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

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 Informática Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil 2 Dipartimento di Informatica Università di Pisa, Pisa, Italy 11th International Conference on Graph Transformation, June 2018 1/29

Introduction Characterization Properties Previous Work Initial Conflicts Conclusions Parallel Independence of Transformations G ρ 1 ρ 2 H 1 H 1 ρ 2 ρ 1 H 2/29

Introduction Characterization Properties Previous Work Initial Conflicts Conclusions Parallel Independence of Transformations G ρ 1 ρ 2 H 1 H 1 ρ 2 ρ 1 H 2/29

Introduction Characterization Properties Previous Work Initial Conflicts Conclusions Parallel Independence of Transformations G ρ 1 ρ 2 H 1 H 1 ρ 2 ρ 1 H Conflict 2/29

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 3/29

Introduction Characterization Properties Previous Work Initial Conflicts Conclusions Background: The dpo Approach l r Rule: ρ = L ֋ K ֌ R Match: m : L ֌ G ρ , m Transformation: G = ⇒ H l r L K R PO PO m k n G D H g h 4/29

Introduction Characterization Properties Previous Work Initial Conflicts Conclusions New Perspective • Previous work based on the standard condition for parallel independence R 1 K 1 L 1 L 2 K 2 R 2 r 1 r 2 l 1 l 2 q 12 q 21 n 1 m 1 n 2 k 1 m 2 k 2 H 1 D 1 G D 2 H 2 g 1 g 2 h 1 h 2 5/29

Introduction Characterization Properties Previous Work Initial Conflicts Conclusions New Perspective • Previous work based on the standard condition for parallel independence R 1 K 1 L 1 L 2 K 2 R 2 r 1 r 2 l 1 l 2 q 12 q 21 n 1 m 1 n 2 k 1 m 2 k 2 H 1 D 1 G D 2 H 2 g 1 g 2 h 1 h 2 • Recently: essential condition for parallel independence (Corradini et al. 2018) • Equivalent to standard condition 5/29

Introduction Characterization Properties Previous Work Initial Conflicts Conclusions New Perspective • Previous work based on the standard condition for parallel independence R 1 K 1 L 1 L 2 K 2 R 2 r 1 r 2 l 1 l 2 q 12 q 21 n 1 m 1 n 2 k 1 m 2 k 2 H 1 D 1 G D 2 H 2 g 1 g 2 h 1 h 2 • Recently: essential condition for parallel independence (Corradini et al. 2018) • Equivalent to standard condition • Goal: review characterization of conflicts under new light 5/29

Introduction Characterization Properties Previous Work Initial Conflicts Conclusions Background: Adhesive Categories Subobjects behave like subsets I A B a ∩ b U a b a ∪ b X 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 a ∩ b U a b a ∪ b X 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 Containment existence of mono A B a ∩ b U a b a ∪ b X 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 Containment existence of mono A B Intersection pullback a ∩ b U a b a ∪ b X 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 Containment existence of mono A B Intersection pullback a ∩ b U a b a ∪ b X 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 Containment existence of mono A B Intersection pullback a ∩ b U a b a ∪ b X 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 Containment existence of mono A B Intersection pullback a ∩ b Union pushout over intersection U a b a ∪ b X 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 Containment existence of mono A B Intersection pullback a ∩ b Union pushout over intersection U a b a ∪ b X 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 Containment existence of mono A B Intersection pullback a ∩ b Union pushout over intersection U a b a ∪ b X 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 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 6/29

Introduction Characterization Properties Previous Work Initial Conflicts Conclusions Background: Set-Valued Functor Categories • Some results not proven for all adhesive categories 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 s � raph = � et � V E � = t 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 s � raph = � et � V E � = t • Limits, colimits, monos and epis are pointwise • Always adhesive 7/29

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 8/29

Introduction Characterization Properties Previous Work Initial Conflicts Conclusions Essential Condition of Parallel Independence Corradini et al. (2018) t 1 t 2 H 1 ⇐ = G = ⇒ H 2 K 1 L 2 L 1 L 2 L 1 K 2 q 12 q 21 p 1 p 2 R 1 K 1 L 1 L 2 K 2 R 2 r 1 r 2 l 1 l 2 n 1 m 1 m 2 n 2 k 1 k 2 H 1 D 1 G D 2 H 2 g 1 g 2 h 1 h 2 9/29

Introduction Characterization Properties Previous Work Initial Conflicts Conclusions Essential Condition of Parallel Independence Corradini et al. (2018) t 1 t 2 H 1 ⇐ = G = ⇒ H 2 K 1 L 2 L 1 L 2 L 1 K 2 q 12 q 21 p 1 p 2 R 1 K 1 L 1 L 2 K 2 R 2 r 1 r 2 l 1 l 2 n 1 m 1 m 2 n 2 k 1 k 2 H 1 D 1 G D 2 H 2 g 1 g 2 h 1 h 2 9/29

Introduction Characterization Properties Previous Work Initial Conflicts Conclusions Essential Condition of Parallel Independence Corradini et al. (2018) t 1 t 2 H 1 ⇐ = G = ⇒ H 2 K 1 L 2 L 1 L 2 L 1 K 2 q 12 q 21 p 1 p 2 R 1 K 1 L 1 L 2 K 2 R 2 r 1 r 2 l 1 l 2 n 1 m 1 m 2 n 2 k 1 k 2 H 1 D 1 G D 2 H 2 g 1 g 2 h 1 h 2 9/29

Introduction Characterization Properties Previous Work Initial Conflicts Conclusions Essential Condition of Parallel Independence Corradini et al. (2018) t 1 t 2 H 1 ⇐ = G = ⇒ H 2 K 1 L 2 L 1 L 2 L 1 K 2 q 12 q 21 p 1 p 2 R 1 K 1 L 1 L 2 K 2 R 2 r 1 r 2 l 1 l 2 n 1 m 1 m 2 n 2 k 1 k 2 H 1 D 1 G D 2 H 2 g 1 g 2 h 1 h 2 • Both morphisms iso ⇒ parallel independence 9/29