On the Essence and Initiality of Conflicts Guilherme Grochau Azzi 1 , - - PowerPoint PPT Presentation

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On the Essence and Initiality of Conflicts Guilherme Grochau Azzi 1 , - - PowerPoint PPT Presentation

Mar 18, 2024 240 likes •1.19k views

Introduction Characterization Properties Previous Work Initial Conflicts Conclusions On the Essence and Initiality of Conflicts Guilherme Grochau Azzi 1 , Andrea Corradini 2 and Leila Ribeiro 1 1 Instituto de Informtica Universidade Federal

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1/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

On the Essence and Initiality of Conflicts

Guilherme Grochau Azzi1, Andrea Corradini2 and Leila Ribeiro1

1Instituto de Informática

Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil

2Dipartimento di Informatica

Università di Pisa, Pisa, Italy

11th International Conference on Graph Transformation, June 2018

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2/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Parallel Independence

  • f Transformations

G H1 H1 H

ρ1 ρ2 ρ2 ρ1

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2/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Parallel Independence

  • f Transformations

G H1 H1 H

ρ1 ρ2 ρ2 ρ1

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2/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Parallel Independence

  • f Transformations

G H1 H1 H

ρ1 ρ2 ρ2 ρ1

Conflict

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3/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Motivation

  • Conflicts capture important information about behaviour
  • Enumerating potential conflicts has many applications
  • Critical pairs or initial conflicts
  • Understanding root causes is often important
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4/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Background: The dpo Approach

Rule: ρ = L

l

֋ K

r

֌ R Match: m : L ֌ G Transformation: G

ρ,m

=⇒ H L K R G D H

m l r k n

PO

g h

PO

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5/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

New Perspective

  • Previous work based on the standard condition for

parallel independence R1 K1 L1 L2 K2 R2 H1 D1 G D2 H2

n1 l1 r1 k1 m1 q12 m2 q21 k2 l2 r2 n2 g1 h1 g2 h2

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5/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

New Perspective

  • Previous work based on the standard condition for

parallel independence R1 K1 L1 L2 K2 R2 H1 D1 G D2 H2

n1 l1 r1 k1 m1 q12 m2 q21 k2 l2 r2 n2 g1 h1 g2 h2

  • Recently: essential condition for parallel independence

(Corradini et al. 2018)

  • Equivalent to standard condition
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5/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

New Perspective

  • Previous work based on the standard condition for

parallel independence R1 K1 L1 L2 K2 R2 H1 D1 G D2 H2

n1 l1 r1 k1 m1 q12 m2 q21 k2 l2 r2 n2 g1 h1 g2 h2

  • Recently: essential condition for parallel independence

(Corradini et al. 2018)

  • Equivalent to standard condition
  • Goal: review characterization of conflicts under new light
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6/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Background: Adhesive Categories

Subobjects behave like subsets I A B U X

a∩b a b a∪b

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6/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Background: Adhesive Categories

Subobjects behave like subsets Lemma (Lack and Sobocinski 2005) In adhesive categories, Sub(X) is distributive lattice I A B U X

a∩b a b a∪b

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6/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Background: Adhesive Categories

Subobjects behave like subsets Lemma (Lack and Sobocinski 2005) In adhesive categories, Sub(X) is distributive lattice Containment existence of mono I A B U X

a∩b a b a∪b

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6/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Background: Adhesive Categories

Subobjects behave like subsets Lemma (Lack and Sobocinski 2005) In adhesive categories, Sub(X) is distributive lattice Containment existence of mono Intersection pullback I A B U X

a∩b a b a∪b

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6/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Background: Adhesive Categories

Subobjects behave like subsets Lemma (Lack and Sobocinski 2005) In adhesive categories, Sub(X) is distributive lattice Containment existence of mono Intersection pullback I A B U X

a∩b a b a∪b

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6/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Background: Adhesive Categories

Subobjects behave like subsets Lemma (Lack and Sobocinski 2005) In adhesive categories, Sub(X) is distributive lattice Containment existence of mono Intersection pullback I A B U X

a∩b a b a∪b

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6/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Background: Adhesive Categories

