A possibilistic-logic-based approach to integrating imprecise and uncertain information

Fuzzy Sets and Systems 113 (2000) 309–322www.elsevier.com/locate/fss

A possibilistic-logic-based approach to integrating impreciseand uncertain information

Jonathan Leea;∗, Kevin F.R. Liub, Weiling Chiangb

a Department of Computer Science & Information Engineering, National Central University, Chungli 32054, Taiwanb Department of Civil Engineering, National Central University, Chungli 32054, Taiwan

Received August 1996; received in revised form January 1998

Abstract

A reasoning mechanism capable of dealing with imprecise and uncertain information is essential for expert systems.In this paper, we propose the use of truth-quali�ed fuzzy propositions as the representation of imprecise and uncertaininformation, where the fuzzy sets embody the intended meaning of imprecise information and the fuzzy truth values serveas the representation of uncertainty for its capability to express the possibility of the degree of truth of a fuzzy proposition.An inference mechanism for fuzzy propositions with fuzzy truth values is developed to serve as a bridge that brings togetherthe possibilistic reasoning and fuzzy reasoning into a hybrid approach to reasoning under uncertainty and imprecision. Thereare three steps involved. First, the fuzzy rules and fuzzy facts with fuzzy truth values are transformed into a set of uncertainclassical propositions with necessity and possibility measures. Second, a possibilistic reasoning called possibilistic entailmentis performed on the set of uncertain classical propositions. Third, we reverse the process in the �rst step to synthesize allthe classical sets obtained in the second step into a fuzzy set, and to compose necessity and possibility pairs to form a fuzzytruth value. c© 2000 Elsevier Science B.V. All rights reserved.

Keywords: Expert systems; Imprecision; Uncertainty; Possibilistic entailment; Fuzzy truth value; Fuzzy reasoning

1. Introduction

Expert systems have become the most visibleand fastest growing branch of Arti�cial Intelligence[12,14,15]. There are two essential components inrule-based expert systems: knowledge base and in-ference engine, which serve the purpose of inferringa useful conclusion from established rules by ex-perts and users’ observed facts. However, certain and

∗ Corresponding author. Tel.: +886 3 422 7151; fax: +886 3422 2681.

E-mail address: [email protected] (J. Lee)

precise knowledge are not always available for hu-man experts to establish a knowledge base; further-more, users’ observations are sometimes uncertainand imprecise. Therefore, an adequate managementof uncertainty and imprecision pervading in the rulebase and the data base of expert systems has becomea signi�cant issue [4].

The distinction between imprecise and uncertaininformation can be best explained by the canonicalform representation (i.e. a quadruple of attribute, ob-ject, value, con�dence) proposed by Dubois and Prade[7,8]. Imprecision implies the absence of a sharpboundary of the value component of the quadruple;

0165-0114/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.PII: S 0165 -0114(98)00039 -6

310 J. Lee et al. / Fuzzy Sets and Systems 113 (2000) 309–322

whereas, uncertainty is related to the con�dencecomponent of the quadruple which is an indication ofour reliance about the information.

In order to perform reasoning for both imprecise anduncertain information, two important issues should beaddressed. First, any improvement of the con�dencelevel for a piece of information can only be achievedat the expense of the speci�city of the information;and vise verse [23,25]. Second, the matching betweena fact and the premise of a rule is not exact, but onlypartial [25]. We have roughly classi�ed the existingapproaches in dealing with both imprecise and uncer-tain information into three categories based on theirtreatments for the two issues.• An uncertainty-quali�ed fuzzy proposition is

translated into a proposition whose con�dencelevel is certain but with less speci�c informa-tion, while partial matching is to modify the in-tended meaning of conclusions. This approachwas advocated by Zadeh [25] and Yager [23].Zadeh proposed three uncertainty-quali�cations forfuzzy propositions: probability-, possibility- andtruth-quali�ers; and Yager focused on certainty-quali�er.

• The degree of partial matching is to in uencethe con�dence level of conclusions, which wasadopted by researchers such as Ogawa et al. [19],Martin-Clouair et al. [17] and Umano [22]. Ogawaet al. combined certainty factors and fuzzy setsto represent uncertain and imprecise informationin an expert system SPERIL-2. Martin-Clouairet al. attached possibility and necessity degreesto fuzzy propositions. Umano employed the fuzzytruth value for the uncertainty-quali�er of fuzzypropositions.

• No partial matching is allowed in Ishizuka et al.[13] and Godo et al. [11]. Ishizuka et al. extendedthe Dempster–Shafer’s evidence theory to fuzzyset in the expert system SPERIL-1. Godo et al.used the fuzzy truth value as uncertainty-quali�erof fuzzy propositions.

Notice that the �rst kind of research results in a com-pletely certain conclusion whose intended meaninghas been changed. On the other hand, the second oneproduces a new con�dence level for a conclusion with-out modifying its intended meaning. The third one canbe viewed as a special case of the second one. It isobvious that these inference strategies are somewhat

limited due to the fact that either the intended meaningis required to be unchanged, or the con�dence levelhas to be completely certain.

