Intermodeling, Queries and Kleisli categories Zinovy Diskin, Tom Maibaum, Krzysztof Czarnecki McMaster University, University of Waterloo

Intermodeling, Queries and Kleisli categories

Zinovy Diskin, Tom Maibaum, Krzysztof Czarnecki

McMaster University,University of Waterloo

NECSIS and MDE

• Network for Engineering of Complex Software-Intensive Systems for Automotive Systems – Eight Canadian research institutions– Three industrial partners (GM Canada, IBM

Canada, and Malina Software)

WADT'12 Query languages are Cartesian monads 2

Intermodeling (termed coined by Guerra et al, at MoDELS’10):

• What: Specification and maintenance of intermodel relations (techniques and tools)

• Why: Modeling is multimodeling

3Query languages are Cartesian monads

M2

M3

M1

R23

R12

R13

R123

WADT'12

Why intermodeling, cont’d

4Query languages are Cartesian monads

M2

M3

M1

R23

R12

R13

R123

WADT'12

Scenarios:• Consistency• Sync/change propagation • Merge

There is much more to R’s than name correspondence!

Contents

• Specifying intermodel relations is difficult• Main features of our framework by examples• Q-mappings at work: Model merge• Formalization via Cartesian monads • Summary: model management via algebra

(or, colors of model management)

Query languages are Cartesian monads 5WADT'12

Specifying intermodel relations is not easy

s:Student name=Mr. Lee

bdate=01/01/1990

a: Woman -Actor name=Ann

age = 21

Model S Model A=

?

semanticrelativism

homonymy

?

?

WADT'12 6

The indirect link problem

The Domain: Ms. Ann Lee, 01/01/1990, a student and an

actor

Query languages are Cartesian monads

“part of”

Contents

• Specifying intermodel relations is difficult• Main features of our framework by examples• Q-mappings at work: Model merge• Formalization via Cartesian monads • Summary: model management via algebra



Query languages are Cartesian monads /30

Indirect links via direct Q-links(Kleisli’s idea)

• Replace indirect links by direct links into derived elements (computed by queries, Q-links)

s: Student name = Ms.Lee bdate = 01/01/90

a: Actor name = Ann age = 22

Model AModel S +

WADT'12

=

=?

/age = 2012-1990 = 22

But first relate metamodels

Query languages are Cartesian monads 13

f Metamodel, AMetamodel S+

Student name: {Ms,Mr} x Str bdate: Int3

/age = Q(bdate) : Int

WADT'12

Actor name = Str age = Int

f: f Model, A: AModel S+: S+

S: Student name = Ms.Lee bdate = 01/01/90 /age = exec Q(01/01/90) = 22: Int

a: Actor name = Ann age = 22


Q-links vs. semantic relativism, 1

Student name: {Mr, Ms} x Str

Metamodel S Actor

name: Str

Metamodel A

Woman Man

?

WADT'12

Common intermodeling jargon: 1-to-M(any) linkWe say indirect 1-M link

/30



Metamodel S Actor name: Str

Metamodel A+

Woman Man

WADT'12

/title = Q1(W-M typing)


/30



Metamodel A+

Woman Man

WADT'12

/title = Q1 (W-M typing) /tname = Q2 (name, /title)

An indirect 1-M link x <---> {y1,…, ym } is replaced by a direct 1-1 q-link x <---> Q(y1,…, ym)



13



Metamodel A+

Woman Man

WADT'12

/title = Q1 (W-M typing) /tname = Q2 (name, title)

m

S A

A+ = Q(A)

mi

Q-links and asymmetry of mappings

m is an ordinary total mapping consisting of pairs of elements.

S is a view of A


14/36

ActorMetamodel A+

Woman Man

WADT'12

/title = Q2 (W-M typing)

m

Derived-derived case. Partial mappings.


Metamodel S+

/title = Q1(name)

S A

S+ = Q1(S)

i1 m2

m1

A+ = Q2(S)

i2


x

x

15/36

ActorMetamodel A+

Woman Man

WADT'12


From partial mappings to spans(Reifying intermodel links)


Metamodel S+

/title = Q1(name)

S A

S+ = Q1(S)

i1

m2

A+ = Q2(S)

i2


Metamodel R Person title: {Mr, Ms}

Rm1

Contents

• Specifying intermodel relations is difficult• Main features of our framework by examples• Q-mappings at work: Model merge • Towards formalization via Cartesian monads • Summary: model management via algebra



General pattern for merge

FASE'12 28/36

M+

A

r1

B

C

D

R1

R2...

r2Colimit

. .

. Color Legend: Black -- given Green --- discovery/heuristics/AI Blue --- pure algebra!

Intermodeling, Queries, and Kleisli categories

…


Merge via colimit: Example

FASE'12 23/36

ActorMetamodel A+

Woman Man



Metamodel S+

/title = Q1(name)

Metamodel R

? (Person) name: {Mr, Ms} x Str

Metamodel M+

Woman Man

/title = Q1 (name) = Q2 (W-M typing)

R2M

Person title: {Mr, Ms}

S2MA2M

R2S R2A


Merge via colimit cont’d

FASE'12 23/36

ActorMetamodel A+

Woman Man



Metamodel S+

/title = Q1(name)

Metamodel R

********

**** R2M

Person title: {Mr, Ms}

S2MA2M

R2S R2A


Metamodel M

Woman Man

Q1 (name) = Q2 (W-M typing)

/30

Another version of intermodel span

Person title: {Mr, Ms} x Str

Metamodel R Actor

Metamodel A

Woman Man

R2A

/MsPerson/MrPerson

FASE'12FASE'12


Metamodel S+

/title = Q1(name)

R2S


Metamodel M+

/title = Q1 (name)

/MsPerson /MrPerson

Q2(/title)

S2M

A2M

Q2(title)

Normalized merge

21/36

Person title: {Mr, Ms} x Str

Metamodel R Actor

Metamodel A

Woman Man

R2A

/MsPerson/MrPerson

FASE'12FASE'12


Metamodel S+

/title = Q1(name)

R2S


Metamodel M

********

Q2(title)


General pattern for merge

FASE'12 28/36

M+

A

r1

B

C

D

R1

R2...

r2Colimit

. .

