Software Design, Modelling and Analysis in UML Lecture 03: Object Constraint Language (OCL)

– 03 – 2011-11-02 – main –

2011-11-02

Prof. Dr. Andreas Podelski, Dr. Bernd Westphal Albert-Ludwigs-Universit¨at Freiburg, Germany

Contents & Goals Last Lecture: • Basic Object System Signature •

System State σ ∈ ΣD S

S

and Structure

D

(Smells like they’re related to class/object diagrams, officially we don’t know yet. . . )

– 03 – 2011-11-02 – Sprelim –

This Lecture: • Educational Objectives: Capabilities for these tasks/questions: • Please explain this OCL constraint. • Please formalise this constraint in OCL. • Does this OCL constraint hold in this system state? • Can you think of a system state satisfying this constraint? • Please un-abbreviate all abbreviations in this OCL expression. • In what sense is OCL a three-valued logic? For what purpose? • How are

D (C) and τC

related?

• Content: • OCL Syntax, OCL Semantics over system states 2/36

– 03 – 2011-11-02 – main –

What is OCL? And What is It Good For?

3/36

What is OCL? How Does it Look Like?

OCL/Beispiel

TeamMember name : String age : Integer

Location

Meeting 2..*

meetings

participants

title : String * numParticipants : Integer start : Date duration: Time

*

move(newStart : Date)

– 03 – 2011-11-02 – Swhatis –

context TeamMember inv: age => 18 context Meeting inv: duration > 0

name : String 1

((C) Prof. Dr. P. Thiemann, http://proglang.informatik.uni-freiburg.de/teaching/swt/2008/)

• OCL: Object Constraint Logic.

4/36

What’s It Good For? • Most prominent: write down requirements supposed to be satisfied by all system states.

– 03 – 2011-11-02 – Swhatis –

Often targeting all alive objects of a certain class.

5/36

What’s It Good For? • Most prominent: write down requirements supposed to be satisfied by all system states. Often targeting all alive objects of a certain class.

– 03 – 2011-11-02 – Swhatis –

• Not unknown: write down pre/post-conditions of methods (Behavioural Features). Then evaluated over two system states.

5/36

What’s It Good For? • Most prominent: write down requirements supposed to be satisfied by all system states. Often targeting all alive objects of a certain class. • Not unknown: write down pre/post-conditions of methods (Behavioural Features). Then evaluated over two system states.

– 03 – 2011-11-02 – Swhatis –

• Common with State Machines: guards in transitions.

5/36

What’s It Good For? • Most prominent: write down requirements supposed to be satisfied by all system states. Often targeting all alive objects of a certain class. • Not unknown: write down pre/post-conditions of methods (Behavioural Features). Then evaluated over two system states.

– 03 – 2011-11-02 – Swhatis –

• Common with State Machines: guards in transitions. • Lesser known: provide operation bodies.

5/36

What’s It Good For? • Most prominent: write down requirements supposed to be satisfied by all system states. Often targeting all alive objects of a certain class. • Not unknown: write down pre/post-conditions of methods (Behavioural Features). Then evaluated over two system states.

– 03 – 2011-11-02 – Swhatis –

• Common with State Machines: guards in transitions. • Lesser known: provide operation bodies. • Metamodeling: the UML standard is a MOF-Model of UML. OCL expressions define well-formedness of UML models (cf. Lecture ∼ 21). 5/36

Plan. • Today:

• The set OCLExpressions(S ) of OCL expressions over

S.

• Given an OCL expression expr , a system state σ ∈ ΣD S , and a valuation of

logical variables β, define

I Jexpr K(σ, β) ∈ {true, false, ⊥}. • Later: use I to define |= ⊆ ΣD S × OCLExpressions(S ).

