A Formal Descriptive Semantics of UML and Its Applications

1. A Formal Descriptive Semantics of UML and Its Applications Hong Zhu Department of Computing and Electronics School of Technology

2. Acknowledgement • The work reported here is based on the outcomes of the collaborative research with • Dr. Ian Bayley, Oxford Brookes University, UK • Ms. Lijun Shan, National University of Defense Technology, China (visitor to Oxford Brookes University funded by China Scholarship Council) • Mr. Richard Amphlett, Oxford Brookes University, UK (Funded by Reinvention Centre as Undergraduate Research Student Scholarship)

3. Outline • What is descriptive semantics? • The concept and motivation • How to specify descriptive semantics? • The formal framework • Formal definition of mappings from diagrams to first order logic (FOPL) • Application to UML class, interaction and state machine diagrams • The relationships between descriptive semantics and functional semantics • Why descriptive semantics are important and useful? • Implementation of the semantics and the tool LANBDES • Applications in formal reasoning about models in FOPL

4. Borrow Instances of the class Book Instances of the class Member What is the meaning of UML diagrams? • The system has two classes called Member and Book, (There are two types of objects called members and books) • There is an association between them, which is called Borrow (Members can borrow books) • The upper limit of borrowing is 10 (Each member can only borrow up to 10 books at any time) • The upper limit for a book to be borrowed is 1 (Each book can only be borrowed by at most 1 member at any time) Member 0..1 0..10 Book Library system

5. Semantics of modelling languages • ‘A model is a set of statement about some system under study’ [Seidwitz, 2003] • The semantics of a modelling language is the satisfaction relationship |=between a model and a system. s|=m Software model, and models in all scientific disciplines A well-formed model in the modelling language. A system in the subject domain D, i.e. universe of systems that the language is modelling. s is an instance of the model m, i.e. s satisfies the statements of the model m

6. Descriptive semantics Describe the system based on a set of basic concepts, such as class, association multiplicity upper bound, etc. Example: The system has two classes called Member and Book Functional semantics Define how the system functions at run time. Example: There are two types of objects called members and books. Two types of semantics • Existing work on UML semantics often occurs as a combination of them. • Example: • Each instance of the Book class can only be borrowed by at most 1 member at any time.

7. Difficulties in the formalisation of UML • Interpretation in different subject domains: • One model can be interpreted in several different subject domains • Library system model: • When interpreted on the subject domain of real world systems, the library of Oxford Brookes University is an instance • When interpreted on the computer software, … • Extension with new concepts: • New basic concepts and constructs can be introduced through extension facilities, such as introduce new stereo types in profile definitions • Uses in different context: • One model may have different meanings in different context

8. Example: • Which of the following is the correct semantics of the model on the left? • There are exactly three different classes such that …; • There are at least three different classes such that …; • Which of the following programs can be considered as satisfied the model? Member Staff Student class Member { … } class Staff extends Member {…} Class Student extends Member {…} class Member { public enum MemberType { Staff, Student } public MemberType TypeOfMember; … } class MScStudent extends Student {…} UML gives no definition on this issue!

9. Overview of our approach This is consistent with the theory of institute proposed by Goguen and Burstall (1992) for formal specification languages.

10. The formal framework Definition 1. (Semantics of a modelling language) A formal semantic definition of a modelling language consists of the following elements. • A signature Sig, which defines a formal logic system; • A set AxmD of axioms about the descriptive semantics, which is in the formal logic system defined by Sig; • A set AxmF of axioms about the functional semantics, which is also in the formal logic systems defined by Sig; • A mapping Tfrom models to a set of formulas in the formal logic system defined by Sig. The formulas are the statements for the descriptive semantics of the model; • A mapping H from models to a set of formulas in the formal logic system defined by Sig. The formulas represent the hypothesis about the context in which the descriptive semantics is interpreted.