In this paper, a possibilistic reasoning called possi-bilistic entailment is proposed to cope with uncertaininformation. We propose the use of fuzzy truth valuesand fuzzy sets for representing uncertain and impre-cise information, respectively. The fuzzy truth valueis adopted for its capability to express the possibilityof the degree of truth of a fuzzy proposition. We de-velop an inference mechanism for fuzzy propositionswith fuzzy truth values. There are three steps involved.First, the fuzzy rules and fuzzy facts with fuzzy truthvalues are transformed into a set of uncertain classicalpropositions with necessity and possibility measuresby means of �-cut. Second, possibilistic entailment isperformed on the set of uncertain classical proposi-tions. Third, we reverse the process in the �rst step tosynthesize all the �-level-sets obtained in the secondstep into a fuzzy set, and to compose necessity andpossibility pairs to form a fuzzy truth value.

The organization of this paper is as follows.A possibilistic reasoning called possibilistic en-tailment is proposed in the next section. Therepresentation of uncertain imprecise propositions andits semantics are de�ned in Section 3. In Section 4,an algorithm for inference strategy is discussed. Re-lated work is described in Section 5. Finally, a sum-mary of our approach and its potential bene�ts aregiven in Section 6.

2. Possibilistic entailment

A classical proposition is true in some possibleworlds and false in the rest of possible worlds. Wemodel our uncertainty about the actual world by de�n-ing a possibility distribution over all possible worlds tospecify the degree of possibility that the actual worldis in each possible world (see Fig. 1). In [5], Dubois,Lang and Prade have formally de�ned the possibilityand necessity measures for classical propositions asfollows. Given a possibility distribution � on the setof possible worlds , the possibility measure of p isde�ned as

�(p) = Sup{�(!) |! |=p};

J. Lee et al. / Fuzzy Sets and Systems 113 (2000) 309–322 311

Fig. 1. Possible worlds semantics for uncertain information.

where ! |= p denotes that p is true in ! (i.e. a pos-sible world in ). The dual necessity measure is thende�ned by

N (p) = Inf{1 − �(!) |! |= @p}:The pair of possibility and necessity measures for aclassical proposition is adopted here for our con�dencelevel about the classical proposition.

To represent uncertain information, we propose arepresentation of possibilistic propositions as follows:

p; (Np;�p);

where p denotes a classical proposition, Np denotesthe lower bounds of necessity measures, and �p de-notes the upper bounds of possibility measures (i.e.N (pi)¿Npi and �(pi)6�pi). A possibilistic propo-sition (p; (Np;�p)) means that the degree of ease tosay that p is false is at most equal to 1 − Np; mean-while, the degree of ease to say that p is true is atmost equal to �p.

The possibilistic reasoning for possibilistic propo-sitions is expressed as follows:

p→ q; (Np→q; �p→q)

p; (Np;�p)

q; (Nq;�q)

(1)

To infer Nq and �q, we propose an approach calledpossibilistic entailment, inspired by Nilsson’s proba-bilistic entailment [18] that helps one deduce a newproposition with associated probability from a baseset of propositions with associated probabilities. In theproposed approach, a semantic tree is used to deter-mine the possible worlds. More speci�cally, a possi-

Fig. 2. A semantic tree.

ble world is the one in which the truth values of allinvolved propositions are consistent in a semantic tree(see Fig. 2). In Fig. 2, inconsistent paths are indicatedby an ×; T and F denote the truth values true andfalse, respectively. This semantic tree shows that thereare four possible worlds:

!1 !2 !3 !4

p true true false falsep→ q true false true trueq true false true false

where the columns give the consistent sets of truthvalues for these three propositions.

We have mentioned that a possibility distributionover all possible worlds is used to model our uncer-tainty about the actual world, the possibility measureof any propositionpi is then reasonably taken to be themaximum of the possibilities of all possible worlds inwhich pi is true. Therefore, we can construct the rela-tionship between the upper bounds of the possibilities(�p;�p→q; �q), the lower bounds of the necessities


(Np; Np→q; Nq) of propositions, and the possibilities ofpossible worlds (�(!i)):

�p1 − Np�p→q

1 − Np→q

�q1 − Nq

¿

1 1 0 00 0 1 11 0 1 10 1 0 01 0 1 00 1 0 1

◦

�(!1)�(!2)�(!3)�(!4)

(2)

where ◦ indicates the composition operator “max-min”and �(!i) denotes the possibility that the actual worldis in possible world!i. The �rst row of the above 6×4matrix gives truth values for p in the four possibleworlds; whereas, the second row indicates the possi-ble worlds where p is false. The right-hand side ofthis inequality means the possibility measures of theinvolved propositions, and the left-hand side of thisinequality expresses the upper bounds of these possi-bility measures.

