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Meta-Modelling in Graphically Extended BNF: Theoretical Foundations and Applications

Meta-Modelling in Graphically Extended BNF: Theoretical Foundations and Applications. Hong Zhu Department of Computing and Electronics Oxford Brookes University Oxford OX33 1HX, UK Email: hzhu@brookes.ac.uk. Outline. Introduction Motivation and Related works Graphic extension of BNF

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Meta-Modelling in Graphically Extended BNF: Theoretical Foundations and Applications

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

  2. 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 at Shanghai Jiaotong University

  3. 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 at Shanghai Jiaotong University

  4. 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 at Shanghai Jiaotong University

  5. 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 at Shanghai Jiaotong University

  6. Overview Graphic Extension of BNF • Formal Meta-Modelling • Meta-notation: GEBNF • Definition of abstract syntax of modelling languages • Predicate logic languages induced from GEBNF syntax • 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 • Theoretical Foundation • Axiomatization of syntax constraints • Formal semantics of GEBNF syntax definitions • Institution of GEBNF meta-models Seminar at Shanghai Jiaotong University

  7. The GEBNF Notation In GEBNF, the abstract syntax of a modelling language is defined as a tuple <R, N, T, S>, 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 ::= X1|X2| …| Xn Y ::= f1: X1 , f2: X2 , …, fn: Xn where • YN, • f1, f2 , …, fn are called field names, • X1, X2, …, Xn are the fields. Seminar at Shanghai Jiaotong University

  8. 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 at Shanghai Jiaotong University

  9. Meanings of the GEBNF Notation Seminar at Shanghai Jiaotong University

  10. 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 at Shanghai Jiaotong University

  11. 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 at Shanghai Jiaotong University

  12. 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 at Shanghai Jiaotong University

  13. Well-Formed Syntax Definitions A syntax definition <R, N, T, S> 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 at Shanghai Jiaotong University

  14. Types Definition Let G=< R, N, T, S> 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 at Shanghai Jiaotong University

  15. Induced Functions Given a well-defined GEBNF syntax G= <R, N, T, S>, we write Fun(G) to denote the set of function symbols derived from the syntax rules as follows • A syntax rule ``A ::= B1 | B2 | …| Bn'' introduces a set of functions IsB1, IsB2, …, IsBn of the type A Bool. • A syntax rule ``A ::= f1:B1,…, fn: Bn'' introduces a set of function symbols fi of type AT(Bi), 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(C1 | … | Cn) = . Seminar at Shanghai Jiaotong University

  16. Example 2: Induced Functions Seminar at Shanghai Jiaotong University

  17. Induced Predicate Logic Language (FL) • From Fun(G), a FL can be defined as usual [Chiswell 2007]. • variables of type t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 at Shanghai Jiaotong University

  18. Definition: Inducted Predicate Logic Let be a set of variables, where xVt are variables of type t. • Each literal constant c of type sT is an expression of type s. • Each element vVt, i.e. variable of type t, is an expression of type tType(G). • e.f is an expression of type t', if f is a function symbol of type tt', e is an expression of type t. • { e(x) | Pred(x) } is an expression of type P(te), if x is a variable of type tx, e(x) is an expression of type te and Pred(x) is a predicate on type tx. • e1e2, e1e_2, and e1-e2 are expressions of type P(t), if e1 and e2 are expressions of type P(t). • eE is a predicate on type t, if e is an expression of type t and E is an expression of type P(t). • e1 = e2 and e1e2 are predicates on type t, if e1 and e2 are expressions of type t. • R(e_1, …, e_n) is a predicate on type t, if e1, …, en are expressions of type t, and R is any n-ary relation symbol on type t. • e1e2 and e1e2 are predicates on type P(t), if e1 and e2 are expressions of type P(t). • 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 a type expression t, x is a variable of type t, and p is a predicate. It is first order, if we restrict type t to be a terminal or non-terminal symbol s Seminar at Shanghai Jiaotong University

  19. Example 3: • 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 at Shanghai Jiaotong University

  20. 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 at Shanghai Jiaotong University

