$ \begin{align*} f(a_0) &= a'_0, f(a'_1) \text{ and } u' = f \circ u. \text{ Particularly} \\ f(u_s(x_1,q,x_0)) &= f((a_1-a_0)q+a_0)) \\ &= u'_s(x_1,q,x_0) = [f(a_1)-f(a_0)]q+f(a_0) \end{align*} $$

$$ \text{Hence } f \text{ is of the form } f(a) = \alpha a + \beta \ \forall a \in I. \qquad \underline{Q.E.D.} $$

Hence, $u_s(.,I)$ is an interval scale.

## 5 - DISCUSSION

These results indicate that it is possible to build utility functions without the independence assumption, only assuming a separability condition. Note that theorems 1 and 2 are valid only for $0 < s < +\infty$. Indeed for $s = 0$ and $s = +\infty$, (12) does not hold. i.e. we can have $x > y$ and $u(x) = u(y)$.
This occurs

$$
- \text{ for } s = +\infty, x = (x_1, p, x_0) > y = (x_1, r, x_0) \text{ and } 0 < p < \min(p, r) = r
$$

$$
- \text{ for } s = 0, \text{ with the same conventions and } 0 < q < 1 + r-p,
$$

$$
\text{we then observe : } u_s(x,q,y) = r = u_s(y), \text{ while } (x,q,y) > y.
$$

In such cases, one may wish to relax axiom U2, changing > into ≥.

This behavior of the utility function means that if x becomes very unlikely (q is small enough), the utility of (x,q,y) cannot be distinguished from that of y (with certainty). Of course this attempt to extend the concept of utility could be considered in the situation when X has no maximal nor minimal elements. Moreover, in this approach we have taken for granted the existence of probabilities of consequences; modern approaches (e.g. [4]) construct both probability and utility functions from purely qualitative arguments. However, as Fine [2] pointed out, to justify additive probability requires the use of complicated or un-natural axioms. A more intuitive setting for extending utility theory in the spirit of separability assumptions may be that of decomposable measures [1] where the additivity axiom of probability measures is generalized by means of a triangular co-norm [6].

REFERENCES

[1] Dubois, D., Prade, H. A class of fuzzy measures based on triangular norms. Int. J. General Systems 8, 43-61, 1982.

[2] - Fine, T. Theories of Probability. Academic Press, 1973.

[3] - Frank, M.J. On the simultaneous associativity of F(x,y) and x+y-F(x,y).
Equationes Math. 19, 194-226, 1979.

[4] - Kranz, D.H., Luce, R.D., Suppes, S., Tversky, A. Foundations of Measurement. Vol. 1, Academic Press, 1971.

[5] - Savage, L.J. The Foundations of Statistics, Dover, 1972.

[6] - Schweizer, B., Sklar, A. Probabilistic Metric Spaces. North-Holland, 1983.

[7] - Von Neumann, J., Morgenstern O. Theory of Games and Economic Behavior, Princeton, Univ. Press, 1953.

A MULTI-LAYER FUZZY B-D TYPE ALGEBRAIC STRUCTURE
IN THE COMPLEX SYSTEM

Yang Bing-ru

Tianjin Science and Technology College, Tianjin,
China

ABSTRACT

The purpose of this article is to seek for a method in the research of the fuzzy control decision in the complex system. That is to study the numerical relations among their variables through examining the logical relationships among the data. Therefore, a multi-layer fuzzy B-D type algebraic system is set up, and with the help of the language method the logical analysis for the complex system is realized.

I. Mathematical Model Setting

The multi-layer logical structure of the complex system $\mathcal{F}$ is shown in the following imitative form:

<figure><img src="image_3.png" /></figure>

Notes:

1. the state space of the input variables.

2. $\mathcal{X}$ is the language variables (or the true-false variables of the language).

3. $\mathcal{X}^n$ is one layer fuzzy variable (input).

4. two layers fuzzy variable.

5. 2 layers fuzzy variable.

6. $y_i^n$ is the fuzzy variable (output).

7. $y_i^n$ is the language variables (or the true-false variables of the language).

8. the state space of the output variables.
This occurs

$$
- \text{ for } s = +\infty, x = (x_1, p, x_0) > y = (x_1, r, x_0) \text{ and } 0 < p < \min(p, r) = r
$$

$$
- \text{ for } s = 0, \text{ with the same conventions and } 0 < q < 1 + r-p,
$$

$$
\text{we then observe : } u_s(x,q,y) = r = u_s(y), \text{ while } (x,q,y) > y.
$$

In such cases, one may wish to relax axiom U2, changing > into ≥.
This behavior of the utility function means that if x becomes very unlikely
(q is small enough), the utility of (x,q,y) cannot be distinguished from
that of y (with certainty). Of course this attempt to extend the concept
of utility could be considered in the situation when X has no maximal nor
minimal elements. Moreover, in this approach we have taken for granted the
existence of probabilities of consequences; modern approaches (e.g. [4])
construct both probability and utility functions from purely qualitative
arguments. However, as Fine [2] pointed out, to justify additive probabi-
lity requires the use of complicated or un-natural axioms. A more intui-
tive setting for extending utility theory in the spirit of separability
assumptions may be that of decomposable measures [1] where the additivi-
ty axiom of probability measures is generalized by means of a triangular
co-norm [6].

REFERENCES

[1] Dubois, D., Prade, H. A class of fuzzy measures based on triangular norms. Int. J. General Systems 8, 43-61, 1982.

[2] - Fine, T. Theories of Probability. Academic Press, 1973.

[3] - Frank, M.J. On the simultaneous associativity of F(x,y) and x+y-F(x,y).
Equationes Math. 19, 194-226, 1979.