Based on the given Np,�p, Np→q and�p→q, we canderive a possibility distribution �̂ which is the upperbound of �:

�̂(!1) = min{�p→q; �p};

�̂(!2) = min{1 − Np→q; �p};

�̂(!3) = min{�p→q; 1 − Np};

�̂(!4) = min{�p→q; 1 − Np}:Thus, the upper bound of the possibility and the lowerbound of the necessity of q are:

Nq = 1 − max{�̂(!2); �̂(!4)}

= min{max[Np→q; 1 −�p];

max[1 −�p→q; Np]};

�q = max{�̂(!1); �̂(!3)}

= max{min[�p→q; �p];

min[�p→q; 1 − Np]}: (3)

If the possibility distribution �̂ is normalized, whichcan only be achieved in the case that �p¡1 and

�p→q¡1 do not exist simultaneously, Eq. (3) thenbecomes

Nq = min{Np→q; Np};(4)

�q =�p→q:

To illustrate this, assume that (p; (0:7; 1:0)) and(p→ q; (0:8; 1:0)), then we can infer (q; (0:7; 1:0)).

However, if the possibility distribution �̂ is sub-normalized (i.e. max{�p;�p→q}¡1, called partiallyinconsistent [5,16]), Nq and �q derived from Eq. (3)are then meaningless since they violate the followingconstraints: (N (q)6�(q)), (N (q)¿ 0 → �(q) = 1)and (�(q)¡ 1 → N (q) =0). For example, given(p; (0:0; 0:3)) and (p→ q; (0:0; 0:4)), (q; (0:6; 0:4))is inferred. Hence, to obtain the consistent necessityand possibility values for p, p → q and q, both �pand �p→q should not be less than 1 simultaneously.

Possibilistic logic has been advocated by Duboisand Prade for more than ten years. In their approach,they viewed a possibility measure �(p) of proposi-tion p as the degree of possibility that p is true, anda necessity measure N (p) as the degree of necessitythat p is true. They have proposed a di�erent repre-sentation for uncertain propositions as follows:

(p; (�p; �p));

where �p is the lower bound of necessity measure and�p is the lower bound of possibility measure, i.e.,

N (p)¿�p; �(p)¿�p

which expresses that p is certainly true at least tothe degree Np, and p is possibly true at least to thedegree �p.

However, to represent a certainly false proposition(i.e. N (p) = 0 and �(p) = 0), their representation(p; (0:0; 0:0)) is not appropriate since it may haveseveral interpretations: (N (p) = 0; �(p) = 0); (N (p)= 0; �(p) = 1); (N (p) = 1; �(p) = 1) and so on.In their approaches, the maximal and minimal valuesare (p; (1:0; 1:0)) (expressing that p is completelycertain) and (p; (0:0; 0:0)) (expressing that we donot know anything about the truth or the falsity) [5],respectively; whereas, our representation• (p; (1:0; 1:0)) means that p is certainly true;• (p; (0:0; 0:0)) expresses that p is certainly false;

and


Fig. 3. The comparison between Dubois, Lang and Prade’s representation and ours. (a) Necessity & possibility measures (N (p); �(p));(b) Dubois, Lang and Prade’s representation (�p; �p); (c) Our representation (Np;�p).

Table 1The comparison between Dubois, Lang and Prade’s methods and our approach

Case Modus Ponens Truth value (N (·); �(·)) Dubois and Pradea Dubois, Lang and Pradeb Our approach

p→ q T (1.0, 1.0) (1.0, 1.0) (1.0, 1.0) (1.0, 1.0)(1) p T (1.0, 1.0) (1.0, 1.0) (1.0, 1.0) (1.0, 1.0)

q T (1.0, 1.0) (1.0, 1.0) (1.0, 1.0) (1.0, 1.0)

p→ q F (0.0, 0.0) (0.0, 0.0) (0.0, 0.0) (0.0, 0.0)(2) p T (1.0, 1.0) (1.0, 1.0) (1.0, 1.0) (1.0, 1.0)

q F (0.0, 0.0) (0.0, 0.0) (0.0, 0.0) (0.0, 0.0)

p→ q T (1.0, 1.0) (1.0, 1.0) (1.0, 1.0) (1.0, 1.0)(3) p F (0.0, 0.0) (0.0, 0.0) (0.0, 0.0) (0.0, 0.0)

q T or F (0.0, 1.0) (0.0, 0.0) (0.0, 0.0) (0.0, 1.0)

p→ q F (0.0, 0.0) (0.0, 0.0) (0.0, 0.0) (0.0, 0.0)(4) p F (0.0, 0.0) (0.0, 0.0) (0.0, 0.0) (0.0, 0.0)

q – – (0.0, 0.0) (0.0, 0.0) –

a See [7,20].b See [5] and Appendix B.

• (p; (0:0; 1:0)) describes the ignorance about thetruth or the falsity (see Fig. 3).