11. Definition 2. (Semantics of a model) Given a semantics definition of a modelling language as in Definition 1, the semantics of a model M under the hypothesisH, written SemH(M), is defined as follows. SemH(M) = AxmDAxmFT(M) H(M) where T(M) and H(M) are the sets of statements obtained by applying the semantic mappings T and H to model M, respectively. The descriptive semantics of a model M under the hypothesisH, written DesSemH(M), is defined as follows. DesSemH(M) = AxmDT(M) H (M)

12. Satisfaction relation Definition 4. (Subject domain) A subject domain Dom of signature Sig with an interpretation Eva is a triple <D, Sig, Eva>, where D is a collection of systems on which the formulas of the logic system defined by Sig can be evaluated according to a specific evaluation rule Eva. The value of a formula f evaluated according to the rule Eva in the context of system sD, written as Eva(f, s), is called the interpretation of the formula f in s. We write s|=Evaf, if a formula f is evaluated to true in a system sD, i.e. s|=Evaf iff Eva(f, s)= true.  Definition 5. (Satisfaction of a model) Let Sig be a given signature and Dom a subject domain of Sig. A system s in Dsatisfies a model M according to a semantic definition SemH(M) if s|= SemH(M), i.e. for all formulas f in SemH(M), s|=f. 

13. A Simplified Metamodel of Class Diagram

14. Signature Mapping The signature mapping S from a metamodel M to a signature  = S (M) is defined by a set of signature rules so that statements representing the descriptive semantics of models are  -sentences of first order predicates. Signature Rules S1. For each metaclass named MC in the metamodel, we define a unary atomic predicate MC(x). S2. For each metaassociation from a metaclass X to a metaclass Y in a metamodel, if MA is the association end on the Y side of the metaassociation, a binary predicates MA(x, y) is defined. S3. For each metaattribute named MAttr of type MT in a metaclass MC in a metamodel, a unary function MAttr(x) is defined with domain MC and range MT. S4. For each enumeration value EV given in an enumeration metaclass ME in a metamodel, a constant EV is defined.

15. Example: • Signature elements: • unary predicates : Class(x), Classifier(x), Generalisation(x) • binary predicates general(x, y) and specific(x, y) Metamodel

16. Translation mapping Translation mapping is defined by a set of translation rules that generate formulas in the signature from a UML model M. Translation Rules T1: Classification of elements. For each identifier id of concrete type MC, a formula in the form of MC(id) is generated. T2: Properties of elements. For each element a in the model and every applicable function MAttr that represents a metaattribute MAttr, a formula in the form of MAttr(a)=v is generated, where v is a’s value on the property. T3: Relationships between elements. For each pair (e1, e2) of elements related by relationship R, a formula in the form of R(e1, e2) is generated to specify the relationship by applying binary predicate R(x1, x2).

17. Example: Metamodel • Signature elements: • unary predicates : Class(x), Classifier(x), Generalisation(x) • binary predicates general(x, y) and specific(x, y) Model • Statements of the model: • Class(Woman), Class(Person), Generalisation(wp) • specific(wp, Woman), general(wp, Person)

18. Interpretation in different contexts • The context in which a model is interpreted can be specified as hypothesis. • A hypothesis can be defined as a rule that maps from a model to formulas in the signature. Hypothesis Rules H1: Distinguishability of elements. A hypothesis that the elements of type MC in the model are all different from each other can be generated as formulas in the form of eiej, for ij {1,2,…,k}. H2: Completeness of elements. A hypothesis on the completeness of elements of type MC can be generated as a formula in the following form. x. MC(x) -> (x = e1)  (x = e2)  …  (x = ek) H3: Completeness of relations. A hypothesis on the completeness of relation R in the model can be generated as a formula in the following form. x1,x2.R(x1,x2)->((x1=e1,1)(x2=e1,2))((x1= e2,1)(x2= e2,2)) … ((x1= en,1)(x2= en,2))

19. Predicates at model level in descriptive semantics. A(x): object x is an instance of class A. It is a predicate at object level. Definition of functional semantics • Functional semantics defines the basic concepts of object orientation • Our approach: axioms in second order predicate logic • For example, • If class A inherits class B, every instance of A is also an instance of B Class(A) Class(B) Inherits(A,B)  x (A(x) B(x))

20. Object and class Axiom 1: Every object must be an instance of a class. x (Object(x) C.(Class(C) C(x))) Axiom 2: Every attribute declared in a class is a property of the class. OwnedAttribute(C, x) HasAttribute(C, x) Axiom 3: Every operation declared in a class is an operation of the class OwnedOperation(C, x) HasOperation(C, x) This formula is now second order because C is a qualified variable range over predicates. Notation: Every class C in the system is represented as a predicate C(x) such that C(a) is true if and only if a is an instance of class C.

