on the theoretical foundation of meta-modelling in graphically extended bnf hong zhu department of...

On the Theoretical Foundation of Meta-Modelling in Graphically Extended BNF

Hong ZhuDepartment of Computing and ElectronicsOxford Brookes UniversityOxford OX33 1HX, UKEmail: [email protected]

Seminar on GEBNF 27 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.


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


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


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.


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?


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


The GEBNF NotationIn 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.


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


Meanings of the GEBNF Notation


Example 1: Directed Graphs

where Graph is the root symbolGraph, Node and Edge are non-terminal symbolsString and Real are terminal symbols


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: EndEnd ::= node: Class, name: [String], mult: [Multiplicity]


Example 2: B. UML Sequence Diagram (subset)

SequenceDiagram ::= lifelines: Lifeline*, msgs: Message*, ordering: Order*

Order::= from: Message, to: MessageLifeline ::=

className: String, objectName: [String], isStatic: Bool, activations: Activation*Activation ::= start, finish: Event, others: Event* Message ::= send, receive: Event, sig: OperationEvent ::= actor: Activation


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.


TypesDefinition

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.


Induced FunctionsGiven 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([email protected])=T(C); T(C*)= P(T(C)); T(C+)=P(T(C)); T(C1 | … | Cn) = .


Example 3: Induced Functions


Induced Predicate Logic Language (FL) From Fun(G), a FL can be defined as

usual [Chiswell 2007]. variables of type TypeGrelations 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


Definition: Inducted Predicate LogicLet 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. e1 e2, e1 e_2, and e1 - e2 are expressions of type P(), if e1 and e2 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(). e1 = e2 and e1 e2 are predicates on type , if e1 and e2 are expressions of type R(e_1, …, e_n) is a predicate on type , if e1, …, en are expressions of type , and R is

any n-ary relation symbol on type . e1 e2 and e1 e2 are predicates on type P(), if e1 and e2 are expressions of type P(). p q, p q, p q, p q and p are predicates, if p and q are predicates. xD.(p) and xD.(p) are predicates, if D is an expression of type P(), x is a variable

of type , and p is a predicate.

)(GType

VV

It is first order, if we restrict type to be a terminal or non-terminal symbol s


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:


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:


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

Examplestrongly connected graphs


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.


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 )


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 ?


FormalisationSituation 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:


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:


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:


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 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.


FormalisationSituation 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:


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:


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:


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.


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:


Algebraic Semantics of GEBNFLet 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-algebra A: {Ax | x NT} 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


Algebra without Junk

A G-algebra A contains no junk, if

|AR|=1

for all sN and all eAs, 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


Example 6: Model as Algebra


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.


Evaluation of Expressions and Predicates


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-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]


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.


Review: CategoryA category C 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.


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.


Syntax MorphismsA 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 opFun(G),

(op: A B) (f(op): m(A) m(B)),where we naturally extend the mapping m to type expressions.


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)


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 (m o n, f o g).

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.


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).


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).


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.


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


Constraint Preserving Syntax Morphisms

DefinitionA syntax morphism from G to H is constraint preserving, if for all constraint cAxiom(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.


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).


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 and are constraint preserving syntax morphisms, so is o provided that they are composable.

We write GEBNF to denote this category.


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 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 op Fun(G), the function op A

is the function f(op) in B.

Denoted by MODG


Example 10: Translation of ModelsConsider 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.


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.


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

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

(Obj, Mor) is a category.

It is denoted by CAT.

GEBNF syntax

Syntax morphism


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 .

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


Review: Institution


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.


Example 11: Truth Invariance under Translation

Both statements are true

Both statements are false


Conclusion

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


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

on the theoretical foundation of meta-modelling in graphically extended bnf hong zhu department of...

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