Their possibilistic reasoning (modus ponens [20]) isexpressed as follows:

p→ q; (�p→q; �p→q)

p; (�p; �p)

q; (�q; �q)

(5)

Dubois et al. have also developed two other ap-proaches to deriving �q and �q given �p, �p, �p→q

and �p→q: (i) to use monotonic unequal relationship

[7,20], then

�q = min(�p; �p→q);

�q = max({

0 �p + �p→q61�p→q �p + �p→q¿1

;

{0 �p + �p→q61�p �p + �p→q¿1

)

and (ii) to use logical consequence technique (Dubois,Lang and Prade [5]). As a comparison, Table 1illustrates some typical cases in which the truthvalues (column three) and their corresponding


Fig. 4. The notion of distance.

necessity and possibility measures (column four) aregiven. The comparison is based on the distance (i.e.de�ned as |�q−�(q)|+ |�q−N (q)| or |�q−�(q)|+ |Nq − N (q)| , see Fig. 4) between the given(N (q); �(q)) and their bounds (upper or lowerbounds) derived in each approach. In case (3), ourapproach coincides with the given necessity and pos-sibility measures, but theirs do not. In case (4), ourapproach will be able to infer inconsistent conclu-sions; whereas, Dubois, Lang and Prade’s approachesstill obtain consistent conclusions.

3. Representing uncertain and impreciseinformation

A classical proposition is true in some possibleworlds and false in the rest of possible worlds, whilea fuzzy proposition p̃ is true with respect to a possibleworld to a degree [9]. We model our uncertainty aboutthe actual world by de�ning a possibility distributionover all possible worlds to specify the degree of pos-sibility that the actual world is in each possible world(see Fig. 5). Esteva et al. [9] have extended Duboisand Prade’s de�nition about the possibility and neces-sity measures of classical propositions to the case offuzzy propositions through fuzzy truth values [3] asfollows. Given a possibility distribution � on the setof possible worlds , the membership function of afuzzy truth value of p̃ is de�ned as

��(p̃ | �)(t) = Sup{�(!) | �p̃(!) = t}; t ∈ [0; 1]; (6)

where �p̃ denotes the fuzzy set of possible worlds ofp̃ in , ! is a possible world in and t is the degree oftruth. �(p̃ | �) can be viewed as the possibility measureof a set of possible worlds in which the truth degreeof p̃ is equivalent to t [9], i.e.,

��(p̃ | �)(t) =��{!∈ | �p̃(!) = t}; t ∈ [0; 1]: (7)

The fuzzy truth value of a fuzzy proposition, repre-senting the possibility of the degree of truth of thefuzzy proposition, can be viewed as our con�dencelevel about the fuzzy proposition.

To represent uncertain imprecise information, wepropose a fuzzy proposition with a fuzzy valuation,denoted as

(p̃; �);

where p̃ is a fuzzy proposition of the form “X is F̃”(i.e. X is a linguistic variable [26] and F̃ is a fuzzy setin a universe of discourse U ), and � is a fuzzy valua-tion. It should be noted that, for every formula (p̃; �)(called a truth-quali�ed fuzzy proposition), we assume�¿�(p̃ | �), which means ��(t) is the upper bound ofthe possibility that p̃ is true to a degree t. The fuzzyset is to represent the intended meaning of impreciseinformation; while, the fuzzy truth value serves as therepresentation of uncertainty for its capability to ex-press the possibility of the degree of truth.

To develop inference rules for truth-quali�ed fuzzypropositions, we treat a truth-quali�ed fuzzy propo-sition as a set of weighted classical propositions, inwhich the weight is represented by using necessityand possibility measures. For the purpose of explain-ing how a set of weighted classical propositions areinduced from a truth-quali�ed fuzzy proposition, we�rst introduce the notion of l-cut.

De�nition 1. The crisp set of elements that belong tothe fuzzy set F̃ to the degree l is called the l-level-set:

F̃ (l),{u∈U | �F̃(u) = l};

where U is the universe of discourse.

Based on De�nition 1, we can derive the followinginequality:

��(t)¿�[p̃(t)]; t ∈ [0; 1]; (8)


Fig. 5. Possible worlds semantics for uncertain imprecise information.

Fig. 6. A fuzzy truth value.

where p̃(t) denotes “X is F̃ (t)”. Thus, Eq. (8) canbe interpreted as: the upper bound of the possibilitymeasure of p̃(t) is ��(t).

It is obvious that a �-level-set F̃� of F̃ is a union ofsome l-level-sets, i.e. F̃� =

⋃l{F̃ (l); ∀l¿�}. There-

fore, the upper bound �p̃� of the possibility of p̃�is equal to the maximum grade of membership oft ∈ [�; 1] in �; the lower bound Np̃� of the necessityof p̃� is equal to the lower bound of the duality ofthe possibility of “X is not F̃�”, which is to takethe di�erence between 1 and the maximum grade ofmembership of t ∈ [0; �) in � (see Fig. 6). These areformally de�ned below.