21. Inheritance Axiom 4: If class A inherits class B, every instance of A is also an instance of B. Class(A) Class(B) Inherits(A,B)  x (A(x) B(x)) Axiom 5: If class A inherits class B, every attribute of B is also an attribute of A Class(A) Class(B) Inherits(A,B)  x (Property(x)  HasAttribute(B, x) HasAttribute(A,x)) Axiom 6: If class A inherits class B, every operation of B is also an operation of A Class(A) Class(B) Inherits(A,B)  x (Operation(x)  HasOperation(B, x) HasOperation(A,x)) Inherits(A,B) = x (Generalisation(x) specific(x, A) general(x, B)

22. Re-definition and Polymorphism Let class A inherits class B. Axiom 7: If A declares attribute a with type TA, then the type of attribute a is TA regardless what is defined in class B. Class(A), Class(B), Inherits(A,B), HasAttribute(A, a), OwnedAttribute(A,a), Type(a,TA)  CurrentType(a, A, TA) Axiom 8: If class A does not declare attribute a, but inherited attribute from B, then the type of attribute a is as in B. Class(A), Class(B), Inherits(A,B), HasAttribute(A, a), OwnedAttribute(A,a) ( CurrentType(a, B, TB) CurrentType(a, A, TB) Notation: CurrentType(a, A, B)means the current type of attributeain classAisB. Note: There are similar axioms for operations.

23. Abstract class • Axiom 9: If class A is abstract, for every instance x of A, there must be a subclass B of A such that x is an instance of B. Class(A) IsAbstract(A)  x.(A(x) B.(Class(B)Inherits(B,A)  B(x))) Notation: IsAbstract(C) == IsAbstract(C, True)

24. A a: B Ea a Eb A B Attribute and association Axiom 10: Class(A) HasAttribute(A, a)  CurrentType(a, A, B)  (x, y. a(x, y)  A(x)  B(y)) Axiom 11: Class(A)  Class(B)  Association(a)  memberEnd(a, Ea)  CurrentType(Ea, A)  memberEnd(a, Eb)  CurrentType(Eb, B)  (x, y. Eb(x, y)  A(x)  B(y))  (x, y. Ea(x, y)  B(x)  A(y)) Notation: For any attribute or association a, a(x,y) denotes x.a=y . This is to keep notation consistent with the notation,e.g. Association(A,B)used in descriptive semanticsderived from metamodel.

25. Multiplicity Axiom 12: Multiplicityof association Association(a) memberEnd(a, Ea) type(Ea, A) memberEnd(a, Eb) type(Eb, B)  upperValue(Eb, m) lowerValue(Eb, n)  (x. A(x) n||{y | Eb(x, y) }||m) Axiom 13: Multiplicity of attributes Class(A)  ownedAttribute(A, a)  type(a, B)  upperValue(a, m)  lowerValue(a, n)  (x. A(x)  n||{y | a(x, y) }||m)

26. Enumeration Axiom 14: Distinguishability of the literal constants Enumeration(A)  ownedLiteral(A, v1)  ownedLiteral(A, v2)  (Identifier(v1) Identifier(v2 )  (v1 v2)) Axiom 15: Type of the literal constants Enumeration(A)  ownedLiteral(A, v) A(v) Axiom 16: Completeness of the enumeration EnumClass(A)  (x.( A(x) ownedLiteral(A, x)))

27. Whole-part relationships Axiom 17: Composite relation (Class(A)  Class(B) Association(C)  memberEnd(C, b) type(b, B)  aggregation(b, composite) ) x. (B(x) !y.(A(y) b(x,y))

28. Combination of functional and descriptive semantics • Applying functional semantics to models • ‘Ordinary’ semantics in first order logic, i.e. the properties that objects of the system at run time must satisfy For example, [Berardi, Cal and Calvanese, 2005], [Kaneiwa and Satoh, 2006], etc. • Applying functional semantics to metamodel • Axioms of models, i.e. the properties satisfied by all models

29. Descriptive semantics Functional semantics Object level semantics Apply functional semantics to model: Example Class(Woman), Class(Person), Generalisation(wp), specific(wp, Woman), general(wp, Person) ,  A, B (Class(A) Class(B) Inherits(A,B)  x(A(x) B(x))) |- x (Woman(x) Person(x)) where Inherits(A,B) = x (Generalisation(x) specific(x, A) general(x, B)

