Lecture 04: OCL Semantics

Transcription

1 Software Design, Modelling and Analysis in UML Lecture 04: OCL Semantics Prof. Dr. Andreas Podelski, Dr. Bernd Westphal main Albert-Ludwigs-Universität Freiburg, Germany Contents & Goals Last Lecture: OCL Syntax This Lecture: Educational Objectives: Capabilities for following tasks/questions. 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? Sprelim Content: OCL Semantics maybe: OCL consistency and satisfiability 2/26

2 OCL Semantics [OMG, 2006] main 3/26 The Task Soclsyn OCL Syntax 1 4: Expressions expr ::= w : τ(w) expr 1 = τexpr 2 : τ τ Bool oclisundefined τ(expr 1 ) : τ Bool expr 1,...,expr n } : τ τ Set(τ) isempty(expr 1 ) : Set(τ) Bool size(expr 1 ) : Set(τ) Int allinstances C : Set(τ C) v(expr 1 ) : τ C τ(v) r 1(expr 1 ) : τ C τ D r 2(expr 1 ) : τ C Set(τ D) Where, given Ë = (Ì,, V,atr), W self } is a set of typed logical variables, w has type τ(w) τ is any type from Ì TB T Set(τ0) τ0 TB T } TB is a set of basic types, in the following we use TB = Bool,Int,String} T = τc C } is the set of object types, Set(τ0) denotes the set-of-τ0 type for τ0 TB T (sufficient because of flattening (cf. standard)) v : τ(v) atr(c), τ(v) Ì, r1 : D0,1 atr(c), r2 : D atr(c), C, D. 7/30 Given an OCL expression expr, a system state σ Σ D S, and a valuation of logical variables β, define I (, ) : OCLExpressions(S) Σ D S (W I(T T B T C )) I(Bool) such that I expr (σ,β) true,false, Bool }. 4/26

3 Basically business as usual... (i) Equip each OCL (!) basic type with a reasonable domain, i.e. define function I with dom(i) = T B 5/26 Basically business as usual... (i) Equip each OCL (!) basic type with a reasonable domain, i.e. define function I with dom(i) = T B (ii) Equip each object type τ C with a reasonable domain, i.e. define function I with dom(i) = τ C (most reasonable: D(C) determined by structure D of S). (iii) Equip each set type Set(τ 0 ) with reasonable domain, i.e. define function I with dom(i) = Set(τ 0 ) τ 0 T B T C } (iv) Equip each arithmetical operation with a reasonable interpretation (that is, with a function operating on the corresponding domains). I with dom(i) = +,,,...}, e.g., I(+) I(Int) I(Int) I(Int) 5/26

4 Basically business as usual... (i) Equip each OCL (!) basic type with a reasonable domain, i.e. define function I with dom(i) = T B (ii) Equip each object type τ C with a reasonable domain, i.e. define function I with dom(i) = τ C (most reasonable: D(C) determined by structure D of S). (iii) Equip each set type Set(τ 0 ) with reasonable domain, i.e. define function I with dom(i) = Set(τ 0 ) τ 0 T B T C } (iv) Equip each arithmetical operation with a reasonable interpretation (that is, with a function operating on the corresponding domains). I with dom(i) = +,,,...}, e.g., I(+) I(Int) I(Int) I(Int) (v) Set operations similar: I with dom(i) = isempty,...} (vi) Equip each expression with a reasonable interpretation, i.e. define function I : Expr Σ D S (W I(T T B T C )) I(Bool)...except for OCL being a three-valued logic, and the iterate expression. 5/26 (i) Domains of Basic Types Recall: T B = Bool,Int,String} We set: I(Bool) := true,false} Bool } I(Int) := Z Int } I(String) :=... String } We may omit index τ of τ if it is clear from context. 6/26

5 (ii) Domains of Object and (iii) Set Types Now we need a structure D of our signature S = (T, C,V,atr). Recall: D assigns an (infinite) domain D(C) to each class C C. Let τ C be an (OCL) object type for a class C C. We set I(τ C ) := D(C) τc } Let τ be a type from T B T C. We set I(Set(τ)) := 2 I(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. 7/26 (iv) Interpretation of Arithmetic Operations Literals map to fixed values: I(true) := true, I(false) := false, I(0) := 0, I(1) := 1,... I(OclUndefined τ ) := τ 8/26