Subobjects behave like subsets Lemma (Lack and Sobocinski 2005) In adhesive categories, Sub(X) is distributive lattice Containment existence of mono Intersection pullback Union pushout over intersection I A B U X

a∩b a b a∪b

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6/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Background: Adhesive Categories

Subobjects behave like subsets Lemma (Lack and Sobocinski 2005) In adhesive categories, Sub(X) is distributive lattice Containment existence of mono Intersection pullback Union pushout over intersection I A B U X

a∩b a b a∪b

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6/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Background: Adhesive Categories

Subobjects behave like subsets Lemma (Lack and Sobocinski 2005) In adhesive categories, Sub(X) is distributive lattice Containment existence of mono Intersection pullback Union pushout over intersection I A B U X

a∩b a b a∪b

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6/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Background: Adhesive Categories

Subobjects behave like subsets Lemma (Lack and Sobocinski 2005) In adhesive categories, Sub(X) is distributive lattice Containment existence of mono Intersection pullback Union pushout over intersection Top is X

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6/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Background: Adhesive Categories

Subobjects behave like subsets Lemma (Lack and Sobocinski 2005) In adhesive categories, Sub(X) is distributive lattice Containment existence of mono Intersection pullback Union pushout over intersection Top is X Bottom usually “empty”, if exists

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7/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Background: Set-Valued Functor Categories

  • Some results not proven for all adhesive categories
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7/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Background: Set-Valued Functor Categories

  • Some results not proven for all adhesive categories
  • We use categories et of functors → et with

natural transformations as arrows (essentially presheaves)

  • Generalizes graphs and graph structures

raph = et = V E

s t

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7/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Background: Set-Valued Functor Categories

  • Some results not proven for all adhesive categories
  • We use categories et of functors → et with

natural transformations as arrows (essentially presheaves)

  • Generalizes graphs and graph structures

raph = et = V E

s t

  • Limits, colimits, monos and epis are pointwise
  • Always adhesive
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8/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Outline

  • 1. Characterize conflict between transformations
  • 2. Useful properties of the characterization
  • 3. Compare with conflict reasons of Lambers, Ehrig, and

Orejas (2008)

  • 4. Relate to initial conflicts
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9/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Essential Condition of Parallel Independence

Corradini et al. (2018) H1

t1

⇐= G

t2

=⇒ H2 K1L2 L1L2 L1K2 R1 K1 L1 L2 K2 R2 H1 D1 G D2 H2

q12 p2 p1 q21 n1 l1 r1 k1 m1 m2 k2 l2 r2 n2 g1 h1 g2 h2

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9/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Essential Condition of Parallel Independence

Corradini et al. (2018) H1

t1

⇐= G

t2

=⇒ H2 K1L2 L1L2 L1K2 R1 K1 L1 L2 K2 R2 H1 D1 G D2 H2

q12 p2 p1 q21 n1 l1 r1 k1 m1 m2 k2 l2 r2 n2 g1 h1 g2 h2

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9/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Essential Condition of Parallel Independence

Corradini et al. (2018) H1

t1

⇐= G

t2

=⇒ H2 K1L2 L1L2 L1K2 R1 K1 L1 L2 K2 R2 H1 D1 G D2 H2

q12 p2 p1 q21 n1 l1 r1 k1 m1 m2 k2 l2 r2 n2 g1 h1 g2 h2

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9/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Essential Condition of Parallel Independence

Corradini et al. (2018) H1

t1

⇐= G

t2

=⇒ H2 K1L2 L1L2 L1K2 R1 K1 L1 L2 K2 R2 H1 D1 G D2 H2

q12 p2 p1 q21 n1 l1 r1 k1 m1 m2 k2 l2 r2 n2 g1 h1 g2 h2

  • Both morphisms iso ⇒ parallel independence
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9/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Essential Condition of Parallel Independence

Corradini et al. (2018) H1

t1

⇐= G

t2

=⇒ H2 K1L2 L1L2 L1K2 R1 K1 L1 L2 K2 R2 H1 D1 G D2 H2

q12 p2 p1 q21 n1 l1 r1 k1 m1 m2 k2 l2 r2 n2 g1 h1 g2 h2

  • Both morphisms iso ⇒ parallel independence
  • Either morphism not iso ⇒ conflict
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9/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Essential Condition of Parallel Independence

