a generalization of termination conditions for partial ...icetcs.ru.is/nwpt2015/slides/rabbi.pdf ·...

A generalization of termination conditions for partialmodel completion

Fazle Rabbi1,2 Yngve Lamo1 Lars Michael Kristensen1

Ingrid Chieh Yu2

1Bergen University College, Bergen, Norway2University of Oslo, Oslo, Norway

[email protected], [email protected], [email protected],[email protected]

October 22, 2015

Fazle Rabbi et al. (HiB, UiO) A generalization of termination conditions October 22, 2015 1 / 60

Introduction: Why do we need Domain-specificlanguages (DSLs)?

DSLs are - tailored to a specific application domain.- used to capture domain-specific knowledge and concepts.- expressive.- easy to use.

Development of domain models require domain specific languages

Figure : MDE focuses on exploiting domain models


Introduction: How to develop DSLs?

conforms to

Figure : How to develop domain specific modelling languages


Introduction: Complexity of developing DSLs

conforms to

Meta-modelConstraints

DSLdevelopment is hard, requiring - language development skill - domain knowledge

Model

Figure : Metamodelling for DSL development


Introduction: Consistency management

Caregiver Department

Ward

caregiversdepartments

Bryan Emergency

Ward10

(a)

(b)

M1

1..**

wards

department1..1

*

caregivers

worksAt *

*

context Wardinv rule: self.caregivers -> includesAll(self.department.caregivers)

M2

Attached OCL constraint

Figure : (a) Model M2, (b) a partial model M1 (not conforming to M2)



conforms to


INCONSIS

TENT ?

Model

Figure : Inconsistent model



conforms to


Completionrules

Apply Completion rules...

Model

Completion rules may 1. automatically fix an inconsistent model2. increase modelling efficiency

Figure : Consistency management by fixing inconsistencies


Challenges

conforms to

Meta-model

Constraints

Completionrules

Apply Completion rules...

Can we reuse completion rules ?

Will it Terminate ?

Can we automatically construct completion rules by processing constraints ?

Will it always find a consistent model ?


Multilevel metamodelling

conforms to

Model

conforms to

Model

conforms to

Model

Metamodel stack

Figure : Multilevel metamodelling

Multilevel metamodelling offers aclean, simple and coherentsemantics for metamodelling[Atkinson and Kuhne, 2001]It is an essential requirement forthe development ofdomain-specific modellinglanguages


Diagram Predicate Framework (DPF) [Rutle, 2010]

ClassData Type AttributeReference

Node Arrow

Typing

Typing

Patient Caregiver Department

Ward

Typing

wardPats

dataAccess

empDeps

depEmps

wardDepswardEmps

[COMP] [COMP]

[INV]

[1.. ]8 [1..1]

Predicate, p

1

(p)

[mult(n,m)]

[inverse]

[composite]

X Y

X Y

X Z

Y

gf

hConstraint

2

3

f

f

g

Semantic interpretation of [composite]:For each instance of (f; g ),there exists an instance of h.




Node Arrow

Typing

Typing


Ward

Typing

wardPats

dataAccess

empDeps

depEmps

wardDepswardEmps

[COMP] [COMP]

[INV]

[1.. ]8 [1..1]

Predicate, p

1

(p)

[mult(n,m)]

[inverse]

[composite]

X Y

X Y

X Z

Y

gf

hConstraint

2

3

f

f

g

Semantic interpretation of [inverse]:For each instance of f thereexists an instance of g or vice




Node Arrow

Typing

Typing


Ward

Typing

wardPats

dataAccess

empDeps

depEmps

wardDepswardEmps

[COMP] [COMP]

[INV]

[1.. ]8 [1..1]

Predicate, p

1

(p)

[mult(n,m)]

[inverse]

[composite]

X Y

X Y

X Z

Y

gf

hConstraint

2

3

f

f

g

Semantic interpretation of [mult(n,m)]:f must have at least n and at most m instances




Ward

Typing,

wardPats

dataAccess

empDeps

depEmps

wardDepswardEmps

[COMP] [COMP]

[INV]

[1.. ]8 [1..1]

