IEEE TRANSACTIONS ON SYSTEMS, MAN AND CYBERNETICS PART B: CYBERNETICS, VOL. 33, NO. 2, APRIL

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1 TRANSACTIONS ON SYSTEMS, MAN AND CYBERNETICS PART B: CYBERNETICS, VOL. 33, NO. 2, APRIL Modeling Uncertainty Reasoning With Possibilistic Petri Nets Jonathan Lee, Senior Member, Kevin F. R. Liu, and Weiling Chiang Abstract Manipulation of perceptions is a remarkable human capability in a wide variety of physical and mental tasks under fuzzy or uncertain surroundings. Possibilistic reasoning can be treated as a mechanism that mimics human inference mechanisms with uncertain information. Petri nets are a graphical and mathematical modeling tool with powerful modeling and analytical ability. The focus of this paper is on the integration of Petri nets with possibilistic reasoning to reap the benefits of both formalisms. This integration leads to a possibilistic Petri nets model ( ) with the following features. A possibilistic token carries information to describe an object and its corresponding possibility and necessity measures. Possibilistic transitions are classified into four types: inference transitions, duplication transitions, aggregation transitions, and aggregation-duplication transitions. A reasoning algorithm, based on possibilistic Petri nets, is also presented to improve the efficiency of possibilistic reasoning and an example related to diagnosis of cracks in reinforced concrete structures is used to illustrate the proposed approach. Index Terms Diagnosis of cracks, possibilistic Petri nets, possibilistic reasoning. I. INTRODUCTION APPROXIMATE reasoning is one of the remarkable human capabilities that manipulate perceptions in a wide variety of physical and mental tasks, whether in fuzzy or uncertain surroundings. To model this remarkable human capability, L.A. Zadeh [37], [38] proposed a new concept of computing with words, which is a methodology in which the objects of computation are words and propositions drawn from a natural language. It provides a basis for a computational theory to imitate how humans make perception-based rational decisions in a fuzzy environment. Nevertheless, besides fuzziness, humans also perform perception-based reasoning well under uncertain circumstances. To deal with uncertain information in reasoning methods, several formalisms have been proposed, such as, certainty factor [35], probabilistic logic [28], Dempster Shafer theory of evidence [32], possibilistic logic [8] and possibilistic Manuscript received April 24, 2001; revised November 7, 2001 and February 28, This research was supported in part by the National Science Council (Taiwan, R.O.C) under Grants NSC E and NSC E , and by the Ministry of Education (Taiwan, ROC) under Grants EX-91-E-FA This paper was recommended by Associate Editor G. Biswas. J. Lee is with the Department of Computer Science and Information Engineering, National Central University, Chungli, Taiwan ( yjlee@selab.csie.ncu.edu.tw). K. F. R. Liu is with the Department of Environmental Engineering, Da-Yeh University, Changhua, Taiwan ( kevinliu@mail.dyu.edu.tw). W. Chiang is with the Department of Civil Engineering, National Central University, Chungli, Taiwan ( chiang@cc.ncu.edu.tw). Digital Object Identifier XXXXXXXXXXX reasoning [21], etc. Consequently, an adequate management of uncertainty for reasoning methods has become a significant issue [1]. To increase the efficiency of rule-based reasoning with uncertain information, two issues are particularly relevant: the possibility of exploiting concurrency and the use of smart control strategies [13]. To achieve these goals, a number of researchers have reported progress toward the modeling of rule-based reasoning with Petri nets [2], [6], [16], [24], [31]. Petri nets are a graphical and mathematical modeling tool to describe and study information processing systems. Petri nets with a powerful modeling and analysis ability are capable of providing a basis for variant purposes, such as knowledge representation [27], [33], reasoning mechanism [2], [31], knowledge acquisition [5], and knowledge verification [36], [39]. For a decade, researchers have enhanced the conventional Petri nets to involve imprecise or uncertain information. We examine these studies related to the modeling of rule-based reasoning with Petri nets (e.g., fuzzy Petri nets [2], [6], [16], [24], [31]), the modeling of probabilistic logic with Petri nets [23] and possibilistic Petri nets [3]. To take fuzzy information into consideration in rule-based reasoning, researchers have advocated the use of fuzzy Petri nets to model fuzzy rule-based reasoning [2], [6], [16], [20], [24], [31]. Looney s fuzzy Petri nets model [24] modified the conventional Petri nets to allow each transition and place attached a fuzzy truth-value. Chen et al. s approach [6] considered not only fuzziness but also uncertainty (i.e., modeled as certainty factors [35]) for representing a fuzzy rule base. They utilized reachability to trace the inference paths through constructing sprouting trees. Bugarin et al. s approach [2] was proposed to model a compositional rule of inference. Their reasoning algorithm generated a variant pattern of the fuzzy Petri net based on the input facts. Konar et al. [16] improved Chen et al. s [6] reasoning