  21. Meta-Modelling in GEBNF + FPL • 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 at Shanghai Jiaotong University

  22. Example 4: The Object Adapter DP • In GoF book: In general, there can be a set of such methods! Indicate where messages can be sent to rather than a component of the pattern! Seminar at Shanghai Jiaotong University

  23. Formal Spec of the Object Adapter DP where predicates are defined using the function symbols induced from the GEBNF definition of UML class diagrams and sequence diagrams. Seminar at Shanghai Jiaotong University

  24. 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 at Shanghai Jiaotong University

  25. 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 at Shanghai Jiaotong University

  26. 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 at Shanghai Jiaotong University

  27. 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 (X1 | …| X | …| Xn) • Axiom: Seminar at Shanghai Jiaotong University

  28. 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 at Shanghai Jiaotong University

  29. 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 (X1 | …| X | …| Xn) • E’(X) is one of the expressions X*, and X+ • Axiom: Seminar at Shanghai Jiaotong University

  30. 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 at Shanghai Jiaotong University

  31. 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 (X1 | …| X | …| Xn) • Axiom: Seminar at Shanghai Jiaotong University

  32. 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 at Shanghai Jiaotong University

  33. 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 at Shanghai Jiaotong University

  34. Example 5 • 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 at Shanghai Jiaotong University

  35. 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 at Shanghai Jiaotong University

  36. Algebraic Semantics of GEBNF Let G= < R, N, T, S> be a GEBNF syntax definition • The signature induced from G: G=(NT, FG), where FG=Fun(G) is the set of function symbols induced from G. • G-algebraA: • {Ax | xNT} of sets • {f | FG }, a set of functions, where if  is of type XY, then f is a function from set AX to the set [[ Y ]], where Seminar at Shanghai Jiaotong University

  37. Algebra without Junk • A G-algebra Acontains no junk, if • |AR|=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AR 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 at Shanghai Jiaotong University

  38. Example 6: Model as Algebra Seminar at Shanghai Jiaotong University

  39. 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 trueinAunder 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 at Shanghai Jiaotong University

  40. Evaluation of Expressions and Predicates Seminar at Shanghai Jiaotong University

  41. Seminar at Shanghai Jiaotong University

  42. 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-modelMM 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 at Shanghai Jiaotong University

  43. 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 at Shanghai Jiaotong University

  44. Review: Category A categoryC consists of • a class Cobj of objects and • a class Cm of morphisms or arrows between objects together with the following three operations: • dom: CmCobj; • codom: CmCobj; • id: CobjCm, 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 idA. Seminar at Shanghai Jiaotong University

  45. There is a partial operation o on Cm, 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) • idA o f = f, if codom(f)= A • g o idA= g, if dom(g) = A. Seminar at Shanghai Jiaotong University

  46. Syntax Morphisms A syntax morphism from G to H, written  :GH, is a pair (m, f) of mappings • m: NGNH and • f: Fun(G)Fun(H) that satisfy the following two conditions: • Root preservation: m(RG)=RH; • 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 at Shanghai Jiaotong University

  47. 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 at Shanghai Jiaotong University

  48. 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 (mon, fog). • Definition (Identity Syntax Morphisms) • IdG is defined as the pair of mappings (idN, idFun(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 at Shanghai Jiaotong University

  49. Review: Functor • Let C, D be two categories. A functor F from C to D consists of two mappings: • an object mapping Fobj : CobjDobj, and • a morphism mapping Fm : CmDm • They have the following properties. • for all morphisms f: AB of category C, Fm(f) : Fobj(A)Fobj(B). • for all morphisms f and g in C, Fm(f o g) = Fm(f) o Fm(g). • for all objects A in category C, Fm(idA) = idFobj(A). Seminar at Shanghai Jiaotong University

  50. 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: • Senobj(G) denotes the set of predicates on the root of G. • Define mapping Senm() from Senobj(G) to Senobj(H): For each predicate p in Senobj(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 at Shanghai Jiaotong University

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