De�nition 2. Atruth-quali�edfuzzyproposition(p̃; �)is equivalent to a set of classical propositions withnecessity and possibility pairs

{(p̃�; (Np̃� ; �p̃�)); �∈ (0; 1]};

where Np̃� denotes the lower bound of the necessitymeasure that p̃� is true; whereas, �p̃� denotes the up-per bound of the possibility measure that p̃� is true,de�ned as

Np̃� = 1 − max{��(t) | t ∈ [0; �)};�p̃� = max{��(t) | t ∈ [�; 1]}:

(9)

The membership function of F̃ can be reconstructedin terms of the set of the characteristic functions �F̃�of its �-level-sets F̃�, i.e.,

�F̃(u) = Sup{� · �F̃�(u) | �∈ (0; 1]}; u∈U: (10)

Reconstruction of � from the set of (Np̃� ; �p̃�) pairsin Eq. (9) is through the use of the principle of min-imum speci�city [6]. The principle says that, given aset of constraints restricting the value of a linguisticvariable, the possibility distribution of the linguisticvariable should be de�ned so as to allocate the maxi-mal degree of possibility to each value, in accordancewith the constraints [5]. Therefore, the least arbitrarychoice among those candidates of �, satisfying Eq. (9)for each pair of (Np̃� ; �p̃�), is the least speci�c solu-tion �(�) (see Fig. 6), i.e., for each �,

��(�)(t) =

{�p̃� if t¿�;

1 − Np̃� if t¡�:(11)

Thus, � can be reconstructed by the followingequation:

��(t) = Inf{��(�)(t) | �∈ (0; 1]}; t ∈ [0; 1]: (12)


4. Reasoning for truth-quali�ed fuzzy propositions

The formulation of the proposed inference rulefor truth-quali�ed fuzzy propositions is expressed asfollows:

p̃→ q̃; �1

p̃′; �2

q̃′; �3

(13)

where p̃, q̃, p̃′, and q̃′ are fuzzy propositions andcharacterized by “X is F̃”, “Y is G̃”, “X is F̃ ′” and“Y is G̃

′”, respectively; �1, �2, and �3 are fuzzy val-

uations for truth values and de�ned by ��1 (t), ��2 (t),and ��3 (t), respectively. F̃ and F̃ ′ are the subsets ofU , while G̃ and G̃

′are the subsets of V.

There are three major steps for deriving q̃′ and �3

of Eq. (13). First, the fuzzy rules and fuzzy facts withfuzzy truth values are transformed into a set of uncer-tain classical propositions with necessity and possibil-ity measures by means of �-cut. Second, possibilisticentailment is performed on the set of uncertain classi-cal propositions. Third, we reverse the process in the�rst step to synthesize all the �-level-sets obtained inthe second step into a fuzzy set, and to compose neces-sity and possibility pairs to form a fuzzy truth value.

Step 1: Transformation. Based on De�nition 2, atruth-quali�ed fuzzy fact is equivalent to a set of clas-sical propositions (the �-level-sets of the fact) withnecessity and possibility pairs. Similarly, a fuzzy rulewith a fuzzy truth value can be viewed as a collec-tion of classical implication relationships (the �-level-sets of the fuzzy relation for this rule) with necessityand possibility pairs. Therefore, Eq. (13) can be trans-formed as follows:

(p̃→ q̃)�; (N(p̃→q̃)� ; �(p̃→q̃)�) (14)

p̃′�; (Np̃′

�; �p̃′

�) �∈ (0; 1] (15)

q̃′�; (Nq̃′� ; �q̃′�) (16)

where

N(p̃→q̃)� = 1 − max{��1 (t) | t ∈ [0; �)};�(p̃→q̃)� = max{��1 (t) | t ∈ [�; 1]};Np̃′

�= 1 − max{��2 (t) | t ∈ [0; �)};

�p̃′�= max{��2 (t) | t ∈ [�; 1]}:

(17)

Step 2: Inference.Computing q̃′�. G̃

′� is computed through the use of

compositional rule of inference, i.e.,

G̃′� = F̃ ′

� ◦ (F̃→ G̃)�; (18)

where ◦ is a composition operator and → denotes animplication operator.

Computing Nq̃′� and �q̃′� . With the help of theprinciple of minimum speci�city [6], Eq. (14) can betransformed into a set of classical propositions withnecessity and possibility pairs as follows:

p̃′� → q̃′�; (Np̃′

�→q̃′�; �p̃′

�→q̃′�) (19)

Np̃′�→q̃′�

and �p̃′�→q̃′�

are de�ned as

Np̃′�→q̃′�

= 1 − max{�(u; v) | (u; v) =∈ F̃ ′� → G̃

′�};

�p̃′�→q̃′�

= max{�(u; v) | (u; v)∈ F̃ ′� → G̃

′�};

(20)

where �(u; v) denotes a possibility distribution overU ×V, derived by means of the principle of minimumspeci�city:

�(u; v) = Inf{��(u; v) | �∈ (0; 1]};

��(u; v) =

{�(p̃→q̃)� ; (u; v)∈ (F̃ → G̃)�;

