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A Finite Satisfiability of UML Class Diagrams with Constrained Class Hierarchy
| Content Provider | Semantic Scholar |
|---|---|
| Author | Ben-Gurion |
| Copyright Year | 2013 |
| Abstract | syntax. The subset of UML2.0 class diagrams considered in this paper includes classes (possibly with attributes), associations and three kinds of constraints: Multiplicity constraints on binary associations, Class hierarchy constraints, and Generalization set (GS) constraints. The formulation is based on the meta-model notion of Property, which is more basic than that of Association [OMG 2007; Alanen and Porres ACM Transactions on Software Engineering and Methodology, Vol. V, No. N, Article A, Pub. date: January YYYY. Finite Satisfiability of UML Class Diagrams with Constrained Class Hierarchy A:5 2008]. The semantics is set-theoretic. In the rest of this paper attributes are omitted as there are no attribute constraints. A class diagram is a tuple 〈C,P,A, props, source, target, Constraint〉 where — C is a set of class symbols. —P is a set of property symbols (sometimes called association ends). Property symbols denote mappings derived from their associations. —A is a set of association symbols. — props : A → P × P is a 1:1 and onto assignment of (unique) properties to association symbols. For a property p, there is a unique a ∈ A, such that props(a) = (p, ∗) or props(a) = (∗, p), where * is a wild card. We write assoc(p) or assoc(p1, p2) for the association of p or of (p1, p2), and props1(a), props2(a) for the two properties of a. — source : P → C and target : P → C are 1:1 mappings of properties to classes such that for an association a with props(a) = (p1, p2), target(p1) = source(p2) and target(p2) = source(p1). In Figure 2, target(p1) = source(p2) = C1 and source(p1) = target(p2) = C2. Fig. 2: Visualization of a binary association, its properties, their source and target classes, and their multiplicities — Constraint is a set of constraints as follows: (1) Multiplicity constraints on properties: mul : P → (N ∪ {0}) × (N ∪ {∗}) assigns multiplicity constraints to property symbols. For simplicity we use a compact symbolic representation, where association a in Figure 2 is denoted a(p1 : C1[ m1, n1], p2 : C2[ m2, n2]). The functions minMul : P → {N ∪ {0}} and maxMul : P → {N∪{∗}} give the minimum and maximum multiplicities assigned to a property, respectively. (2) Class-hierarchy: A non-circular binary relationship ≺ on the set of class symbols: ≺ ⊆ C × C. Henceforth we use the notation C2 ≺ C1, where C1 is the superclass and C2 is the subclass (also called direct-descendant). The weak version of≺ is denoted , which is “≺ or equal“. The reflexive transitive closure of ≺ is called the descendant relation, and denoted ≺∗. Its irreflexive version is denoted ≺. Figure 3a shows the concrete (visual) syntax of a class hierarchy constraint. (3) Generalization-set (GS) constraints: GS is an (n + 1)-ary relationship on C, for n ≥ 2. An element 〈C,C1, . . . , Cn〉 in GS must satisfy: For i, j = 1..n (1) C 6= Ci; (2) Ci 6= Cj ; (3) Ci ≺ C. C is called the superclass and the Ci-s are called the subclasses. Elements of GS maybe associated with a constraint const ∈ {〈disjoint〉, 〈overlapping〉, 〈complete〉, 〈incomplete〉, 〈disjoint, complete〉, 〈disjoint, incomplete〉, 〈overlapping, complete〉, 〈overlapping, incomplete〉}. We use the symbolic representation GS(C,C1, . . . , Cn; const) for GS constraints. Note that an unconstrained GS is redundant, as it specifies only class hierarchy constraints. Figure 3b shows the concrete (visual) syntax of a GS constraint. Henceforth we omit the term “symbol”, and refer just to classes, associations, properties. A class diagram CD′ is a sub-diagram of a class diagram CD, denoted CD′ ≤ CD, if its classes, associations and constraints belong to CD. ACM Transactions on Software Engineering and Methodology, Vol. V, No. N, Article A, Pub. date: January YYYY. A:6 M. Balaban and A. Maraee |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://www.cs.bgu.ac.il/~modeling/wp-content/uploads/2011/04/paper2.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
