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

Axiom:

Hong Zhu

Department of Computing and Electronics

Oxford Brookes University

Oxford OX33 1HX, UK

Email: [email protected]

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

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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

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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.

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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?

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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

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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.
- RN 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

- YN,
- f1, f2 , …, fn are called field names,
- X1, X2, …, Xn are the fields.

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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 YNT.
- Y*, Y+ and [Y] are expressions, if YNT.
- Exp(Y)@Z.f is an expression, where
- Y, ZN
- 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

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Meanings of the GEBNF Notation

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Example 1: Directed Graphs

where

- Graph is the root symbol
- Graph, Node and Edge are non-terminal symbols
- String and Real are terminal symbols

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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]

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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

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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 XN, there is one and only one syntax rule sS that defines X.

- Reachability
For each non-terminal symbol XN, X is reachable from the root R.

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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 sTN, 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.
- 12 is a type, called a function type from 1 to 2, if 1 and 2 are types.

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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 AT(Bi), where T(B) is defined as follows.
- T(C)=C, if CTN;
- T([C])=T(C);
- T([email protected])=T(C);
- T(C*)=P(T(C));
- T(C+)=P(T(C));
- T(C1 | … | Cn) = .

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Example 2: Induced Functions

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Induced Predicate Logic Language (FL)

- From Fun(G), a FL can be defined as usual [Chiswell 2007].
- variables of type tType(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

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Definition: Inducted Predicate Logic

Let be a set of variables, where xVt are variables of type t.

- Each literal constant c of type sT is an expression of type s.
- Each element vVt, i.e. variable of type t, is an expression of type tType(G).
- e.f is an expression of type t', if f is a function symbol of type tt', 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.
- e1e2, e1e_2, and e1-e2 are expressions of type P(t), if e1 and e2 are expressions of type P(t).
- eE 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 e1e2 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.
- e1e2 and e1e2 are predicates on type P(t), if e1 and e2 are expressions of type P(t).
- pq, pq, pq, pq and p are predicates, if p and q are predicates.
- xD.(p) and xD.(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

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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:

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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:

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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

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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!

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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.

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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 AB,

- 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.
xA . (x.g)

where means undefined.

- for each non-optional function symbol g, we require it satisfying the following condition.

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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 AP(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.
xA . (x.g)

- for each non-empty repetition function symbol g, we require it satisfying the following condition.

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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 ?

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- 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:

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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:

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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+

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Axiomatization of Syntax Constraints (4)

Referential Occurrences

Consider two syntax definitions of a non-terminal symbol A

A ::= …, f: B, … . (1)

A ::= ... , f’: [email protected] , … . (2)

- Similarity:
- Both functions f and g have the type AB,

- 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.

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Situation 1: Single reference to single element

Y ::= ... , g: E(X), … . (1)

Z ::= …, f: [email protected], … . (2)

where E(X) is one of the expressions X, [ X ], and (X1 | …| X | …| Xn)

- Axiom:

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Situation 2: Single reference to set of elements

Y ::= ... , g: E(X), … . (1)

Z ::= …, f: [email protected], … . (2)

where E(X) is one of the expressions

X*, and X+

- Axiom:

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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:

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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.

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- 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:

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Algebraic Semantics of GEBNF

Let G= < R, N, T, S> be a GEBNF syntax definition

- The signature induced from G: G=(NT, FG), where FG=Fun(G) is the set of function symbols induced from G.
- G-algebraA:
- {Ax | xNT} of sets
- {f | FG }, a set of functions, where if is of type XY, then f is a function from set AX to the set [[ Y ]], where

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Algebra without Junk

- A G-algebra Acontains no junk, if
- |AR|=1
- for all sN and all eAs, we can define a function f : RP(s) in PL such that for some mAR we have ef(m).

There is one and only one root element

Every element in a model must be accessible from the root

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Example 6: Model as Algebra

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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.

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Evaluation of Expressions and Predicates

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Valid Models

- A G–algebra A without junk is a syntactically valid model with respect to G, if for all pAxiom(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]

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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.

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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: CmCobj;
- codom: CmCobj;
- id: CobjCm,
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 : AB.
- For each object A, id(A) is the identity morphism that its domain and codomain are A. id(A) is also written as idA.

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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.

