Do model mappings really map On derived information in model management

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Do model mappings really map? On derived
information in model management

• Zinovy Diskin
• School of Computing, Queens University
• Kingston, Canada

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Q What is model mapping?
Schema mappings are specifications that describe
the relationships between schemas at a high
level. These specifications are typically given
in a logical formalism that captures the
interaction between schemas at a logical level
without spelling out implementation details
relevant to the physical level. - Phokion
Kolaitis. Invited Talk at ACM PODS2005

• Mapping (in the math sense) between models?
  (early MMt papers, 2000-02)
• Relation (math) between models? (2003-current)
• Relation (math) between sets of instances of the
  models? (05-current)
• Correspondence between models?
• Anything talking about anything in-between models.

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Lewis Carroll about the issue
When I use a word, Humpty Dumpty said, in a
rather scornful tone, it means just what I choose
it to mean, neither more nor less. The
question is, said Alice, whether you can make
words mean so many different things. The
question is, said Humpty Dumpty, which is to be
master - that's all. lt.gt When I make a word do
a lot of work like that, said Humpty Dumpty, I
always pay it extra.
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Humpty Dumpty Syndrome

• Patients model mapping,
• model management,
• association (in UML),
• semantics, formalization,
• Etiology informal/semi-formal interpretations
  with some elementary concepts missing.
• Treatment build an adequate formal framework,
  disassemble complex notions into elementary units
  and accurately re-assemble them again

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Modeling engineering and mathematical
Structure of Engineering Models (models and
operations/relations over them)
M-modeling
Structure of Mathematical Models
E-modeling
Model11
Model12
Model 1
Model13
Reality (say, a bridge)
Model 2
Model21
Model22


E-mechanics
M-mechanics
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Back to model mappings

• Practical situations of correspondence between
  models often involve derived info elements of
  one model correspond to elements that can be
  derived over another model rather than
  immediately belong to it.
• How to model and manage such situation in an
  intelligent way?

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An early attempt to work with derived
info(Bernstein, 2003)
m1
m1
namel-name f-name
age currentYear - b-date.year
The problem is how to compose mappings with
annotated expressions
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Solution suggested by categorical algebra

• Place operation expressions in models rather than
  in mappings. In more detail, augment models with
  required derived elements and consider mappings
  between models augmented with derived elements
  (so called Kleisly mappings).
• We will consider two examples. One is from model
  merge and the other is model translation.

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Model merge a generic pattern
Corresp. Model, R

r1 (e1 ? f1)
r12 (/e14 ? f12)
Model B
Model A
r2 (/e4 ? f2)
f1
f3
e3
e1
f12
f23
f2
q
e2
/e14
/e4
Model derQAB
Model derQBA
e3
r1
/r12
f3
f23
e2
/r2
Model derQBA ?R derQAB
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Meta-schema of merge
match

matching
merge
join
merge
normalization
normalization
Color Legend green means heuristic,
blue means automatic
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Example extracting ER-diagrams from
SQL-tables(simplified)
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Model translation (MT) MT-programming (on the
left) via PB (pull-back) (right)
Source model S Source metamodel MS Target
metamodel MT
Source model Metamodel mapping, MT ? MS
Transformation Engine
Transformation Spec (rules),
PB-algorithm
Target model
Trace mapping
Target model
Trace mapping
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MT in universal (not elementwise)
terms(specification vs. implementation)
m
T
u!
?
m
T
S
(1) Definition (T,?,m) PB(?, m)
?
?

MT
MS
m
(2) Theorem an elementwise implementation of
def(1) T (e,y)? S x MT e. ? y.m
Th. (2) gives rise to a procedure implementing
specification (1)
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MT-via-PB separation of concerns
Procedural part
Declarative part
m
derQS
T
S
PB (retyping)
?
?
algExp (query exec)
?Q
MS
derQMS
MT
m
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Does PB works? Yes, if we use proper (Kleisly)
mappings to derived elements.
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How essential are derived elements?

• Relational metamodel augmented with derived
  elements to interpret ER-metamodel.

Semantics of data is hidden in the application
code.
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Delicate issues to be addressed

• automation (algebra) and heuristics in model
  management
• does algebra do what we really need?
• basic-vs.-derived a pseudo-conflict between
  views
• derived information and normalization.

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Summary

• Kleisly mappings provide a convenient way to
  handle derived information issues in MMt
• They appear quite naturally in model match, merge
  and translation tasks
• They allow us to consider many seemingly
  different notions of model mapping in a unified
  and mathematically justified way.