N

UML ϕ ∈ OCL

CD, SD

expr

S , SD

E

OCL

Diagram

S

Model

Structure Diagram

S

= (T , C , V, atr ), SM

Class Diagram

(ΣD S , AS , →SM ) = M

Instances

– 03 – 2011-11-02 – Swhatis –

CD, SM

W

Com ponent Diagram

Activity Diagram

Object Diagram

B = (QSD , q0 , AS , →SD , FSD ) Com posite Structure Diagram

(cons 0 ,Snd 0 )

Deploym ent Diagram

Package Diagram

Interactio Diagram

(cons 1 ,Snd 1 )

(σ0 , ε0 ) −−−−−−−−→ (σ1 , ε1 ) −−−−−−−−→ . . . G = (N, E, f )

Mathematics

Sequence Diagram

6/36 Com m unica

– 03 – 2011-11-02 – main –

(Core) OCL Syntax [OMG, 2006]

7/36

OCL Syntax 1/4: Expressions expr ::= w

S

= (T , C , V, atr ),

• W ⊇ {self C | C ∈ C } is a set of typed logical variables, w has type τ (w)

: τ (w)

: τ × τ → Bool | expr 1 =τ expr 2 | oclIsUndefinedτ (expr 1 ) : τ → Bool

– 03 – 2011-11-02 – Soclsyn –

Where, given

• τ is any type from T ∪ TB ∪ TC ∪ {Set(τ0 ) | τ0 ∈ TB ∪ TC }

• TB is a set of basic types, in the following we use τ × · · · × τ → Set(τ ) TB = {Bool , Int, String} Set(τ ) → Bool • TC = {τC | C ∈ C } is the set of object types, Set(τ ) → Int • Set(τ 0 ) denotes the Set(τC ) set-of-τ0 type for τ0 ∈ TB ∪ TC τC → τ (v) (sufficient because of τC → τD “flattening” (cf. standard))

| {expr 1 , . . . ,expr n } | isEmpty(expr 1 ) | size(expr 1 ) | allInstancesC

: : : :

| v(expr 1 ) | r1 (expr 1 ) | r2 (expr 1 )

: : : τC → Set(τD )

• v : τ (v) ∈ atr (C), τ (v) ∈ T , • r1 : D0,1 ∈ atr (C), • r2 : D∗ ∈ atr (C), • C, D ∈ C .

8/36

OCL Syntax: Notational Conventions for Expressions • Each expression

ω(expr 1 , expr 2 , . . . , expr n ) : τ1 × · · · × τn → τ may alternatively be written (“abbreviated as”) • expr 1 . ω(expr 2 , . . . , expr n ) if τ1 is an object type, i.e. if τ1 ∈ TC . • expr 1 -> ω(expr 2 , . . . , expr n ) if τ1 is a collection type

(here: only sets), i.e. if τ1 = Set(τ0 ) for some τ0 ∈ TB ∪ TC . • Examples: (self : τC ∈ W ;

v, w : Int ∈ V ;

r1 : D0,1 , r2 : D∗ ∈ V )

– 03 – 2011-11-02 – Soclsyn –

• self . v • self . r1 . w • self . r2 -> isEmpty

9/36

OCL Syntax 2/4: Constants, Arithmetical Operators For example:

– 03 – 2011-11-02 – Soclsyn –

expr ::= . . . | true, false | expr 1 {and, or, implies} expr 2 | not expr 1 | 0, −1, 1, −2, 2, . . . | OclUndefined | expr 1 {+, −, . . . } expr 2 | expr 1 {iterate(w1 : τ1 ; w2 : τ2 = expr 2 | expr 3 ) or, with a little renaming,

expr ::= · · · | expr 1 ->iterate(iter : τ1 ; result : τ2 = expr 2 | expr 3 ) where • expr 1 is of a collection type (here: a set Set(τ0 ) for some τ0 ), • iter ∈ W is called iterator, gets type τ1

(if τ1 is omitted, τ0 is assumed as type of iter ) – 03 – 2011-11-02 – Soclsyn –

• result ∈ W is called result variable, gets type τ2 , • expr 2 in an expression of type τ2 giving the initial value for result,

(‘OclUndefined’ if omitted) • expr 3 is an expression of type τ2

in which in particular iter and result may appear. 11/36

Iterate: Intuitive Semantics (Formally: later) expr ::= expr 1 ->iterate(iter : τ1 ; result : τ2 = expr 2 | expr 3 )

τ0 hlp = hexpr 1 i; τ2 result = hexpr 2 i; while (!hlp.empty()) do τ1 iter = hlp.pop(); result = hexpr 3 i; – 03 – 2011-11-02 – Soclsyn –

od Note: In our (simplified) setting, we always have expr 1 : Set(τ1 ) and τ0 = τ1 . In the type hierarchy of full OCL with inheritance and oclAny, they may be different and still type consistent. 12/36

Abbreviations on Top of Iterate expr ::= expr 1 ->iterate(w1 : τ1 ; w2 : τ2 = expr 2 | expr 3 )

expr 1 ->forAll(w : τ1 | expr 3 )

•

is an abbreviation for expr 1 ->iterate(w: τ1 ; w1 : Bool = true | w1 ∧ expr 3 ).