1

John Bryan Emergency

Ward10

:wardEmps

:wardDeps:wardPats

I

Figure : Conformance checking




Ward

Typing,

wardPats

dataAccess

empDeps

depEmps

wardDepswardEmps

[COMP] [COMP]

[INV]

[1.. ]8 [1..1]

1

John Bryan Emergency

Ward10

:wardEmps

:wardDeps:wardPats

I

g

X

Y

f

hZ

0([composite]) 1

Bryan Emergency

Ward10

:wardEmps:wardDeps

*

1*

PB

Figure : Pullback αΣ0 ([composition])ι∗←− O∗

δ∗1−−→ I of αΣ0 ([composition])

δ1−−→ S ι

←− I


An inconsistent instance

Figure : An inconsistent instanceFazle Rabbi et al. (HiB, UiO) A generalization of termination conditions October 22, 2015 15 / 60

Diagrammatic model completion

Diagrammatic model completion is based on completion rulesCompletion rules are typed coupled transformation rulesType graphs of completion rules are not changed by thetransformationCompletion rules are linked to predicatesCompletion rules are applied to a partial model to correctinconsistenciesWe use the standard double-pushout (DPO) approach[Ehrig, 2006] for defining completion rules.



Diagrammatic model completion is based oncompletion rules

Completion rules are typed coupledtransformation rules

Type graphs of completion rules are notchanged by the transformation

Completion rules are linked to predicates

Completion rules are applied to a partial modelto correct inconsistencies

We use the standard double-pushout (DPO)approach [Ehrig, 2006] for defining completionrules.

Figure : A completion rule









X

Y

Z

g f

h

[comp]

([composite])

LHS (L)N RHS (R)

Typing

x

y

z

:f

:h

:g

x

y

z

:f:g

x

y

z

:f

:h

:g

x

y

z

:f:g

Glue (K)

Figure : A transformation rule is linked tothe [composite] predicate









X

Y

Z

g f

h

[COMP]

([composite])

Typing

x:X z:Z

y:Y

:f:g

:h

Figure : A completion rule



Table : Completion scheme for predicates and completion rules of S1

Cp ζ(Cp ) Interpretation

[inv-com〉[inv]

X Yg

f

[INV]

([inverse])

Typing

x:X y:Y

:g

:f

X Yg

f

[INV]

([inverse])

Typing

x:X y:Y

:g

:f

derive an edge y:g−−→ x (if it does not

exist) from the existence of an edge

x :f−→ y or vice versa.

[comp-com〉[comp]

X

Y

Z

g f

h

[COMP]

([composite])

Typing

x:X z:Z

y:Y

:f:g

:h

derive an edge x :h−−→ z (if it does

not exist) from the existence of edges

x:g−−→ y and y :f

−→ z.


Application of a completion rule

(p)

N

N

(p)

L

L

(p)

K

K

(p)

R

R

id id id

n k l



(p)

N

N

(p)

L

L

(p)

K

K

(p)

R

R

id id id

n k l

S

S

i

Si-1

i



(p)

N

N

(p)

L

L

(p)

K

K

(p)

R

R

id id id

n k l

S

S

i

Si-1

im (p)

L

L

S

S

i

Si-1

i

m

=

Match of a completion rule



(p)

N

N

(p)

L

L

(p)

K

K

(p)

R

R

id id id

n k l

S

S

i

Si-1

im (p)

L

L

S

S

i

Si-1

i

m

=

Match of a completion rule

A match (δ,m) is given by an atomic constraint δ : αΣ(p)→ Si−1 and a match m : L → Si such that the constraint δ and

match m together with typing morphisms ιL : L → αΣ(p) and ιSi: Si → Si−1 constitute a commuting square: ιL ; δ = m; ιSi



(p)

N

N

(p)

L

L

(p)

K

K

(p)

R

R

id id id

n

S

S

i

Si-1

imq

=

(δ,m) |= NAC if there does not exists an injective morphism q : N → Si with n; q = m such that the typing morphisms

ιN : N → αΣ(p) and ιSi: Si → Si−1 constitute a commuting square ιN ; δ = q; ιSi

.



