80 K. Xiao, S. Chen, and X. Chen • When adding new KP to the syllabus, the material reflecting this KP must be added into the courseware also. • When the KP does not longer present in the syllabus, any material reflecting this KP must be deleted in the courseware. Meanwhile, the KP reflected in the remain- ing material must be enough for the students to learn the syllabus. In order to deal with these cases, this paper discusses the relationships among KP and the relationships among material. On the basis of them, it provides material Petri net (MPN) to validate some material property. It is organized as follows. Section 2 describes the KP net and clarifies the relationships among KP in the syllabus. Based on KP net, Section 3 presents the material net. Section 4 introduces Petri net and gives the definition of MPN. Then, it researchs some material property by the MPN. In the section 5, it draws a conclusion and presents the future work. 2 Knowledge Point 2.1 Overview of KP The whole knowledge in a book consists of some KP, and these KP distribute from the various charpters. It is essence that various material reflecting KP is learned in the e- learning. In order to describe material, we must give some related conception of KP firstly. Well-known, the KP is an teaching unit including knowledge in the learning process. From the conception of KP, we know that KP is demarcated according to require- ment. It brings forth a question about how to divide knowledge into KP properly? It is the basic principle of the knowledge demarcation that the partial completeness must be ensured[4]. For example, one chapter in the book may be a large KP. The contents of chapter may be divided into some sections also. A section of in the chapter may be divided into some less KP again. However, if KP is continuously divided, it would not reflect concrete knowledge contents lastly. Hence, we have two conceptions: • Atom KP: if the knowledge of the KP is further splitted, there would not exist any completeness knowledge contents. This KP is called as Atom KP. • Composite KP: if the knowledge of the KP may be divided into less KP reflecting knowledge contents, this KP is called as Composite KP. If the KP C consists of the KP A and KP B, it is called as the parent of KP A and KP B. KP C is a Composite KP. The KP A and the KP B are called as children of KP C. All of children of one KP are called as brother KP each other. 2.2 Relationships among KP The parent-child, the association and the parallel are three relationships among KP: • parent-child relationship: it is the relationship between a Composite KP and its children KP. • association relationship: it is the relationship that a KP may be learned directly after another KP have been mastered. • parallel relationship: if the relationship between two KP is not the parent-child relationship and the association relationship, it is a parallel relationship. Course Material Research Based on Perti Net 81 In the association relationship among KP, if a KP must be mastered directly before another KP is learned, it is called as the Ancestor KP for another KP. if a KP may be learned directly after another KP has been learned, it is called as the Descendant KP for another KP. 2.3 KP Net Diagram The relationships among KP are described with a net structure[4]. The Composite KP can be divided into some little KP. On the grounds of parent-child relationship, if a material reflects in an Atom KP, it will also reflect in the parent KP of this atom KP. So, on the discussion below, we assume that all KP should be atomistic. An arc is expressed as the association relationship among KP. The arrowhead di- rects from the Ancestor KP to the Descendant KP. If a KP has an arc pointing at it, this arc is called as In-arc for this KP. If a KP has an arc backing at it, this arc is called as Out-arc for this KP. Definition 1 KP net: it is the net that is made up of the KP and the relationships of them. The node means the KP and the arc means the association relationship of the KP. In an ordinary way, the KP net is an directed acyclic graph. See figure 1. B D E C A Fig. 1. KP Net At the figure 1, A, B, C, D and E means some Atom KP in the courseware. The rela- tions of A and B, A and C, B and E, B and D, C and D are the association relationships. A is the ancestor B and C. B and C are the descendant of A. Likewise, D is descendant of B and C. B and C are the ancestor of D. the relationship between B and C is the parallel, and the relationship between D and E is the parallel also. 3 Material 3.1 Material Set Material is a media that can reflect some KP independently. It may be a document, picture, sound, etc. The relationship between the material X and the KP A, B can be expressed as X={A,B}. The expression X={A,B} means that the material X reflects the KP set, which includes KP A and B. 82 K. Xiao, S. Chen, and X. Chen The relationship between the material and the KP is many to many. One material may reflect with much KP, and one KP may be reflected with much material. The material reflecting one KP can be learned only if learner masters the Ancestor KP of this KP. Further, learner must master the Ancestor KP of this Ancestor KP. Obvi- ously, the KP set, which includes the Descendant KP and the Ancestor KP, is redun- dant. It might be simplified. The algorithm is described as follows: Step 1. selecting any element in the KP set described by material; Step 2. searching all paths from starting KP to this KP; Step 3. in the KP set reflected in the material, deleting all elements in the paths except for itself; Step 4. selecting the remain element in the KP set, and repeating step 2. and