Prob Totorial 1ans

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IEICE TRANS. INF. & SYST., VOL.E82–D, NO.5 MAY 1999955

PAPER Special Issue on Multiple-Valued Logic and Its Applications

Interval and Paired Probabilities for Treating Uncertain

Events

Yukari YAMAUCHI†, Nonmember and Masao MUKAIDONO†, Member

SUMMARY When the degree of intersections A∩B of eventsA,B is unknown arises a problem: how to evaluate the proba-bility P(A ∩ B) and P(A ∩B) from P(A) and P(B). To treatrelated problems two models of valuation: interval and pairedprobabilities are proposed. It is shown that the valuation corre-sponding to the set operations ∩ (intersection), ∪ (union) and ∼(complement) can be described by the truth functional ∧ (AND),∨ (OR) and ∼ (negation) operations in both models. The proba-bilistic AND and OR operations are represented by combinationsof Kleene and Lukasiewicz operations, and satisfy the axioms of

MV (multiple-valued logic)-Algebra except the complementarylaws.key words: interval probability

1. Introduction

Representing uncertain knowledge has been a very ac-tive research area in Artificial Intelligence, since, in lifeknowledge is rarely exact and certain. Some attemptsto use probability theory in this area revealed certainunpleasant effects. One of them is the loss of flexibil-ity and another is the loss of functionality in probabil-ity operations. In probability theory, since an event is

given as a set, the probability of a compound event cannot be uniquely determined from the probabilities of itscomponent events.

Other attempts to find structures weaker thanprobability to represent uncertainty are Dempster-Shafer’s evidence theory (DS theory) [1], possibility the-ory [2]. The main difference from the probability theoryis that the evidence and possibility theories do not sat-isfy additivity. Instead of the standard point-valuedprobability, Giang suggested the interval probability [3]given by a pair of lower and upper bounds. He, basedon the semantic model of probability for sentences pro-posed by Nilsson, shows that the functionality is re-tained if one use his probability.

In this paper, the interval probability and thepaired probability are introduced in order to evaluateprobabilities P(A ∩ B) and P(A ∪ B) from P(A) andP(B). Here, the semantic models (so as the intersec-tion A ∩ B of events A, B) are assumed to be unknown.

In both models three probability operations: p∧,

p∨ and

p∼ are introduced corresponding to the set operations

Manuscript received September 10, 1998.Manuscript revised November 17, 1998.

†The authors are with the Department of Computer Sci-ence, Meiji University, Kawasaki-shi, 214–0033 Japan.

∩, ∪ and - (complement) over the event space. It isshown that the probabilistic AND and the probabilis-tic OR operations can be represented by combinationsof Kleene and Lukasiewicz [4] operations defined in theclosed interval [0, 1]. The set of probabilistic opera-tions proposed in this paper satisfies the axioms of MV(multiple-valued logic)-Algebra[5] except the comple-mentary laws.

2. Preliminaries of Probability Theory

In a probability sample space Ω an event is a subset of Ω. We denote the set of all events (the power set of Ω)by F . A mapping P : F from a closed interval [0, 1]is called a probability measure on (Ω, F ) and satisfiesP(A) ≥ 0 for all A ∈ F , P(Ω) = 1 and P(Ø) = 0 andif A ∩ B = Ø then P(A ∪ B) = P(A) + P(B).

The third property is called the additivity of prob-ability measure. Note that the degree of membershipof a sample e to an event A is given either 1 or 0 here.And the probability for a compound event is given only

for the compound of disjoint events.In the case that the events A and B are inde-

pendent to each other, P(A ∩ B) and P(A ∪ B) areinterpreted truth-functionally as P(A) × P(B) andP(A) + P(B) − P(A ∩ B) respectively. However, a cru-cial point here is that from P(A) and P(B), P(A ∩ B)and P(A ∪ B) are not determined when the degree of intersection of A and B is unknown. In order to retainthis loss of truth functionality, the operations for inter-val and paired probabilities are proposed in the sequel.