algorithm to deal with the case that there exists intermediate places with multi-input arcs. Scarpelli et al. [31] were devoted to high-level fuzzy Petri nets for modeling fuzzy reasoning based on a compositional rule of inference. Lee et al. [20] proposed a fuzzy Petri nets approach to modeling fuzzy rule-based reasoning which brings the possibilistic entailment and fuzzy reasoning together to handle uncertain and imprecise information. Researchers have made some progress embedding uncertainty models into Petri nets for different purposes. For example, stochastic Petri nets [15], [22] is a class of Petri nets in which the firing times are considered as random variables, and a probability distribution over all transition firing times is formed. Lin et al. [23] investigated the use of Petri nets to deal with the size problem of the possible world matrix in /03$

2 2 TRANSACTIONS ON SYSTEMS, MAN AND CYBERNETICS PART B: CYBERNETICS, VOL. 33, NO. 2, APRIL 2003 Nilsson s probabilistic logic [28]. Cardoso et al. [3] proposed a possibilistic Petri nets model that combined possibility theory and Petri net theory to lead to a tool for qualitative representation of uncertain knowledge about a system state. They used possibility distributions over all places and tokens to display the uncertainty about possible locations of a token before receiving certain information. Although the attempt has been made by these researchers to embed the uncertainty model into Petri nets, little emphasis has been put on how to model uncertainty reasoning in rule-based expert systems with Petri nets. In this paper, we combine a high-level Petri nets model with possibilistic reasoning as possibilistic Petri Nets to model uncertainty with possibilistic information. Possibilistic reasoning [21], inspired by Nilsson s probabilistic entailment [28], is an uncertainty reasoning for classical propositions weighted by the lower bounds of necessity measures and the upper bounds of possibility measures. In, a possibilistic token carries information to describe a proposition and the corresponding possibility and necessity measures. Four types of possibilistic transitions, inference, aggregation, duplication, and aggregation duplication transitions, are introduced to fulfill the mechanism of possibilistic reasoning. The inference transitions perform possibilistic reasoning; duplication transitions duplicate a possibilistic token to several tokens representing the same proposition, possibility, and necessity measures; aggregation transitions combine several possibilistic tokens with the same classical proposition; and aggregation duplication transitions combine aggregation transitions and duplication transitions. A reasoning algorithm based on possibilistic Petri nets is also outlined to improve the efficiency of possibilistic reasoning. An example related to diagnosis of cracks in reinforced concrete structures is used as the illustration to our approach. The results through not only match Chao et al. s conclusions [4], but also provide more information because the explanation of diagnosis process served by token movement and the confidence level associated with possible causes can be used as a way of justification on whether to take the diagnosis into account or not. The organization of this paper is as follows. A general formalism of possibilistic Petri nets and its hierarchical representation are proposed in the next section. A possibilistic reasoning based on possibilistic entailment is introduced in Section III. In Section IV, a knowledge representation of possibilistic reasoning and a reasoning algorithm are developed through the use of possibilistic Petri net. An application of possibilistic Petri nets to diagnosis of cracks in reinforced concrete structures is used in Section V as an illustration. Finally, a summary of our approach and its potential benefits are given in Section VI. II. POSSIBILISTIC PETRI NETS In this section, are defined for modeling possibilistic systems and are used as a knowledge representation for possibilistic information. A. Petri Nets A Petri net is a directed, weighted, bipartite graph consisting of two kinds of nodes, called places ( ) and transitions ( ), where arcs are either from a place to a transition, or from a transition to a place [29]. Murata has formally defined Petri nets as a 5-tuple [26]: is a finite set of places, is a finite set of transitions, set of arcs,, where is a is a weight function and is the initial marking. A marking is a -vector,, where denotes the number of the tokens in place. The evolution of markings, used to simulate the dynamic behavior of a system, is based on the firing rule, such as: a transition is enabled if each input place is marked with at least tokens, where is the weight of the arc from to ; an enabled transition may or may not be fired; and a firing of an enabled transition removes tokens from each input place of and adds tokens to each output place of, where is the weight of the arc from to. Some notations are introduced as follows: denotes the input places of, denotes the output places of, denotes the input transitions of and denotes the output transitions of. B. The simple view of a system concentrates on two primitive concepts: events and conditions. An event is an action which