(p)

N

N

(p)

L

L

(p)

K

K

(p)

R

R

id id id

n

S

S

i

Si-1

im

S'i

Si-1

S*i

Si-1

m*q (PO1) (PO2)=


Example: Application of a completion rule


Ward

Patient

empDeps

depEmps

wardDeps

Bryan Emergency

:wardEmps

:wardDeps

dataAccess

wardEmps

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John

:wardPatsSi

con

stra

int,

injective morphism, m

Ward10

X Z

Y f

g

h

y :f:g

x z

X

Y

Z

g f

h

[comp]

([composite])

LHS (L)N RHS (R)

Typing

x

y

z

:f

:h

:g

x

y

z

:f:g

x

y

z

:f

:h

:g

x

y

z

:f:g

Glue (K)




Ward

Patient

empDeps

depEmps

wardDeps

Bryan Emergency

:wardEmps

:wardDeps

dataAccess

wardEmps

wardPats

John

:wardPatsSi

con

stra

int,


Ward10

X Z

Y f

g

h

y :f:g

x z

X

Y

Z

g f

h

[comp]

([composite])

LHS (L)N RHS (R)

Typing

x

y

z

:f

:h

:g

x

y

z

:f:g

x

y

z

:f

:h

:g

x

y

z

:f:g

Glue (K)

injective morphism, q




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depEmps

wardDeps

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:wardDeps

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:wardPatsSi

con

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int,


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X Z

Y f

g

h

y :f:g

x z

X

Y

Z

g f

h

[comp]

([composite])

LHS (L)N RHS (R)

Typing

x

y

z

:f

:h

:g

x

y

z

:f:g

x

y

z

:f

:h

:g

x

y

z

:f:g

Glue (K)


:empDeps




Ward

Patient

empDeps

depEmps

wardDeps

Bryan Emergency

:wardEmps

:wardDeps

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:wardPatsSi

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int,


Ward10

X Yf

:fx y

LHS (L)N RHS (R)

Typing

Glue (K)

:empDeps

X Y

g

f

[INV]

([inverse])

x y

:f

:g

x y

:f

x y

:f

x y

:f

:g

g




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empDeps

depEmps

wardDeps

Bryan Emergency

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X Yf

:fx y

LHS (L)N RHS (R)

Typing

Glue (K)


:empDeps

X Y

g

f

[INV]

([inverse])

x y

:f

:g

x y

:f

x y

:f

x y

:f

:g

g




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depEmps

wardDeps

Bryan Emergency

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X Yf

:fx y

LHS (L)N RHS (R)

Typing

Glue (K)


:empDeps

X Y

g

f

[INV]

([inverse])

x y

:f

:g

x y

:f

x y

:f

x y

:f

:g

g

:g:depEmps




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empDeps

depEmps

wardDeps

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:wardEmps

:wardDeps

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:wardPatsSi

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:empDeps

:depEmps:dataAccess


Example: Application of a completion rule that deleteselements


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X

Y

f

f1x

y

LHS (L) RHS (R)

Typing

Glue (K)

:empDeps

X Yf [1..1]

([mult[n,m]])

x:X y:Yf1:f

:depEmps

z:Yf2:f

x:X y:Yf1:f

z:Y

x:X y:Yf1:f

z:Y

Cardiologyz

:wardDepsf2


Example: Application of a completion rule that deleteselements


Ward

Patient

empDeps

depEmps

wardDeps

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X

Y

f

f1x

y

LHS (L) RHS (R)

Typing

Glue (K)

:empDeps

X Yf [1..1]

([mult[n,m]])

x:X y:Yf1:f

:depEmps

z:Yf2:f

x:X y:Yf1:f

z:Y

x:X y:Yf1:f

z:Y

Cardiologyz


Termination criteria

Based on the principles adapted from layered graph grammars [Ehrig, 2006].Completion rules are distributed across different layers.Rules of a layer are applied as long as possible before going to the next layer.We generalize the layer conditions from [Ehrig, 2006] allowing deleting and non-deletingrules to reside in the same layer as long as the rules are loop-free.

Generalized layered approach

........

r1 r2 r3 ....