2.1 A Motivation — Numerical Probability

Given the probabilities P(A) = a, P(B) = b for eventsA, B (a, b ∈ [0, 1]), a compound event such as “A and Bhappen,” is the intersection of the sets A and B whosesize depends on the degree of intersection. P(A ∩ B)takes the minimum value when there is the least inter-section between A and B (either 0: Fig. 1 (a), or thepossible least intersections: Fig. 1 (b)) and the max-imum value when there is the most (either B ⊆ A:Fig. 2 (a) or A ⊆ B: Fig. 2 (b)). Thus the minimum andthe maximum values of P(A ∩ B) are max(0, a + b − 1)and min(a, b), respectively.

Similarly, a compound event such as “A or B hap-pens,” is the union of the sets A and B whose size de-



956IEICE TRANS. INF. & SYST., VOL.E82–D, NO.5 MAY 1999

Fig.1 Minimum P(A∩ B) = max(0, a + b− 1).

Fig. 2 Maximum P(A ∩B) = min(a, b).

Fig. 3 Minimum P(A ∪ B) = max(a, b).

Fig. 4 Maximum P(A ∪B) = min(1, a + b).

pends on the degree of intersection. P(A ∪ B) takes theminimum value max(a, b) when there is the most inter-section between A and B (when P(A ∩ B) = min(a, b))and the maximum value min(1, a + b) when there is theleast (when P(A ∩ B) = max(0, a + b − 1)) as shownin the Figs. 3 and 4. Thus the minimum and the maxi-

mum values of P(A ∪ B) are max(a, b) and min(1, a+b),respectively.

For our purpose we introduce notations borrowedfrom multiple-valued logic algebras. For a, b ∈ [0, 1]define Lukasiewicz’s and Kleene’s AND operations, re-spectively, by

a b = max(0, a + b − 1)

anda ∧ b

= min(a, b).Similarly, define Lukasiewicz’s and Kleene’s OR opera-tions, respectively, by

a ⊕ b = min(1, a + b)

and

a ∨ b = max(a, b).

Then the above discussion is summarized as

a b ≤ P(A ∩ B) ≤ a ∧ b

a ∨ b ≤ P(A ∪ B) ≤ a ⊕ b.

Inspired by the above observation, in order to re-tain the truth functionality of the probability opera-tions for compound events, the probability operations

AND and OR, denoted as p

∧ and p

∨ respectively, assignthe interval probabilities to the events A∩B and A∪B.

Definition 2.1.1: Given numerical probabilitiesP(A) = a and P(B) = b, the probability AND opera-

tion p

∧ and the probability OR operation p

∨ assign theclosed intervals of probability for the compound eventsof A and B as

P(A) p

∧ P(B)

=[a b, a ∧ b] (1)

P(A) p

∨ P(B) = [a ∨ b, a ⊕ b] (2)

when the degree of intersection is unknown.

The probability negation operationp∼ assigns a nu-

merical probability 1 − P(A) for P(∼A) since the oper-ation does not involve a compound event. (Hereafter,we use notation ∼ a for 1 − a.)

P(∼A) =p∼a

= ∼ a (3)

Example: if P(A) = 0.7 and P(B) = 0.8 then

P(A)

p

∧ P(B) = [max(0, 0.7 + 0.8 − 1),

min(0.7, 0.8)]

= [0.5, 0.7]

P(A) p

∨ P(B) = [max(0.7, 0.8), min(1, 0.7 + 0.8)]

= [0.8, 1].

3. Interval Probability

3.1 Interval Probability

Use of the interval probability defined in the previoussection retains the truth functionality of the probabil-ity operations. However, the domain of the numericalprobabilities is not closed as an interval, so a general-ization of the operations on the interval probabilities isneeded.

Given the set of all interval-value I , I = [a, b] |0 ≤ a ≤ b ≤ 1, a mapping IP: F → I is called aninterval probability . An interval probability IP(E ) canbe interpreted as the scope of P(E ), i.e. IP(E ) = [a, b]means a ≤ P(E ) ≤ b.

Let the interval probabilities be given as IP(A) =[a1, a2], IP(B) = [b1, b2] for events A, B. P(A ∩ B)



YAMAUCHI and MUKAIDONO: INTERVAL AND PAIRED PROBABILITIES FOR TREATING UNCERTAIN EVENTS957

Fig. 5 P(A ∩B).