step 3; After all elements in the KP set reflected in material are selected, the KP set, which only includes the surplus elements, is the simplest KP set of this material. Example 1: To figure 1, Supposed the KP sets reflected in some material as follows. X1={A}, X2={A}, X3={A,B}, X4={A,B,C,D}, X5={A,C}, X6={A,B,C}, X7={E}. After simplified, the KP sets is described as follows. X1={A}, X2={A}, X3={B}, X4={D}, X5={C}, X6={B,C},X7={E}. Because there are much material reflecting the same KP set, we concern a material set rather than single material. Definition 2 material set (MS): if all elements in a material set can replace with each other when they reflect KP, this set can be called as a material set. About example 1, the MS are X12={X1,X2}, X3={X3}, etc. All material in the MS reflect the same KP set. Below, we do not discuss the single material except for the MS. 3.2 Relationship among Material Set Because all material are equal when they reflect KP, there does not exist the parent- child relationship among material. The association and the parallel are two relation- ships among MS. • association relationship: it is the relationship that the MS may be learned di- rectly after another MS have been mastered. • parallel relationship: if the relationship between two MS is not association rela- tionship, it is parallel relationship. In the association relationship, if a MS must be mastered directly beore another MS is learned, it is called the Ancestor MS for another MS. if a MS may be learned directly after another MS has been mastered, it is called the Descendant MS for another MS. It is relative about the Ancestor MS and the Descendant MS. If MS A is the ances- tor of MS B, MS B is the descendant of MS A also. 3.3 MS Net An arc is expressed for the association relationship among MS. The arrowhead directs from the Ancestor MS to the Descendant MS. If a MS has an arc pointing at it, the arc is called as In-arc for this MS. If a MS has an arc backing at it, the arc is called as Out-arc for this MS. The figure 2 is the MS net graph of the example 1. Course Material Research Based on Perti Net 83 A X12 X3 X5 X6 X4 X7 B BC C D E Fig. 2. MS Net Definition 3 MS net: it is the net that is made up of the MS and the relationships among them. The node means MS and the arc means the association relationship of MS. A MS may be ancestor and descendant of itself also. MS net is an cyclic directed graph. At the figure 2, X12, X3, X4, X5, X6 and X7 means some MS in the courseware. A, B, C, D, E and BC means KP reflected in MS. The relationships of X12 and X3, X12 and X5, X3 and X6, X5 and X6, X5 and X7, X3 and X4, X5 and X4, X6 and X4 are the association relationship. X12 is the ancestor of X3 and X5. X3 and X5 are the descen- dant of X12. etc. 4 The MS Specialty Research 4.1 Material Petri Net (MPN) It is an effective system model verification tools for Petri net (PN)[9,10,11]. Definition 4 Petri net: it includes six items, PN=(P,T,F,K,W,M 0 ) P: is place set. T: is transition set (P T≠ ∅, P∩T= ∅). F ⊆(P ×T) (T×P): is flow relation. K: defines the maximum token in a place. W: defines the weighted coefficient in token. M 0 : is a start label. Definition 5 path: is a transition sequence σ =M 0 t 1 M 1 t 2 M 2 …t n M n in Petri net. It is for short σ =t 1 t 2 …t n , and called as trigger sequence of transition. Definition 6 accessibility: if it has a sequence transition t 1 , t 2 , …, t n from M 0 to M n in PN, it can be say that M n is accessibility from M 0 . Definition 7 Supposed N=(P,T,F,K,W,M 0 ) is Petri net. N is MPN, if and only if two conditions is satisfied as follows: 84 K. Xiao, S. Chen, and X. Chen • there are two special places, which are M start and M stop . M start place is the beigin- ing, and M stop is the end. • There are a special transition t stop , all places can arrive at M stop only via t stop . Definition 8 material accessibility: if it has a sequence transition t 1 , t 2 , …, t stop from M start to M stop in the MPN, it can be say that M stop is the material accessibility from M start . 4.2 Transformation from MS Net Graph to MPN On the procession of the learning in the courseware, if the MS has been learned, which means that the KP reflected in this MS has been mastered, other KP in the sequence may be learned continuously. Therefore, the MS may be treat as place (P), and the KP may be treat as transition (T). The maximum of token (K) in the place is the element numbers in the MS. In order to describe the relationship of MS in MPN, some definition must be introduced. • Stop KP set: it is a set where all KP had been learned. • Stop MS: it is a virtual MS included all KP that have been finished. • Void KP: is a virtual KP without any real knowledge. the Void KP transits without fail. • Void MS: is a virtual MS without reflecting any real KP. The learning of Void MS does not need any premise. The transformation about four special relationships of MS are given through the figure 3. X3 X6 BC C B X5 X5 A X3 X12 (a) (b) t stop E D X stop X3 X4 D BC C B X5 (d) (c) Fig. 3. Four Special Relationship in the MPN . reflecting KP is learned in the e- learning. In order to describe material, we must give some related conception of KP firstly. Well-known, the KP is an teaching unit including knowledge in the. selecting any element in the KP set described by material; Step 2. searching all paths from starting KP to this KP; Step 3. in the KP set reflected in the material, deleting all elements in the. Step 4. selecting the remain element in the KP set, and repeating step 2. and step 3; After all elements in the KP set reflected in material are selected, the KP set, which only includes the