takes place in the system. A condition is a predicate or logical description of the state of the systems [29]. A typical interpretation of Petri nets is to view a place as a condition, a transition as the causal connectivity of conditions (an event), and a token in a place as a fact used to claim the truth of the condition associated with the place. The confidence about the connectivity of conditions and the facts could be uncertain, however. To take the uncertain situations into account, we formally define a possibilistic Petri nets as Definition 1: A is defined as a five-tuple is a finite set of places, where place represents a classical proposition. is a finite set of possibilistic transitions, where represent the connectivity of places, denotes the lower bounds of necessity measures and denotes the upper bounds of possibility measures to represent the uncertainty about the connectivity between places. is a set of arcs. is a weight function. is the initial marking, where is the number of tokens in. Each token is associated with a pair of necessity and possibility measures ( ) (called a possibilistic token) and inserted into the related place that represents a classical proposition. To simulate the dynamic behavior of a possibilistic system, a marking in a is changed according to the firing rule: a firing of an enabled possibilistic transition removes the possibilistic tokens from each input place of, adds a new token (1)

3 LEE et al.: MODELING UNCERTAINTY REASONING WITH POSSIBILISTIC PETRI NETS 3 to each output place of and the necessity and possibility measures attached to the new token will be computed based on possibilistic systems. (In this paper, the possibilistic system is considered as possibilistic reasoning. The mapping of to possibilistic reasoning is described in Section IV.) A simple example of is illustrated in Fig. 1. C. Hierarchical Structures of To overcome the complexity arising from large sizes of, a hierarchical structure of is developed. In a hierarchical structure, is divided into several hierarchies that contain smaller which may, or may not, include other hierarchies. As illustrated in Fig. 2(a), a hierarchy is drawn as a double-lined square to connect places viewed as a transition. The hierarchical in Fig. 2(a) is considered a main hierarchy, and contains three hierarchies,,, and. In this example, the main hierarchy is at the top level of the hierarchical structure and,, and are at the second level. The connections between hierarchies are achieved by defining importing and exporting places (see Fig. 2). That is, an exporting place w.r.t. a hierarchy is defined as a place that is connected to the hierarchy by an arc from the place to the hierarchy; meanwhile, an importing place with respect to a hierarchy is defined as a place connected to the hierarchy by an arc from the hierarchy to the place. In this hierarchical, places, and are the exporting place w.r.t. hierarchy and place is the importing places w.r.t. hierarchy. In the hierarchy, places and are the importing places w.r.t. and place is the exporting place w.r.t.. When a token is inserted into the place in, it will be transited into hierarchy and added to place in hierarchy. Similarly, once a token enters into place in, it will be sent into hierarchy, and reaches the place in. After firing the transitions in, the computations of possibilistic reasoning are triggered and the token arrives at the place in and then enters the place in. There are two main benefits by having a hierarchical structure for : 1) the notion of hierarchy makes the handling of complex systems through decomposition easy and 2) a hierarchical Petri net facilitates the reusability, namely, each hierarchy can be considered as a reuse unit. III. POSSIBILISTIC REASONING The distinction between imprecise and uncertain information is best explained by the canonical form representation (i.e., a quadruple of attribute, object, value, confidence) proposed by Dubois and Prade [9], [10]. Imprecision implies the absence of a sharp boundary of the value component of the quadruple; whereas, uncertainty is related to the confidence component of the quadruple, which is an indication of our reliance about the information. Dubois and Prade [9] have proposed the possibility and necessity measures as an uncertainty model for classical propositions. In this paper, we propose a representation for possibilistic information and an inference mechanism called possibilistic entailment based on Dubois and Prade s possibility and necessity measures. Fig. 1. Illustration of a possibilistic Petri net: (a) Before firing t. (b) After firing t. Fig. 2. (a) Hierarchical structure of PPN; (b) hierarchy H : modeling two rules; (c) hierarchy H : modeling one rule; and (d) hierarchy H : modeling one rule. A. Uncertainty Representation A classical proposition is true in some possible worlds and false in the rest. We model our uncertainty about the actual world by defining a possibility distribution over all possible worlds to specify the degree of possibility that the actual world is in each possible world. As shown in Fig. 3, the possibility of the possible world to be the actual world is and these possibilities form a possibility distribution. A classical proposition is true in possible worlds, and, and false in the rest of possible worlds. Given a possibility distribution on the set of possible worlds, Dubois et al. [8] have formally defined the possibility, and necessity measures, for a classical proposition as where denotes that is true in (i.e., a possible world in ). In Fig. 3, the possibility measure of is and the necessity measure is. The possibility measure means the degree of ease to say that is true; meanwhile, the necessity measure is a duality of possibility measure. ( ) denotes the degree of ease to say that is false. The pair of possibility and necessity measures for a classical proposition is adopted here for our confidence level about the classical proposition. (2)

