A Formal Semantics of UML Sequence Diagram - UNU-IIST - United ...

¡CampaignManagergetName()¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡listCampaigns():ClientcheckCampaignBudget()* getCampaignDetails():Campaign* getCost()* getCost():AdvertgetOverheads()Figure 8. Illegal sequence diagramoperations, i.e. method calls, while a class diagramsdetermines the following variables on which sequencediagram operates:• for every class, a class variable that takes valuesof sets of objects of the concept;• for every object of a concept, a variable for eachattribute of the concept;• for every association, an association variablethat take values of sets of links (i.e. pairs) betweenobjects of the associated classes.2. Due to the inheritance mechanism, the effect of a usecase on a variable depends on its current type duringexecution, rather than its originally declared type.3. As in imperative languages, a state of a variable is itscurrent value. An object is represented as a finite tuplethat records its identity, current type, and the values ofits attributes.In summary, a model of an OO design is a system S =(α, Φ, Θ, P ) where• P consists of a set of operations obtained from a numberof sequence diagrams.• α identifies the variables on which the operations in Poperate and it is determined by the class diagram andthe input and output parameters of the methods in thesequence diagram.• The invariant Φ formally models the invariant constraint,such as multiplicities and business rules. Thepair (α, Φ) thus gives the formalization of the classmodel.• Θ is a condition to be established when starting up thesystem.5 Formal Semantics of Sequence DiagramWe consider both static and dynamic semantics of sequencediagrams. In an abstract syntactic form, a wellformedsequence diagram corresponds to an ordered hierarchicaltree structure. The static semantics of a sequencediagram is to check whether it is consistent with the classdiagram declaration. This means that the static semantics isalways dependent on a given class diagram ∆. Meanwhile,the dynamic semantics is defined as a sequence compositionof the first layer system messages of its ordered tree.When a message is executed, it must be consistent with systemstate, i.e., object diagram and the state diagrams of therelevant objects.5.1 Static semantics of sequence diagramGiven a class diagram ∆, let CN be the set of classnames in ∆, AN set of association names, Ass the set ofall associations each of which is represented in the form< C 1 , A, C 2 >, where C 1 , C 2 ∈ CN and A ∈ CN, andmethod(C) be the set of all the methods of class C in ∆.Obviously, for a given messagemsg = (ob i : C i , ob j : C j , g → m(), order)we can define its static semantics as follows.S s [smg]] def= C i ∈ CN ∧ C j ∈ CN∧∃A ∈ CN. < C i , A, C j >∈ Ass ∧ m ∈ Method(C j )The static semantics of a sequence diagram is defined asthe conjunction of all its messages in the structure tree.5.2 Dynamic Semantics of MessageExecution of a message makes system change from onestate s to another s ′ . Based on Hoare and He’s UnifyingTheories of Programming [9], we can define a message actionas first order logic formula on a pair of system states(s, s ′ ). Before giving the semantics of messages, we introducesome useful auxiliary functions: for states s and s ′ ,• s(C) : the set of objects of class C in state s.• s(ob): the corresponding state of object ob in systemstate s, i.e. the values of the attributes of ob.6