30. Descriptive semantics applied to metamodel Functional semantics Axiom of models Apply functional semantics to metamodel MetaClass(Class), MetaClass(Classifier), MetaGeneralisation(cc), specific(cc, Class), general(cc, Classifier) ,  A, B.(Class(A) Class(B) Inherits(A,B)  x (A(x) B(x))) |-x (Class(x) Classifier(x)) where Inherits(A,B) = x (MetaGeneralisation(x) specific(x, A) general(x, B)

31. Implementation of functional semantics The functional semantics for the OO concepts used in UML class diagram are defined as a set of rules that maps from class diagram (metamodel) to axioms (formulas of the -sentences) and implemented directly. Axiom rules: A1: Completeness of classification. Let MC1, MC2, …, MCn be the set of concrete metaclasses in a metamodel. We have an axiom x. MC1(x) MC2(x) … MCn(x) A2: Disjointness of classification. Let MC1, MC2, …, MCn be the set of concrete metaclasses in a metamodel. For each pair of different concrete metaclasses MCiand MCj, ij, we have an axiom x. MCi(x)  ¬ MCj(x). A3: Logical implication of inheritance. For a generalisation relation from metaclass MA to MB in a metamodel, we have an axiom x. MA(x)  MB(x). Corresponding to Axiom 4 of functional semantics

32. Axiom Rules (Continue) A4: Completeness of specialisations. Let MA be a metaclass in a metamodel and MB1, MB2, …, MBk be the set of metaclasses specialising MA. We have an axiom x. MA(x) MB1(x) MB2(x)  … MBk(x). A5: Types of parameters of predicates. For each binary predicate MA(x, y) derived from an association from metaclass MC1 to MC2 in a metamodel, we have an axiom x, y. MA(x, y)  MC1(x)  MC2(y). A6: Domain and range of functions. For each function MAttr(x) derived from a metaattribute MAttr of type MT in a metaclass MC, we have an axiom x,y. MC(x)  (MAttr(x) = y)  MT(y). A7: Multiplicity of binary predicate. For each binary predicate MA(x, y) derived from an association from metaclass MC1 to MC2 in a metamodel, let Mul be the multicity value specified on the association end MA, we have axioms in the following form. If Mul = 0..1: x, y, z. MC1(x)  MA(x, y)  MA(x, z)  (y = z) If Mul = 1 or unspecified: x. MC1(x)  y. MA(x, y) and x, y, z. MC1(x)  MA(x, y)  MA(x, z) (y = z) If Mul = 1..*: x. MC1(x)   y. MA(x, y) If Mul = 2..*: x. MC1(x)   y, z. MA(x, y)  MA(x, z)  (yz) If Mul = 0..2: x, y, z, u. MC1(x)  MA(x, y)  MA(x, z)  MA(x, u)  (y = z)  (y = u)  (u = z) Corresponding to Axiom 10 Corresponding to Axiom 11 Corresponding to Axiom 12

33. Axiom Rules (continue) A8: Multiplicity of function. For each function MAttr(x) derived from a metaattribute MAttr of type MT in a metaclass MC, let Mul be the multicity value of the metaattribute MAttr, we have axioms: If Mul = 0..1: x, y, z. MC(x)  (MAttr(x) = y)  (MAttr(x) = z) -> (y = z) If Mul = 1: x. MC(x) ->  y. (MAttr(x) = y) and x, y, z. MC(x)  (MAttr(x) = y) (MAttr(x) = z) -> (y = z) If Mul = 1..*: x. MC(x) ->  y. (MAttr(x) = y) A9: Distinguishability of the literal constants. For each pair of different literal values a and b of an enumeration type, we have an axiom a b. A10: Type of the literal constants. For each enumeration value a defined in an enumeration metaclass ME, we have an axiom in the form of ME(a) stating that the type of a is ME. A11: Completeness of the enumeration. An enumeration type only contains the listed literal constants as its values, hence for each enumeration metaclass ME with literal values a1, a2, …, ak, we have an axiom in the form of x. ME(x) -> (x = a1)  (x = a2) … (x = ak). Axiom Rule 12: Well-formedness rules. For each WFR formally specified in OCL, we have a corresponding axiom in the first order language. Corresponding to Axiom 13 Corresponding to Axiom 14 Corresponding to Axiom 15 Corresponding to Axiom 16

34. Strict metamodelling principle • Axiom rules also contain ‘hypothesis’ on the uses of class diagrams as metamodels, such as the strictmeta-modelling principle, which isto ensure that a metamodel is a well-defined abstract syntax of modelling language. Strict Metamodelling: In an n-level modelling architecture M0, M1, …, Mn, every element of an Mm-level model must be an instance-of exactlyone element of an Mm+1-level model, for all 0 m < n-1, and any relationship other than the instance-of relationship between two elements X and Y implies that level(X ) = level(Y). Corresponding to axiom rules A1, A2 and A4.