Fig. 6 P(A ∪B).

attains the minimum value a1 b1, when P(A) = a1,

P(B) = b1 (when the probability for each event takesthe minimum value) and the intersection of A, B is atthe least. It attains the maximum value a2 ∧ b2, whenP(A) = a2, P(B) = b2 (when the probability for eachevent takes the maximum value) and the intersection of A, B is at the most (Fig. 5). Similarly, P(A ∪ B) attainsthe minimum value a1 ∨ b1 when P(A) = a1, P(B) = b1

and the intersection of A, B is at the most. It attainsthe maximum value a2 ⊕ b2, when P(A) = a2, P(B) =b2 and the intersection of A, B is at the least (Fig. 6).

Definition 3.1.1: Given interval probabilities IP(A)

and IP(B), the probability AND operation p

∧, the prob-

ability OR operation p∨ and the probability negation op-

erationp∼ assign the interval probabilities for the com-

pound events of A and B and the negation of A asfollows.

IP(A) p∧ IP(B)

= [a1 b1, a2 ∧ b2], (4)

IP(A) p∨ IP(B)

= [a1 ∨ b1, a2 ⊕ b2], (5) p∼IP(A)

= [∼a2, ∼a1]. (6)

The above definition leads to the following identities.

IP(A ∩ B) := IP(A) p

∧ IP(B),

IP(A ∪ B) := IP(A) p∨ IP(B),

IP(∼A) :=p∼IP(A).

The set of interval probability operations p∨,

p∧,

p∼

has truth functionality and the domain of interval prob-abilities is closed under it. Although [1, 1] and 1 areformally different, they represent the same geometricobject (the point 1). In such cases we will put the sign, i.e. [1, 1] 1.

Note, that the interval probability is inclusive of numerical probability as a special case, since P(A) =a [a, a] = IP(A).

3.2 Algebraic Properties of Interval Probability Oper-ations

The set of interval probability operations p∨,

p∧,

p∼ and

constants [0, 0], [1, 1] satisfy the axioms of MV Algebra1 ∼6.

1. commutative laws

A p∨B = B

p∨A, A

p∧B = B

p∧A.

2. associative laws

A p

∨(B p

∨C )= (A p

∨B) p

∨C, A p

∧(B p

∧C ) = (A p

∧B) p

∧C .3. maximum element

A p∨1 = 1, A

p∧1 = A.

4. minimum element

A p∨0 = A, A

p∧0 = 0.

5. double negation law p∼(

p∼A) = A.

6. de Morgan laws p∼(A

p∨B) =

p∼A

p∧ p∼B,

p∼(A

p∧B) =

p∼A

p∨ p∼B.

Instead of the following

7. complementary laws

A +p∼A = 1, A ×

p∼A = 0,

it satisfies the Kleene laws.

7’. Kleene laws

A p

∨ p∼A

p

∨ B p

∧ p∼B = A

p

∨ p∼A,

(A p∨ p∼A)

p∧ B

p∧ p∼B = B

p∧ p∼B.

As it holdsA ∧ B = (A ⊕ ∼B) BA ∨ B = (A ∼B) ⊕ B,

the set of Lukasiewicz operations , ⊕ together withthe operations ∼ is able to represent Kleene operations∧, ∨. So the Lukasiewicz(MV) Algebra , ⊕, ∼ isinclusive of the Kleene Algebra ∧, ∨, ∼. However, theset of interval probability operations can not representKleene operations.

From the algebraic standpoint, it is natural to

compare the two system of operations p∨,

p∧,

p∼ and

∨, ∧, ⊕, , ∼ which is defined component-wisely as

follows A∨B ≡ [a1 ∨b1, a2 ∨b2], A∧B ≡ [a1 ∧b1, a2 ∧b2],A ⊕ B ≡ [a1 ⊕ b1, a2 ⊕ b2], A B ≡ [a1 b1, a2 b2],∼A ≡ [∼a2, ∼a1].

Lemma 3.2.1: Each of the two systems of operations

p

∨, p

∧,p∼ and ∨, ∧, ⊕, , ∼ defined over intervals can

not represent the other system.