4 4 TRANSACTIONS ON SYSTEMS, MAN AND CYBERNETICS PART B: CYBERNETICS, VOL. 33, NO. 2, APRIL 2003 Fig. 3. Possible worlds semantics for possibilistic information. To represent uncertain information, we have proposed a representation of possibilistic propositions as follows [21]: (3) where denotes a classical proposition, denotes the lower bounds of necessity measures and denotes the upper bounds of possibility measures (i.e., and ). A possibilistic proposition ( ) means that the degree of ease to say that is false is at most equal to ; meanwhile, the degree of ease to say that is true is at most equal to. B. Possibilistic Entailment The possibilistic reasoning for possibilistic propositions is expressed as follows [21]: To infer and, we have proposed an approach called possibilistic entailment [21], inspired by Nilsson s probabilistic entailment [28] that helps one deduce a new proposition with associated probability from a base set of propositions with associated probabilities. In the proposed approach, a semantic tree is used to determine the possible worlds. More specifically, a possible world is the one in which the truth-values of all involved propositions are consistent in a semantic tree (see Fig. 4). In Fig. 4, inconsistent paths are indicated by an ; and denote the truth-values and, respectively. This semantic tree shows that there are four possible worlds. (4) (5) Fig. 4. Semantic tree. sibilities ( ), the lower bounds of the necessities ( ) of propositions and the possibilities of possible worlds ( ). where indicates the composition operator - and denotes the possibility that the actual world is in possible world. The first row of the above 6 4 matrix gives truthvalues for in the four possible worlds; whereas, the second row indicates the possible worlds where is false. The right-hand side of this inequality means the possibility measures of the involved propositions and the left-hand side of this inequality expresses the upper bounds of these possibility measures. Based on the given,,, and, we can derive a possibility distribution which is the upper bound of Thus, the upper bound of the possibility, and the lower bound of the necessity of are (6) (7) (8) where the columns denote the four possible worlds and give the consistent sets of truth-values for these three propositions. We have mentioned that a possibility distribution over all possible worlds is used to model our uncertainty about the actual world, the possibility measure of any proposition is then reasonably taken to be the maximum of the possibilities of all possible worlds in which is true. Therefore, we can construct the relationship between the upper bounds of the pos- If the possibility distribution is normalized, which can only be achieved in the case that and do not exist simultaneously, (8) then becomes To illustrate this, assume that ( ) and ( ), then we can infer ( ). (9)

5 LEE et al.: MODELING UNCERTAINTY REASONING WITH POSSIBILISTIC PETRI NETS 5 However, if the possibility distribution is subnormalized (i.e.,, called partially inconsistent [8], [19]), and, derived from (8), are then meaningless since they violate the following constraints: ( ), ( ) and ( ). For example, given ( ) and ( ), ( ) is inferred. Hence, to obtain the consistent necessity and possibility values for, and, both and should not be less than 1 simultaneously. A more general formulation of the inference rule which includes multiple antecedents of a rule is expressed as (10) at the bottom of the page. where and are classical propositions and ( ) is viewed as a classical proposition ;, and are the lower bounds of necessity measures; and,, and are the upper bounds of possibility measures. After performing the possibilistic reasoning, we can derive the following conclusions: where becomes (11) and. In the case that and do not exist simultaneously, (11) then (12) When several rules having a same conclusion are fired, these inferred conclusions with different confidence levels, for example ( )( ), should be aggregated as a conclusion ( ). This aggregation can be viewed as a disjunction; we then obtain and Possibilistic logic has been advocated by Dubois and Prade for more than ten years. In their approach, they viewed a possibility measure of proposition as the degree of possibility that is true and a necessity measure as the degree of necessity that is true. A comparison between their approach and ours is discussed in our previous work [21]. IV. POSSIBILISTIC REASONING AND Many researchers have devoted themselves to modeling rulebased reasoning via Petri nets [2], [6], [16], [24], [31]. There are