– 03 – 2011-11-02 – Soclsyn –

(To ensure confusion, we may again omit all kinds of things, cf. [OMG, 2006]).

expr 1 ->Exists(w : τ1 | expr 3 )

• Similar:

13/36

OCL Syntax 4/4: Context

context ::= context w1 : τ1 , . . . , wn : τn inv : expr where w ∈ W and τi ∈ TC , 1 ≤ i ≤ n, n ≥ 0.

context w1 : C1 , . . . , wn : Cn inv : expr is an abbreviation for allInstancesC1 -> forAll(w1 : C1 | ... – 03 – 2011-11-02 – Soclsyn –

allInstancesCn -> forAll(wn : Cn | expr ) ... ) 14/36

Context: More Notational Conventions • For

context self : τC inv : expr we may alternatively write (“abbreviate as”) context τC inv : expr

• Within the latter abbreviation, we may omit the “self ” in expr , i.e. for

self .v

and

self .r

v

and

r

15/36

Examples (from lecture “Softwaretechnik 2008”) OCL/Beispiel

TeamMember name : String age : Integer

Location

Meeting 2..*

meetings

participants

title : String * numParticipants : Integer start : Date duration: Time

*

name : String 1

move(newStart : Date)

context TeamMember inv: age => 18

– 03 – 2011-11-02 – Soclsyn –

context Meeting inv: duration > 0

16/36

((C) Prof. Dr. P. Thiemann, http://proglang.informatik.uni-freiburg.de/teaching/swt/2008/)

– 03 – 2011-11-02 – Soclsyn –

we may alternatively write (“abbreviate as”)

OCL/Mehr Navigation/Beispiele

TeamMember name : String age : Integer

Meeting 2..*

meetings

participants

title : String * numParticipants : Integer start : Date duration: Time

Location *

name : String 1

move(newStart : Date)

context Meeting inv: self.participants->size() = numParticipants

context Location

– 03 – 2011-11-02 – Soclsyn –

inv: name="Lobby" implies meeting->isEmpty()

((C) Prof. Dr. P. Thiemann, http://proglang.informatik.uni-freiburg.de/teaching/swt/2008/)

Examples (from lecture “Softwaretechnik 2008”)

17/36

Example (from lecture “Softwaretechnik 2008”)

Location

Meeting

TeamMember name : String age : Integer

2..*

meetings

participants

title : String * numParticipants : Integer start : Date duration: Time

*

name : String 1

move(newStart : Date)

• context Meeting inv : participants -> iterate(i : TeamMember ; n : Int = 0 | n + i . age)

– 03 – 2011-11-02 – Soclsyn –

/participants -> size() > 25

18/36

“Not Interesting” Among others: • Enumeration types • Type hierarchy • Complete list of arithmetical operators • The two other collection types Bag and Sequence • Casting • Runtime type information • Pre/post conditions

(maybe later, when we officially know what an operation is)

– 03 – 2011-11-02 – Soclsyn –

• ...

19/36

– 03 – 2011-11-02 – main –

OCL Semantics [OMG, 2006]

20/36

OCL Syntax 1/4: Expressions

The Task

expr ::= w

Where, given

: τ (w)

– 03 – 2010-10-27 – Soclsyn –

: τ × τ → Bool | expr 1 =τ expr 2 | oclIsUndefinedτ (expr 1 ) : τ → Bool

– 03 – 2011-11-02 – Soclsem –

S

= (T , C , V, atr ),

• W ⊇ {self } is a set of typed logical variables, w has type τ (w) • τ is any type from T ∪ TB ∪ TC ∪ {Set(τ0 ) | τ0 ∈ TB ∪ TC }

• TB is a set of basic types, in the following we use TB = {Bool , Int, String} τ × · · · × τ → Set(τ ) • TC = {τC | C ∈ C } is the Set(τ ) → Bool set of object types, Set(τ ) → Int • Set(τ0 ) denotes the Set(τC ) set-of-τ0 type for τ0 ∈ TB ∪ TC τC → τ (v) (sufficient because of “flattening” (cf. standard)) τC → τD

| {expr 1 , . . . ,expr n } | isEmpty(expr 1 ) | size(expr 1 ) | allInstancesC

: : : :

| v(expr 1 ) | r1 (expr 1 ) | r2 (expr 1 )

: : : τC → Set(τD )

• v : τ (v) ∈ atr (C), τ (v) ∈ T , • r1 : D0,1 ∈ atr (C), • r2 : D∗ ∈ atr (C), • C, D ∈ C .