Proof: It is enough for the proof if we find out onecounter example.

∨, ∧, ⊕, , ∼ can not represent p

∨, p

∧,p∼:

Let A and B be two numerical values a and b (A= [a,a] and B = [b,b]). Then the result of the




operations ∨, ∧, ⊕, , ∼ are numerical values, asshown bellow. That is, the results of any itera-tions of ∨, ∧, ⊕, , ∼ to A and B are numericalvalues.

A ∨ B = [a, a] ∨ [b, b] = [a ∨ b, a ∨ b] = a ∨ b,

A ∧ B = [a, a] ∧ [b, b] = [a ∧ b, a ∧ b] = a ∧ b,A ⊕ B = [a, a] ⊕ [b, b] = [a ⊕ b, a ⊕ b] = a ⊕ b,

A B = [a, a] [b, b] = [a b, a b] = a b,

∼A = ∼[a, a] = [∼a, ∼a] = ∼a.

On the other hand, interval values are generated

from the set of operations p

∨, p

∧,p∼ on numerical

values a and b as shown bellow.

A p

∨B = [a, a] p

∨[b, b] = [a ∨ b, a ⊕ b],

A p

∧B = [a, a] p

∧[b, b] = [a b, a ∧ b].

Thus, the above interval truth values can not begenerated when a ∨ b = a ⊕ b or a b = a ∧ bfrom the set of operations ∨, ∧, ⊕, , ∼ which isclosed under the numerical values.

p∨,

p∧,

p∼ can not represent ∨, ∧, ⊕, , ∼:

The set of interval truth values A =[0, 1], [1, 1/2], [1/2, 1/2], [1/2, 1] is closed under

the operation p

∨, p

∧,p∼ as shown in the following

tables.p

∧ [0,1/2] [1/2,1/2] [1/2,1] [0,1][0,1/2] [0,1/2] [0,1/2] [0,1/2] [0,1/2]

[1/2,1/2] [0,1/2] [0,1/2] [0,1/2] [0,1/2][1/2,1] [0,1/2] [0,1/2] [0,1] [0,1]

[0,1] [0,1/2] [0,1/2] [0,1] [0,1]p

∨ [0,1] [0,1/2] [1/2,1/2] [1/2,1][0,1] [0,1] [0,1] [1/2,1] [1/2,1]

[0,1/2] [0,1] [0,1] [1/2,1] [1/2,1][1/2,1/2] [1/2,1] [1/2,1] [1/2,1] [1/2,1]

[1/2,1] [1/2,1] [1/2,1] [1/2,1] [1/2,1]

[0,1] [0,1/2] [1/2,1/2] [1/2,1]p∼ [0,1] [1/2,1] [1/2,1/2] [0,1/2]

On the other hand, the minimum closed set in-

cluding A under the operations ∨, ∧, ⊕, , ∼is [0, 1], [1, 1/2], [1/2, 1/2], [1/2, 1], [0, 0], [1, 1] as

shown easily. That is, p

∨, p

∧,p∼ can not gen-

erate [0, 0] and [1, 1] from A but ∨, ∧, ⊕, , ∼

can. This shows that p∨,

p∧,

p∼ can not represent

∨, ∧, ⊕, , ∼.

4. Paired Probability

4.1 Paired Probability

Now, we would like to consider a different model called

Fig. 7 Paired event A = (A1,A2).

Fig. 8 Paired events and a triangle diagram.

the paired probability which has the form of a pairof probabilities (optimistic case and pessimistic case)for an event. In the interval probability, the interval

value is considered as the scope of numerical probabil-ity for an event. In the paired probability, the min-imum (lower) and maximum (upper) probabilities aregiven for lower and upper events of a paired event whichhas two boundaries. A paired event is interpreted asA = (A1, A2) such that A1, A2 ∈ F and A1 ⊆ A2

(Fig. 7). Thus a paired probability PP(A) should beinterpreted as (P(A1), P(A2)).

Given the set of all paired-values P , P = (x1, x2) |0 ≤ x1 ≤ x2 ≤ 1, and the set of all paired-events F P ,F P = (X 1, X 2) | X 1 ⊆ X 2, X 1, X 2 ∈ F, a mappingPP:F P → P is called the paired probability measure on(Ω, F P ).