several reasons behind the computational paradigm for rulebased reasoning on Petri net theory. Petri nets achieve the structuring of knowledge within rule bases which express the relationships among rules, and also help experts to construct and modify rule bases. Petri net s graphic nature provide visualization of dynamic behavior of rule-based reasoning. Petri nets make it easier to design an efficient reasoning algorithm. Petri net s analytic capability provides a basis for developing knowledge verification technique. Petri nets can model the underlying relationship of concurrency among rules activation, an important aspect where real-time performance is crucial. A. Knowledge Representation The three key components in uncertain rule-based reasoning: propositions, uncertain rules, and uncertain facts, can be formulated as places, possibilistic transitions, and possibilistic tokens, respectively. The mapping between possibilistic reasoning and is described as follows. Places: Places correspond to classical propositions. The classical propositions that attached to places represent conditions. Possibilistic tokens: A possibilistic token represents an uncertain fact. A pair of necessity and possibility measures are attached to possibilistic tokens to represent our confidence level about the observed facts. Possibilistic transitions: Possibilistic transitions are classified into four types: inference, aggregation, duplication, and aggregation-duplication transitions. The inference transitions represent the uncertain rules, the aggregation transitions are designed to aggregate the conclusion parts of rules which have the same classical propositions, the duplication transitions are used to duplicate possibilistic tokens to avoid the conflict problem and the aggregation-duplication transitions link the same classical propositions. These are formally defined below. Type 1: Inference Transition ( ): An inference transition serves as modeling of an uncertain rule. An uncertain rule having multiple antecedents is represented as (13).... (10)

6 6 TRANSACTIONS ON SYSTEMS, MAN AND CYBERNETICS PART B: CYBERNETICS, VOL. 33, NO. 2, APRIL 2003 where and are classical propositions. In Fig. 5, after firing the inference transition, the tokens will be removed from the input places of, a new token will be deposited into the output place of and a pair of necessity and possibility measures attached to the new token are derived by the possibilistic reasoning (see Section III-B). Type 2: Aggregation Transition ( ): An aggregation transition is used to aggregate the conclusions of several uncertain rules which have a same classical proposition and to link the antecedent of an uncertain rule which also have the same classical proposition. For example, there are uncertain rules with the same classical proposition in the conclusions, denoted as Fig. 5. Modeling possibilistic reasoning through PPN; (a) before firing inference transition t, (b) after firing t. (14) In Fig. 6, after firing the aggregation transition, the tokens in input places of will be removed, a new token will be deposited into the output place of and a pair of necessity, and possibility measures attached to the new token are derived by disjunction (see Section III-B). It should be noticed that is dead if one of its input places never received a token. To avoid deadlock in aggregation transitions, we assume that for each source place, a token will be inserted into and a pair of necessity and possibility (0,1) is attached to represent ignorance if no fact refers to the proposition in place. Type 3: Duplication Transition ( ): The purpose of duplication transitions is to avoid the conflict by duplicating the token. For example, there are uncertain rules with a same classical proposition in the antecedents, denoted as (15) They are linked by a duplication transition shown in Fig. 7. After firing the duplication transition, the tokens in the input place of will be removed, new tokens will be added into the output places of and a pair of necessity and possibility measures attached to the new tokens are not changed. Type 4: Aggregation-Duplication Transition ( ): An aggregation-duplication transition is a combination of an aggregation transition and a duplication transition (see Fig. 8). It is used to link all the same classical propositions. For example, there are uncertain rules with a same classical proposition in the conclusions and uncertain rules with the same classical proposition in the antecedents, denoted as (16) They are linked by an aggregation-duplication transition shown in Fig. 8. After firing the aggregation-duplication transition, the tokens in the input places of will be removed, new tokens will be deposited into the output places of and the pair of necessity and possibility measures attached to the new tokens are derived by disjunction. Fig. 6. Modeling the aggregation of conclusion by an aggregation transition: (a) Before firing the aggregation transition t. (b) After firing t. Fig. 7. Modeling the duplication of a possibilistic token through PPN: (a) before firing the duplication transition t, (b) after firing t. B. Reasoning Algorithm To consider the efficiency of, an algorithm of transition firing is proposed based the notion of extended markings. An extended marking, denoted as, isa -vector, where is the number of possibilistic tokens. The elements of denoted by are called extended places, defined as (17) where represents a place; is a classical proposition; ( ) is a pair of necessity and possibility measures attached to the token in place ; is the output transition of. From an extended places, we know that: (a) the pair of necessity and possibility measures attached to the token in (i.e., (, )), (b) the other tokens needed to fire (i.e., ), (c) what kind of computation to carry out after firing (i.e., the type of ) and (d) where to go for the new tokens after firing (i.e., ). The algorithm is described as follows:

7 LEE et al.: MODELING UNCERTAINTY REASONING WITH POSSIBILISTIC PETRI NETS 7 Fig. 8. Modeling the aggregation-duplication of possibilistic tokens through PPN. (a) Before firing the aggregation-duplication transition t. (b) After firing t. 6. Delete the element and each from the current extended marking. 7. Repeat step 3 to step 6 until no element is in the current extended marking. 8. Repeat step 2 to step 7 until all output transitions in the current extended marking are not fired. 9. Send the final extended marking to the upper-leveled hierarchy. Algorithm 1: (Possibilistic Petri Nets) 1. Get the initial extended marking, which consists of all source places. 2. For each, set a current extended marking and the next extended marking. 3. Select an element of the current extended marking, 4. (a) If is an inference transition and the extended place of each exists in, then infer the extended place of by possibilistic reasoning. (b) Else if is a duplication transition, then infer the extended place of each by duplication. (c) Else if is an aggregation transition and the extended place of each exists in, then infer the extended place of by disjunction. (d) Else if is an aggregation-duplication transition and the extended place of each exists in, then infer the extended place of each by disjunction. 5. (a) If the output transition of is fired, then insert the inferred extended place into the next extended marking. (b) Else if the output transition of is a hierarchy, then insert the extended place into the hierarchy and wait for the final extended marking of the hierarchy to be inserted into the current extended marking. (c) Else insert this element each the next extended marking. and into Basically, the algorithm is used to manage the evolution of extended marking. First, it defines the current and next extended markings (step 2). Second, it determines which type of the transitions in the current extended marking are and decides whether the transitions can be fired or not (step 4). Finally, it deletes the extended places from the current extended marking if their output transitions are fired and inserts the output extended places of these fired transitions into next extended marking or next hierarchies. The procedure is repeated until no distinction can be made between the current and next extended markings. The following is an example to illustrate how to apply the proposed and the reasoning algorithm to model possibilistic rule-based reasoning. A in Fig. 2 contains three hierarchies,, and. Hierarchies,, and represent small rule bases. where is a classical proposition. There are three uncertain facts such as ( ), ( ), and ( ). After performing the proposed reasoning algorithm, the evolution of extended fuzzy marking is shown in Table I. The conclusions inferred by the reasoning algorithm are ( ) and ( ). V. DIAGNOSIS OF CRACKS IN RC STRUCTURES Reinforced concrete structures can be found everywhere, including buildings, bridges, highways, dams, pipe systems and many other structures. Cracking is the most common defect and can be discovered in any kind of concrete structures. Most concrete structures will develop cracks sooner or later and ultimately affect the performance of the structural elements. Some of the cracks indicate the minor deterioration of concrete while some others may lead to the structural failure. To properly maintain and repair concrete structures, diagnosing the causes of cracks is an important but complicated task. Traditionally, an engineer who has closely studied these problems over years, could pick up heuristically patterns to link the nature of the cracking to its possible causes. However, a lack of complete understanding about the mechanism of cracking usually results in the lack of confidence about the possible causes. In this example, we apply

8 8 TRANSACTIONS ON SYSTEMS, MAN AND CYBERNETICS PART B: CYBERNETICS, VOL. 33, NO. 2, APRIL 2003 TABLE I EVOLUTION OF EXTENDED MARKING TABLE II CAUSES OF CRACKING (CHAO et al. [4]) our possibilistic Petri nets model to mimic the uncertainty reasoning for crack diagnosis. According to Chao et al. s investigation [4], there are four primary causes resulting in cracking: 1) causes concerning quality of concrete material; 2) causes related to construction of concrete structures; 3) causes associated with environmental factors; 4) causes connected to applied loading. Each primary cause can be further divided into several subcauses for detailed possible causes (see Table II). The symptom of cracks occurred in concrete structures can be described by the following characteristics: 1) the development time after casting 