# On the Theoretical Foundation of Meta-Modelling in Graphically Extended BNF Hong Zhu Department of Computing and Electronics Oxford Brookes University.

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On the Theoretical Foundation of Meta-Modelling in Graphically Extended BNF Hong Zhu Department of Computing and Electronics Oxford Brookes University Oxford OX33 1HX, UK Email: hzhu@brookes.ac.uk

Seminar on GEBNF2 7 Sept 2010 Acknowledgement H. Zhu, The Institution of Meta-Modelling in Graphically Extended BNF, in Preparation. H. Zhu, On The Theoretical Foundation of Meta-Modelling in Graphically Extended BNF and First Order Logic, Prof. of TASE 2010, IEEE CS Press, Aug. 2010. H. Zhu and L. Shan, ``Well-formedness, consistency and completeness of graphic models,'' Proc. of UKSIM'06, Apr. 2006, pp. 47-- 53.

Seminar on GEBNF3 7 Sept 2010 Outline Introduction Motivation and Related works Graphic extension of BNF Predicate logic induced from GEBNF syntax Semantics of GEBNF Axiomatisation of syntactic constraints Algebraic semantics of GEBNF Institution theory of GEBNF Meta-Models Future works

Seminar on GEBNF4 7 Sept 2010 Introduction Modelling To represent a system or a set of systems under study at a high level of abstraction. A model is a set of statements about the system or a set of systems under study Meta-modelling? To model models To define a set of models that have certain features

Seminar on GEBNF5 7 Sept 2010 Meta-Modelling in MDSD defines modelling languages syntax: usually at the abstract syntax level semantics: usually in the form of a set of basic concepts underlying the models and their interrelationships example: the meta-model for UML imposes restrictions on an existing modelling language a subset of the syntactically valid models example: design patterns-a meta-model whose instances conform to the design pattern extends an existing meta-model introducing new concepts and their relationship to the existing ones example: platform specific models - by introducing platform specific model elements Aspect-oriented modelling - by extending UML meta-model with basic concepts of aspect-orientation, such as crosscut points, etc.

Seminar on GEBNF6 7 Sept 2010 Related Work General purpose MM Languages MOF and UML class diagram + OCL Special Purpose MM Languages (for example: design patterns) LePUS (Gasparis et al. 2008; Eden 2001, 2002) RBML (France et al. 2004) DPML (Maplesden at el., 2001, 2002) PDL (Albin-Amiot, et al. 2001) Problems of graphic meta-modelling Expressiveness? Rigorous in semantics? Readability?

Seminar on GEBNF7 7 Sept 2010 Overview: Formal Meta-Modelling Meta-notation: GEBNF for the definition of abstract syntax of modelling languages Formal Theory of Meta-Modelling Predicate logic languages induced from GEBNF syntax definitions Formal semantics of GEBNF syntax definitions Institution of GEBNF meta-models: Meta-models as formal specification languages Applications Specification of a non-trivial subset of UML Specification of consistency and completeness constraints of UML Specification of design patterns all 23 in GoF Both structural and behavioural features Graphic Extension of BNF

Seminar on GEBNF8 7 Sept 2010 The GEBNF Notation In GEBNF, the abstract syntax of a modelling language is defined as a tuple, where N is a finite set of non-terminal symbols, T is a finite set of terminal symbols. Each terminal symbol represents a set of atomic elements that may occur in a model. R N is the root symbol and S is a finite set of syntax rules in one of the forms of Y ::= X 1 | X 2 | … | X n Y ::= f 1 : X 1, f 2 : X 2, …, f n : X n where Y N, f 1, f 2, …, f n are called field names, X 1, X 2, …, X n are the fields.