[4] - Kranz, D.H., Luce, R.D., Suppes, S., Tversky, A. Foundations of Measurement. Vol. 1, Academic Press, 1971.

[5] - Savage, L.J. The Foundations of Statistics, Dover, 1972.

[6] - Schweizer, B., Sklar, A. Probabilistic Metric Spaces. North-Holland, 1983.

[7] - Von Neumann, J., Morgenstern O. Theory of Games and Economic Behavior, Princeton, Univ. Press, 1953.

A MULTI-LAYER FUZZY B-D TYPE ALGEBRAIC STRUCTURE
IN THE COMPLEX SYSTEM

Yang Bing-ru

Tianjin Science and Technology College, Tianjin,
China

ABSTRACT

The purpose of this article is to seek for a method in the research of the fuzzy control decision in the complex system. That is to study the numerical relations among their variables through examining the logical relationships among the data. Therefore, a multi-layer fuzzy B-D type algebraic system is set up, and with the help of the language method the logical analysis for the complex system is realized.

I. Mathematical Model Setting

The multi-layer logical structure of the complex system $\mathcal{F}$ is shown in the following imitative form:

<figure><img src="image_3.png" /></figure>

Notes:

1. the state space of the input variables.

2. $\mathcal{X}$ is the language variables (or the true-false variables of the language).

3. $\mathcal{X}^n$ is one layer fuzzy variable (input).

4. two layers fuzzy variable.

5. 2 layers fuzzy variable.

6. $y^n$ is the fuzzy variable (output).

7. $y$ is the language variables (or the true-false variables of the language).

8. the state space of the output variables.
In the above model, for the horizontal levels the main object of the study is the relation between the layers of the language variable $\mathcal{X}_i$ and the fuzzy variable $x_i^j$, or they could be considered as two typical layers; and the relation between the i layer and the i+1 (i=1,...,k-1) fuzzy variables is the reappearance and wideness of these two typical layers. For the vertical layers the main object of the study is the logical relations among the several internal fuzzy variables (called the B layer) of the same language variable $\mathcal{X}_i$ in the input state space, as well as the logical relations among the fuzzy variables (called the D layer) of the different species in the different language variables.

Practically the above modular form is set up according to the concept called the turning mapping.

**Definition1.** Let us call mapping $\mathcal{X} \to \mathcal{U}$ be the turning mapping, if $\mathcal{U}$ is the field theory of the basic variables, $\mathcal{W}$ is the word set reflecting the corresponding changing state of the given state space and $\mathcal{V} \subseteq \mathcal{U}$, there is a unique word in correspondence, that is, $f(u)=w$; vice versa.

For example, considering the test of the vinylox formalism as a complex system, the following table show how the above steps are carried out:

<table><caption>Table1. The state words(fuzzy variables)</caption><thead><tr><th>S</th><th>x<sub>i</sub><sup>(1)</sup></th><th>x<sub>i</sub><sup>(2)</sup></th><th>x<sub>i</sub><sup>(3)</sup></th><th>x<sub>i</sub><sup>(4)</sup></th><th>x<sub>i</sub><sup>(5)</sup></th><th>x<sub>i</sub><sup>(6)</sup></th></tr></thead><tbody><tr><td>x<sub>i</sub><sup>(1)</sup></td><td>y<sub>i</sub><sup>(1)</sup></td><td>y<sub>i</sub><sup>(2)</sup></td><td>y<sub>i</sub><sup>(3)</sup></td><td>y<sub>i</sub><sup>(4)</sup></td><td>y<sub>i</sub><sup>(5)</sup></td><td>y<sub>i</sub><sup>(6)</sup></td></tr><tr><td>x<sub>i</sub><sup>(2)</sup></td><td>y<sub>i</sub><sup>(1)</sup></td><td>y<sub>i</sub><sup>(2)</sup></td><td>y<sub>i</sub><sup>(3)</sup></td><td>y<sub>i</sub><sup>(4)</sup></td><td>y<sub>i</sub><sup>(5)</sup></td><td>y<sub>i</sub><sup>(6)</sup></td></tr><tr><td>x<sub>i</sub><sup>(3)</sup></td><td>y<sub>i</sub><sup>(1)</sup></td><td>y<sub>i</sub><sup>(2)</sup></td><td>y<sub>i</sub><sup>(3)</sup></td><td>y<sub>i</sub><sup>(4)</sup></td><td>y<sub>i</sub><sup>(5)</sup></td><td>y<sub>i</sub><sup>(6)</sup></td></tr></tbody></table>

T(minute) — reaction time.

S(g/l) — density of formaldehyde.

K(gram molecule) — degree of formalism.

x<sub>i</sub><sup>(1)</sup>: very short,...; x<sub>i</sub><sup>(2)</sup>: not thick, ...;

y<sub>i</sub><sup>(1)</sup>: low, ... the meanings of the words could be given in a corresponding consistent function.