6 (iv) Interpretation of Arithmetic Operations Literals map to fixed values: I(true) := true, I(false) := false, I(0) := 0, I(1) := 1,... I(OclUndefined τ ) := τ Boolean operations (defined point-wise for x 1,x 2 I(τ)): true, if x 1 τ x 2 and x 1 = x 2 I(= τ )(x 1,x 2 ) := false, if x 1 τ x 2 and x 1 x 2, otherwise Bool 8/26 (iv) Interpretation of Arithmetic Operations Literals map to fixed values: I(true) := true, I(false) := false, I(0) := 0, I(1) := 1,... I(OclUndefined τ ) := τ Boolean operations (defined point-wise for x 1,x 2 I(τ)): true, if x 1 τ x 2 and x 1 = x 2 I(= τ )(x 1,x 2 ) := false, if x 1 τ x 2 and x 1 x 2, otherwise Bool Integer operations (defined point-wise for x 1,x 2 I(Int)): x 1 +x 2, if x 1 x 2 I(+)(x 1,x 2 ) :=, otherwise 8/26

7 (iv) Interpretation of Arithmetic Operations Literals map to fixed values: I(true) := true, I(false) := false, I(0) := 0, I(1) := 1,... I(OclUndefined τ ) := τ Boolean operations (defined point-wise for x 1,x 2 I(τ)): true, if x 1 τ x 2 and x 1 = x 2 I(= τ )(x 1,x 2 ) := false, if x 1 τ x 2 and x 1 x 2, otherwise Bool Integer operations (defined point-wise for x 1,x 2 I(Int)): x 1 +x 2, if x 1 x 2 I(+)(x 1,x 2 ) :=, otherwise Note: There is a common principle. Namely, the interpretation of an operation ω : τ 1...τ n τ is a function I(ω) : I(τ 1 ) I(τ n ) I(τ) on corresponding semantical domain(s). 8/26

8 (iv) Interpretation of OclIsUndefined The is-undefined predicate (defined point-wise for x I(τ)): true, if x = τ I(oclIsUndefined τ )(x) := false, otherwise 9/26 (v) Interpretation of Set Operations Basically the same principle as with arithmetic operations... Let τ T B T C. Set comprehension (x 1,...,x n I(τ)): I(} τ n)(x 1,...,x n ) := x 1,...,x n } for all n N 0 Empty-ness check (x I(Set(τ))): true I(isEmpty τ )(x) := false Bool, if x =, if x = Set(τ), otherwise Counting (x I(Set(τ))): I(size τ )(x) := x if x Set(τ) and Int otherwise 10/26

9 (vi) Putting It All Together Soclsyn OCL Syntax 1 4: Expressions isempty(expr 1 ) : Set(τ) Bool size(expr 1 ) : Set(τ) Int allinstances C : Set(τ C) v(expr 1 ) : τ C τ(v) r 1(expr 1 ) : τ C τ D r 2(expr 1 ) : τ C Set(τ D) Where, given Ë = (Ì,, V,atr), W self } is a set of typed expr ::= w : τ(w) logical variables, w has type τ(w) For example: τ is any type from Ì TB T expr 1 = τexpr 2 : τ τ Bool Set(τ0) τ0 TB T } expr ::=... oclisundefined τ(expr 1 ) : τ Bool TB is a set of basic types, in true, false : Bool expr 1,...,expr n } the following we use : τ τ Set(τ) TB = Bool,Int,String} expr 1 and, or, implies} expr 2 : Bool Bool Bool T = τc C } is the set of object types, Set(τ0) denotes the set-of-τ0 type for τ0 TB T (sufficient because of flattening (cf. standard)) v : τ(v) atr(c), τ(v) Ì, Generalised notation: r1 : D0,1 atr(c), r2 : D atr(c), C, D Soclsyn OCL Syntax 2 4: Constants, Arithmetical Operators 7/30 not expr 1 : Bool Bool 0, 1, 1, 2, 2,... : Int OclUndefined : τ expr 1 +,,... } expr 2 : Int Int Int expr 1 <,,... } expr 2 : Int Int Bool expr ::= ω(expr 1,...,expr n ) with ω +,,... } : τ 1 τ n τ OCL Syntax 3 4: Iterate expr ::= expr 1 ->iterate(w 1 : τ 1 ; w 2 : τ 2 = expr 2 expr 3 ) or, with a little renaming, expr ::= expr 1 ->iterate(iter : τ 1; result : τ 2 = expr 2 expr 3 ) where expr 1 is of a collection type (here: a set Set(τ 0) for some τ 0), OCL Syntax 4 4: Context context ::= context w 1 : τ 1,...,w n : τ n inv : expr where w W and τ i T, 1 i n, n 0. 11/26 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, β(w) I(τ(w)). 12/26