Seminar on GEBNF9 7 Sept 2010 Field Expression Each field can be an expression: C is an expression, if C is a literal constant of a terminal symbol, such as a string or number. Y is an expression, if Y N T. Y*, Y + and [Y] are expressions, if Y N T. Exp(Y)@Z.f is an expression, where Y, Z N f is a field name in the definition of Z Y is the type of f field in Z's definition Exp(Y) is an expression of Y only using the above rules Referential occurrence

Seminar on GEBNF10 7 Sept 2010 Meanings of the GEBNF Notation

Seminar on GEBNF11 7 Sept 2010 Example 1: Directed Graphs where Graph is the root symbol Graph, Node and Edge are non-terminal symbols String and Real are terminal symbols

Seminar on GEBNF12 7 Sept 2010 Example 2: A. UML Class Diagram (subset) ClassDiagram ::= classes: Class+, assocs: Rel*, inherits: Rel*, compag: Rel* Class ::= name: String, attrs: Property*, opers: Operation* Operation ::= name: String, params: Parameter*, isAbstract: [Bool], isQuery: [Bool], isLeaf: [Bool], isNew: [Bool], isStatic: [Bool] Parameter ::= name: [String], type: [Type], direction: [ParaDirKind], mult: [Multiplicity] ParaDirKind ::=in | inout | out | return Multiplicity ::=lower: [Natural], upper: [Natural | *] Property ::= name: String, type: Type, isStatic: [ Bool], mult: [ Multiplicity] Rel ::= name: [String], source: End, end: End End ::= node: Class, name: [String], mult: [Multiplicity]

Seminar on GEBNF13 7 Sept 2010 Example 2: B. UML Sequence Diagram (subset) SequenceDiagram ::= lifelines: Lifeline*, msgs: Message*, ordering: Order* Order::= from: Message, to: Message Lifeline ::= className: String, objectName: [String], isStatic: Bool, activations: Activation* Activation ::= start, finish: Event, others: Event* Message ::= send, receive: Event, sig: Operation Event ::= actor: Activation

Seminar on GEBNF14 7 Sept 2010 Well-Formed Syntax Definitions A syntax definition in GEBNF is well-formed if it satisfies the following two conditions. Completeness For each non-terminal symbol X N, there is one and only one syntax rule s S that defines X. Reachability For each non-terminal symbol X N, X is reachable from the root R.

Seminar on GEBNF15 7 Sept 2010 Types Definition Let G= be a GEBNF syntax definition. The set of types of G, denoted by Type(G), is defined inductively as follows. For all s T N, s is a type, which is called a basic type. P( ) is a type, called the power type of, if is a type. 1 +… + n is a type, called the disjoint union of 1, …, n for n>1, if 1 … n are types. We also write to denote 1 +…+ n. 1 2 is a type, called a function type from 1 to 2, if 1 and 2 are types.

Seminar on GEBNF16 7 Sept 2010 Induced Functions Given a well-defined GEBNF syntax G =, we write Fun ( G ) to denote the set of function symbols derived from the syntax rules as follows A syntax rule `` A ::= B 1 | B 2 | …| B n '' introduces a set of functions IsB 1, IsB 2, …, IsB n of the type A Bool. A syntax rule `` A ::= f 1 : B 1,…, f n : B n '' introduces a set of function symbols f i of type A T ( B i ), where T ( B ) is defined as follows. T ( C ) =C, if C T N ; T ([ C ]) =T ( C ); T ( C @ Z. f ) =T ( C ); T ( C *) = P ( T ( C )); T ( C +) =P ( T ( C )); T ( C 1 | … | C n ) =.

Seminar on GEBNF17 7 Sept 2010 Example 3: Induced Functions

Seminar on GEBNF18 7 Sept 2010 Induced Predicate Logic Language (FL) From Fun ( G ), a FL can be defined as usual [Chiswell 2007]. variables of type Type G relations and operators on sets, relations and operators on basic data types denoted by terminal symbols, equality logic connectives or, and, not, implication and equivalence, quantifiers for all and exists

