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Towards Typechecking for Model Transformations by Monadic 2 nd -Order Logic. Kazuhiro Inaba ( 稲葉 一浩 ) @ NII, BiG Team Nov 16, 2009 Changsha 3 rd Bi-Trans in ABC. Checking Models Every Time. Model Transformation F. Model A. Model F(A). Typechecking. Typechecking. Metamodel M1.

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towards typechecking for model transformations by monadic 2 nd order logic

Towards Typechecking for Model Transformations by Monadic 2nd-Order Logic

Kazuhiro Inaba (稲葉 一浩) @ NII, BiG Team

Nov 16, 2009

Changsha

3rd Bi-Trans in ABC

checking models every time
Checking Models Every Time

ModelTransformation

F

Model

A

Model

F(A)

Typechecking

Typechecking

Metamodel

M1

Metamodel

M2

checking transformation only once
Checking TransformationOnly Once!

ModelTransformation

F

“Any model satisfying M1 always outputs by F a model of M2!”

Checkingof

ModelTransformation

Metamodel

M1

Metamodel

M2

our approach use logic
Our Approach: Use Logic

ModelTransformation

F

Covert to

Logic

Formula

Metamodel

M2

Metamodel

M1

“output satisfies M2”

Valid! (True for any model!)

Solver

“Input and Output are related by F”

&

“input satisfies M1”

agenda
Agenda
  • From Model Transformation to Logic
    • MSO Logic, Graphs, Schemas and UnCAL
  • Validation of MSO Logic Formula
  • Conclusion and Future Work
    • Application to Bidirectional Transformation
monadic 2 nd order logic mso
Monadic 2nd-Order Logic (MSO)

connected(x,y) :=∃set P. (x ∈ P & y ∈ P & ∀u,v.(u∈P & edge(u,v)⇒ v∈P) & …)

  • MSO is a
    • Usual 1st order predicate logic
      • Boolean ops and quantifiers: ¬, ∧, ∨, ∀, ∃
    • …extended with “set-quantifications”: ∀set∃set
      • E,g,.,
graph model
Graph (Model)

a

c

a

b

c

d

  • We regard models as edge-labeled graphs

 i.e., we consider the problem of typechecking for graph transformation

schema metamodel
Schema (Metamodel)

class Customer {

reference email [1-*] : String;

reference order [0-*] : Order;

}

class Order {

reference no [1-*] : Int;

reference order_of [1-*] : Customer;

}

Only [0-*] and [1-*] is allowed

Subset of KM3 [Jouault&Bezivin ‘06]

converting schema to mso
Converting Schema to MSO

Customer

class Customer {

reference email [1-*] : String;

reference order [0-*] : Order; }

email

order

“a@b.c”

order

Order

Order

∀e: label_Customer(e) ⇒ ∃set C1, C2. outgoing(e, C1∪C2) & |C1|≧1 & ∀f∈C1. (label_email(f) & …) & |C2|≧0 & ∀f∈C2. (label_order(f) &∃g. outgoing(f, {g}) & label_Order(g))

transformation language
Transformation Language

rec(λ($L, _). // for each edge if $L = a then {d: &} // if label=a change to d else if $L = c then & // if label=c, delete

else {$L: &} // otherwise, keep unchanged)

  • UnCAL [Buneman&Fernandez&Suiciu ‘00]
    • Internal Graph Algebra of GRoundTram
    • Based on “Structural Recursion” on graphs
converting uncal to mso
Converting UnCAL to MSO

rec(λ($L, _). if $L = a then {d: &} else if $L = c then &

else {$L: &} )

d

a

b

b

d

d

ε

c

outgraph_label_d(e)⇔ ingraph_label_a(e) ∨ ingraph_label_d(e)∧outgraph_label_a(e) ⇔ false

∧outgraph_label_b(e) ⇔ ingraph_label_b(e)

∧outgraph_label_c(e) ⇔ false

∧outgraph_label_ε(e) ⇔ ingraph_label_c(e)

revisited our approach
[Revisited] Our Approach

ModelTransformation

F

Covert to

Logic

Formula

Metamodel

M2

Metamodel

M1

“output satisfies M2”

Valid! (True for any graph!)

Solver

“Input and Output are related by F”

&

“input satisfies M1”

bad news
Bad News

Valid! (True for any graph!)

Theorem [Trakhtenbrot 50]:Validness property is undecidable on graphs, even for 1st-order logic.

not so bad news
Not-so-bad News
  • MSO validness is decidable on trees. [Thatcher&Wright68]
    • Also decidable on tree-likegraphs (bounded tree-width),but it’s too tree-like
    • Models are general graphs!
good news
Good News

a

c

UnCAL Transformation

Unfolding

Unfolding

a

a

a

c

c

c

・・・ ∞

・・・ ∞

UnCAL is bisimulation-generic. [BFS ‘00]

our approach infinite trees
Our Approach:Infinite Trees

Decidable!

UnCAL and our schema do not distinguish a graph and its unfolded infinite tree

Our MSO formulas are valid (true on all graphs) ifftrue on all infinite trees

our approach infinite trees to finite trees
Our Approach:Infinite Trees to Finite Trees

c

c

c

c

・・・ ∞

c

c

c

c

c

・・・

Bad: Deciding MSO on infinite trees is costy

Good: Considering only Finite Cuts suffices

slide18

Type Correctness

Metamodel

Metamodel

Transformation

Validness of MSO Formula on Graphs

Validness of MSO Formula on Infinite Trees

Validness of MSO Formula on Finite Trees

Checked by the Well-known Solver MONA

ΦM1&ΦF⇒ΦM2

ΦM1&ΦF⇒ΦM2

Φ’M1&Φ’F⇒Φ’M2

slide19
Demo

(For showing the taste of the system)

summary
Summary

Typechecking is reduced to MSO Validness

Not by restricting to tree-like graphs, butby exploiting the Bisimulation-Genericity of UnCAL

ongoing work
Ongoing Work
  • To Finish Implementation 
    • Performance Improvements
    • Supporting larger class of translation
  • Application to Bidirectional Transformation
    • “Updatability”: the following formula defines the set of outputs having corresponding inputs

Φ’M1&Φ’F&Φ’M2