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Syntax Morphisms

A syntax morphism from G to H, written :GH, is a pair (m, f) of mappings

- m: NGNH and
- f: Fun(G)Fun(H)
that satisfy the following two conditions:

- Root preservation: m(RG)=RH;
- Type preservation: for all opFun(G),
(op: AB)(f(op): m(A)m(B)),

where we naturally extend the mapping m to type expressions.

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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 = (GraphMap, NodeCity, EdgeRoute),
- f = ( nodescities, edgesroutes,
namename, weightpopulation,

toarrive, fromdepart,

weightdistance)

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The Category of GEBNF

- Definition: (Composition of syntax morphisms)
- Assume that =(m, f): GH and =(n, g):HJ 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.

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Review: Functor

- Let C, D be two categories. A functor F from C to D consists of two mappings:
- an object mapping Fobj : CobjDobj, and
- a morphism mapping Fm : CmDm

- They have the following properties.
- for all morphisms f: AB 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).

- for all morphisms f: AB of category C,

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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 opFun(G) in predicate p is replaced by the function symbol f(op).

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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 PLAR.

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- Theorem:
The pair (Senobj, Senm) is a functor from category GEB to the category SET of sentences in the corresponding induced FLs.

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Constraint Preserving Syntax Morphisms

- Definition
A syntax morphism m from G to H is constraint preserving, if for all constraint cAxiom(G) we have that Axiom(H) |-Senm(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.

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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 PLAR according to the syntax morphism.
- For instance, the following constraint c on directed graph
c=gGraph.(g.nodes)

is translated into

c’=gMap.(g.cities)

according to the syntax morphism.

- It is easy to see that Axiom(AR)|-c' because c'Axiom(AR).

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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, IdG is constraint preserving.
- If m and n are constraint preserving syntax morphisms, so is m o n provided that they are composable.

- We write GEBNF to denote this category.

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Translation of Models

Denoted by MODG

- 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 MODH to category MODG is defined as follows.

Let B |MODH|. We define an G-algebra A:

- For each sNG, As = Bm(s);
- For each function symbol opFun(G), the function op A is the function f(op) in B.

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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.

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- Theorem:
For each syntax morphism =(m,f) from G to H, the mapping U from objects of category MODH to the objects of category MODG and its naturally induced mapping on homomorphisms is a functor from MODH to MODG.

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It is denoted by CAT.

Syntax morphism

- Theorem:
Let Obj={ MODG | G |GEB| } and

Mor={U | ||GEB}||}.

(Obj, Mor) is a category.

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Model Translation Functor

- Definition:
- Define MODobj: |GEB| |CATop| as
MODobj(G)= Mod(G);

- Define MODm: ||GEB|| ||CATop}|| as
MODm(u)= Uop

where for an arrow : a b, op is the inverse arrow of .

- Define MODobj: |GEB| |CATop| as
- Theorem:
MOD=(MODobj,MODm) is a functor from GEB to CATop.

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Review: Institution

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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.

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Example 11: Truth Invariance under Translation

- Both statements are true
- Both statements are false

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Conclusion

The GEBNF approach to meta-modelling can lay its theoretical foundation on the basis of mathematical logic and the theory of institutions.

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Future Work

- Application to model transformation
- 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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References

- Zhu, H. and Shan, L. Well-formedness, consistency and completeness of graphic models, Proc. of UKSIM'06, pp.47~53 , Apr. 2006.
- Bayley, I. and Zhu, H. Specifying behavioural features of design patterns in first order logic, Proc. of COMPSAC'08, pp. 203~210, Aug. 2008.
- Zhu, H., Bayley, I., Shan, L. and Amphlett, R. Tool support for design pattern recognition at model level, Proc. of COMPSAC'09, pp. 228~233 , July 2009.
- Shan, L. and Zhu, H. Semantics of metamodels in UML, Proc. of TASE'09. pp.55~62, July 2009.
- Bayley, I. and Zhu, H. Formal specication of the variants and behavioural features of design patterns, Journal of Systems and Software, vol. 83, no. 2, pp.209~221, Feb. 2010.
- Zhu, H. On The Theoretical Foundation of Meta-Modelling in Graphically Extended BNF and First Order Logic, Prof. of TASE 2010, pp95-104, Aug. 2010.

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