## II. E-D type calculus system

### (1) D-layer calculus:

The discussion is limited in $\mathcal{X}_i$ (the rest may be inferred by ann-ology), and every $x_i^j$ is explained in the meanings of the language variables. Assume $\mathcal{X}_i = (x_i^{(1)}, x_i^{(2)}, \dots, x_i^{(k)}$, including three operators $\oplus, \ominus, \otimes$ and elements $\oplus, \ominus, \otimes$ and so on. As every variable is in its definite occasion, take and only take one in the r-state and stipulate:

**Definition2.** Let $x_i^{(j)} \in \int_0^1 \mathcal{U}_i \mathcal{U}_i^*(u)/u$ , then

$$x_i^{(1)} \oplus x_i^{(2)} \triangleq \left\{ \begin{array}{ll} \int_0^1 0/u & (p \neq q) \\ \int_0^1 (\mathcal{U}_i \mathcal{U}_i^*(u) \wedge \mathcal{U}_i \mathcal{U}_i^*(u)) / u & (p=q) \end{array} \right.$$

$$x_i^{(3)} \oplus x_i^{(4)} \triangleq \int_0^1 (\mathcal{U}_i \mathcal{U}_i^*(u) \vee \mathcal{U}_i \mathcal{U}_i^*(u) / u)$$

$$\overline{x_i^{(5)}} \triangleq \int_0^1 \{ \mathcal{U}_i \mathcal{U}_i^*(u) \vee \dots \vee \mathcal{U}_i \mathcal{U}_i^*(u) \vee \dots \vee \mathcal{U}_i \mathcal{U}_i^*(u) \} / u$$

$$0_i \triangleq \int_0^1 0/u$$

$$1_i \triangleq \int_0^1 \{ \mathcal{U}_i \mathcal{U}_i^*(u) \vee \dots \vee \mathcal{U}_i \mathcal{U}_i^*(u) \} / u$$

In it, $\oplus \in \mathcal{U}_i(\mathcal{U}_i)$ is the field theory of $\mathcal{X}_i$, $\vee \in \mathcal{Max}$, $\wedge \in \mathcal{Min}$.

From the law of the equivalent power, the simple expressive formula or the above can be obtained:

$$x_i^{(p)} \oplus x_i^{(q)} \triangleq \left\{ \begin{array}{ll} 0_i & (p \neq q) \\ x_i^{(p)} & (p=q) \end{array} \right.$$

$$x_i^{(m)} \oplus x_i^{(n)} \triangleq \left\{ \begin{array}{ll} \int_0^1 (\mathcal{U}_i \mathcal{U}_i^*(u) \vee \mathcal{U}_i \mathcal{U}_i^*(u)) / u & (p \neq q) \\ x_i^{(m)} & (p=q) \end{array} \right.$$

$$\overline{x_i^{(s)}} \triangleq x_i^{(a)} \oplus \dots \oplus x_i^{(s)} \oplus \dots \oplus x_i^{(m)} \triangleq \bigoplus_{j \neq p} x_i^{(j)}$$

$$1_i \triangleq x_i^{(a)} \oplus \dots \oplus x_i^{(m)} \triangleq \bigoplus_{j=1}^{k} x_i^{(j)}$$

Its clear that $0_i$ and $1_i$ are two special elements different from $x_i^{(j)}$ ($j=1,...,r$). They are stipulated as =

<table><caption>Table 2</caption><thead><tr><th>Θ (Θ)</th><th>0<sub>i</sub></th><th>1<sub>i</sub></th></tr></thead><tbody><tr><td>0<sub>i</sub></td><td>0<sub>i</sub>(0<sub>i</sub>)</td><td>0<sub>i</sub>(1<sub>i</sub>)</td></tr><tr><td>1<sub>i</sub></td><td>0<sub>i</sub>(1<sub>i</sub>)</td><td>1<sub>i</sub>(1<sub>i</sub>)</td></tr></tbody></table>

Note: Here only the continuous model is discussed, as for the discrete model it is on the analogy of this.

### (2) D-layer calculus:

In the calculus of the fuzzy variables in the different species, every $x_i^{(j)}, x_j^{(k)}, \dots$ is explained in the meanings of the true-false variables of the boolean language. It is provided that these three operators $\oplus, \wedge, -$ are "and", "or" and "non", the logical meanings respectively. The fuzzy variables in different language variables are linked by these operators. The output variables are expressed by the specific logical relations among them.

It is ordered that $\gamma(x_i^{(j)}) = \int_0^1 \mathcal{U}_i \mathcal{U}_i^*(u) \vee \gamma(x_j^{(k)}) = \int_0^1 \mathcal{U}_j \mathcal{U}_j^*(u)$, in it, $\gamma_i \in \mathcal{U}_i \mathcal{U}_i^*(u)$ is the points in $[0,1]$, and $\mathcal{U}_i \mathcal{U}_i^*(u)$ is the classes of their qualifications in $\gamma(x_i^{(j)})$.
In the above model, for the horizontal levels the main object of the study is the relation between the layers of the language variable $\mathcal{X}_i$ and the fuzzy variable $x_i^j$, or they could be considered as two typical layers; and the relation between the i layer and the i+1 (i=1,...,k-1) fuzzy variables is the reappearance and wideness of these two typical layers. For the vertical layers the main object of the study is the logical relations among the several internal fuzzy variables (called the B layer) of the same language variable $\mathcal{X}_i$ in the input state space, as well as the logical relations among the fuzzy variables (called the D layer) of the different species in the different language variables.

Practically the above modular form is set up according to the concept called the turning mapping.

**Definition1.** Let us call mapping $\mathcal{X} \to \mathcal{U}$ be the turning mapping, if $\mathcal{U}$ is the field theory of the basic variables, $\mathcal{W}$ is the word set reflecting the corresponding changing state of the given state space and $\mathcal{V}$ is $\mathcal{U}$, there is a unique word in correspondence, that is, $f(u)=w$; vice versa.

