Software Design, Modelling and Analysis in UML Lecture 05: Object - PDF document

Software Design, Modelling and Analysis in UML Lecture 05: Object Diagrams, OCL Consistency 2013-11-06 – 05 – 2013-11-06 – main – Prof. Dr. Andreas Podelski, Dr. Bernd Westphal Albert-Ludwigs-Universit¨ at Freiburg, Germany Contents & Goals Last Lecture: • OCL Semantics This Lecture: • Educational Objectives: Capabilities for following tasks/questions. • What is an object diagram? What are object diagrams good for? • When is an object diagram called partial? What are partial ones good for? • When is an object diagram an object diagram (wrt. what)? • Is this an object diagram wrt. to that other thing? • How are system states and object diagrams related? • What does it mean that an OCL expression is satisfiable? • When is a set of OCL constraints said to be consistent? • Can you think of an object diagram which violates this OCL constraint? – 05 – 2013-11-06 – Sprelim – • Content: • Object Diagrams • Example: Object Diagrams for Documentation • OCL: consistency, satisfiability 2 /31

Where Are We? – 05 – 2013-11-06 – main – 3 /31 You Are Here. N UML W E S = ( T , C , V, atr ) , SM S , SD CD , SM ϕ ∈ OCL CD , SD S ✔ D Model S , → SM ) S , → SD , F SD ) S , A expr ✔ ✔ M = (Σ B = ( Q SD , q 0 , A ✔ Instances ( cons 0 , Snd 0 ) π = ( σ 0 , ε 0 ) − − − − − − − − → ( σ 1 , ε 1 ) · · · w π = (( σ i , cons i , Snd i )) i ∈ N u 0 ! ! Mathematics – 05 – 2013-11-06 – Spostmap – G = ( N, E, f ) ! ! OD UML 4 /31

Object Diagrams – 05 – 2013-11-06 – main – 5 /31 Graph Definition. A node labelled graph is a triple G = ( N, E, f ) consisting of • vertexes N , • edges E , • node labeling f : N → X , where X is some label domain, – 05 – 2013-11-06 – Sod – 6 /31

D be a structure of signature S = ( T , C , V, atr ) D Object Diagrams S a system state. Definition. Let and σ ∈ Σ D ( C ) finite , Then any graph G = ( N, E, f ) where • nodes are identities (not necessarily alive), i.e. C }} × N, N ⊂ • edges correspond to “links” of objects, i.e. E ⊆ N × { v : τ ∈ V | τ ∈ { C 0 , 1 , C ∗ | C ∈ ∪ ( V � ( D ( T ) ∪ D ( C ∗ ))) ∀ ( u 1 , r, u 2 ) ∈ E : u 1 ∈ dom( σ ) ∧ u 2 ∈ σ ( u 1 )( r ) , • objects are labelled with attribute valuations and non-alive identities marked with “X”, i.e. X = { X } ˙ – 05 – 2013-11-06 – Sod – ∀ u ∈ N ∩ dom( σ ) : f ( u ) ⊆ σ ( u ) ∀ u ∈ N \ dom( σ ) : f ( u ) = { X } is called object diagram of σ . 7 /31 D ( C ) finite, ∪ ( V � ( D ( T ) ∪ D ( C ∗ ))) Graphical Representation of Object Diagrams S = ( { Int } , { C } , { v 1 : Int , v 2 : Int , r : C ∗ } , { C �→ { v 1 , v 2 , r }} ) . X = { X } ˙ N ⊂ E ⊂ N × V 0 , 1 , ∗ × N , u 1 ∈ dom( σ ) ∧ u 2 ∈ σ ( u 1 )( r ) , f ( u ) ⊆ σ ( u ) or f ( u ) = { X } • Assume • Consider σ = { u 1 �→ { v 1 �→ 1 , v 2 �→ 2 , r �→ { u 2 }} , u 2 �→ { v 1 �→ 3 , v 2 �→ 4 , r �→ ∅}} S and any D with D ( Int ) ⊇ { 1 , 2 , 3 , 4 } . • Then G = ( N, E, f ) = ( { u 1 , u 2 } , { ( u 1 , r, u 2 ) } , { u 1 �→ { v 1 �→ 1 , v 2 �→ 2 } , u 2 �→ { v 1 �→ 3 , v 2 �→ 4 }} , is an object diagram of σ wrt. – 05 – 2013-11-06 – Sod – 8 /31