1 − N(p̃→q̃)� ; (u; v) =∈ (F̃ → G̃)�:

(21)

Therefore, for each �∈ (0; 1], we have

p̃′� → q̃′�; (Np̃′

�→q̃′�; �p̃′

�→q̃′�)

p̃′�; (Np̃′

�; �p̃′

�)

q̃′�; (Nq̃′� ; �q̃′�)

(22)

The possibilistic entailment in Section 2 is then ap-plied to Eq. (22) to obtain the upper bound of the pos-sibility measure and the lower bound of the necessitymeasure of q̃′� (if max{�p̃′

�→q̃′�; �p̃′

�}= 1):

Nq̃′� = min{Np̃′� → q̃′�

; Np̃′�};

�q̃′� =�p̃′�→q̃′�

:(23)

Step 3: Composition. Based on Eq. (10), the con-struction of membership function of G̃

′is performed

by

�G̃′(v) = Sup{� · �G̃′�(v) | �∈ (0; 1]}; v∈V: (24)


Meanwhile, the construction of �3 is calculated byEq. (12):

��3 (t) = Inf{��3(�)(t) | �∈ (0; 1]}; t ∈ [0; 1]; (25)

where

��3(�)(t) ={�q̃′� if t¿�;1 − Nq̃′� if t¡�:

(26)

The following example illustrates how one appliesthe algorithm to compute the meaning-modi�ed con-clusion and its associated con�dence level based onthe given rule and fact. In this example, “Sup-min” andG�odel are chosen as the composition operator and theimplication operator, respectively (see Appendix A).

Rule: (A father is tall →His son is quite tall ; very true)

Fact: (Paul is quite tall ; more or less true)Query: What can we say about the height of

Paul’s son?

where tall, quite tall, true, very true, and more or lesstrue are de�ned as (see Fig. 7)

� tall(u) =

1; u¿190;(u− 170)=20; 1706u6190;0; u6170;

�quite tall(u) = [� tall(u)]2;

� true(t) = t;

� very true(t) = t2;

�more or less true(t) =√t:

Applying our algorithm to the example, the answer is

Conclusion: (Paul’s son is quite tall ;more or less true):

Notice that, in Eq. (13), if “Sup-min” is chosenas the composition operator, and both �1 and �2 are“true” which is de�ned by its membership function,�true(t) = t for all t ∈ [0; 1], �3 is then “true”. That is,our proposed inference rule reduces to

p̃→ q̃; true

p̃′; trueq̃′; true

It should also be noted that if both p̃ and p̃′ are equalto a classical proposition p, and both q̃ and q̃′ are

equal to a classical proposition q, and each �i (i= 1–3)reduces to ��i(0) which means the possibility of falsity(i.e. the duality of the necessity of truth) and ��i(1)which means the possibility of truth, our inference rulebecomes (if max{�p̃′

�→q̃′�; �p̃′

�}= 1):

p→ q; (Np→q; �p→q)

p; (Np;�p)q; (min{Np→q; Nq}; �p→q)

To summarize, our proposed algorithm is not only ageneralization of Zadeh’s generalized modus ponens(called fuzzy reasoning), but also an uncertain reason-ing for classical propositions with necessity and pos-sibility pairs (see Fig. 8).

5. Related work

A great deal of work concerned reasoning for eitherimprecise or uncertain information in expert systems[12,14,15], but only a few advocated the integration ofuncertainty and imprecision. In order to perform rea-soning for both imprecise and uncertain information,two important issues should be addressed:• Any improvement of the con�dence level for a

piece of information can only be achieved at theexpense of the speci�city of the information; andvise verse [23,25]. For example, “It is fairly truethat John is young” is semantically equivalent to“It is true that John is fairly young”.

• The matching between a fuzzy fact and the premiseof a fuzzy rule is not exact, but only partial [25].

We have roughly classi�ed the existing approaches indealing with both imprecise and uncertain informationinto three categories based on their treatments for thetwo issues.

Certain and meaning-modi�ed conclusion. Basedon fuzzy reasoning, an uncertainty-quali�ed fuzzyproposition is translated into a proposition whosecon�dence level is certain but with less speci�c in-formation, while partial matching is to modify theintended meaning of the conclusion. This approachwas advocated by Zadeh [25] and Yager [23]: Zadehproposed three uncertainty-quali�cations for fuzzypropositions: probability-, possibility- and truth-quali�ers. However, instead of developing the infer-ence mechanism for these uncertainty-quali�ed fuzzy


Fig. 7. The membership functions of some fuzzy sets.

Fig. 8. Our inference rule is a generalization of fuzzy reasoning and possibilistic reasoning.

proposition, Zadeh used these quali�ers to modifythe meaning of fuzzy propositions. Therefore, anuncertainty-quali�ed fuzzy fact and an uncertainty-quali�ed rule were translated into the semanticallyequivalent propositions whose con�dence levelswere certain but less speci�c. Hence, this approachcould be viewed as a generalization of fuzzy reason-ing. Following Zadeh’s work, Yager focused on thecertainty-quali�er.