For example, considering the test of the vinyron formalism as a complex system, the following table show how the above steps are carried out:

<table><caption>Table1. The state words(fuzzy variables)</caption><thead><tr><th>S</th><th>x<sub>i</sub><sup>(1)</sup></th><th>x<sub>i</sub><sup>(2)</sup></th><th>x<sub>i</sub><sup>(3)</sup></th><th>x<sub>i</sub><sup>(4)</sup></th><th>x<sub>i</sub><sup>(5)</sup></th><th>x<sub>i</sub><sup>(6)</sup></th></tr></thead><tbody><tr><td>x<sub>i</sub><sup>(1)</sup></td><td>y<sub>i</sub><sup>(1)</sup></td><td>z<sub>i</sub><sup>(1)</sup></td><td>y<sub>i</sub><sup>(2)</sup></td><td>y<sub>i</sub><sup>(3)</sup></td><td>y<sub>i</sub><sup>(4)</sup></td><td>y<sub>i</sub><sup>(5)</sup></td></tr><tr><td>x<sub>i</sub><sup>(2)</sup></td><td>y<sub>i</sub><sup>(2)</sup></td><td>y<sub>i</sub><sup>(3)</sup></td><td>y<sub>i</sub><sup>(2)</sup></td><td>y<sub>i</sub><sup>(4)</sup></td><td>y<sub>i</sub><sup>(5)</sup></td><td>y<sub>i</sub><sup>(5)</sup></td></tr><tr><td>x<sub>i</sub><sup>(3)</sup></td><td>y<sub>i</sub><sup>(3)</sup></td><td>y<sub>i</sub><sup>(3)</sup></td><td>y<sub>i</sub><sup>(3)</sup></td><td>y<sub>i</sub><sup>(4)</sup></td><td>y<sub>i</sub><sup>(5)</sup></td><td>y<sub>i</sub><sup>(5)</sup></td></tr></tbody></table>

T(minute) — reaction time.

S(g/l) — density of formaldehyde.

K(gram molecule) — degree of formalizing.

x<sub>i</sub><sup>(1)</sup>: very short,...; x<sub>i</sub><sup>(2)</sup>: not thick, ...;

y<sub>i</sub><sup>(1)</sup>: low, ... the meanings of the words could be given in a corresponding consistent function.

## II. E-D type calculus system

### (1) D-layer calculus:

The discussion is limited in $\mathcal{X}_i$ (the rest may be inferred by ann-ology), and every $x_i^j$ is explained in the meanings of the language variables. Assume $\mathcal{X}_i = (x_i^{(1)}, x_i^{(2)}, ..., x_i^{(k)}$, including three operators $\oplus, \ominus, \wedge$ and elements $\oplus, \wedge$ and $\ominus$) and so on. As every variable is in its definite occasion, take and only take one in the r-state and stipulate:

**Definition2.** Let $x_i^{(k)} \in \int_0^1 \mathcal{U}_i \mathcal{U}_i^*(u)/u$ , then

$$x_i^{(k)} \oplus x_i^{(k)} \triangleq \left\{ \begin{array}{ll} \int_0^1 0/u & (p \neq q) \\ \int_0^1 (\mathcal{U}_i \mathcal{U}_i^*(u) \wedge \mathcal{U}_i \mathcal{U}_i^*(u)) / u & (p=q) \end{array} \right.$$

$$x_i^{(k)} \oplus x_i^{(k)} \triangleq \int_0^1 (\mathcal{U}_i \mathcal{U}_i^*(u) \vee \mathcal{U}_i \mathcal{U}_i^*(u)) / u$$

$$\overline{x_i^{(k)}} \triangleq \int_0^1 \{ \mathcal{U}_i \mathcal{U}_i^*(u) \vee \dots \vee \mathcal{U}_i \mathcal{U}_i^*(u) \vee \dots \vee \mathcal{U}_i \mathcal{U}_i^*(u) \} / u$$

$$0_i \triangleq \int_0^1 0/u$$

$$1_i \triangleq \int_0^1 \{ \mathcal{U}_i \mathcal{U}_i^*(u) \vee \dots \vee \mathcal{U}_i \mathcal{U}_i^*(u) \} / u$$

In it, $\oplus \in \mathcal{U}_i(\mathcal{U}_i)$ is the field theory of $\mathcal{X}_i$, $\vee \in \mathcal{Max}$, $\wedge \in \mathcal{Min}$.

From the law of the equivalent power, the simple expressive formula or the above can be obtained:

$$x_i^{(k)} \oplus x_i^{(k)} \triangleq \left\{ \begin{array}{ll} 0_i & (p \neq q) \\ x_i^{(k)} & (p=q) \end{array} \right.$$

$$x_i^{(k)} \oplus x_i^{(k)} \triangleq \left\{ \begin{array}{ll} \int_0^1 (\mathcal{U}_i \mathcal{U}_i^*(u) \vee \mathcal{U}_i \mathcal{U}_i^*(u)) / u & (p \neq q) \\ x_i^{(k)} & (p=q) \end{array} \right.$$

$$\overline{x_i^{(k)}}^* \triangleq x_i^{(k)} \oplus \dots \oplus x_i^{(k)} \oplus \dots \oplus x_i^{(k)} \triangleq \bigoplus_{j \neq p} x_i^{(j)}$$

$$1_i \triangleq x_i^{(k)} \oplus \dots \oplus x_i^{(k)} \triangleq \bigoplus_{j=1}^{k} x_i^{(j)}$$

Its clear that $0_i$ and $1_i$ are two special elements different from $x_i^{(k)}$ ($j=1,...,r$). They are stipulated as =:

<table><caption>Table 2</caption><thead><tr><th>0 (•)</th><th>0<sub>i</sub></th><th>1<sub>i</sub></th></tr></thead><tbody><tr><td>0<sub>i</sub></td><td>0<sub>i</sub>(0<sub>i</sub>)</td><td>0<sub>i</sub>(1<sub>i</sub>)</td></tr><tr><td>1<sub>i</sub></td><td>0<sub>i</sub>(1<sub>i</sub>)</td><td>1<sub>i</sub>(1<sub>i</sub>)</td></tr></tbody></table>

Note: Here only the continuous model is discussed, as for the discrete model it is on the analogy of this.