Let PP(A) and PP(B) be paired probabilities(a1, a2) and (b1, b2). We adopt a triangle diagramdrawn with the representative three points (0,0), (1,1)and (0,1) to represent the distribution of paired prob-abilities. A point in the diagram represents a pairedprobability as exemplified in Fig. 8. A partial order ≤between paired probabilities is represented in horizon-tal direction. That is PP(A) ≤ PP(B) in Fig.8.

We define compound events of A = (A1, A2) andB = (B1, B2) by

A ∩ B = (A1 ∩ B1, A2 ∩ B2),

A ∪ B = (A1 ∪ B1, A2 ∪ B2).

We call the first and second components the lower andupper limit events of A∩B (A∪B). As each componentis now indexed by two events we can apply the ideaof the interval of probabilities (Definition 2.1.1) to itsevaluation. Thus define

PP(A ∩ B) = (P(A1)

p∧P(B1), P(A2)

p∧P(B2))

PP(A ∪ B) = (P(A1)

p∨P(B1), P(A2)

p∨P(B2)).

The first and second components of PP(A ∩ B) arecalled lower and upper limits of PP(A ∩ B). They aregiven as interval probabilities




Fig.9 Possible PP(A ∩B).

Fig. 10 Example of contradiction.

P(A1) p

∧P(B1) = [a1 b1, a1 ∧ b1],

P(A2) p

∧P(B2) = [a2 b2, a2 ∧ b2].

The set of paired probabilities represented by thetwo intervals of PP(A ∩ B) is shown as a rectangle (anaggregate of paired probabilities) in the triangle dia-gram in Fig. 9 whose minimum paired probability is

PP(A B) : (a1 b1, a2 b2), and maximum pairedprobability is PP(A ∧ B) : (a1 ∧ b1, a2 ∧ b2).

The highest point in the triangle diagram repre-sents a paired probability (the minimum P(A1 ∩ B1),the maximum P(A2 ∪ B2)) However the combinationof the minimum A1 ∩ B1 and maximum A2 ∪ B2 some-times contradicts A1 ⊆ A2 or B1 ⊆ B2 as exemplifiedin Fig.10.

Since A1 ⊆ A2 and B1 ⊆ B2 holds for every pairedevent, the difference between P(A1 ∩ B1) (P(A1 ∪ B1))and P(A2 ∪ B2) (P(A2 ∪ B2)) is limited to less than orequal to (a2 − a1) + (b2 − b1).

Lemma 4.1.1: The maximum difference between thelower and upper limits of PP(A ∩ B) is less than orequal to (a2 − a1) + (b2 − b1).

Proof:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B), (7)

P(A ∩ B) = P(A) + P(B) − P(A ∪ B), (8)

A1 ⊆ A2, B1 ⊆ B2 (9)

holds for every events.

P(A1 ∪ B1) = P(A1) + P(B1) − P(A1 ∩ B1) from (7)

P(A1 ∪ B1) ≤ P(A2 ∪ B2) since A1 ∪ B1 ⊆ A2 ∪ B2

Fig. 1 1 P(A1 ∪ B1) = a1 + b1 − α1.

Fig. 12 Possible arrangement of P(A2 ∪ B2).

Fig. 13 Maximum P(A2 ∩ B2).

from (9)

P(A2 ∩ B2) = P(A2) + P(B2) − P(A2 ∩ B2) from (8)

≤ P(A2) + P(B2) − (P(A1) + P(B1)

− P(A1 ∩ B1))

Thus

P(A2 ∩ B2) − P(A1 ∩ B1)

≤ P(A2) + P(B2) − (P(A1) + P(B1))

≤ (a2 − a1) + (b2 − B1)

In the following we will explain the proof of Lemma 4.1.1 in detail by using figures and numericalvalues. Let the lower limit P(A1 ∩B1) of PP(A ∩ B) beα1 ∈ [a1 b1, a1 ∧ b1], then P(A1 ∪ B1) is a1 + b1 − α1

(Fig. 11). Since P(A2 ∪ B2) is bigger than or equalto P(A1 ∪ B1), its range is restricted to [a1 + b1 −α1, a2 + b2 − α1] (Fig.12). The maximum P(A2 ∩ B2)is a2 + b2 − (a1 + b1 − α1) (Fig. 13).