7/30

• Given an OCL expression expr , a system state σ ∈ ΣD S , and a valuation of logical variables β, define I J · K( · , · ) : OCLExpressions(S ) × ΣD S × (W → I(T ∪ TB ∪ TC )) → I(Bool ) such that I Jexpr K(σ, β) ∈ {true, false, ⊥Bool }. 21/36

Basically business as usual... (i) Equip each OCL (!) basic type with a reasonable domain, i.e. define function I on TB ⊂ dom(I)

(iv) Equip each arithmetical operation with a reasonable interpretation (that is, with a function operating on the corresponding domains). I.e. define function I on {+, −, ≤, . . . } ⊂ dom(I),

(ii) Equip each object type τC with a reasonable domain, i.e. define function I on τC ⊂ dom(I)

– 03 – 2011-11-02 – Soclsem –

(most reasonable: D (C) as determined by structure

e.g., I(+) ∈ I(Int) × I(Int) → I(Int) (v) Set operations similar: Define function I on {isEmpty, . . . } ⊂ dom(I)

D

of

S ).

(iii) Equip each set type Set(τ0 ) with reasonable domain, i.e. define function I on {Set(τ0 ) | τ0 ∈ TB ∪ TC } ⊂ dom(I)

(vi) Equip each expression with a reasonable interpretation, i.e. define function I on I : Expr × ΣD S × (W → I(T ∪ TB ∪ TC )) → I(Bool)

...except for OCL being a three-valued logic, and the “iterate” expression. 22/36

(i) Domains of Basic Types Recall: • TB = {Bool , Int, String}

We set: • I(Bool ) := {true, false} ∪ {⊥Bool } • I(Int) := Z ∪ {⊥Int } • I(String) := . . . ∪ {⊥String }

– 03 – 2011-11-02 – Soclsem –

We may omit index τ of ⊥τ if it is clear from context.

23/36

(ii) Domains of Object and (iii) Set Types • Now we need a structure • Recall:

D

D

of our signature

assigns an (infinite) domain

S

= (T , C , V, atr ).

D (C) to each class C ∈ C .

• Let τC be an (OCL) object type for a class C ∈ • We set

C.

I(τC ) := D (C) ∪˙ {⊥τC }

– 03 – 2011-11-02 – Soclsem –

• Let τ be a type from TB ∪ TC . • We set

I(Set(τ )) := 2I(τ ) ∪˙ {⊥Set(τ ) } Note: in the OCL standard, only finite subsets of I(τ ). But infinity doesn’t scare us, so we simply allow it. 24/36

Basically business as usual... (i) Equip each OCL (!) basic type with a reasonable domain, i.e. define function I on TB ⊂ dom(I)

(iv) Equip each arithmetical operation with a reasonable interpretation (that is, with a function operating on the corresponding domains). I.e. define function I on {+, −, ≤, . . . } ⊂ dom(I),

(ii) Equip each object type τC with a reasonable domain, i.e. define function I on

e.g., I(+) ∈ I(Int) × I(Int) → I(Int) (v) Set operations similar: Define function I on

τC ⊂ dom(I)

– 03 – 2011-11-02 – Soclsem –

(most reasonable: D (C) as determined by structure

{isEmpty, . . . } ⊂ dom(I)

D

of

S ).

(iii) Equip each set type Set(τ0 ) with reasonable domain, i.e. define function I on

(vi) Equip each expression with a reasonable interpretation, i.e. define function I on

{Set(τ0 ) | τ0 ∈ TB ∪ TC } ⊂ dom(I)

I : Expr × ΣD S × (W → I(T ∪ TB ∪ TC )) → I(Bool)

...except for OCL being a three-valued logic, and the “iterate” expression. 25/36

(iv) Interpretation of Arithmetic Operations • Literals map to fixed values: I(true) := true,

I(false) := false,

I(0) := 0,

I(1) := 1, . . .