. Color Legend: Black -- given Green --- discovery/heuristics/AI Blue --- pure algebra!


…

Three colors of model management

Discovering (heuristics + AI): Model diff/matching/alignment

Execution (pure algebra): Model merge/update propagation

Mixture: Weaving Green and Blue in one algorithm/module often results in Red

FASE'12 Intermodeling, Queries, and Kleisli categories


But…

• Algebraic techniques work well if the universe of models and Q-mappings between them is a category

• We need to define composition of Q-mappings• It is not trivial

24/36FASE'12


Q-mappings as Kleisli mappings

WADT'12 27/36

A B

Q(B)

fiQ

A B*******

m = (Q, f)A B

m

A B

Q(B)

fiQ

Thick (extensional) notation

Thin (formula) notation

Ordinary mappings

Q-mappings


Problem of Q-mapping composition

26/36

BD

Qf(AD) Qg(BD)

CDfD gD

gSfS

AS BS

Qf(AS )

CSfS

gS

Qg(BS )

CBAf g

qExeSchemas (data & query/view definitions) S stands for “the schema part”

Instances (data and query execs) Subindex D stands for “the Data part”

Typing mapping (t is “the typing part”)

?

?At

Bt

CtqExe

AD

AS BS CSMetamodels (or schemas)WADT'12

Models and their maps

Qf(At)Qg(Bt)


Solution: first, we manage syntax (the easy part)

AD BD

Qf(AD) Qg(BD)

CDfD gD

AS BS

Qf (AS)

CSfS

gS

Qg#( Qf (AS))

fS#

subst Qg (BS)

typing

CBAf g

WADT'12

Schemas (data & query/view definitions) S stands for “the schema part”

Instances (data and query execs) Subindex D stands for “the Data part”

?


Solution: second, we manage semantics(the difficult part)

AD BD

Qf(AD) Qg(BD)

CDfD gD

AS BS

Qf (AS)

CSfS

gS

Qg#( Qf (AS))

fS#

subst Qg (BS)

typing

CBAf g

WADT'12

Schemas (data & query/view definitions) S stands for “the schema part”

Instances (data and query execs) Subindex D stands for “the Data part” Qg

#(Qf(AD))

qExe

fD#

monot

Fortunately, many useful queries are monotonic (e.g., SPJ)


Some formal details1) Simple (no-queries) model translation from one metamodel to

another (retyping) can be specified by a categorical construct called fibration, p: Mod -->MMod (by def).

2) A monotonic query language is specified by a monad over the fibration 1), Q: Mod-->Mod, Qdef: MMod-->MMod, p:Q-->Qdef. This monad is Cartesian! (by def)

3) Models and Q-mappings form the Kleisli category of the monad in 2)• Corollary. If the cat of models and simple mappings is closed

wrt. merge, then the cat of models and Q-mappings is closed wrt. merge as well.

4) Theorem. Model translations determined by Q-mappings (= view mechanism) again form a fibration. That is, querying followed by retyping is compositional. 33/36 FASE'12


Summary• A (vast?) majority of intermodel relationships

can be specified by Q-mappings• With Kleisli construction, models and Q-

mappings between them form a category, i.e., a universe of well-defined algebraic operations

• Model management scenarios include two basic stages:– Discovery of Q-mappings between models;– Execution of an algebraic operation over models and

mappings• Weaving green into blue often results in red

WADT'12


THE END

WADT'12 /30


s:Student name=Mr. Lee a: Woman -Actor

name=Ann

Model S Model A

WADT'12 9

Solution 1: Annotate indirect links with correspondence rules

(= expressions attached to links)

If s = a, then if s.name.title = Mr, then a.class = Man, else a.class = Woman

=

Conflict!


Solution 1: another sample

s:Student name=Mr. Lee bdate=01/01/1990

a: Woman -Actor name=Ann age = 21

Model S Model A=

WADT'12 10

if s = a, then if s.bdate.month&day current.month&day then a.age = 2012 s.bdate.year - 1, else a.age = 2012 – p.bdate.year

Conflict!


Problems of Solution 1

M1 M2

• Partially ad hoc• Or, use something universal, say, OCL or FOL, but• Network structure is collapsed into one big model • Mapping composition problem (well recognized

in the DB literature, Bernstein et al.)WADT'12

M3 …R12 R23 …

… … …R13 ?


Grand statement

• Many (all?) 1-M indirect links are nothing but hidden 1-1 Q-links:

A basic element of one model can be derived in another model and vv.

• Semantic relativism = Basic-Derived correspondence (= Q-links)

WADT'12 /30


Grand indeed?

• What’s about M-M links, or the derived-derive case?

WADT'12 /30

Documents

Intermodeling, Queries and Kleisli categories Zinovy Diskin, Tom Maibaum, Krzysztof Czarnecki McMaster University, University of Waterloo