D ( C ) finite, ∪ ( V � ( D ( T ) ∪ D ( C ∗ ))) Graphical Representation of Object Diagrams S = ( { Int } , { C } , { v 1 : Int , v 2 : Int , r : C ∗ } , { C �→ { v 1 , v 2 , r }} ) . X = { X } ˙ N ⊂ E ⊂ N × V 0 , 1 , ∗ × N , u 1 ∈ dom( σ ) ∧ u 2 ∈ σ ( u 1 )( r ) , f ( u ) ⊆ σ ( u ) or f ( u ) = { X } • Assume • Consider σ = { u 1 �→ { v 1 �→ 1 , v 2 �→ 2 , r �→ { u 2 }} , u 2 �→ { v 1 �→ 3 , v 2 �→ 4 , r �→ ∅}} S and any D with D ( Int ) ⊇ { 1 , 2 , 3 , 4 } . • Then G = ( N, E, f ) = ( { u 1 , u 2 } , { ( u 1 , r, u 2 ) } , { u 1 �→ { v 1 �→ 1 , v 2 �→ 2 } , u 2 �→ { v 1 �→ 3 , v 2 �→ 4 }} , is an object diagram of σ wrt. • We may equivalently (!) represent G graphically as follows: u 1 : C u 2 : C – 05 – 2013-11-06 – Sod – r v 1 = 1 v 1 = 3 v 2 = 2 v 2 = 4 8 /31 UML Notation for Object Diagrams mandatory : id class optional � v 1 = d 1 “compartment” . . �� . optional v n = d n � r optional – 05 – 2013-11-06 – Sod – : id class 9 /31

D ( C ) finite, ∪ ( V � ( D ( T ) ∪ D ( C ∗ ))) Object Diagrams: More Examples X = { X } ˙ N ⊂ E ⊂ N × V 0 , 1 , ∗ × N , u 1 ∈ dom( σ ) ∧ u 2 ∈ σ ( u 1 )( r ) , f ( u ) ⊆ σ ( u ) or f ( u ) = { X } σ = { 1 C �→ { p �→ ∅ , n �→ { 5 C }} , 5 C �→ { p �→ ∅ , n �→ ∅} , 1 D �→ { x �→ 23 }} vs. • ( ∅ , ∅ , ∅ ) 5 C : C 1 C : C n 1 D : D • n = ∅ p = ∅ x = 23 p = ∅ 1 C : C n 5 C : C 1 D : D • x = 23 – 05 – 2013-11-06 – Sod – • 1 C : C 5 C : C 1 D : D 1 C : C x 5 C : C 1 D : D • x = 23 10 /31 Complete vs. Partial Object Diagram D S . Definition. Let G = ( N, E, f ) be an object diagram of system state σ ∈ Σ We call G complete wrt. σ if and only if • G is object complete, i.e. • G consists of all alive objects, i.e. N = dom( σ ) , • G is attribute complete, i.e. T -typed attributes, • G comprises all “links” between alive objects, i.e. if u 2 ∈ σ ( u 1 )( r ) for some u 1 , u 2 ∈ dom( σ ) and r ∈ V , T ∪{ r �→ ( σ ( u )( r ) \ N ) | r ∈ V : σ ( u )( r ) \ N � = ∅} then ( u 1 , r, u 2 ) ∈ E , and T } . T := { v : τ ∈ V | τ ∈ • each node is labelled with the values of all i.e. for each u ∈ dom( σ ) , – 05 – 2013-11-06 – Sod – f ( u ) = σ ( u ) | V where V Otherwise we call G partial. 11 /31