Corradini et al. (2018) H1

t1

⇐= G

t2

=⇒ H2 K1L2 L1L2 L1K2 R1 K1 L1 L2 K2 R2 H1 D1 G D2 H2

q12 p2 p1 q21 n1 l1 r1 k1 m1 m2 k2 l2 r2 n2 g1 h1 g2 h2

  • Both morphisms iso ⇒ parallel independence
  • Either morphism not iso ⇒ conflict
  • K1L2 → L1L2 not iso ⇒ t1 disables t2
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10/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Example: Conflict

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10/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Example: Conflict

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10/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Example: Conflict

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10/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Example: Conflict

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11/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Determining the Root Cause

  • Useful concept: initial pushout over f : X → Y

B X C Y

b f f c

  • “Categorical diff” for a morphism
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11/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Determining the Root Cause

  • Useful concept: initial pushout over f : X → Y

B X C Y

b f f c

  • “Categorical diff” for a morphism
  • Context c : C ֌ Y contains “modified stuff”
  • Boundary b : B ֌ C contains “points of contact”
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12/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Example: Initial Pushout

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12/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Example: Initial Pushout

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12/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Example: Initial Pushout

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13/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Conflict and Disabling Essences

Definition Given transformations (t1, t2) : H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2: B1 C1 K1L2 L1L2 R1 K1 L1 L2 K2 R2 H1 D1 G D2 H2

b1 c1 q12 p2 p1 n1 l1 r1 k1 m1 m2 k2 l2 r2 n2 g1 h1 g2 h2

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13/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Conflict and Disabling Essences

Definition Given transformations (t1, t2) : H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2: B1 C1 K1L2 L1L2 R1 K1 L1 L2 K2 R2 H1 D1 G D2 H2

b1 c1 q12 p2 p1 n1 l1 r1 k1 m1 m2 k2 l2 r2 n2 g1 h1 g2 h2

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13/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Conflict and Disabling Essences

Definition Given transformations (t1, t2) : H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2: B1 C1 K1L2 L1L2 R1 K1 L1 L2 K2 R2 H1 D1 G D2 H2

b1 c1 q12 p2 p1 n1 l1 r1 k1 m1 m2 k2 l2 r2 n2 g1 h1 g2 h2

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13/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Conflict and Disabling Essences

Definition Given transformations (t1, t2) : H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2:

  • Disabling essence for (t1, t2) is c1 ∈ Sub(L1L2)

B1 C1 K1L2 L1L2 R1 K1 L1 L2 K2 R2 H1 D1 G D2 H2

b1 c1 q12 p2 p1 n1 l1 r1 k1 m1 m2 k2 l2 r2 n2 g1 h1 g2 h2

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13/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Conflict and Disabling Essences

Definition Given transformations (t1, t2) : H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2:

  • Disabling essence for (t1, t2) is c1 ∈ Sub(L1L2)
  • Disabling essence for (t2, t1) is c2 ∈ Sub(L1L2)

B1 C1 K1L2 L1L2 R1 K1 L1 L2 K2 R2 H1 D1 G D2 H2

b1 c1 q12 p2 p1 n1 l1 r1 k1 m1 m2 k2 l2 r2 n2 g1 h1 g2 h2

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13/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Conflict and Disabling Essences

Definition Given transformations (t1, t2) : H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2:

  • Disabling essence for (t1, t2) is c1 ∈ Sub(L1L2)
  • Disabling essence for (t2, t1) is c2 ∈ Sub(L1L2)
  • Conflict essence for (t1, t2) is c = c1 ∪ c2

B1 C1 K1L2 L1L2 R1 K1 L1 L2 K2 R2 H1 D1 G D2 H2

b1 c1 q12 p2 p1 n1 l1 r1 k1 m1 m2 k2 l2 r2 n2 g1 h1 g2 h2

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14/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Example: Parallel Independence

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14/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Example: Parallel Independence

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14/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Example: Parallel Independence

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14/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Example: Parallel Independence

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14/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Example: Parallel Independence

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14/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Example: Parallel Independence

No conflict =⇒ no element caused a conflict

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15/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Empty Essences

Recall: bottom subobject generalizes “emptiness”

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15/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Empty Essences

Recall: bottom subobject generalizes “emptiness” Consider (t1, t2) : H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2

Theorem The conflict essence for (t1, t2) is ⊥ ∈ Sub(L1L2) if and only if t1 and t2 are parallel independent.