### (2) D-layer calculus:

In the calculus of the fuzzy variables in the different species, every $x_i^{(k)}, x_j^{(k)}, \dots$ is explained in the meanings of the true-false variables of the boolean language. It is provided that these three operators $\oplus, \wedge, =$, are "and", "or" and "non", the logical meanings respectively. The fuzzy variables in different language variables are linked by these operators. The output variables are expressed by the specific logical relations among them.

It is ordered that $\vee_i(x_i^{(k)}) = \int_0^1 \mathcal{U}_i \mathcal{U}_i^*(u) \vee \dots \vee \mathcal{U}_i \mathcal{U}_i^*(u)$, in it, $\vee_i$ is the points in $[0,1]$, and $\mathcal{U}_i$ is the classes of their qualifications in $\vee_i(x_i^{(k)})$.
assume $X$ in the language value of the Boolean language variable $\mathcal{F}$.
$T$ is that of the Boolean true and false language variables $\mathcal{F}$, we shall have the homomorphic mapping $\nu$ from the $\mathcal{F}$ dictionary to the $\mathcal{F}$ dictionary, so $\nu(x)=T$.

For $x_1^i \in \mathcal{F}$ and $x_2^j \in \mathcal{F}$ the following regulations are given:

**Definition 3.** Let $\nu(x_1^i) \equiv \int_0^1 \alpha_i / \nu_i$, $\nu(x_2^j) \equiv \int_0^1 \beta_j / \nu_j$, then

$$x_1^{(i)} \cdot x_2^{(j)} \equiv \int_0^1 (\alpha_i \wedge \beta_j) / (\nu_i \wedge \nu_j) \\ x_1^{(i)} + x_2^{(j)} \equiv \int_0^1 (\alpha_i \wedge \beta_j) / (\nu_i \vee \nu_j) \\ \overline{\Sigma}^{(i)} \equiv \int_0^1 \alpha_i / (1 - \nu_i)$$

Here $\overline{\Sigma}^{(i)}$ and $\overline{\Sigma}^{(j)}$ are two different expressions of the meanings of words. In order to tell one from another its better to call the former (for the whole system $\mathcal{F}$) the whole complement and the latter (for $\mathcal{F}$) the relative complement. In it

$$\wedge \equiv \text{conjunction} \quad (\text{be different from } \wedge)$$

$$\vee \equiv \text{disjunction} \quad (\text{be different from } \vee)$$

$$\mu_T(\nu) = \begin{cases} 0 & \text{if } \nu \le \frac{\underline{a}}{1-\underline{a}} \\ 2 \left( \frac{\nu-\underline{a}}{1-\underline{a}} \right)^2 & \text{if } \underline{a} \le \nu \le \frac{\underline{a}+1}{2} \\ 1-2 \left( \frac{\nu-1}{1-\underline{a}} \right)^2 & \text{if } \frac{\underline{a}+1}{2} \le \nu \le 1 \end{cases}$$

In the D-layer calculus, there are two elements 0 and 1, they are two special true-false values (corresponding to the maximal and minimal elements of the lattice constructed by the fuzzy subset of the interval $(0,1)$).

$$0 \equiv \int_0^1 \alpha_i / \nu_i, \quad 1 \equiv \int_1^1 \alpha_i / \nu_i$$

For 0 and 1 the regulation is as follows:

<table><caption>Table 3</caption><thead><tr><th>.(+)</th><th>0</th><th>1</th></tr></thead><tbody><tr><td>0</td><td>0(0)</td><td>0(1)</td></tr><tr><td>1</td><td>0(1)</td><td>1(1)</td></tr></tbody></table>

### (3) B-D type calculus system:

The B-D type calculus is mingled with the B-layer calculus and the D-layer calculus. For the calculus of the whole complex system, it is the development of the basic calculuses mentioned above.

Through the B-D type calculus not only the fuzzy formula $F$ could be simplified, but also the degree of truth and accuracy of the given logical and numerical relations by $F$ could be obtained.

For example, from the above vinyton formaling test tables it can be got that

$$\begin{align*} \mathcal{F}^{(i)} &= x_1^{(i)} \cdot x_2^{(i)} + x_3^{(i)} \cdot x_4^{(i)} + x_5^{(i)} \cdot x_6^{(i)} + x_7^{(i)} \cdot x_8^{(i)} \\ (\mathcal{F})^{(i)} &= x_1^{(i)} \cdot x_2^{(i)} + x_3^{(i)} \cdot (x_4^{(i)} \oplus x_5^{(i)} \oplus x_6^{(i)}) \\ (\mathcal{G})^{(i)} &= x_1^{(i)} \cdot x_2^{(i)} + x_3^{(i)} \cdot 1 \\ (\mathcal{H})^{(i)} &= x_1^{(i)} \cdot x_2^{(i)} + x_3^{(i)} \end{align*}$$

(Here the calculus character on which the deformation is based will be worked out below).

The consequent conclusion is that the reaction time should be considerably long ($x_1^{(i)}$) in order to let the degree of the formaling reach the highest point ($y^{(i)}$), or the reaction time is long ($x_2^{(i)}$) while the density of formaldehyde is medium ($x_3^{(i)}$). The language true and false value of $y^{(i)}$ expressive formula can be calculated, so the corresponding degree of faithfulness to the control decision provided by the above conclusion can be obtained.