2) crack depth, 3) crack regularity 4) whether cracks appear only in a concrete member or throughout the overall structures 5) the type of member in which cracks occur 6) crack patterns 7) crack locations The description of crack characteristics is listed in Table III. The first four characteristics are employed to determine the primary possible causes and the related subcauses are decided by the rest of the characteristics. The 477 rules representing heuristically patterns to link the characteristics of cracks with possible causes were elicited from published literatures (Chao et al. [4], Chiang et al. [7], Furuta et al. [11], [12], [34], Ross et al. [14], [30], Krauthammer et al. [17], and Miyamoto et al. [18], [25]). To overcome the complexity arising from large sizes of the rule base, the whole rule base is partitioned into 34 modules: four modules related to primary causes and 30 modules related to subcauses. As was mentioned previously, Petri net s graphical representation can help us construct and modify rule bases. Therefore, a hierarchical TABLE III CHARACTERISTICS OF CRACKS (CHAO et al. [5]) structure of for diagnosing cracks in RC structures is established based on these modules. For example, the possibilistic Petri net in hierarchy, shown in Fig. 9, represents four rules in module related to subcause acids and sulfate attacks ( ): To demonstrate the use of, three cases adopted from Chao et al. s work [4] are considered. Case 1: In a RC structure, fine cracks occurred on the slab surface three days after casting. The cracks were random in nature and had no regularity. The observed symptoms and the associated confidence levels can be summarized in Table IV. It should be noted that the necessity and possibility are assigned to be zero if a characteristic does not occurr. After performing the reasoning algorithm in Section IV-B, the confidence levels of four possible primary causes are inferred and shown in Table V. The result indicates that the most possible primary cause is referred to the quality of the concrete material (, Table II) with 0.62 degree of necessity. The hierarchy related to subcauses of

9 LEE et al.: MODELING UNCERTAINTY REASONING WITH POSSIBILISTIC PETRI NETS 9 TABLE IV OBSERVED SYSTEMS AND THE ASSOCIATED CONFIDENCE LEVELS FOR CASE 1 TO 3 Fig. 9. Possibilistic Petri net for diagonising acids and sulfate attack. material is also triggered and the reasoning algorithm infers a conclusion as follows (all confidence levels for other subcauses,,, and are (0,0)). Abnormal setting of cement paste ( ), (0.75,1.0) Drying and shrinkage of concrete ( ), (0.75,1.0). Thus, the most possible subcauses resulting in cracking in this case are abnormal setting of cement paste (, Table II) and drying and shrinkage of concrete (, Table II) with 0.75 degree of necessity. Case 2: In a RC structure, cracks were found in wall surfaces one year after casting. The cracks were very deep and formed a regular pattern, like an X shape, near the center of the wall. The observed symptoms and the associated confidence levels can be summarized in Table IV. The confidence levels of four possible primary causes can be inferred and shown in Table V by applying the reasoning algorithm. The result indicates that the most possible primary cause is referred to applied loading (, Table II) with 0.75 degree of necessity. The hierarchy related to subcauses of material is also triggered and the reasoning algorithm infers a conclusion as follows (all confidence levels for other subcauses,, and are (0,0)). Earthquake force ( ) (0.75,1.0). Thus, the most possible subcauses resulting in cracking in this case are abnormal earthquake force (, Table II) with 0.75 degree of necessity. Case 3: In a RC structure, honeycombing occurred on the member surface seven days after casting. The cracks were deep and had no regularity. The observed symptoms and the associated confidence levels can be summarized in Table IV. After performing the reasoning algorithm in Section IV-B, the confidence levels of four possible primary causes are inferred and shown in Table V. The result indicates that the most possible primary cause is referred to construction of concrete structure TABLE V PRIMARY CAUSES AND THE INFERRED CONFIDENCE LEVELS FOR CASE 1 TO 3 (, Table II) with 0.73 degree of necessity. The hierarchy related to subcauses of material is also triggered and the reasoning algorithm infers the conclusion as follows (all confidence levels for other subcauses,,,,,,,,,, and are (0,0)). Segregation during placement ( ), (0.75,1.0) Under or overvibrated during concrete placing ( ), (0.75,1.0). Thus, the most possible subcauses resulting in cracking in this case are segregation during placement (, Table II) and undervibrated or overvibrated during concrete placing (, Table II) with 0.75 degree of necessity. This example demonstrates that the results match Chao et al. s conclusions derived from fuzzy relation based pattern recognition model [4]. The three cases are typical in civil engineering. Both our results or Chao et al. s match the civil engineers judgment. Moreover, our approach offers several benefits as follows. Knowledge representation through make experts easy to develop and modify rule bases, which can achieve the structuring of knowledge within rule bases. Furthermore, it makes easy the users to realize the knowledge by identifying the relationships among rules. s mathematical modeling capability makes system developers easier to design an efficient reasoning algorithm. The underlying relationship of concurrency among rule activations can also be modeled by the real-time performance. to enhance