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16/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Extension

  • Same transformation in “larger context”

G H G H

f t f′ t

≡ L K R G D H G D H

m l k r n f d g h f′ g h

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16/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Extension

  • Same transformation in “larger context”

G H G H

f t f′ t

≡ L K R G D H G D H

m l k r n f d g h f′ g h

  • Lower pushouts ensure t behaves like t
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17/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Essence Inheritance

Theorem If extension diagrams below exist, (t1, t2) and (t1, t2) have the same disabling and conflict essences. H1 G H2 H1 G H2

f t1 t2 t1 t2

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17/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Essence Inheritance

Theorem If extension diagrams below exist, (t1, t2) and (t1, t2) have the same disabling and conflict essences. H1 G H2 H1 G H2

f t1 t2 t1 t2

C L1 C L2

p1◦c p2◦c

  • p1◦c

p2◦c

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17/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Essence Inheritance

Theorem If extension diagrams below exist, (t1, t2) and (t1, t2) have the same disabling and conflict essences. H1 G H2 H1 G H2

f t1 t2 t1 t2

C L1 C L2

p1◦c p2◦c

  • p1◦c

p2◦c

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17/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Essence Inheritance

In categories of set-valued functors (also graphs, typed graphs...) Theorem If extension diagrams below exist, (t1, t2) and (t1, t2) have the same disabling and conflict essences. H1 G H2 H1 G H2

f t1 t2 t1 t2

C L1 C L2

p1◦c p2◦c

  • p1◦c

p2◦c

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17/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Essence Inheritance

In categories of set-valued functors (also graphs, typed graphs...) Theorem If extension diagrams below exist, (t1, t2) and (t1, t2) have the same disabling and conflict essences. H1 G H2 H1 G H2

f t1 t2 t1 t2

C L1 C L2

p1◦c p2◦c

  • p1◦c

p2◦c

Conflicts are preserved and reflected by extension.

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18/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Disabling Reasons

Essences are not the first proposed characterization Given transformations (t1, t2) : H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2 Definition (Lambers, Ehrig, and Orejas 2008) The disabling reason L1 ← S1 → L2 for (t1, t2) Bl1 Cl1 S1 K1 L1 L2 G

l1 bl1 cl1

  • 1

s12 b∗ l1 m1 m2

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SLIDE 62

18/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Disabling Reasons

Essences are not the first proposed characterization Given transformations (t1, t2) : H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2 Definition (Lambers, Ehrig, and Orejas 2008) The disabling reason L1 ← S1 → L2 for (t1, t2) is obtained from the initial pushout over l1, Bl1 Cl1 S1 K1 L1 L2 G

l1 bl1 cl1

  • 1

s12 b∗ l1 m1 m2

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18/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Disabling Reasons

Essences are not the first proposed characterization Given transformations (t1, t2) : H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2 Definition (Lambers, Ehrig, and Orejas 2008) The disabling reason L1 ← S1 → L2 for (t1, t2) is obtained from the initial pushout over l1, then pullback of (m1 ◦ cl1, m2). Bl1 Cl1 S1 K1 L1 L2 G

l1 bl1 cl1

  • 1

s12 b∗ l1 m1 m2

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18/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Disabling Reasons

Essences are not the first proposed characterization Given transformations (t1, t2) : H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2 Definition (Lambers, Ehrig, and Orejas 2008) The disabling reason L1 ← S1 → L2 for (t1, t2) is obtained from the initial pushout over l1, then pullback of (m1 ◦ cl1, m2). Conflict condition: There is no b∗ making diagram commute. Bl1 Cl1 S1 K1 L1 L2 G

l1 bl1 cl1

  • 1

s12 b∗ l1 m1 m2

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18/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Disabling Reasons