Based on this, the field theory intervals can be divided more precisely and finely, B-D type calculus is carried out again, and moved into the deeper layers, then the "ideal state" is approached by and by the optimum control is realized.

### III. The axiomatized structure of the B-D type calculus theory

**Definition 4.** Let $\mathfrak{B}$ is the set of some fuzzy objects, it has two definite elements $\mathfrak{U}$ and $\mathfrak{F}$, and $\mathfrak{U} \neq \mathfrak{F}$. If three algebraic calculuses $\Psi$, $\Lambda$ and $\Xi$ are defined, then for $\forall \mathfrak{A}, \mathfrak{B}, \mathfrak{C} \in \mathfrak{B}$, the following 4 axioms are reasonable:

$$(\text{Law of commutation}) \qquad \begin{array}{l} \mathfrak{A} \Psi \mathfrak{B} = \mathfrak{B} \Psi \mathfrak{A} \\ \mathfrak{A} \cap \mathfrak{B} = \mathfrak{B} \cap \mathfrak{A} \end{array}$$

$$(\text{Law of distribution}) \qquad \begin{array}{l} \mathfrak{A} \Psi (\mathfrak{B} \cap \mathfrak{C}) = (\mathfrak{A} \Psi \mathfrak{B}) \cap (\mathfrak{A} \Psi \mathfrak{C}) \\ \mathfrak{A} \cap (\mathfrak{B} \Psi \mathfrak{C}) = (\mathfrak{A} \cap \mathfrak{B}) \Psi (\mathfrak{A} \cap \mathfrak{C}) \end{array}$$

$$(\text{Law of identity}) \qquad \begin{array}{l} \mathfrak{A} \Psi \Phi = \mathfrak{A} \\ \mathfrak{A} \cap \mathfrak{U} = \mathfrak{A} \end{array}$$

$$(\text{Law of complementary}) \qquad \begin{array}{l} \mathfrak{A} \Psi \overline{\mathfrak{A}} = \mathfrak{U} \\ \mathfrak{A} \cap \overline{\mathfrak{A}} = \Phi \end{array}$$

Then $\langle \mathfrak{B}, \Psi, \Lambda, \Xi, \mathfrak{U}, \Phi \rangle$ form a fuzzy Boolean algebra.

### (1) B-layer calculus theory:

From the definition 2 the following properties can be induced:

$$(1a) \quad x_1^{(i)} \oplus x_2^{(i)} = x_1^{(i)}$$

$$(1b) \quad x_1^{(i)} \oplus x_2^{(i)} = x_1^{(i)}$$

$$(2a) \quad x_1^{(i)} \oplus x_2^{(i)} = x_1^{(i)} \oplus x_2^{(i)}$$

$$(2b) \quad x_1^{(i)} \oplus x_2^{(i)} = x_1^{(i)} \oplus x_2^{(i)}$$

$$(3a) \quad x_1^{(i)} \oplus (x_2^{(i)} \oplus x_3^{(i)}) = (x_1^{(i)} \oplus x_3^{(i)}) \oplus x_2^{(i)}$$

$$(4a) \quad x_1^{(i)} \oplus (x_2^{(i)} \oplus x_3^{(i)}) = (x_1^{(i)} \oplus x_3^{(i)}) \oplus (x_2^{(i)} \oplus x_3^{(i)})$$

$$(4b) \quad x_1^{(i)} \oplus (x_2^{(i)} \oplus x_3^{(i)}) = (x_1^{(i)} \oplus x_3^{(i)}) \oplus (x_2^{(i)} \oplus x_3^{(i)})$$

$$(5a) \quad x_1^{(i)} \oplus 0_i = x_1^{(i)}$$

$$(5b) \quad x_1^{(i)} \oplus 1_i = x_1^{(i)}$$

$$(6a) \quad x_1^{(i)} \oplus \overline{x_1^{(i)}} = 1_i$$

$$(6b) \quad x_1^{(i)} \oplus \overline{x_1^{(i)}} = 0_i$$
assume $X$ in the language value of the Boolean language variable $\mathcal{F}$.
$T$ is that of the Boolean true and false language variables $\mathcal{F}$, we shall have the homomorphic mapping $\nu$ from the $\mathcal{F}$ dictionary to the $\mathcal{F}$ dictionary, so $\nu(x)=T$.

For $x_1^i \in X$ and $x_2^j \in Y$ the following regulations are given:

**Definition 3.** Let $\nu(x_1^i, \dots, x_i^i) \equiv \frac{\alpha_i}{\nu_1}, \nu(x_2^j, \dots, x_i^j) \equiv \frac{\alpha_j}{\nu_2}$, then

$$x_1^{(i)} \cdot x_2^{(j)} \equiv \int_0^1 (\alpha_i \wedge \beta_j) / (\nu_1 \wedge \omega_j) d\omega_j$$ $$x_1^{(i)} + x_2^{(j)} \equiv \int_0^1 (\alpha_i \wedge \beta_j) / (\nu_1 \vee \omega_j) d\omega_j$$ $$\overline{\Sigma}^{(i)} \equiv \int_0^1 \alpha_i / (1 - \nu_i) d\alpha_i$$

