Intermodeling, Queries and
Kleisli categories
Zinovy Diskin, Tom Maibaum, Krzysztof Czarnecki
McMaster University,
University of Waterloo
NECSIS and MDE
• Network for Engineering of Complex SoftwareIntensive 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
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Intermodeling
(termed coined by Guerra et al, at MoDELS’10):
• What: Specification and maintenance of intermodel
relations (techniques and tools)
• Why: Modeling is multimodeling
R12
M2
M1
R123
R13
WADT'12
R23
M3
Query languages are Cartesian monads
3
Why intermodeling, cont’d
Scenarios:
• Consistency
• Sync/change
propagation
• Merge
R12
M2
M1
R123
R13
R23
M3
There is much more to R’s
than name correspondence!
WADT'12
Query languages are Cartesian monads
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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)
WADT'12
Query languages are Cartesian monads
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Specifying intermodel relations is not easy
The Domain: Ms. Ann Lee,
01/01/1990, a student and an
actor
=
Model S
s:Student
name=Mr. Lee
?
?
bdate=01/01/1990
Model A
a: Woman -Actor
name=Ann
age = 21
?
WADT'12
Query languages are Cartesian monads
The indirect
link problem
6
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)
WADT'12
Query languages are Cartesian monads
7
Indirect links via direct Q-links
(Kleisli’s idea)
• Replace indirect links by direct links into derived
elements (computed by queries, Q-links)
Model S +
s: Student
name = Ms.Lee
bdate = 01/01/90
/age = 2012-1990
= 22
WADT'12
=
?
=
Model A
a: Actor
name = Ann
age = 22
Query languages are Cartesian monads
8
But first relate metamodels
Metamodel S+
f
Student
name: {Ms,Mr} x Str
bdate: Int3
/age = Q(bdate) : Int
Model S+: S+
S: Student
name = Ms.Lee
bdate = 01/01/90
/age = exec Q(01/01/90)
= 22: Int
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Metamodel, A
Actor
name = Str
age = Int
f: f
Model, A: A
a: Actor
name = Ann
age = 22
Query languages are Cartesian monads
13
Q-links vs. semantic relativism, 1
Metamodel S
Student
name: {Mr, Ms} x Str
?
Metamodel A
Actor
name: Str
Woman
Man
Common intermodeling jargon: 1-to-M(any) link
We say indirect 1-M link
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Query languages are Cartesian monads
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Q-links vs. semantic relativism, 2
Metamodel S
Student
name: {Mr, Ms} x Str
Metamodel A+
Actor
name: Str
/title = Q1(W-M typing)
Woman
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Query languages are Cartesian monads
Man
11
Q-links vs. semantic relativism, 3
Metamodel S
Student
name: {Mr, Ms} x Str
Metamodel A+
Actor
name: Str
/title = Q1 (W-M typing)
/tname = Q2 (name, /title)
Woman
Man
An indirect 1-M link
x <---> {y1,…, ym }
is replaced by a direct 1-1 q-link
x <---> Q(y1,…, ym)
WADT'12
Query languages are Cartesian monads
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Q-links and asymmetry of mappings
m
Metamodel S
Student
name: {Mr, Ms} x Str
Metamodel A+
Actor
name: Str
/title = Q1 (W-M typing)
/tname = Q2 (name, title)
S is a view of A
S
Woman
A
m
Man
i
A+ = Q(A)
m is an ordinary total mapping
consisting of pairs of elements.
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Query languages are Cartesian monads
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Derived-derived case. Partial mappings.
Metamodel S+
Student
name: {Mr, Ms} x Str
m
Metamodel A+
Actor
x
/title = Q2 (W-M typing)
/title = Q1(name)
Woman
x
S
i1
Man
A
m2
S+ = Q1(S)
i2
A+ = Q2(S)
m1
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Query languages are Cartesian monads
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From partial mappings to spans
(Reifying intermodel links)
Metamodel S+
Student
name: {Mr, Ms} x Str
Metamodel A+
Actor
Metamodel R
Person
title: {Mr, Ms}
/title = Q1 (W-M typing)
/title = Q1(name)
Woman
S
m1
i1
A
m2
i2
S+ = Q1(S)
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R
Man
A+ = Q2(S)
Query languages are Cartesian monads
15/36
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
(or, colors of model management)
WADT'12
Query languages are Cartesian monads
16
General pattern for merge
B
R1
A
...
…
r1
R2
C
r2
. . .
M
D
+
Color Legend:
Black -- given
Green --- discovery/heuristics/AI
Blue --- pure algebra!