35. Semantics of Interaction and State Machine • Same signature, axiom and formula mappings are applied to the meta-models of UML state machine and interaction diagrams. • Additional Axiom Rules for inter-metamodel connections Axiom Rules A10: Cross metamodel association and inheritance. For each cross metamodel inheritance from metaclass MA to external metaclass MB, we have an axiom in the form of x. MA(x) -> MB(x). A2’: Completeness of specialisations across metamodels. Let MA be a metaclass depicted in two metamodels MM1and MM2. Let metaclasses MB1, MB2, …, MBk be the set of metaclasses that specialise MA in metamodel MM1, and MC1, MC2, …, MCp be the set of metaclasses that specialise MA in metamodel MM2. We have the following axiom when a model is defined by MM1 and MM2. x. MA(x) -> MB1(x)  … MBk(x) MC1(x)  … MCp(x)

36. A Simplified Metamodel of Interaction Diagram

37. A Simplified Metamodel of State Machine

38. The Tool LAMBDES LAMBDES stands for Logic Analyser of Model/Metamodel Based on Descriptive Semantics

39. Applications of Descriptive Semantics Definition 3. (Properties of a model) Let SemH(M) be the semantics of a model M. M has a property P (represented as a formula in the logic system defined by Sig) under the semantics definition SemH(M) and the hypothesis H, if and only if AxmDAxmFT(M) H(M) |-P in the formal logic system. Similarly, we say that M has a property P in descriptive semantics, if and only ifAxmDT(M) H(M) |-P in the formal logic system.  • Consistency checking of UML models • Validation of consistency constraints for UML models • Consistency checking of UML meta-models • Conformance checking of designs to design patterns • Consistency checking of specification of design patterns

40. Consistency checking of UML models Definition 6. (Logical consistency) Let SemH(M) = AxmDAxmFT(M)H(M) be the semantics of a model M. Model M is said to be logically inconsistent in the semantic definition SemH(M) if SemH(M)|-false; otherwise, we say that the model is logically consistent.  Definition 7. (Consistent interpretation of formulas in a subject domain) Let Dom=<D, Sig, Eva> be a subject domain as defined in Definition 4. The interpretation of formulas in signature Sig is consistent with first order logic if and only if for all formulas q and p1, p2, …, pk that p1, p2, …, pk |- q, and for all systems s in D that Eva(pi, s) =true for i=1,2,…, k,we always have Eva(q, s) =true. 

41. Validity of consistency checking Theorem 1. (Unsatisfiability of inconsistent model) A model M that is logically inconsistent in the semantic definition SemH(M) is not satisfiable on any subject domain whose interpretation of formulas is consistent with first order logic. • Inconsistent model => it cannot be implemented (not satisfiable) • Consistency model => not necessarily implementable • Other issues effect satisfiability: • Property of the subject domain: e.g. whether the programming language is powerful enough to implement • Non-logic property of the model: e.g. whether the system is feasible

42. Validation of consistency constraints • Consistency constraints: • Logic statements about models, • e.g. ‘a life line must represent an instance of a class’ x, y, z. Lifeline(x) represent(x,y) type(y, z) ->Class(z) Definition 8. (Consistency w.r.t. consistency constraints) Given a set of consistency constraints C={c1, c2, …, cn}, the consistency of a model M with respect to the constraints C under the semantics definition SemH(M) is the consistency of the set U = SemH(M) C of formulas. In particular, we say that a model fails on a specific constraint ck, if SemH(M) is consistent, but SemH(M) {ck} is not.  • Results of experiments: • A number of sample UML models were checked to be consistent. • Mutants of the models were checked and detected inconsistency in mutants.