Seminar on GEBNF19 7 Sept 2010 Definition: Inducted Predicate Logic Let be a set of variables, where x V are variables of type. Each literal constant c of type s T is an expression of type s. Each element v V, i.e. variable of type, is an expression of type Type ( G ). e. f is an expression of type ', if f is a function symbol of type ', e is an expression of type { e ( x ) | Pred ( x ) } is an expression of type P( e ), if x is a variable of type x, e ( x ) is an expression of type e and Pred(x ) is a predicate on type x. e 1 e 2, e 1 e _ 2, and e 1 - e 2 are expressions of type P ( ), if e 1 and e 2 are expressions of type P ( ). e E is a predicate on type, if e is an expression of type and E is an expression of type P ( ). e 1 = e 2 and e 1 e 2 are predicates on type, if e 1 and e 2 are expressions of type R ( e _ 1, …, e _ n ) is a predicate on type, if e 1, …, e n are expressions of type, and R is any n -ary relation symbol on type. e 1 e 2 and e 1 e 2 are predicates on type P ( ), if e 1 and e 2 are expressions of type P ( ). p q, p q, p q, p q and p are predicates, if p and q are predicates. x D.( p ) and x D. ( p ) are predicates, if D is an expression of type P ( ), x is a variable of type, and p is a predicate. It is first order, if we restrict type to be a terminal or non-terminal symbol s

Seminar on GEBNF20 7 Sept 2010 Example 4: (A) Functions and Predicates Function: the set of nodes in a graph g that have no weight associated with them Predicate: node x reaches node y in a graph g :

Seminar on GEBNF21 7 Sept 2010 Example 4: (B) Constraints on UML In a sequence diagram, every message must start an activation Every message to an activation must be for an operation of a concrete class: If a message is for a static operation, then the lifeline must be a class lifeline; but if a message is for a non- static operation, the lifeline must be an object lifeline: Every class in the class diagram must appear in the sequence diagram:

Seminar on GEBNF22 7 Sept 2010 Meta-Modelling in GEBNF + FL The Approach Defining the abstract syntax of a modelling language in GEBNF Defining a predicate p such that the required subset of models are those that satisfy the predicate Example strongly connected graphs

Seminar on GEBNF23 7 Sept 2010 Axiomatization of Syntax Constraints (1) Optional Elements Consider two syntax definitions of a non-terminal symbol A A ::= …, f: [B], …. (1) A ::=..., g: B, …. (2) Similarity: Both functions f and g have the type A B, Difference: For (1), an occurrence of an element of type B in an element of type A is optional, i.e. f is a partial function For (2), an occurrence of an element of type B is not optional. i.e. g is a total function. Formalisation: for each non-optional function symbol g, we require it satisfying the following condition. x A. ( x. g ) where means undefined.

Seminar on GEBNF24 7 Sept 2010 Axiomatization of Syntax Constraints (2) Non-Empty Repetitions Consider two syntax definitions of a non-terminal symbol A A ::= …, f: B*, …. (1) A ::=..., g: B +, …. (2) Similarity: Both functions f and g have the type A P ( B), Difference: For (1), the set of element of type B in an element of type A is a set (can be the empty set), For (2), the set of element of type B in an element of type A is a non- empty set. Formalisation: for each non-empty repetition function symbol g, we require it satisfying the following condition. x A. ( x. g )

Seminar on GEBNF25 7 Sept 2010 Axiomatization of Syntax Constraints (3) Creative Occurrences Consider two syntax definitions of non-terminal symbols Y and Z that both contains non-terminal symbol X Y ::= …, f: E(X), …. (1) Z ::=..., g: E(X), …. (2) Both functions f and g have elements of type X as components. What are the relationships between these elements of type X ?

Seminar on GEBNF26 7 Sept 2010 Formalisation Situation 1: Two simple type creative occurrences Y ::= …, f: E(X), …. (1) Z ::=..., g: E(X), …. (2) where E ( X ) and E ( X ) is one of the expressions X, [ X ], and ( X 1 | …| X | …| X n ) Axiom:

Seminar on GEBNF27 7 Sept 2010 Situation 2: Two set type creative occurrences Y ::= …, f: E(X), …. (1) Z ::=..., g: E(X), …. (2) where E ( X ) and E ( X ) is one of the expressions X*, and X +, Axiom: or simply:

Seminar on GEBNF28 7 Sept 2010 Situation 3: Two creative occurrences of different types Y ::= …, f: E(X), …. (1) Z ::=..., g: E(X), …. (2) where E ( X ) is one of the expressions X, [ X ], and ( X 1 | …| X | …| X n ) E ( X ) is one of the expressions X*, and X + Axiom:

Seminar on GEBNF29 7 Sept 2010 Axiomatization of Syntax Constraints (4) Referential Occurrences Consider two syntax definitions of a non-terminal symbol A A ::= …, f: B, …. (1) A ::=..., f: B@C.g, …. (2) Similarity: Both functions f and g have the type A B, Difference: For (1), the element of type B in an element of type A is a new element in the model, For (2), the element of type B in an element of type A is an existing element in the model, already in C.g.

Seminar on GEBNF30 7 Sept 2010 Formalisation Situation 1: Single reference to single element Y ::=..., g: E(X), …. (1) Z ::= …, f: X@Y.g, …. (2) where E ( X ) is one of the expressions X, [ X ], and ( X 1 | …| X | …| X n ) Axiom:

Seminar on GEBNF31 7 Sept 2010 Situation 2: Single reference to set of elements Y ::=..., g: E(X), …. (1) Z ::= …, f: X@Y.g, …. (2) where E ( X ) is one of the expressions X*, and X + Axiom:

Seminar on GEBNF32 7 Sept 2010 Situation 3: Set reference to set of elements Y ::=..., g: E(X), …. (1) Z ::= …, f: E(X)@Y.g, …. (2) where E ( X ) and E ( X ) are one of the expressions X*, and X + Axiom:

Seminar on GEBNF33 7 Sept 2010 Example 5 (A) There are two referential occurrences of non-terminal symbols in the GEBNF syntax definition of directed graphs. Thus, the functions to and from must satisfy the following conditions.

Seminar on GEBNF34 7 Sept 2010 Example 5 (B) Definition: Let G be any well-formed syntax definition in GEBNF. We write Axiom ( G ) to denote the set of constraints derived from G according to the above rules. Example: Axiom ( DG ) contains the following axioms:

Seminar on GEBNF35 7 Sept 2010 Algebraic Semantics of GEBNF Let G= be a GEBNF syntax definition The signature induced from G : G = (N T, F G ), where F G =Fun(G) is the set of function symbols induced from G. G -algebra A : {A x | x N T} of sets {f | F G }, a set of functions, where if is of type X Y, then f is a function from set A X to the set [[ Y ]], where

Seminar on GEBNF36 7 Sept 2010 Algebra without Junk A G -algebra A contains no junk, if | A R |=1 for all s N and all e As, we can define a function f : R P ( s ) in PL such that for some m A R we have e f ( m ). There is one and only one root element Every element in a model must be accessible from the root

Seminar on GEBNF37 7 Sept 2010 Example 6: Model as Algebra

Seminar on GEBNF38 7 Sept 2010 Satisfaction of Constraints Assignment in an G -algebra A a mapping from the set V of variables to the elements of the algebra. Evaluation of an expression e under an assignment is written [[ e ]]. (See paper for definition ) A predicate p is true in A under assignment written A |= p, if [[ p ]] = true. A predicate p is true in A and written A |= p, if for all assignments in A, A |= p.

Seminar on GEBNF39 7 Sept 2010 Evaluation of Expressions and Predicates

Seminar on GEBNF40 7 Sept 2010

Seminar on GEBNF41 7 Sept 2010 Valid Models A G –algebra A without junk is a syntactically valid model with respect to G, if for all p Axiom ( G ), we have that A |= p. Let MM=(G, p) be a meta-model G is a GEBNF syntax definition and p is a predicate in the FOL induced from G. The semantics of the meta-model MM is a subset of syntactically valid models of G that satisfy the predicate p. This is the standard treatment of predicate logic in the model theory of mathematical logics. [Chiswell 2007]

Seminar on GEBNF42 7 Sept 2010 Institution of Meta-Models We prove the following statements, thus the institution structure of meta-models GEBNF syntax definitions + Syntax morphisms form a category Valid models (i.e. algebras) of GEBNF syntax definitions + homomorphisms between algebras form a category Translation of FL sentences through syntax morphisms is a functor Translation of valid models through syntax morphisms is a functor Translations of sentences and models through a syntax morphism are truth invariant J. A. Goguen and R. M. Burstall, Institutions: Abstract model theory for specification and programming, J. ACM, vol.39, no.1, pp. 95--146, 1992.