Layer 1

Layer 2

r1 r7 ....

......

..

r2 r3 ....

r1 r2 r3 ....





Layer 1

Layer 2

r1 r7 ....

S0

......

..

r2 r3 ....

r1 r2 r3 ....





Layer 1

Layer 2

r1 r7 ....

S0 S1

......

..

r1

r2 r3 ....

r1 r2 r3 ....





Layer 1

Layer 2

r1 r7 ....

S0 S1 S2

......

..

r1 r1

r2 r3 ....

r1 r2 r3 ....





Layer 1

Layer 2

r1 r7 ....

S0 S1 S2 S3 .... Sn

......

..

r1 r1 r1 r7

r2 r3 ....

r1 r2

S0 S7 S8 S9 .... Sm

r1 r7 r7 r7

..............

r3 ....





Layer 1

Layer 2

r1 r7 ....

S0 S1 S2 S3 .... Sn

......

..

r1 r1 r1 r7

r2 r3 ....

Sn S21 S22 S23 .... St

r2 r2 r3 r3

r1 r2

S0 S7 S8 S9 .... Sm

r1 r7 r7 r7

..............

Sm S33 S34 S35 .... Sv

r2 r3 r3 r3

..............

r3 ....



Our generalized layered approach is based on a necessary condition (C1∨C2∨C3) for looping,where:

C1: A rule ri that creates an element x of type t does not have a NAC that forbids theexistence of element of type t .

C2: If a rule ri creates an element x of type t and has a NAC that forbids the existence ofelement of type t , then there exists a rule rj that deletes an element of type t .

C3: If a rule ri deletes an element x of type t , then there exists a rule rj that creates anelement of type t .



Lemma 1. (C1 ∨ C2 ∨ C3) is a necessary condition for looping of a set of rules at layer k

Proof: Let G0 = Si be an initial graph typed by Si−1 where Si ,Si−1 are finite graphs. Let Rk be afinite set of rules at layer k . A rule r ∈ Rk can either

1 creates an element x of type t , wherea r does not have a NAC that forbids the existence of element of type t .

orb r has a NAC that forbids the existence of element of type t .and/or

2 deletes an element x′ of type t ′




Proof:A rule r ∈ Rk can either




Consider case 1.(a):The rule r has finite number of injective matches cr = {(δ,m) | (δ,m) is a match for G0 B Si−1}.For each injective match of L → G0, application of r creates an element x of type t .The rule can be applied indefinitely in a loop during the derivation process of layer k since theapplication of rule r does not decreases the number of matches.Therefore C1 is a necessary condition for looping.








Consider case 1.(b):The rule r has finite number of injective matches cr = {(δ,m) | (δ,m) is a match forG0 B Si−1 and (δ,m) |= NAC}. For each injective match of L → G0, application of r creates anelement x of type t .Therefore, the application of rule r decreases the number of matches.In order to apply r indefinitely in a loop during the derivation process of layer k , elements of typet must be deleted.Therefore C2 is a necessary condition for looping.








Consider the first case 2:The rule r has finite number of injective matches cr = {(δ,m) | (δ,m) is a match forG0 B Si−1 and (δ,m) |= NAC}.In order to apply r indefinitely in a loop during the derivation process, new elements of type t ′

must be created.Therefore C3 is a necessary condition for looping.



Corollary 1. A sufficient condition for loop-free rules in layer k is the negation of (C1 ∨ C2 ∨ C3)


r1

r2r3

r5r4

r6

r7

r8 .........

Rk

Rules that can beapplied indefinitelywithout stopping





r1

r2r3

r5r4

r6

r7

r8 .........

Rk

Rules that can beapplied indefinitelywithout stopping

Rules that satisfy the necessary condition





r1

r2r3

r5r4

r6

r7

r8 .........

Rk

Rules that always terminates

Rules that satisfy the necessary condition

Rules that satisfy the sufficient condition


Termination criteriaWe propose a loop detection algorithm that is based on the following sufficient conditions for loopfreeness. Let Rk be the set of rules of a layer k .