10 (vi) Putting It All Together... expr ::= w ω(expr 1,...,expr n ) allinstances C v(expr 1 ) r 1 (expr 1 ) r 2 (expr 1 ) expr 1 ->iterate(v 1 : τ 1 ; v 2 : τ 2 = expr 2 expr 3 ) I w (σ,β) := β(w) I ω(expr 1,...,expr n ) (σ,β) := I(ω)(I expr 1 (σ,β),...i expr n (σ,β)) I allinstances C (σ,β) := dom(σ) D(C) Note: in the OCL standard, dom(σ) is assumed to be finite. Again: doesn t scare us. 13/26 (vi) Putting It All Together... expr ::= w ω(expr 1,...,expr n ) allinstances C v(expr 1 ) r 1 (expr 1 ) r 2 (expr 1 ) expr 1 ->iterate(v 1 : τ 1 ; v 2 : τ 2 = expr 2 expr 3 ) Assume expr 1 : τ C for some C C. Set u 1 := I expr 1 (σ,β) D(τ C ). σ(u 1 )(v), if u 1 dom(σ) I v(expr 1 ) (σ,β) :=, otherwise u, if u 1 dom(σ) and σ(u 1 )(r 1 ) = u} I r 1 (expr 1 ) (σ,β) :=, otherwise σ(u 1 )(r 2 ), if u 1 dom(σ) I r 2 (expr 1 ) (σ,β) :=, otherwise (Recall: σ evaluates r 2 of type C to a set) 13/26

11 (vi) Putting It All Together... expr ::= w ω(expr 1,...,expr n ) allinstances C v(expr 1 ) r 1 (expr 1 ) r 2 (expr 1 ) expr 1 ->iterate(v 1 : τ 1 ; v 2 : τ 2 = expr 2 expr 3 ) I expr 1 ->iterate(v 1 : τ 1 ; v 2 : τ 2 = expr 2 expr 3 ) (σ,β) I expr := 2 (σ,β), if I expr 1 (σ,β) = iterate(hlp,v 1,v 2,expr 3,σ,β ), otherwise where β = β[hlp I expr 1 (σ,β),v 2 I expr 2 (σ,β)] and iterate(hlp,v 1,v 2,expr 3,σ,β ) I expr := 3 (σ,β [v 1 x]), if β (hlp) = x} I expr 3 (σ,β ), if β (hlp) = X x} and X where β = β [v 1 x,v 2 iterate(hlp,v 1,v 2,expr 3,σ,β [hlp X])] 13/26

12 (vi) Putting It All Together... expr ::= w ω(expr 1,...,expr n ) allinstances C v(expr 1 ) r 1 (expr 1 ) r 2 (expr 1 ) expr 1 ->iterate(v 1 : τ 1 ; v 2 : τ 2 = expr 2 expr 3 ) I expr 1 ->iterate(v 1 : τ 1 ; v 2 : τ 2 = expr 2 expr 3 ) (σ,β) I expr := 2 (σ,β), if I expr 1 (σ,β) = iterate(hlp,v 1,v 2,expr 3,σ,β ), otherwise where β = β[hlp I expr 1 (σ,β),v 2 I expr 2 (σ,β)] and iterate(hlp,v 1,v 2,expr 3,σ,β ) I expr := 3 (σ,β [v 1 x]), if β (hlp) = x} I expr 3 (σ,β ), if β (hlp) = X x} and X where β = β [v 1 x,v 2 iterate(hlp,v 1,v 2,expr 3,σ,β [hlp X])] Quiz: Is (our) I a function? 13/26 References main 25/26

13 main [Cabot and Clarisó, 2008] Cabot, J. and Clarisó, R. (2008). UML-OCL verification in practice. In Chaudron, M. R. V., editor, MoDELS Workshops, volume 5421 of Lecture Notes in Computer Science. Springer. [Cengarle and Knapp, 2001] Cengarle, M. V. and Knapp, A. (2001). On the expressive power of pure OCL. Technical Report 0101, Institut für Informatik, Ludwig-Maximilians-Universität München. [Cengarle and Knapp, 2002] Cengarle, M. V. and Knapp, A. (2002). Towards OCL/RT. In Eriksson, L.-H. and Lindsay, P. A., editors, FME, volume 2391 of Lecture Notes in Computer Science, pages Springer-Verlag. [Flake and Müller, 2003] Flake, S. and Müller, W. (2003). Formal semantics of static and temporal state-oriented OCL constraints. Software and Systems Modeling, 2(3): [Jackson, 2002] Jackson, D. (2002). Alloy: A lightweight object modelling notation. ACM Transactions on Software Engineering and Methodology, 11(2): [OMG, 2006] OMG (2006). Object Constraint Language, version 2.0. Technical Report formal/ [OMG, 2007a] OMG (2007a). Unified modeling language: Infrastructure, version Technical Report formal/ [OMG, 2007b] OMG (2007b). Unified modeling language: Superstructure, version Technical Report formal/ /26