Here $\overline{\Sigma}^{(i)}$ and $\overline{\alpha}^{(i)}$ are two different expressions of the meanings of words. In order to tell one from another its better to call the former (for the whole system $\mathcal{F}$) the whole complement and the latter (for $\mathcal{F}$) the relative complement. In it

$$\wedge \equiv \text{conjunction} \quad (\text{be different from } \wedge)$$

$$\vee \equiv \text{disjunction} \quad (\text{be different from } \vee)$$

$$\mu_T(\nu) = \begin{cases} 0 & \text{if } \nu \le \frac{\alpha}{\alpha+1} \\ 2 \left( \frac{\nu-\alpha}{\alpha-1} \right)^2 & \text{if } \frac{\alpha}{\alpha+1} \le \nu \le \frac{\alpha+1}{2} \\ 1-2 \left( \frac{\nu-1}{\alpha-1} \right)^2 & \text{if } \frac{\alpha+1}{2} \le \nu \le 1 \end{cases}$$

In the D-layer calculus, there are two elements 0 and 1, they are two special true-false values (corresponding to the maximal and minimal elements of the lattice constructed by the fuzzy subset of the interval (0,1)).

$$0 \equiv \int_0^1 \alpha/\omega \quad , \quad 1 \equiv \int_1^1 1/\omega$$

For 0 and 1 the regulation is as follows:

<table><caption>Table 3</caption><thead><tr><th>.(+)</th><th>0</th><th>1</th></tr></thead><tbody><tr><td>0</td><td>0(0)</td><td>0(1)</td></tr><tr><td>1</td><td>0(1)</td><td>1(1)</td></tr></tbody></table>

**(3) B-D type calculus system:**

The B-D type calculus is mingled with the B-layer calculus and the D-layer calculus. For the calculus of the whole complex system, it is the development of the basic calculuses mentioned above.

Through the B-D type calculus not only the fuzzy formula $F$ could be simplified, but also the degree of truth and accuracy of the given logical and numerical relations by $F$ could be obtained.

For example, from the above vinyron formaling test tables it can be got that

$$\begin{align*} y_i^{(n)} &= x_1^{(n)} \cdot x_2^{(n)} + x_3^{(n)} \cdot x_4^{(n)} + x_5^{(n)} \cdot x_6^{(n)} + \dots + x_{i-1}^{(n)} \cdot x_{i+1}^{(n)} \cdot x_{i+2}^{(n)} \cdot x_{i+3}^{(n)} \cdot x_{i+4}^{(n)} \cdot x_{i+5}^{(n)} \cdot x_{i+6}^{(n)} \\ (\bigwedge) x_1^{(n)} \cdot x_2^{(n)} \cdot x_3^{(n)} \cdot x_4^{(n)} \cdot (x_5^{(n)} \oplus x_6^{(n)} \oplus x_7^{(n)}) \\ (\bigwedge) x_1^{(n)} \cdot x_2^{(n)} \cdot x_3^{(n)} \cdot x_4^{(n)} \cdot (x_5^{(n)} \oplus x_6^{(n)} \oplus x_7^{(n)}) \\ (\bigwedge) x_1^{(n)} \cdot x_2^{(n)} \cdot x_3^{(n)} \cdot x_4^{(n)} \cdot x_5^{(n)} \cdot x_6^{(n)} \cdot x_7^{(n)} \end{align*}$$

(Here the calculus character on which the deformation is based will be worked out below).

The consequent conclusion is that the reaction time should be considerably long ($x_1^{(n)}$) in order to let the degree of the formaling reach the highest point ($y_1^{(n)}$), or the reaction time is long ($x_2^{(n)}$) while the density of formaldehyde is medium ($x_3^{(n)}$). The language true and false value of $y_1^{(n)}$ expressive formula can be calculated, so the corresponding degree of faithfulness to the control decision provided by the above conclusion can be obtained.

Based on this, the field theory intervals can be divided more precisely and finely, B-D type calculus is carried out again, and moved into the deeper layers, then the "ideal state" is approached by and by the optimum control is realized.

**III.** The axiomatized structure of the B-D type calculus theory

**Definition 4.** Let $\mathfrak{B}$ is the set of some fuzzy objects, it has two definite elements $\mathfrak{U}$ and $\mathfrak{F}$, and $\mathfrak{U} \neq \mathfrak{F}$. If three algebraic calculuses $\Psi$, $\mathcal{A}$ and $\mathcal{B}$ are defined, then for $\forall \mathfrak{A}, \mathfrak{B}, \mathfrak{C} \in \mathfrak{B}$, the following 4 axioms are reasonable:

$$(\text{Law of commutation}) \qquad A \Psi B = B \Psi A \\ A \cap B = B \cap A$$

$$(\text{Law of distribution}) \qquad A \Psi (B \cap C) = (A \Psi B) \cap (A \Psi C) \\ A \cap (B \Psi C) = (A \cap B) \Psi (A \cap C)$$

$$(\text{Law of identity}) \qquad A \Psi \Phi = \mathfrak{A} \\ \mathfrak{A} \cap \mathfrak{U} = \mathfrak{A}$$

$$(\text{Law of complementary}) \qquad A \Psi \bar{\mathfrak{A}} = \mathfrak{U} \\ \mathfrak{A} \cap \bar{\mathfrak{A}} = \Phi$$

Then $\langle \mathfrak{B}, \Psi, \mathcal{A}, \mathfrak{U}, \Phi \rangle$ form a fuzzy Boolean algebra.