I(OclUndefinedτ ) := ⊥τ

– 03 – 2011-11-02 – Soclsem –

• Boolean operations (defined point-wise for x1 , x2 ∈ I(τ )):   , if x1 6= ⊥τ 6= x2 and x1 = x2 true I(=τ )(x1 , x2 ) := false , if x1 6= ⊥τ 6= x2 and x1 = 6 x2   ⊥Bool , otherwise • Integer operations (defined point-wise for x1 , x2 ∈ I(Int)): ( x1 + x2 , if x1 6= ⊥ 6= x2 I(+)(x1 , x2 ) := ⊥ , otherwise

Note: There is a common principle. Namely, the interpretation of an operation ω : τ1 × . . . τn → τ is a function I(ω) : I(τ1 )×· · ·×I(τn ) → I(τ ) on corresponding semantical domain(s). 26/36

(iv) Interpretation of OclIsUndefined • The is-undefined predicate (defined point-wise for x ∈ I(τ )):

true false

, if x = ⊥τ , otherwise

– 03 – 2011-11-02 – Soclsem –

I(oclIsUndefinedτ )(x) :=

(

27/36

(v) Interpretation of Set Operations Basically the same principle as with arithmetic operations... Let τ ∈ TB ∪ TC . • Set comprehension (x1 , . . . , xn ∈ I(τ )):

I({}τn )(x1 , . . . , xn ) := {x1 , . . . , xn } for all n ∈ N0

– 03 – 2011-11-02 – Soclsem –

• Empty-ness check (x ∈ I(Set(τ ))):

  true I(isEmptyτ )(x) := ⊥Bool   false

, if x = ∅ , if x = ⊥Set(τ ) , otherwise

• Counting (x ∈ I(Set(τ ))):

I(sizeτ )(x) := |x| if x 6= ⊥Set(τ ) and ⊥Int otherwise 28/36

(vi) Putting It All Together: Semantics of Expressions • Task: Given OCL expression expr , system state σ ∈ ΣD S , and valuation β, define

I J · K( · , · ) : OCLExpressions(S ) × ΣD S × (W → I(T ∪ TB ∪ TC )) → I(Bool)

such that I Jexpr K(σ, β) ∈ {true, false, ⊥Bool }. OCL Syntax 1/4: Expressions expr ::= w

Where, given

S

OCL = (T , C , V, atr ),

Syntax 2/4: Constants, Arithmetical Operators

• W ⊇ {self } is a set of typed logical variables, w has type τ (w)

For example:

: τ (w)

• τ is any type from T ∪ TB ∪ TC ∪ {Set(τ0 ) | τ0 ∈ TB ∪ TC } expr ::= . . . • TB is a set of basic types, in | true, false the following we use | expr 1 {and, or, implies} expr 2 TB = {Bool , Int, String} τ × · · · × τ → Set(τ ) | not expr 1 • TC = {τC | C ∈ C } is the Set(τ ) → Bool set of object types, | 0, −1, 1, −2, 2, . . . Set(τ ) → Int • Set(τ0 ) denotes the | OclUndefined Set(τC ) set-of-τ0 type for | expr 1 {+, −, . . . } expr 2 τ0 ∈ TB ∪ TC τC → τ (v) | expr 1 {iterate(w1 : τ1 ; w2 : τ2 = expr 2 | expr 3 ) or, with a little renaming,

expr ::= · · · | expr 1 ->iterate(iter : τ1 ; result : τ2 = expr 2 | expr 3 )

context ::= context w1 : τ1 , . . . , wn : τn inv : expr where w ∈ W and τi ∈ TC , 1 ≤ i ≤ n, n ≥ 0.

29/36 where

Preliminaries: Valuations of Logical Variables • Recall: we have typed logical variables (w ∈) W , τ (w) is the type of w. • By β, we denote a valuation of the logical variables, i.e. for each w ∈ W ,

– 03 – 2011-11-02 – Soclsem –

β(w) ∈ I(τ (w)).

30/36

(vi) Putting It All Together... expr ::= w | ω(expr 1 , . . . , expr n ) | allInstancesC | v(expr 1 ) | r1 (expr 1 ) | r2 (expr 1 ) | expr 1 ->iterate(v1 : τ1 ; v2 : τ2 = expr 2 | expr 3 )

• I Jw K(σ, β) := β(w) • I Jω(expr 1 , . . . , expr n )K(σ, β) := I(ω)(I Jexpr 1 K(σ, β), . . . I Jexpr n K(σ, β)) • I JallInstancesC K(σ, β) := dom(σ) ∩ D (C)

– 03 – 2011-11-02 – Soclsem –

Note: in the OCL standard, dom(σ) is assumed to be finite. Again: doesn’t scare us.