Complete vs. Partial Examples T ∪ { r �→ ( σ ( u )( r ) \ N ) | σ ( u )( r ) \ N } • N = dom( σ ) , if u 2 ∈ σ ( u 1 )( r ) , then ( u 1 , r, u 2 ) ∈ E , • f ( u ) = σ ( u ) | V Complete or partial? σ = { 1 C �→ { p �→ ∅ , n �→ { 5 C }} , 5 C �→ { p �→ ∅ , n �→ ∅} , 1 D �→ { x �→ 23 }} 5 C : C 1 C : C n 1 D : D • n = ∅ p = ∅ x = 23 p = ∅ 1 C : C n 5 C : C 1 D : D • – 05 – 2013-11-06 – Sod – x = 23 • 1 C : C 5 C : C 1 D : D 12 /31 Complete/Partial is Relative • Claim: • Each finite system state has exactly one complete object diagram. D • A finite system state can have many partial object diagrams. S | G is an object diagram of σ } • Each object diagram G represents a set of system states, namely G − 1 := { σ ∈ Σ • Observation : If somebody tells us , that a given (consistent) object diagram G is complete , we can uniquely reconstruct the corresponding system state. In other words: G − 1 is then a singleton. – 05 – 2013-11-06 – Sod – 13 /31

T Corner Cases – 05 – 2013-11-06 – main – 14 /31 Closed Object Diagrams vs. Dangling References Find the 10 differences! (Both diagrams shall be complete.) 1 C : C n 5 C : C 1 C : C n 5 C : C p = { 1 C } p = { 7 C } Definition. Let σ be a system state. We say attribute v ∈ V 0 , 1 , ∗ has a dangling reference in object u ∈ dom( σ ) if and only if the attribute’s value comprises an object which is not alive in σ , i.e. if σ ( u )( v ) �⊂ dom( σ ) . We call σ closed if and only if no attribute has a dangling reference in – 05 – 2013-11-06 – Sodsconf – any object alive in σ . 15 /31

Closed Object Diagrams vs. Dangling References Find the 10 differences! (Both diagrams shall be complete.) 1 C : C n 5 C : C 1 C : C n 5 C : C p = { 1 C } p = { 7 C } Definition. Let σ be a system state. We say attribute v ∈ V 0 , 1 , ∗ has a dangling reference in object u ∈ dom( σ ) if and only if the attribute’s value comprises an object which is not alive in σ , i.e. if σ ( u )( v ) �⊂ dom( σ ) . We call σ closed if and only if no attribute has a dangling reference in – 05 – 2013-11-06 – Sodsconf – T -typed attribute/value pairs only. any object alive in σ . Observation : Let G be the (!) complete object diagram of a closed system state σ . Then the nodes in G are labelled with 15 /31 S = ( { Int } , { C } , { n, p : C ∗ } , { C �→ { n, p }} ) . Special Notation • • Instead of 1 C : C n 5 C : C we want to write 1 C : C n 5 C : C p = ∅ p = ∅ or p p 1 C : C n 5 C : C | | – 05 – 2013-11-06 – Sodsconf – to explicitly indicate that attribute p : C ∗ has value ∅ (also for p : C 0 , 1 ). 16 /31

Aftermath We slightly deviate from the standard (for reasons): • In the course, C 0 , 1 and C ∗ -typed attributes only have sets as values . UML also considers multisets, that is, they can have n u 1 : C u 2 : C n (This is not an object diagram in the sense of our definition because of the requirement on the edges E . Extension is straightforward but tedious.) • We allow to give the valuation of C 0 , 1 - or C ∗ -typed attributes in the values compartment . • Allows us to indicate that a certain r is not referring to another object. – 05 – 2013-11-06 – Sodsconf – • Allows us to represent “dangling references”, i.e. references to objects which are not alive in the current system state. • We introduce a graphical representation of ∅ values. 17 /31 The Other Way Round – 05 – 2013-11-06 – main – 18 /31