Essences are not the first proposed characterization Given transformations (t1, t2) : H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2 Definition (Lambers, Ehrig, and Orejas 2008) The disabling reason L1 ← S1 → L2 for (t1, t2) is obtained from the initial pushout over l1, then pullback of (m1 ◦ cl1, m2). Conflict condition: There is no b∗ making diagram commute. Conflict reason is union of relevant disabling reasons. Bl1 Cl1 S1 K1 L1 L2 G

l1 bl1 cl1

  • 1

s12 b∗ l1 m1 m2

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19/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Comparing Reasons and Essences

  • Non-empty reasons exist even with parallel

independence

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19/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Comparing Reasons and Essences

  • Non-empty reasons exist even with parallel

independence

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19/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Comparing Reasons and Essences

  • Non-empty reasons exist even with parallel

independence

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19/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Comparing Reasons and Essences

  • Non-empty reasons exist even with parallel

independence

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19/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Comparing Reasons and Essences

  • Non-empty reasons exist even with parallel

independence

  • Isolated boundary nodes (Lambers, Born, et al. 2018)
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19/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Comparing Reasons and Essences

  • Non-empty reasons exist even with parallel

independence

  • Isolated boundary nodes (Lambers, Born, et al. 2018)
  • Inheritance also doesn’t hold
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20/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Essence ⊆ Reason

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20/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Essence ⊆ Reason

Remark Conflict reason determines s ∈ Sub(L1L2). L1 S L1L2 G L2

m1 s2 s1 s p1 p2 m2

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20/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Essence ⊆ Reason

Remark Conflict reason determines s ∈ Sub(L1L2). L1 S L1L2 G L2

m1 s2 s1 s p1 p2 m2

Theorem If c ∈ Sub(L1L2) is disabling essence and s ∈ Sub(L1L2) disabling reason, then c ⊆ s. The same holds if c is conflict essence and s conflict reason.

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21/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Initial Conflicts

  • We now understand individual conflicting

transformations

  • We want overview of potential conflicts for rules
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21/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Initial Conflicts

  • We now understand individual conflicting

transformations

  • We want overview of potential conflicts for rules
  • Lambers, Born, et al. (2018) proposed initial conflicts

(w.r.t extension) J1 I J2 H1 G H2 H1 G H2

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22/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Initial Conflicts

  • Initial conflicts are subset of critical pairs, often much

smaller!

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22/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Initial Conflicts

  • Initial conflicts are subset of critical pairs, often much

smaller!

  • Initial conflicts capture all conflicts ⇐⇒ every

transformation pair is extension of some initial transformation pair

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22/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Initial Conflicts

  • Initial conflicts are subset of critical pairs, often much

smaller!

  • Initial conflicts capture all conflicts ⇐⇒ every

transformation pair is extension of some initial transformation pair

  • But: no categorical construction yet
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23/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Constructing Initial Transformation Pairs

Conflict essences and initial transformation pairs are closely related (in categories of set-valued functors)

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23/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Constructing Initial Transformation Pairs

Conflict essences and initial transformation pairs are closely related (in categories of set-valued functors) Theorem Given H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2, the pushout of its conflict essence determines its initial transformation pair. C R1 K1 L1 L2 K2 R2 J1 E1 I E2 J2 H1 D1 G D2 H2

p1◦c p2◦c r1 l1 m1 m1 m2 m2 l2 r2

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23/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Constructing Initial Transformation Pairs

Conflict essences and initial transformation pairs are closely related (in categories of set-valued functors) Theorem Given H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2, the pushout of its conflict essence determines its initial transformation pair. C R1 K1 L1 L2 K2 R2 J1 E1 I E2 J2 H1 D1 G D2 H2

p1◦c p2◦c r1 l1 m1 m1 m2 m2 l2 r2

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23/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Constructing Initial Transformation Pairs

Conflict essences and initial transformation pairs are closely related (in categories of set-valued functors) Theorem Given H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2, the pushout of its conflict essence determines its initial transformation pair. C R1 K1 L1 L2 K2 R2 J1 E1 I E2 J2 H1 D1 G D2 H2

p1◦c p2◦c r1 l1 m1 m1 m2 m2 l2 r2

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SLIDE 84

23/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Constructing Initial Transformation Pairs