FASE'12
Intermodeling, Queries, and Kleisli categories
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Merge via colimit: Example
Metamodel S+
Student
name: {Mr, Ms} x Str
R2S
Metamodel R
Person
title: {Mr, Ms}
R2A
/title = Q2 (W-M typing)
/title = Q1(name)
R2M
S2M
Metamodel A+
Actor
Woman
Metamodel M+
? (Person)
name: {Mr, Ms} x Str
Man
A2M
/title = Q1 (name)
= Q2 (W-M typing)
Woman
FASE'12
Man
Intermodeling, Queries, and Kleisli categories
23/36
Merge via colimit cont’d
Metamodel S+
Student
name: {Mr, Ms} x Str
R2S
Metamodel R
Person
title: {Mr, Ms}
R2A
/title = Q2 (W-M typing)
****
/title = Q1(name)
S2M
Metamodel A+
Actor
R2M
Woman
Metamodel M
? (Person)
name: {Mr, Ms} x Str
Woman
Man
A2M
Man
Q1 (name) =
Q2 (W-M typing)
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Another version of intermodel span
Metamodel S+
R2S Metamodel R
Student
Person
name: {Mr, Ms} x Str
title: {Mr, Ms} x Str
/title = Q1(name)
Q2(title)
/MrPerson
R2A
Metamodel A
Actor
Woman
Man
/MsPerson
Metamodel M+
? (Person)
name: {Mr, Ms} x Str
/title = Q1 (name)
Q2(/title)
/MrPerson
FASE'12
/MsPerson
20
Normalized merge
R2S Metamodel R
Metamodel S+
Student
Person
name: {Mr, Ms} x Str
title: {Mr, Ms} x Str
/title = Q1(name)
Q2(title)
/MrPerson
R2A
Metamodel A
Actor
Woman
Man
/MsPerson
Metamodel M
? (Person)
name: {Mr, Ms} x Str
FASE'12
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General pattern for merge
B
R1
A
...
…
r1
R2
C
r2
. . .
M
D
+
Color Legend:
Black -- given
Green --- discovery/heuristics/AI
Blue --- pure algebra!
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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
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Q-mappings as Kleisli mappings
Thick (extensional)
notation
A
Ordinary
mappings
A
B
f
Thin (formula)
notation
B
f
iQ
Q(B)
Q(B)
m = (Q, f)
Q-mappings
WADT'12
A
*******
B
iQ
A
Query languages are Cartesian monads
m
B
27/36
Problem of Q-mapping composition
Models and their maps
A
f
AD
Instances (data and query execs)
Subindex D stands for “the Data
part”
Typing mapping
(t is “the typing part”)
fD
Qf(AD)
At
fS
Qf(AS )
Schemas (data &
query/view definitions)
S stands for “the schema part”
Metamodels (or schemas)
WADT'12
AS
BD
fS
C
gD
CD
Qg(BD)
?
Qf(At)
AS
g
B
Ct
Bt
Qg(Bt)
BS
?
gS
CS
Qg(BS )
BS
Query languages are Cartesian monads
gS
CS
26/36
Solution: first, we manage syntax
(the easy part)
f
A
B
AD
BD
Qf(AD)
Instances (data and query execs)
Subindex D stands for “the Data
part”
typing
AS
Schemas (data &
query/view definitions)
S stands for “the schema part”
WADT'12
fD
C
gD
?
fS
g
Qg(BD)
BS
Qf (AS)
Qg#( Qf (AS))
CD
gS
Qg (BS)
f S#
Query languages are Cartesian monads
CS
Solution: second, we manage semantics
(the difficult part)
Fortunately, many useful
queries are monotonic
(e.g., SPJ)
f
A
B
AD
BD
Schemas (data &
query/view definitions)
S stands for “the schema part”
WADT'12
CD
Qg(BD)
Qg#(Qf(AD))
AS
C
gD
Qf(AD)
Instances (data and query execs)
Subindex D stands for “the Data
part”
typing
fD
g
fS
f D#
BS
Qf (AS)
Qg#( Qf (AS))
gS
Qg (BS)
f S#
Query languages are Cartesian monads
CS
Some formal details
1) 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 Qmappings 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.
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Intermodeling, Queries, and Kleisli categories
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Summary
• A (vast?) majority of intermodel relationships
can be specified by Q-mappings
• With Kleisli construction, models and Qmappings 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
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Query languages are Cartesian monads
THE END
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Query languages are Cartesian monads
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Solution 1:
Annotate indirect links with correspondence rules
(= expressions attached to links)
Model S
s:Student
name=Mr. Lee
=
Conflict!
Model A
a: Woman -Actor
name=Ann
If s = a, then
if s.name.title = Mr,
then a.class = Man,
else a.class = Woman
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Query languages are Cartesian monads
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Solution 1: another sample
Model S
s:Student
name=Mr. Lee
bdate=01/01/1990
=
Conflict!
Model A
a: Woman -Actor
name=Ann
age = 21
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
WADT'12
Query languages are Cartesian monads
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Problems of Solution 1
R12
M1
…
R23
M2
…
…
M3
…
…
R13 ?
• 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
Query languages are Cartesian monads
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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)
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Grand indeed?
• What’s about M-M links, or the derived-derive
case?
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