The maximum difference between the lower andupper limits of PP(A ∩ B) is (a2 + b2) − (a1 + b1 −α1) − α1 = (a2 − a1) + (b2 − b1) thus not depending onα1 (Fig.14). Notice that is less than or equal to 1 sinceP(A1 ∩ B1) and P(A2 ∩ B2) are in [0,1].

Since PP(A ∩ B) can not be uniquely definedas a paired probability, we adopt the paired inter-val probability to represent it. The paired inter-val probability PPI (A) should be interpreted as thepossible distribution of paired probability and repre-sented by the minimum and maximum paired prob-ability (a1, a2), (a3, a4) with the maximum differencebetween lower and upper limit, maxA, denoted as




Fig. 14 The set of PP(A ∩B).

Fig. 15 Paired probability operations.

PPI (A)

= a1 a2

a3 a4

[maxA] (a1 ≤ a2, a3 ≤ a4, a1 ≤

a3, a2 ≤ a4 and maxA ∈ [0, a4 − a1]). Note, the corre-spondence to IP(A1) = [a1, a3], and IP(A2) = [a2, a4].

Given the set of all paired-interval-value P I , amapping PPI : F P → P I is called a paired interval

probability measure on (Ω, F P ).

Definition 4.1.1: Given paired probabilities PP(A)

and PP(B), the paired probability operation p

∧ and p

∨ assign the paired interval probabilities for the com-pound events of A and B and the probability negation

operationp∼ assigns a paired probability as,

PP(A) p∧ PP(B)

=

a1 b1a2 b2

a1 ∧ b1 a2 ∧ b2

[∗1]

, (10)

PP(A) p∨ PP(B)

=

a1 ∨ b1 a2 ∨ b2

a1 ⊕ b1a2 ⊕ b2

[∗2]

. (11)

∼PP(A) = (∼ a2, ∼ a1). (12)

∗1 =min((a2 ∧ b2)−(a1 b1), (a2 − a1)⊕(b2 − b1))

∗2 =min((a2 ⊕ b2)−(a1 ∨ b1), (a2 − a1)⊕(b2 − b1))

Examples are shown in Fig. 15.

4.2 Paired Interval Probability

The domain of paired probabilities is not closed under

the set of probability operations p

∨, p

∧,p∼. In the sub-

sequent part, these operations are generalized so thatthe domain of the paired interval probabilities is closedunder them.

Given the paired events A and B, the paired inter-val probabilities of A and B are written as PPI (A) =

a1 a2

a3 a4

[maxA]

, PPI (B) =

b1 b2

b3 b4

[maxB].

PP(A ∩ B) attains the minimum value, (a1 b1, a2 b2), and the maximum value, (a3 ∧b3, a4 ∧b4), as shown

Fig. 16 Minimum and maximum PP(A ∩B).

Fig. 1 7 PPI (A ∩B).

Fig. 18 Minimum and maximum PP(A ∪B).

Fig. 1 9 PPI (A∪B).

in Figs. 16 and 17.Similarly, PP(A ∪ B) attains the minimum value,

(a1 ∨ b1, a2 ∨ b2), and the maximum, (a3 ⊕ b3, a4 ⊕ b4),as shown in Figs. 18 and 19.

The following definitions are based on these obser-vations.

Definition 4.2.1: Given two paired interval proba-

bilities PPI (A) =

a1 a2

a3 a4

[maxA]

, PPI (B) =

b1 b2b3 b4

[maxB]

, the paired interval probability op-

erations p

∨, p

∧,p∼ assign the paired interval probabilities

for the paired events A ∩ B, A ∪ B and ∼A as followswhen the degree of intersection is unknown.