31/36

(vi) Putting It All Together... expr ::= w | ω(expr 1 , . . . , expr n ) | allInstancesC | v(expr 1 ) | r1 (expr 1 ) | r2 (expr 1 ) | expr 1 ->iterate(v1 : τ1 ; v2 : τ2 = expr 2 | expr 3 )

Assume expr 1 : τC for some C ∈ C . Set u1 := I Jexpr 1 K(σ, β) ∈ D (τC ). ( σ(u1 )(v) , if u1 ∈ dom(σ) • I Jv(expr 1 )K(σ, β) := ⊥ , otherwise

– 03 – 2011-11-02 – Soclsem –

( u , if u1 ∈ dom(σ) and σ(u1 )(r1 ) = {u} • I Jr1 (expr 1 )K(σ, β) := ⊥ , otherwise ( σ(u1 )(r2 ) • I Jr2 (expr 1 )K(σ, β) := ⊥

, if u1 ∈ dom(σ) , otherwise

(Recall: σ evaluates r2 of type C∗ to a set)

31/36

(vi) Putting It All Together... expr ::= w | ω(expr 1 , . . . , expr n ) | allInstancesC | v(expr 1 ) | r1 (expr 1 ) | r2 (expr 1 ) | expr 1 ->iterate(v1 : τ1 ; v2 : τ2 = expr 2 | expr 3 )

• I Jexpr 1 ->iterate(v1 : τ1 ; v2 : τ2 = expr 2 | expr 3 )K(σ, β) ( , if I Jexpr 1 K(σ, β) = ∅ I Jexpr 2 K(σ, β) := ′′ iterate(˜ v1 , v1 , v2 , expr 3 , σ, β ) , otherwise

where β ′ = β[˜ v1 7→ β(v1 ) \ {x}, v1 7→ x, v2 7→ I Jexpr 3 K(σ, β)], x ∈ β(˜ v1 )

Quiz: Is (our) I a function? Not if the outcome depends on order of choice of the x! 31/36

Example

Location

Meeting

TeamMember name : String age : Integer

2..*

meetings

participants

title : String * numParticipants : Integer start : Date duration: Time

*

move(newStart : Date)

context TeamMember inv: age => 18

– 03 – 2011-11-02 – Soclsem –

context Meeting inv: duration > 0

name : String 1

((C) Prof. Dr. P. Thiemann, http://proglang.informatik.uni-freiburg.de/teaching/swt/2008/

– 03 – 2011-11-02 – Soclsem –

where β ′′ = β[˜ v1 7→ I Jexpr 1 K(σ, β) \ {x}, v1 7→ x, v2 7→ I Jexpr 2 K(σ, β)] and ( β(v2 ) , if β(˜ v1 ) = ∅ iterate(˜ v1 , v1 , v2 , expr 3 , σ, β) = ′ iterate(˜ v1 , v1 , v2 , expr 3 , σ, β ) , otherwise

32/36

– 03 – 2011-11-02 – main –

Outlook on Type Theory

33/36

Well-Typedness... • Note: in the definition of I, we have silently assumed that expressions

are well-typed. • Which is somewhat clear from the typed syntax. For instance,

context C inv : r -> size() + 1 is “obviously” well-typed, while context C inv : r + 1

– 03 – 2011-11-02 – Soutlook –

is not (if r : D∗ ).

• In Lecture 06:

A precise definition of well-typed expressions using basic type theory. Why so late? Consider visibility of attributes in one go.

34/36

– 03 – 2011-11-02 – main –

References

35/36

References [OMG, 2006] OMG (2006). Object Constraint Language, version 2.0. Technical Report formal/06-05-01. [OMG, 2007a] OMG (2007a). Unified modeling language: Infrastructure, version 2.1.2. Technical Report formal/07-11-04. [OMG, 2007b] OMG (2007b). Unified modeling language: Superstructure, version 2.1.2. Technical Report formal/07-11-02.

– 03 – 2011-11-02 – main –

[Warmer and Kleppe, 1999] Warmer, J. and Kleppe, A. (1999). The Object Constraint Language. Addison-Wesley.

36/36