Uncertain and meaning-retained conclusion. Thedegree of partial matching is to in uence the con�-dence level of the conclusion, which was adopted

by researchers such as Ogawa et al. [19], Martin-Clouair et al. [17] and Umano [22]: Ogawa et al.extended certainty factor to the case of fuzzy proposi-tions for representing uncertain imprecise informationin SPERIL-2, an expert system for damage assessmentof existing structures. Obviously, Ogawa’s extensionwas not based on the fuzzy reasoning but on the un-certain reasoning. Martin-Clouair and Prade designeda special inference mechanism where the rule was hy-brid (i.e. the rule had fuzzy premise but only preciseconclusion, and its uncertainty was expressed as ne-cessity and possibility measures), but the fuzzy fact


had to be completely certain. The degree of partialmatching was viewed as the uncertainty of the premiseof the rule, possibilistic reasoning was then adoptedto obtain the necessity and possibility degrees for theprecise conclusion. Hence, this special inference ruleis neither an extension of fuzzy reasoning or a general-ization of uncertain reasoning. Umano et al. chose thefuzzy truth value as the uncertainty-quali�er for fuzzypropositions. Partial matching was completely trans-formed into the fuzzy truth value of the conclusion.

Uncertain and meaning-retained conclusion; nopartial matching. No partial matching is allowed inIshizuka et al. [13] and Godo et al. [11]: Ishizukaet al. [13] extended the Dempster–Shafer’s evidencetheory to fuzzy set in an expert system SPERIL-1.They introduced conditional Dempster–Shafer ba-sic probabilities m(·|·) as the con�dence level forrules. The upper and lower probabilities of facts andconclusions which were derived by the bodies ofevidence for or against these facts and conclusionswere de�ned as the �nal con�dence level. The pre-sence of fuzzy sets was just used to reduce thecon�dence level. This inference mechanism was obvi-ously not a generalization of fuzzy reasoning, althoughit could reduce to evidence theory if fuzzy sets reducedto classical ones. Godo et al. used the fuzzy truthvalue as uncertainty-quali�er for fuzzy propositions.Although they claimed that they have extended fuzzylogic (i.e. multi-valued logic) to so-called fuzzy truth-valued logic, partial matching was not allowed forthe sake of simpli�cation. Hence, strictly speaking,their inference rule was not a generalization of fuzzyreasoning. A comparison between our approach andother related work is summarized in Table 2.

6. Conclusion

We propose truth-quali�ed fuzzy propositions asa representation of uncertain imprecise information,since the fuzzy truth value is capable of express-ing the possibility of the degree of truth of a fuzzyproposition. An uncertain imprecise proposition isinterpreted as a set of classical propositions withnecessity and possibility pairs. Based on the interpre-tation, we also develop an algorithm for reasoningwith uncertain imprecise propositions. There are threesteps involved. First, the fuzzy rules and fuzzy facts

with fuzzy truth values are transformed into a set ofuncertain classical propositions with necessity andpossibility measures by means of �-cut. Second, pos-sibilistic entailment is performed on the set of un-certain classical propositions. Third, we reverse theprocess in the �rst step to synthesize all the �-level-sets obtained in the second step into a fuzzy set, andto compose necessity and possibility pairs to form afuzzy truth value. As a result, a generalized modusponens for truth-quali�ed fuzzy propositions has beendeveloped.

Our approach o�ers two important advantages.• Our approach does not impose any restriction on

the inference, that is, the intended meaning is notrequired to be unchanged; meanwhile, the con�-dence level can be partially certain.

• The proposed inference is not only a generalizationof Zadeh’s generalized modus ponens but also anuncertain reasoning for classical propositions withnecessity and possibility pairs (called possibilisticentailment). Therefore, it serves as a bridge thatbrings together the possibilistic entailment and thefuzzy reasoning into a hybrid approach to reasoningunder uncertainty and imprecision.Our future research plan will consider the following

tasks: (1) to extend the proposed reasoning to fuzzyPetri nets, and (2) to apply it to a practical case ofdamage assessment of bridges.

Appendix A. Selecting operators

Fuzzy reasoning is represented as follows:

If X is F̃ then Y is G̃X is F̃

′

Y is G̃′(27)

where X and Y are linguistic variables; F̃ and F̃ ′ arefuzzy sets of U ; G̃ and G̃′ are fuzzy sets of V. In theframework of compositional rule of inference, G̃′ iscomputed by

G̃′ = F̃ ′ ◦ (F̃→ G̃); (28)

where ◦ is a composition operator and → means animplication operator. Selection of operators is an im-portant issue for calculating G̃′. To determine these


Table 2The comparison between related work and our approach

Imprecision Uncertainty A generalization of A generalization ofmodel model fuzzy reasoning uncertain reasoning