**(1) B-layer calculus theory:**

From the definition 2 the following properties can be induced:

$$(1a) \quad x_1^{(i)} \oplus x_2^{(i)} = x_3^{(i)}$$

$$(1b) \quad x_1^{(i)} \ominus x_2^{(i)} = x_3^{(i)}$$

$$(2a) \quad x_1^{(i)} \oplus x_2^{(i)} = x_3^{(i)} \ominus x_4^{(i)}$$

$$(2b) \quad x_1^{(i)} \ominus x_2^{(i)} = x_3^{(i)} \oplus x_4^{(i)}$$

$$(3a) \quad x_1^{(i)} \oplus (x_2^{(i)} \ominus x_3^{(i)}) = (x_4^{(i)} \ominus x_5^{(i)}) \oplus x_6^{(i)}$$

$$(4a) \quad x_1^{(i)} \ominus (x_2^{(i)} \oplus x_3^{(i)}) = (x_4^{(i)} \oplus x_5^{(i)}) \ominus (x_6^{(i)} \oplus x_7^{(i)})$$

$$(4b) \quad x_1^{(i)} \ominus (x_2^{(i)} \ominus x_3^{(i)}) = (x_4^{(i)} \ominus x_5^{(i)}) \ominus (x_6^{(i)} \oplus x_7^{(i)})$$

$$(5a) \quad x_1^{(i)} \ominus 0_i = x_2^{(i)}$$

$$(5b) \quad x_1^{(i)} \oplus 1_i = x_2^{(i)}$$

$$(6a) \quad x_1^{(i)} \ominus \overline{x_2^{(i)}} = 1_i$$

$$(6b) \quad x_1^{(i)} \oplus \overline{x_2^{(i)}} = 0_i$$
From (2a), (2b), (4a), (4b), (5a), (5b), (6a), (6c), the following theorem can be obtained :
Theorem 1. < $\mathcal{L}_1, \mathcal{O}_1, \mathcal{O}_2, \dots, \mathcal{O}_n, \mathcal{I}_1, \dots, \mathcal{I}_n >$ (i=1,...,n) form a fuzzy DeMorgan algebra.

Some other properties can be induced :

(7a) $x_i^{(p)} \oplus 1_i = 1_i$

(7b) $x_i^{(p)} \oplus o_i = o_i$

(3b) $x_i^{(p)} \oplus ( x_i^{(q)} \oplus x_i^{(r)} ) = ( x_i^{(p)} \oplus x_i^{(q)} ) \oplus x_i^{(r)}$

(8) The uniqueness of $o_i$ and $1_i$.

(9) The uniqueness of $\overline{x_i^{(p)}}$ complementation for $x_i^{(p)}$.

(10) $\overline{o_i^*} = 1_i, \overline{1_i^*} = 0_i$

(11) $\overline{x_i^*} = x_i^{(p)}$

(12a) $x_i^{(p)} \oplus ( x_i^{(q)} \oplus x_i^{(r)} ) = x_i^{(q)}$

(12b) $x_i^{(p)} \oplus ( x_i^{(q)} \oplus x_i^{(r)} ) = x_i^{(r)}$

(13a) $\overline{x_i^{(p)} \oplus x_i^{(q)}} = \overline{x_i^{(p)} \oplus x_i^{(r)}}$

(13b) $\overline{x_i^{(p)} \oplus x_i^{(r)}} = \overline{x_i^{(p)} \oplus x_i^{(q)}}$

(2) D-layer calculus theory :

From the definition 3 it is can be proved that.

(1a)' $x_i^{(s)} + x_j^{(s)} = x_j^{(s)} + x_i^{(s)}$

(1b)' $x_i^{(s)} \cdot x_j^{(s)} = x_j^{(s)} \cdot x_i^{(s)}$

(2a)' $x_h^{(r)} + ( x_i^{(s)} \cdot x_j^{(s)} ) = ( x_h^{(r)} + x_i^{(s)} ) \cdot ( x_i^{(s)} + x_j^{(s)} )$

(2b)' $x_h^{(r)} \cdot ( x_i^{(s)} + x_j^{(s)} ) = ( x_h^{(r)} \cdot x_i^{(s)} ) + ( x_h^{(r)} + x_i^{(s)} )$

(3a)' $x_i^{(s)} + \overline{x_j^{(s)}} \neq 1$

(3b)' $x_i^{(s)} \cdot \overline{x_j^{(s)}} \neq 0$

(4a)' $\overline{x_i^{(s)} + x_j^{(s)}} = \overline{x_i^{(s)} \cdot x_j^{(s)}}$

(4b)' $\overline{x_i^{(s)} \cdot x_j^{(s)}} = \overline{x_i^{(s)} + x_j^{(s)}}$

The following theorem can be obtained :

Theorem 2. < $\mathcal{L}, \mathcal{O}, \mathcal{O}, \mathcal{O}, 0, 1 >$ form a fuzzy DeMorgan algebra.

(3) Conclusion :

Based on the B-D type calculus theory < $\mathcal{P}$, $\cdot$($\mathcal{O}$), $\cdot$($\mathcal{P}$), $\cdot$($-$), $\mathcal{O}$, $(\mathcal{O}_1), 1, (1_i)$ > algebraic system is a multi-layer fuzzy B-D type algebraic system, a hard-and-soft algebraic system ;it has become the algebraic model of the multi-layer logical structure in the above system.

In this algebraic model, based on the above calculus characters worked out, the given B-D type fuzzy formula (function) can be turned into the standard form and simplified, also the true-false value of the language can be estimated, therefore the logical analysis on the complex system can be better carried out.

REFERENCES

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2. Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems 1(1978), 3-28.

3. Zadeh, L.A., Fuzzy logic as a basis for the management of uncertainty in expert systems Fuzzy sets and systems 11 (1983c), 199-227.

4. Negoita, C.V., and Raiescu, D.A., Applications of Fuzzy sets to systems analysis, 1975.

5. Dubois, D. and prade, H. (1980), Fuzzy sets and systems, theory and applications, Academic press. Inc., New York.

6. S.C.Kleene, Introduction to Metamathematics, p.334, Van Nostrand, New York, (1952).