If a rule ri ∈ Rk creates an element x of type t , then ri must have an element of type t in itsNAC,

If a rule ri ∈ Rk creates an element x of type t , then there is no rule in rj ∈ Rk that deletesan element of type t ,

If a rule ri ∈ Rk deletes an element of type t , then there is no rule in rj ∈ Rk that creates anelement of type t


X Y

f

g

x y

:f

:g

[inv]


Termination criteriaWe propose a loop detection algorithm that is based on the following sufficient conditions for loopfreeness. Let Rk be the set of rules of a layer k .

If a rule ri ∈ Rk creates an element x of type t , then ri must have an element of type t in itsNAC,If a rule ri ∈ Rk creates an element x of type t , then there is no rule in rj ∈ Rk that deletesan element of type t ,If a rule ri ∈ Rk deletes an element of type t , then there is no rule in rj ∈ Rk that creates anelement of type t


X Y

f

g

x y

:f

:g

[inv] X Yf

h

x y:f

[r1]

z

:h

z

Figure : These rules may produce a non-terminating situation if they areexecuted in the same layer


Termination criteriaIf a rule ri ∈ Rk creates an element x of type t , then ri must have an element of type t in itsNAC,If a rule ri ∈ Rk creates an element x of type t , then there is no rule in rj ∈ Rk that deletesan element of type t ,If a rule ri ∈ Rk deletes an element of type t , then there is no rule in rj ∈ Rk that creates anelement of type t


Ae1

e2

constraint,

X

f

X Y

g

f

[p1]

([p1])

x y

:f

:gg

BY

r1

NAC(e2) R(e2) NAC(e1) R(e1) NAC(A) R(A) NAC(B) R(B)

r1

C

e3

NAC(C)R(C)NAC(e3) R(e3)




Ae1

e2

constraint,

X

f

X Y

g

f

[p1]

([p1])

x y

:f

:gg

BY

r1


r1

C

e3





Ae1

e2

constraint,

X

f

X Y

g

f

[p1]

([p1])

x y

:f

:gg

BY

r1


C

e3





Ae1

e2

constraint,

X

f

X Y

g

f

[p1]

([p1])

x y

:f

:gg

BY

r1


r1

C

e3


X Y

g

f

[p2]

([p2])

x y

:f

:g

r2

Z

h

z

:h

r2 X

Z

h



Theorem 1. (termination of loop-free rules). An empty table obtained by loop free rule detectionanalysis for a set of rules E implies that the execution of E will terminate for any finite size initialgraph.


Conclusion and Future work

SummaryCompletion rules are defined as coupled graph transformationrulesCompletion rules are reusableGeneralized termination analysis is based on layered approachWe have Implemented a proof-of-concept of the proposedapproach

Future WorkImprove performance of the transformation systemAutomatically construct completion rules by processing constraintsDevelop concrete graphical syntaxSupport collaborative development


References

Atkinson, C. and Kuhne, T. (2001).

The essence of multilevel metamodeling.In Proceedings of the 4th International Conference on The Unified Modeling Language, Modeling Languages, Concepts,and Tools, UML; ’01, pages 19–33, London, UK, UK. Springer-Verlag.

Zinovy Diskin, Uwe Wolter:

A Diagrammatic Logic for Object-Oriented Visual Modeling. Electr. Notes Theor. Comput. Sci. 203(6): 19-41 (2008)

Adrian Rutle:

Diagram Predicate Framework: A Formal Approach to MDE. PhD thesis, Department of Informatics, University of Bergen,Norway, November 2010.

Gabriele Taentzer, Florian Mantz, Thorsten Arendt, Yngve Lamo:

Customizable Model Migration Schemes for Meta-model Evolutions with Multiplicity Changes. MoDELS 2013: 254-270

Hartmut Ehrig, Karsten Ehrig, Ulrike Prange, Gabriele Taentzer:

Fundamentals of Algebraic Graph Transformation. Monographs in Theoretical Computer Science. An EATCS Series,Springer 2006, ISBN 978-3-540-31187-4.

Annegret Habel, Reiko Heckel, Gabriele Taentzer:

Graph Grammars with Negative Application Conditions. Fundam. Inf., 1996 vol. 26, pp. 287–313.


The End


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