Zadeh [25] Fuzzy set Uncertainty-quali�ers Yes NoYager [23] Fuzzy set Certainty Yes NoOgawa et al. [19] Fuzzy set Certainty factor No YesMartin-Clouair et al. [17] Fuzzy set Possibility and necessity No NoUmano [22] Fuzzy set Fuzzy truth value No YesIshizuka et al. [13] Fuzzy set Evidence theory No YesGodo et al. [11] Fuzzy set Fuzzy truth value No YesOur approach Fuzzy set Fuzzy truth value Yes Yes

operators, the following criteria should be satis�ed forthe case of single rule [10]:• Criterion 1: If F̃ ′ is F̃ then G̃′ is G̃.• Criterion 2: If F̃ ′ is very F̃ then G̃′ is G̃.• Criterion 3: If F̃ ′ is more or less F̃ then G̃′ is more

or less G̃.• Criterion 4: If F̃ ′ is not F̃ then G̃′ is unknown.

In this paper, “Sup-min” [24] is chosen as thecomposition operator since it satis�es the followingequality [1,2]:

G̃′� = (F̃ ′ ◦ (F̃→ G̃))� = F̃ ′

� ◦ (F̃→ G̃)�: (29)

Based on Eq. (28), Eq. (24) becomes

�G̃′(v) = maxu

min{�F̃′(u); �F̃→ G̃(u; v)};(u; v)∈U ×V:

Under “sup-min” composition operator, eight im-plication operators are candidates. An implication op-erator, a→ b, is de�ned by a function of a and b. Theeight implication operators are listed as below:

G �odel a→ b=

{1; a6b;

b; a¿b;

Lukasiewicz a→ b= min(1; 1 − a+ b);

Kleene–Dienes a→ b= max(1 − a; b);

Mamdani a→ b= min(a; b);

Zadeh a→ b= max(1 − a;min(a; b));

Goguen a→ b=

{1; a= 0;

min(1; b=a) otherwise;

Yager a→ b= ba;

Probabilistic a→ b= a · b:Table 3 indicates that G�odel implication operator

is a good one under Sup-min composition operator.Furthermore, Turksen et al. [21] have pointed out thatif “min” is chosen as the combination operator un-der G�odel implication and Sup-min composition, andthe membership functions of rules satisfy certain con-strains described in [21], then the criterion (i.e. F ′ =Finfers G′ =G) for the case of multiple rules is satis-�ed. Hence, in this paper, we select “Sup-min” andG�odel as the composition and implication operators,respectively.

Appendix B. Dubois, Lang and Prade’s possibilisticlogic

In Dubois, Lang and Prade’s approach, they as-sume !1 = (p; q); !2 = (p;@ q); !3 = (@p; q);!4 = (@p;@ q), then four cases in Table 1 are de-scribed as follows:

Case (1):

max{�(!1); �(!3); �(!4)}= 1

�(!2) = 0

max{�(!1); �(!2)}= 1

max{�(!3); �(!4)}= 0

max{�(!1); �(!2); �(!3); �!4}= 1


Table 3The comparison between some implication operators under Sup-min composition

G�odel Lukasiewicz Kleene–Dienes Mamdani Zadeh Goguen Yager Probabilistic

Criterion 1√ √

Criterion 2√ √

Criterion 3√ √ √

Criterion 4√ √ √ √ √ √

⇒

�(!1) = 1

�(!2) = 0

�(!3) = 0

�(!4) = 0

⇒ N (q) = 1; �(q) = 1 ⇒ q; (1:0; 1:0):

Case (2):

max{�(!1); �(!3); �(!4)}¿0

�(!2)61

max{�(!1); �(!2)}= 1

max{�(!3); �(!4)}= 0

max{�(!1); �(!2); �(!3); �(!4)}= 1

⇒

�(!1) = 1; �(!2)61or�(!1)61 (¿0); �(!2) = 1

�(!3) = 0

�(!4) = 0

⇒N (q)¿0; �(q) = 1orN (q) = 0; �(q)¿0

⇒ q; (0:0; 0:0):

Case (3):

max{�(!1); �(!3); �(!4)}= 1

�(!2) = 0

max{�(!1); �(!2)}¿0

max{�(!3); �(!4)}61

max{�(!1); �(!2); �(!3); �(!4)}= 1

⇒

�(!1)¿0

�(!2) = 0

�(!3)61 (¿0)

�(!4)61

⇒ N (q)¿0; �(q)¿0 ⇒ q; (0:0; 0:0):

Case (4):

max{�(!1); �(!3); �(!4)}¿0

�(!2)61

max{�(!1); �(!2)}¿0

max{�(!3); �(!4)}61

max{�(!1); �(!2); �(!3); �(!4)}= 1

⇒

�(!1)¿0

�(!2)61

�(!3)61 (¿0)

�(!4)61

⇒ N (q)¿0; �(q)¿0 ⇒ q; (0:0; 0:0):

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A possibilistic-logic-based approach to integrating imprecise and uncertain information