Software specification in CASL - The Common Algebraic Specification - PowerPoint PPT Presentation

Software specification in CASL - The Common Algebraic Specification Language Till Mossakowski, Lutz Schr¨ oder January 2007

Semantics of CASL basic specifications (recalled)

Semantics of CASL basic specifications (recalled) 3 The CASL logic (institution) • Signatures: a signature provides the vocabulary T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 3 The CASL logic (institution) • Signatures: a signature provides the vocabulary • Signature morphisms: for extending and renaming signatures T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 3 The CASL logic (institution) • Signatures: a signature provides the vocabulary • Signature morphisms: for extending and renaming signatures • Models: interpret the vocabulary of a signature with mathematical objects (sets, functions, relations) T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 3 The CASL logic (institution) • Signatures: a signature provides the vocabulary • Signature morphisms: for extending and renaming signatures • Models: interpret the vocabulary of a signature with mathematical objects (sets, functions, relations) • Sentences (formulae): for axiomatizing models denote true or false in a given model T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 3 The CASL logic (institution) • Signatures: a signature provides the vocabulary • Signature morphisms: for extending and renaming signatures • Models: interpret the vocabulary of a signature with mathematical objects (sets, functions, relations) • Sentences (formulae): for axiomatizing models denote true or false in a given model • Terms: parts of sentences, denote data values T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 3 The CASL logic (institution) • Signatures: a signature provides the vocabulary • Signature morphisms: for extending and renaming signatures • Models: interpret the vocabulary of a signature with mathematical objects (sets, functions, relations) • Sentences (formulae): for axiomatizing models denote true or false in a given model • Terms: parts of sentences, denote data values • Satisfaction of sentences in models T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 4 CASL many-sorted signatures • a set S of sorts, T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 4 CASL many-sorted signatures • a set S of sorts, • an S ∗ × S -indexed set ( TF w,s ) w,s ∈ S ∗ × S of total operation symbols, T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 4 CASL many-sorted signatures • a set S of sorts, • an S ∗ × S -indexed set ( TF w,s ) w,s ∈ S ∗ × S of total operation symbols, • an S ∗ × S -indexed set ( PF w,s ) w,s ∈ S ∗ × S of partial operation symbols, such that TF w,s ∩ PF w,s = ∅ , T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 4 CASL many-sorted signatures • a set S of sorts, • an S ∗ × S -indexed set ( TF w,s ) w,s ∈ S ∗ × S of total operation symbols, • an S ∗ × S -indexed set ( PF w,s ) w,s ∈ S ∗ × S of partial operation symbols, such that TF w,s ∩ PF w,s = ∅ , • an S ∗ -indexed set ( P w ) w ∈ S ∗ of predicate symbols Signature morphisms map these components in a compatible way T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 5 Example signatures • Σ Nat = ( { Nat } , { 0 : Nat, succ : Nat − → Nat } , { pre : Nat − → ? Nat } , ∅ ) • ( { Elem } , ∅ , ∅ , { : Elem ∗ Elem } ) < • ( { Elem, List } , { Nil : Elem, Cons : Elem ∗ List − → List } , ∅ , ∅ ) T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 6 CASL many-sorted models For a many-sorted signature Σ = ( S, TF , PF , P ) a many-sorted model M ∈ Mod (Σ) consists of • a non-empty carrier set s M for each sort s ∈ S (let w M denote the Cartesian product s M 1 × · · · × s M n when w = s 1 . . . s n ), T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 6 CASL many-sorted models For a many-sorted signature Σ = ( S, TF , PF , P ) a many-sorted model M ∈ Mod (Σ) consists of • a non-empty carrier set s M for each sort s ∈ S (let w M denote the Cartesian product s M 1 × · · · × s M n when w = s 1 . . . s n ), • a partial function f M from w M to s M for each function symbol f ∈ TF w,s or f ∈ PF w,s , the function being required to be total in the former case, T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 6 CASL many-sorted models For a many-sorted signature Σ = ( S, TF , PF , P ) a many-sorted model M ∈ Mod (Σ) consists of • a non-empty carrier set s M for each sort s ∈ S (let w M denote the Cartesian product s M 1 × · · · × s M n when w = s 1 . . . s n ), • a partial function f M from w M to s M for each function symbol f ∈ TF w,s or f ∈ PF w,s , the function being required to be total in the former case, • a predicate p M ⊆ w M for each predicate symbol p ∈ P w . T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 7 Example Σ Nat -models • Nat M = I N , 0 M =0, suc M ( x ) = x + 1 , � x − 1 , x > 0 pre M ( x ) = undefined, otherwise T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 7 Example Σ Nat -models • Nat M = I N , 0 M =0, suc M ( x ) = x + 1 , � x − 1 , x > 0 pre M ( x ) = undefined, otherwise • Nat N = I N ∪ {∞} , 0 N =0, � ∞ , if x = ∞ suc N ( x ) = x + 1 , otherwise , � x − 1 , if 0 < x � = ∞ pre N ( x ) = undefined , otherwise T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 7 Example Σ Nat -models • Nat M = I N , 0 M =0, suc M ( x ) = x + 1 , � x − 1 , x > 0 pre M ( x ) = undefined, otherwise • Nat N = I N ∪ {∞} , 0 N =0, � ∞ , if x = ∞ suc N ( x ) = x + 1 , otherwise , � x − 1 , if 0 < x � = ∞ pre N ( x ) = undefined , otherwise • Nat T = {∗} , 0 T = ∗ , suc T ( ∗ ) = ∗ , pre T ( ∗ ) = ∗ T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 7 Example Σ Nat -models • Nat M = I N , 0 M =0, suc M ( x ) = x + 1 , � x − 1 , x > 0 pre M ( x ) = undefined, otherwise • Nat N = I N ∪ {∞} , 0 N =0, � ∞ , if x = ∞ suc N ( x ) = x + 1 , otherwise , � x − 1 , if 0 < x � = ∞ pre N ( x ) = undefined , otherwise • Nat T = {∗} , 0 T = ∗ , suc T ( ∗ ) = ∗ , pre T ( ∗ ) = ∗ T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 8 • Nat K = I N , 0 N = K , suc K ( x ) = x , � y, if TM x outputs y on input x pre K ( x ) = undefined , otherwise T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 8 • Nat K = I N , 0 N = K , suc K ( x ) = x , � y, if TM x outputs y on input x pre K ( x ) = undefined , otherwise • Nat F = I N , 0 F ( x ) = 0 , suc F ( f )( x ) = f ( x ) + 1 , N → I pre F ( f ) undefined for each f T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 9 CASL many-sorted terms Given a signature Σ and a variable system ( X s ) s ∈ S , the set of terms is defined inductively as follows: • variables x ∈ X s are terms of sort s T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 9 CASL many-sorted terms Given a signature Σ and a variable system ( X s ) s ∈ S , the set of terms is defined inductively as follows: • variables x ∈ X s are terms of sort s • applications f w,s ( t 1 , . . . , t n ) is a term of sort s , if f ∈ TF w,s ∪ PF w,s and t i is a term of sort s i , w = s 1 . . . s n . T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 10 Semantics of terms Given a Σ -model and a variable valuation ν : X − → M , the semantics ν # of terms is defined as follows: • variables ν # ( x ) = ν ( x ) T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 10 Semantics of terms Given a Σ -model and a variable valuation ν : X − → M , the semantics ν # of terms is defined as follows: • variables ν # ( x ) = ν ( x ) • applications ν # ( f w,s ( t 1 , . . . , t n )) = f M w,s ( ν # ( t 1 ) , . . . , ν # ( t n )) if all components are defined (undefined otherwise) T.Mossakowski, L. Schr¨ oder: Casl ; January 2007

Semantics of CASL basic specifications (recalled) 11 CASL formulae The set of (Σ , X ) -formulae is defined inductively as follows: • strong equations t 1 = t 2 T.Mossakowski, L. Schr¨ oder: Casl ; January 2007