PPI (A) p

∧PPI (B) =

a1 b1a2 b2

a3 ∧ b3 a4 ∧ b4

[∗∗1]

, (13)

PPI (A) p∨PPI (B)

=

a1 ∨ b1 a2 ∨ b2

a3 ⊕ b3a4 ⊕ b4

[∗∗2]

, (14)

p∼PPI (A)

=

∼ a4∼ a3

∼ a2∼ a1

[maxA]

. (15)




∗∗1 = min((a4 ∧ b4) − (a1 b1), maxA ⊕ maxB)

∗∗2 = min((a4 ⊕ b4) − (a1 ∨ b1), maxA ⊕ maxB)

Numerical, interval and paired probabilities canbe regarded as the special cases of the paired inter-val probabilities. Thus the paired interval probabilities

can be regarded as a general form of the probabilities.The domain of the paired interval probabilities is closedunder the set of paired interval probability operations

p∨,

p∧,

p∼.

The set of paired interval probability operations

p∨,

p∧,

p∼ satisfies exactly the same axioms of the oper-

ations on interval probabilities. Algebraic structures of

< I ; p∨,

p∧,

p∼, ∨, ∧ > and < P ;

p∨,

p∧,

p∼, ∨, ∧ > must be

similar to each other, and should be interesting topicsfor further discussions.

5. Conclusion

For representing uncertain knowledge using probabil-ity, two models of probabilities, the interval and pairedprobabilities, and their operations are introduced. Theoperations for these models are defined by combinationsof both Kleene and Lukasiewicz operations that repre-sent Kleene Algebra and MV Algebra, respectively. Inthis paper we examine the algebraic properties of thesetwo models.

In application these models may have a drawbacksuch that the results become more wide range, however,have a good point such that the domain of multiple

valued logic 0, 1/m, 2/m, . . . , (m−1)/m, 1 is closedin both algebras like the algebra of interval truth valuesin[6].

Further discussion of the interval probability maybe related to approximated reasoning in Artificial Intel-ligence. The models and operations for paired probabil-ity may be applied to the rough sets, fuzzy measure [7]and data mining in Artificial Intelligence.

References

[1] G. Shafer, “A mathematical theory of evidence,” PrincetonUniv. Press, 1976.

[2] D. Dubois and H. Prade, “Belief revision and update in nu-

merical formalisms: An overview with new results for thepossibilistic framework,” Procs. IJCAI’93, 1993.

[3] P.H. Giang, “Representation of uncertain belief using intervalprobability,” Proc. twenty-seventh International Symposiumon Multiple-Valued Logic, IEEE, 1997.

[4] M. Mukaidono, “Fuzzy Logic (Kouza Fuzzy — vol.4),”Nikkan Kougyo Press, 1993.

[5] C.C. Chang, “Algebraic analysis of many-valued logics,”Trans. of the American Mathematical Society, vol.88, 1958.

[6] M. Mukaidono, “Algebraic structures of interval truth val-ues in fuzzy logic,” Proc. Sixth International Conferenceon Fuzzy Systems (FUZZ-IEEE’97), vol.II, pp.699–705, July1997.

[7] M. Sugeno and T. Murofushi, “Fuzzy Measure (Kouza Fuzzy— vol.3),” Nikkan Kougyo Press, 1993.

Yukari Yamauchi was born in Japanon 1973. She received the B.E. degreein Liberal Arts and M.E. degree in Nat-ural Science from International ChristianUniversity, in 1996 and 1998, respectively.She is currently a Ph.D. student in the De-partment of Computer Science, Meiji Uni-

versity. Her main research interests are inmultiple-valued logic, fuzzy logic, and itsapplications, rough sets and data mining,modal logic and approximated reasoning

in Artificial Intelligence.

Masao Mukaidono was born inJapan on 1942. He received the B.E.,M.E. and Ph. D. degree in electrical en-gineering from Meiji University, in 1965,1967 and 1970, respectively. He is cur-rently a Professor in the Department of Computer Science, School of Science and

Technology, Meiji University. His mainresearch interests are in multiple-valuedlogic, fuzzy set theory and its applica-tions, fault tolerant computing, and com-

puter aided logic design. Dr. Mukaidono was president of JapanSociety of Fuzzy Theory and Systems, and member of the IEEEComputer Society, the Information Processing Society of Japanand Japanese Society of Artificial Intelligence.

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Prob Totorial 1ans