Contents & Goals Last Lecture: • Behavioural Features • State Machines Variation Points Software Design, Modelling and Analysis in UML “...shall be usable...” for UML • Inheritance in UML: concrete syntax • Liskov Substitution Principle — desired semantics Lecture 21: Inheritance II This Lecture: • Educational Objectives: Capabilities for following tasks/questions. • What’s the Liskov Substitution Principle? • What is late/early binding? 2014-02-05 • What is the subset, what the uplink semantics of inheritance? • What’s the effect of inheritance on LSCs, State Machines, System States? – 21 – 2014-02-05 – Sprelim – • What’s the idea of Meta-Modelling? – 21 – 2014-02-05 – main – – 21 – 2014-02-05 – main – Prof. Dr. Andreas Podelski, Dr. Bernd Westphal • Content: Albert-Ludwigs-Universit¨ at Freiburg, Germany • Two approaches to obtain desired semantics • The UML Meta Model 2 /74 3 /74 Easy: Static Typing Static Typing Cont’d C 1 C 2 itsC1 x : Int x : Int � � signal � � E f ( Int ) : Int f ( Int ) : Int C C 1 C 2 x : Int x : Int D 1 D 2 � � signal � � E itsD1 Given : f ( Int ) : Int f ( Int ) : Int x : Bool � � signal � � F f ( Float ) : Int Excursus: Late Binding of Behavioural Features D 1 D 2 Wanted : x : Bool � � signal � � F f ( Float ) : Int • x > 0 also well-typed for D 1 Notions (from category theory): • assignment itsC1 := itsD1 being well-typed • invariance , • itsC1 .x = 0 , itsC1 .f (0) , itsC1 ! F • covariance , being well-typed (and doing the right thing). • contravariance . Approach : We could call, e.g. a method, sub-type preserving , if and only if it • Simply define it as being well-typed, – 21 – 2014-02-05 – Sstatic – – 21 – 2014-02-05 – Sstatic – – 21 – 2014-02-05 – main – adjust system state definition to do the right thing. • accepts more general types as input ( contravariant ), • provides a more specialised type as output ( covariant ). This is a notion used by many programming languages — and easily type-checked. 4 /74 5 /74 6 /74
Late Binding Late Binding in the Standard and Programming Lang. What transformer applies in what situation? (Early (compile time) binding.) • In the standard , Section 11.3.10, “CallOperationAction”: f overridden in D f not overridden in D “ Semantic Variation Points C value C The mechanism for determining the method to be invoked as a of m e C s o f () : Int Back to the Main Track: “...tell the difference...” for UML f () : Int someC/ result of a call operation is unspecified.” [OMG, 2007b, 247] C 0 someD someD D D • In C ++ , f () : Int someC -> f () C :: f () C :: f () u 1 • methods are by default “(early) compile time binding”, • can be declared to be “late binding” by keyword “ virtual ”, someD -> f () C :: f () D :: f () u 2 • the declaration applies to all inheriting classes. someC -> f () C :: f () D :: f () u 2 • In Java , What one could want is something different: (Late binding.) • methods are “late binding”; u 1 someC -> f () C :: f () C :: f () • there are patterns to imitate the effect of “early binding” – 21 – 2014-02-05 – Slatebind – – 21 – 2014-02-05 – Slatebind – someD -> f () D :: f () D :: f () u 2 – 21 – 2014-02-05 – main – Exercise : What could have driven the designers of C ++ to take that approach? someC -> f () C :: f () C :: f () u 2 Note : late binding typically applies only to methods , not to attributes . (But: getter/setter methods have been invented recently.) 7 /74 8 /74 9 /74 With Only Early Binding... Difficult: Dynamic Subtyping Sub-Typing Principles Cont’d C • In the standard, Section 7.3.36, “ Operation ”: f ( Int ) : Int • ...we’re done (if we realise it correctly in the framework). “ Semantic Variation Points • Then [...] When operations are redefined in a specialization, rules regarding D invariance , covariance , or contravariance of types and preconditions • if we’re calling method f of an object u , • C :: f and D :: f are type compatible , determine whether the specialized classifier is substitutable for its more • which is an instance of D with C � D but D is not necessarily a sub-type of C . f ( Int ) : Int general parent. Such rules constitute semantic variation points with respect to redefinition of operations.” [OMG, 2007a, 106] • via a C -link, • Examples : (C ++ ) • then we (by definition) only see and change the C -part. • So, better: call a method sub-type preserving , if and only if it • We cannot tell whether