10 10 TRANSACTIONS ON SYSTEMS, MAN AND CYBERNETICS PART B: CYBERNETICS, VOL. 33, NO. 2, APRIL 2003 s graphic nature provides the visualization of the dynamic behavior of possibilistic rule-based reasoning. Therefore, the explanation of diagnosis process served by token movement and the confidence level associated with possible causes can be used as a way of justification on whether to take the diagnosis into account or not. VI. CONCLUSION Possibilistic Petri nets and a reasoning algorithm are proposed for modeling uncertainty reasoning with possibilistic information. Three key components in our possibilistic reasoning: propositions, possibilistic rules and possibilistic facts, can be formulated as places, possibilistic transitions and possibilistic tokens, respectively. In terms of possibilistic reasoning, possibilistic transitions are classified into four types: inference transitions that perform possibilistic reasoning; duplication transitions that duplicate a possibilistic token to several tokens, representing the same proposition, possibility, and necessity measures; aggregation transitions combine several possibilistic tokens with the same classical proposition; and aggregation-duplication transitions combine aggregation transitions and duplication transitions. A reasoning algorithm based on also presented to improve the efficiency of possibilistic reasoning. Our approach offers the following advantages. serves as a bridge that brings the possibilistic entailment and Petri nets together into a hybrid approach to modeling possibilistic reasoning. We stick to Petri net s firing rule that tokens will be removed from the input places of a transition after the transition fired and Petri net s original semantics of markings and incidence matrixes. 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11 LEE et al.: MODELING UNCERTAINTY REASONING WITH POSSIBILISTIC PETRI NETS 11 [31] H. Scarpelli, F. Gomide, and R. Yager, A reasoning algorithm for high level fuzzy Petri nets, Trans. Fuzzy Syst., vol. 4, pp , Aug [32] G. Shafer, Mathematical theory of evidence, in Mathematical Theory of Evidence. Princeton, NJ: Princeton Univ. Press, [33] T. Shimura, J. Lobo, and T. Murata, An extended Petri net model for normal logic programs, Trans. Knowledge Data Eng., vol. 7, pp , Feb [34] N. Shiraishi, H. Furuta, M. Umano, and K. Kawakami, An expert system for damage assessment of a reinforced concrete bridge deck, Fuzzy Sets Syst., vol. 44, pp , [35] E. H. Shortliffe, Computer Based Medical Consultations: MYCIN. New York: Elsevier, [36] C. Wu and S. Lee, Enhanced high-level Petri nets with multiple colors for knowledge verification/validation of rule-based expert systems, Trans. Systems, Man, Cybern. B, vol. 27, pp , Sept [37] L. A. Zadeh, Fuzzy logic = computing with words, Trans. Syst., Man, Cybern., vol. 4, pp , May [38], From computing with numbers to computing with words From manipulation of measurements to manipulation of perceptions, Trans. Circuits Syst. I, vol. 45, pp , Jan [39] A. K. Zaidi and A. H. Levis, Validation and verification of decision making rules, Automatica, vol. 33, no. 2, pp , Jonathan Lee(SM WHAT YEARS?) received the Ph.D. degree in computer science from Texas A&M University, College Station, TX in He is a Professor and Chairman in the Department of Computer Science and Information Engineering at National Central University, Chungli, Taiwan, R.O.C. His research interests include software engineering with computational intelligence, agent-based software engineering and soft computing. He is on the editorial boards of Fuzzy Sets and Systems, Fuzzy Optimization and Decision Making, and International Journal of Artificial Intelligence. He is a member of the ACM and AAAI. He is currently the vice president of IFSA and the president of CFSAT. Kevin F. R. Liu received the B.S. degree in 1992 and the Ph.D. degree in 1998, both in civil engineering from National Central University, Chungli, Taiwan, R.O.C. Currently, he is an Assistant Professor in the Department of Environmental Engineering at Da-Yeh University, Changhua, Taiwan. His research focuses on the application of artificial intelligence techniques to the civil engineering problems. Areas of interest include soft computing, fuzzy set theory, neural networks, and Petri nets. Weiling Chiang received the B.S. degree from the Department of Civil Engineering, National Taiwan University, Taiwan R.O.C., and the M.S. and Ph.D. degrees from Stanford University, Stanford, CA. He is currently a Professor in the Department of Civil Engineering, National Central University, Taiwan, R.O.C. His research interests include uncertainty management that involves applications of fuzzy theory and probabilistic method to civil engineering problems, applying newly developed methodology such as neural networks, catastrophe theory and evidence theory to various civil-engineering-related problems, risk management, optimal control, and bridge management.