Conflict essences and initial transformation pairs are closely related (in categories of set-valued functors) Theorem Given H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2, the pushout of its conflict essence determines its initial transformation pair. C R1 K1 L1 L2 K2 R2 J1 E1 I E2 J2 H1 D1 G D2 H2

p1◦c p2◦c r1 l1 m1 m1 m2 m2 l2 r2

slide-85
SLIDE 85

23/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Constructing Initial Transformation Pairs

Conflict essences and initial transformation pairs are closely related (in categories of set-valued functors) Theorem Given H1

ρ1,m1

⇐= G

ρ2,m2

=⇒ H2, the pushout of its conflict essence determines its initial transformation pair. C R1 K1 L1 L2 K2 R2 J1 E1 I E2 J2 H1 D1 G D2 H2

p1◦c p2◦c r1 l1 m1 m1 m2 m2 l2 r2

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SLIDE 86

24/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Overview

Conflict Conflict Essence Conflict Reason Critical Pair Essential Critical Pair Initial Conflict

represented by unique is a has unique uniquely determines represented by unique is a is a represented by unique has unique contained in uniquely determines

Available for:

Adhesive Categories et raphT

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SLIDE 87

24/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Overview

Conflict Conflict Essence Conflict Reason Critical Pair Essential Critical Pair Initial Conflict

represented by unique is a has unique uniquely determines represented by unique is a is a represented by unique has unique contained in uniquely determines

Available for:

Adhesive Categories et raphT

slide-88
SLIDE 88

24/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Overview

Conflict Conflict Essence Conflict Reason Critical Pair Essential Critical Pair Initial Conflict

represented by unique is a has unique uniquely determines represented by unique is a is a represented by unique has unique contained in uniquely determines

Available for:

Adhesive Categories et raphT

Open Problem: in all adhesive categories?

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SLIDE 89

25/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Conclusions

  • Essential condition allowed powerful characterization for

root causes of conflicts

  • Lots of future work!
  • Constraints and application conditions
  • Compare with notions of granularity (Born et al. 2017)
  • Attributed graphs and other adhesive categories
  • Sesqui-Pushout and AGREE
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SLIDE 90

26/29 Introduction Characterization Properties Previous Work Initial Conflicts Conclusions

Thank you!

Questions?

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SLIDE 91

27/29 References Subobjects

References I

Born, Kristopher et al. (2017). “Granularity of Conflicts and Dependencies in Graph Transformation Systems”. In: ICGT.

  • Vol. 10373. LNCS. Springer, pp. 125–141. doi:

10.1007/978-3-319-61470-0_8. url: https://doi.org/10.1007/978-3-319-61470-0_8. Corradini, Andrea et al. (2018). “On the Essence of Parallel Independence for the Double-Pushout and Sesqui-Pushout Approaches”. In: Graph Transformation, Specifications, and

  • Nets. Vol. 10800. LNCS. Springer, pp. 1–18. doi:

10.1007/978-3-319-75396-6_1. url: https://doi.org/10.1007/978-3-319-75396-6_1. Lack, Stephen and Pawel Sobocinski (2005). “Adhesive and quasiadhesive categories”. In: ITA 39.3, pp. 511–545. doi: 10.1051/ita:2005028. url: https://doi.org/10.1051/ita:2005028.

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SLIDE 92

28/29 References Subobjects

References II

Lambers, Leen, Kristopher Born, et al. (2018). “Initial Conflicts and Dependencies: Critical Pairs Revisited”. In: Graph Transformation, Specifications, and Nets. Vol. 10800. LNCS. Springer, pp. 105–123. doi: 10.1007/978-3-319-75396-6_6. url: https://doi.org/10.1007/978-3-319-75396-6_6. Lambers, Leen, Hartmut Ehrig, and Fernando Orejas (2008). “Efficient Conflict Detection in Graph Transformation Systems by Essential Critical Pairs”. In: ENTCS 211, pp. 17–26. doi: 10.1016/j.entcs.2008.04.026. url: https://doi.org/10.1016/j.entcs.2008.04.026.

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SLIDE 93

29/29 References Subobjects

Notes