u is a C or an D instance. (i) accepts more input values ( contravariant ), int C::f(int) { int D::f(int) { (ii) on the old values , has fewer behaviour ( covariant ). So we immediately also have behavioural/dynamic subtyping. vs. return 0; return 1; Note : This (ii) is no longer a matter of simple type-checking! } ; } ; • And not necessarily the end of the story: • One could, e.g. want to consider execution time. – 21 – 2014-02-05 – Ssubtyping – – 21 – 2014-02-05 – Ssubtyping – – 21 – 2014-02-05 – Ssubtyping – • Or, like [Fischer and Wehrheim, 2000], relax to “fewer observable int C::f(int) { int D::f(int x) { behaviour”, thus admitting the sub-type to do more work on inputs. return (rand() % 2); vs. return (x % 2); } ; } ; Note : “testing” differences depends on the granularity of the semantics. • Related : “has a weaker pre-condition,” ( contravariant ), “has a stronger post-condition.” ( covariant ). 10 /74 11 /74 12 /74
� Ensuring Sub-Typing for State Machines Towards System States C Wanted : a formal representation of “if C � D then D ‘ is a ’ C ”, that is, • In the CASE tool we consider, multiple classes (i) D has the same attributes and behavioural features as C , and in an inheritance hierarchy can have state machines. D (ii) D objects (identities) can replace C objects. • But the state machine of a sub-class cannot be drawn from scratch. Domain Inclusion Semantics We’ll discuss two approaches to semantics: • Instead, the state machine of a sub-class can only be obtained by • Domain-inclusion Semantics (more theoretical ) applying actions from a restricted set to a copy of the original one. Roughly (cf. User Guide, p. 760, for details), • add things into (hierarchical) states, • add more states, • attach a transition to a different target (limited). • They ensure , that the sub-class is a behavioural sub-type of the super • Uplink Semantics (more technical ) – 21 – 2014-02-05 – Ssubtyping – – 21 – 2014-02-05 – Ssubtyping – class. (But method implementations can still destroy that property.) – 21 – 2014-02-05 – main – • Technically, the idea is that (by late binding) only the state machine of the most specialised classes are running. By knowledge of the framework, the (code for) state machines of super-classes is still accessible — but using it is hardly a good idea... 13 /74 14 /74 15 /74 S = ( T , C , V, atr , E , F, mth , ⊳ ) be a signature. S wrt. D is a type-consistent mapping A Domain Inclusion Structure Domain Inclusion System States D ( C ) � → ( D ( T ) ∪ D ( C 0 , 1 ) ∪ D ( C ∗ ))) Preliminaries: Expression Normalisation v : Int D D ( C ) , Recall : Now : a system state of Let C • we want to allow, e.g., “context D inv : v < 0 ”. D ( τ ) if v : τ , τ ∈ T or τ ∈ { C ∗ , C 0 , 1 } . 0 , 1 v : Int σ : → ( V n Now a structure • we assume fully qualified names , e.g. C :: v . • [ as before ] maps types, classes, associations to domains, that is, for all u ∈ dom( σ ) ∩ Intuitively, v shall denote the “ most special more general ” C :: v according to ⊳ . D • [ for completeness ] methods to transformers, • [ as before ] σ ( u )( v ) ∈ • [ as before ] indentities of instances of classes not (transitively) related by To keep this out of typing rules, we assume that the following normalisation • [ changed ] dom( σ ( u )) = � C : D ( C ) � D ( D ) . C 0 � C atr ( C 0 ) , generalisation are disjoint, has been applied to all OCL expressions and all actions. Example : • [ changed ] the indentities of a super-class comprise all identities of • Given expression v (or f ) in context of class D , as determined by, e.g. C sub-classes, i.e. • by the (type of the) navigation expression prefix, or 0 , 1 � x : Int ∀ C ∈ n • by the class, the state-machine where the action occcurs belongs to, C ⊳ D • similar for method bodies, – 21 – 2014-02-05 – Sdomincl – – 21 – 2014-02-05 – Sdomincl – – 21 – 2014-02-05 – Sdomincl – • normalise v to ( = replace by) C :: v , D x : Int • where C is the greatest class wrt. “ � ” such that y : Int • C � D and C :: v ∈ atr ( C ) . Note : the old setting coincides with the special case ⊳ = ∅ . If no (unique) such class exists, the model is considered not well-formed ; the Note : the old setting still coincides with the special case ⊳ = ∅ . expression is ambiguous. Then: explicitly provide the qualified name . 16 /74 17 /74 18 /74
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