Seminar on GEBNF43 7 Sept 2010 Review: Category A category C consists of a class C obj of objects and a class C m of morphisms or arrows between objects together with the following three operations: dom: C m C obj ; codom : C m C obj ; id : C obj C m, where for all morphisms f, dom(f ) =A is called the domain of the morphism f ; codom(f)=B the codomain, and the morphism f is from object A=dom ( f ) to object B=codom ( f ), written f : A B. For each object A, id ( A ) is the identity morphism that its domain and codomain are A. id(A ) is also written as id A.

Seminar on GEBNF44 7 Sept 2010 There is a partial operation o on C m, called composition of morphisms. The composition of morphisms f and g, written f o g, is defined, if dom ( f ) = codom ( g ). The result of composition f o g is a morphism from dom ( g ) to codom ( f ). The composition operation has the following properties. For all morphisms f, g, and h, ( f o g ) o h = f o ( g o h ) id A o f = f, if codom(f) = A g o id A = g, if dom ( g ) = A.

Seminar on GEBNF45 7 Sept 2010 Syntax Morphisms A syntax morphism from G to H, written : G H, is a pair (m, f) of mappings m : N G N H and f : Fun( G ) Fun( H ) that satisfy the following two conditions: Root preservation: m(R G ) = R H ; Type preservation: for all op Fun(G), (op : A B) (f(op) : m(A) m(B)), where we naturally extend the mapping m to type expressions.

Seminar on GEBNF46 7 Sept 2010 Example 7: Syntax Morphism GEBNF Syntax Definition AR: Map::= cities: City+, routes: Route* City::= name, country: :String, population: Real Route::= depart, arrive: City, distance: Real, flights: TimeDay* Syntax Morphism from DG to AR: m = (Graph Map, Node City, Edge Route), f = ( nodes cities, edges routes, name name, weight population, to arrive, from depart, weight distance)

Seminar on GEBNF47 7 Sept 2010 The Category of GEBNF Definition: (Composition of syntax morphisms) Assume that =(m, f) : G H and =(n, g) : H J be syntax morphisms. The composition of to, written o, is defined as (m o n, f o g). Definition (Identity Syntax Morphisms) Id G is defined as the pair of mappings ( id N, id Fun ( G ) ) Lemma: The above definitions are sound Theorem: Let Obj be the set of well-formed GEBNF syntax definitions, Let Mor be the set of syntax morphisms on Obj. ( Obj, Mor ) is a category. It is denoted by GEB in the sequel.

Seminar on GEBNF48 7 Sept 2010 Review: Functor Let C, D be two categories. A functor F from C to D consists of two mappings: an object mapping F obj : C obj D obj, and a morphism mapping F m : C m D m They have the following properties. for all morphisms f : A B of category C, F m (f) : F obj (A) F obj (B). for all morphisms f and g in C, F m (f o g) = F m (f ) o F m (g ). for all objects A in category C, F m (id A ) = id F obj (A).

Seminar on GEBNF49 7 Sept 2010 Translation of Sentences Given a syntax morphism =(m, f) from GEBNF defined modelling language G to H, we can define a translation between the PLs induced from them. Definition: Sen obj ( G ) denotes the set of predicates on the root of G. Define mapping Sen m ( ) from Sen obj ( G ) to Sen obj (H ): For each predicate p in Sen obj ( G ), Each variable v of type in predicate p is replaced by a variable v' of type m ( ). Each op Fun(G ) in predicate p is replaced by the function symbol f(op ).

Seminar on GEBNF50 7 Sept 2010 Example 8: Translation of Sentence Consider the syntax morphism defined in Example 7. The reaches predicate defined in Example 4 can be translated into the following sentence in PL AR.