43. Definition of validity and effectiveness Definition 9. (Validity of consistency constraints) Let AxmD and AxmF be the sets of axioms for descriptive semantics and functional semantics, respectively. A set C={c1, c2, …, cn} of consistency constraints is descriptively valid if AxmDC is logically consistent. The set C of consistency constraints is functionally valid AxmDAxmFC is logically consistent.  Definition 10. (Effectiveness of consistency constraints) Let A be a set of semantics axioms. A set C={c1, c2, …, cn} of consistency constraints is logicallyineffective with respect to the set A of axioms if A|-C.  Results of Experiments: A set of 5 consistency constraints were validated by using LAMBDES tool. They were proved valid and effective.

44. Consistency checking of meta-models Definition 11 (Inconsistency of meta-model). A meta-model M is inconsistent if AxmD(M) is logically inconsistent. Experiments: • Subjects: Simplified UML, UML 2.0 metamodel, AspectJ profile • Findings: • Simplified metamodel: consistent • UML 2.0: • 16 inconsistencies due to voilation of strict meta-modelling • 1 inconsistently as abstract and concrete metaclasses in different metamodel class diagrams. • AspectJ: • Incomplete definition of 7 meta-classes • 2 inconsistency due to violation of strict meta-modelling

45. Conformance of designs to patterns • Patterns are meta-models (rather than models) • A pattern specifies a set of models that have common structural and behavioural properties • A pattern can be specified as a predicate on models in first order logic (Bayley and Zhu 2007, 2008)

46. Translation of pattern specification • A specification of design pattern is translated into a specification of systems in sig-formulas Template Method Specification formula(exists([ %Components: xAbstractClass, xTemplateMethod, xOthers ], and( %Static conditions: Class(xAbstractClass), ownedOperation(xAbstractClass, xTemplateMethod), ownedOperation(xAbstractClass,xOthers), isLeaf(xTemplateMethod,bTrue), not(equal(xTemplateMethod,xOthers)), isLeaf(xOthers,bFalse) %Dynamic conditions: callsHook(xTemplateMethod,xOthers) ))).

47. Validity of conformance checking Definition 12 (Correctness of translation) Let p be a predicate on models, p' be a predicate on systems. The predicate p' is a correct translation of p, if for all models m, we have that m |- p iff sD. (s |= (Sem(m) p’). Theorem Let P be a pattern and Spec(P) be a specification of the pattern. Suppose that Spec'(P) is a correct translation of Spec(P). For all models m, if Sem(m) Spec'(P) is true in FOL, then, for all systems sD, s |= m implies s |= Spec'(P).

48. Experiments • Specification of patterns: • 23 design patterns in GoF book are specified in first order logic on models • Translation: • The specifications are translated into first order logic in LAMBDES syntax • Experiment 1: checking specifications Consistency • 23 specifications of patterns were checked for consistency with the axioms • Results: all specification are consistent • Experiment 2: Testing conformance checking ability • 23 designs represented in UML diagrams are created according to GOF; • Each design is checked against all 23 patterns using LAMBDES • Results: • 100% recognition of designs as instances of intended pattern • 22% of false positive recognitions (including time-out, which is 5%) • Experiment 3: Testing modelling tool’s correctness • 23 design created using StarUML templates • Each design is checked against all 23 patterns using LAMBDES • Results: 61% is not recognised, Overall fault detecting rate: 81%

49. Conclusion and comparison • Separation of functional semantics from descriptive semantics can simplify the formal semantic definition of UML and it is scalable • Applicable for all types of diagrams defined in meta-model uniformly • Applicable to multiple views defined by multiple meta-models, and addressed the extendability issues • Addressed the issue due to flexibility in the uses of modelling language in different development context • Addressed the issue for different interpretation of modelling languages in different subject domains • Reasoning about models in first order logic can be feasible and useful in model drive software development • Can be automated by tools such as LAMBDES + SPASS + StarUML • Can reason about properties that are not possible for semantics at object level, such as • The conformance of designs to patterns, • The validation of consistency constraints, • The consistency of meta-model, etc.

50. Future work • Further development of the axioms in second order predicate logic for functional semantics; • Theoretical analysis of the logic properties of the semantics definition • Soundness of the rules: yes • Consistency of the rules: yes • Completeness of the rules: • In what sense? • How to prove? • Case studies on reasoning about other properties of designs, such as • Platform specific models, platform independence, etc. • Transformation of models,