A Formal Descriptive Semantics of Uml and Its Applications

ClassXX +TemplateMethod()+Others()ConcreteClassXX Figure 1.19 Example design instance in template method pattern The following theorem states that if we can prove [[M ]] → Spec′(P ) inFOPL for model M and pattern P , then every system that is an instance ofM must conform to pattern P . The proof is omitted for the sake of space.Theorem 1.2 Suppose that Spec′(P ) is a correct translation of the formalspecification Spec(P ) of pattern P . For all models M , if [[M ]] ⇒Spec′(P ) istrue in FOPL, then, for all systems s ∈ D, s |= M and M |= Spec(P ) implys |= Spec′(P ). We have translated the specifications given in [5] for all 23 design pat-terns in the GoF book into LAMBDES format. They are stored in a patternspecification repository. The conjecture generator of the LAMBDES tool isimplemented to enable the proof (or disproof) of the conformance of a UMLdesign model to a pattern. We have also conducted an experiment with theLAMBDES tool on its ability to recognise patterns in design instances. Theexperiment results show that the false negative error rate (for rejecting a pat-tern it should accept) is 0% while the false positive error rate (for accepting apattern it should reject) is below 22%. Details of the experiment are omittedhere for the sake of space, and will be reported separately. 1.4.5 Logic analysis of design patterns It is worth noting that the specification of a design pattern may contain errors.The conditions to satisfy the pattern may be in conflict with the semantics ofthe modelling language, or they may be in conflict with each other. Such logicerrors can be detected by using LAMBDES tool and SPASS theorem prover. 26 UML DESCRIPTIVE SEMANTICS AND APPLICATIONS In particular, let Spec(P ) be a specification of a pattern P . If AxmD ∪Spec(P ) false, we can conclude that Spec(P ) contains such errors.In the development of the pattern specification repository, using LAMB-DES and SPASS, we have proved that for all specifications of design patternsP in the repository, AxmD ∪ Spec(P ) false. So, all the specifications inour repository are consistent with the axioms of descriptive semantics.Another application of LAMBDES and SPASS in the logic analysis of de-sign patterns is to prove relations between patterns, for example, to prove onepattern is a specialisation of another. In [4], it is argued that the relationshipthat a design pattern P is a specialisation of pattern Q can be written asSpec(P ) → Spec(Q). Such a relationship can be formally proved by usingLAMBDES and SPASS to infer that AxmD ∪ Spec(P ) Spec(Q). In thecontext of descriptive semantics, we can now prove the following property ofthe pattern specialisation relation.Theorem 1.3 Let Dom be a subject domain that is consistent with FOPL.If AxmD ∪ Spec(P ) Spec(Q), then, for all systems x ∈ Dom, if x is aninstance of P then x is also an instance of pattern Q, i.e. ∀x·(x |= Spec(P ) →x |= Spec(Q)).