Seminar on GEBNF51 7 Sept 2010 Theorem: The pair ( Sen obj, Sen m ) is a functor from category GEB to the category SET of sentences in the corresponding induced FLs.

Seminar on GEBNF52 7 Sept 2010 Constraint Preserving Syntax Morphisms Definition A syntax morphism from G to H is constraint preserving, if for all constraint c Axiom ( G ) we have that Axiom ( H ) Sen ( c ). Informally, constraint preserving means that the syntax constraints that GEBNF syntax definition G imposes on models are all satisfied by the modelling language defined by H when the notations in G is translated into notations in H.

Seminar on GEBNF53 7 Sept 2010 Example 9: Constraint Preserving Syntax Morphism Consider the syntax morphism given in Example 7. It is constraint preserving because for each constraint c in Axiom(DG ), we can prove that Axiom ( AR ) c ', where c' is the translation of c into PL AR according to the syntax morphism. For instance, the following constraint c on directed graph c = g Graph.( g.nodes ) is translated into c = g Map.( g.cities ) according to the syntax morphism. It is easy to see that Axiom ( AR ) c ' because c' Axiom ( AR ).

Seminar on GEBNF54 7 Sept 2010 Constraint Preservation Sub-Category Theorem The set of well-formed GEBNF syntax definitions as objects and the set of constraint preserving syntax morphisms between them as morphisms form a category This category is a full sub-category of GEB, because the following statements are true. For all well-formed GEBNF syntax definition G, Id G is constraint preserving. If and are constraint preserving syntax morphisms, so is o provided that they are composable. We write GEBNF to denote this category.

Seminar on GEBNF55 7 Sept 2010 Translation of Models Lemma: The models in any given modeling language defined by a GEBNF syntax definition G is a category, where the morphisms are the homomorphisms between the models (i.e. the algebras). Definition: For each syntax morphism =( m, f ) from G to H, the mapping U from category MOD H to category MOD G is defined as follows. Let B | MOD H |. We define an G -algebra A : For each s N G, A s = B m ( s ); For each function symbol op Fun(G ), the function op A is the function f(op ) in B. Denoted by MOD G

Seminar on GEBNF56 7 Sept 2010 Example 10: Translation of Models Consider the model of AR shown in (a). It can be translated into the model of directed graph shown in (b) when the syntax morphism defined in Example 7 is applied.

Seminar on GEBNF57 7 Sept 2010 Theorem: For each syntax morphism =(m, f) from G to H, the mapping U from objects of category MOD H to the objects of category MOD G and its naturally induced mapping on homomorphisms is a functor from MOD H to MOD G.

Seminar on GEBNF58 7 Sept 2010 Theorem: Let Obj ={ MOD G | G | GEB | } and Mor ={ U | || GEB }||}. ( Obj, Mor ) is a category. It is denoted by CAT. GEBNF syntax Syntax morphism

Seminar on GEBNF59 7 Sept 2010 Model Translation Functor Definition: Define MOD obj : | GEB | | CAT op | as MOD obj (G)= Mod(G); Define MOD m : || GEB || || CAT op }|| as MOD m (u) = U op where for an arrow : a b, op is the inverse arrow of. Theorem: MOD=(MOD obj, MOD m ) is a functor from GEB to CAT op.

Seminar on GEBNF60 7 Sept 2010 Review: Institution

Seminar on GEBNF61 7 Sept 2010 GEBNF Institution Theorem: The tuple ( GEB, MOD, Sen, |=) is an institution. In particular, we have that: The truth of a sentence is invariant under the translation of sentence and the models.

Seminar on GEBNF62 7 Sept 2010 Example 11: Truth Invariance under Translation Both statements are true Both statements are false

Seminar on GEBNF63 7 Sept 2010 Conclusion The GEBNF approach to meta-modelling can lay its theoretical foundation on the basis of mathematical logic and the theory of institutions.

Seminar on GEBNF64 7 Sept 2010 Future Work Developing software tools to support meta-modelling in GEBNF Application of theory to facilitate a meta- model extension mechanism Investigation into the theoretical foundation of meta-models in UML class diagrams and OCL

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