Transformation of Corecursive Graphs Towards M -Adhesive Categories - PowerPoint PPT Presentation

Transformation of Corecursive Graphs Towards M -Adhesive Categories of Corecursive Graphs Julia Padberg 10.2.2017 Padberg Transformation of Corecursive Graphs 10.2.2017 1

Motivation Table of Contents 1 Motivation 2 Node- and Edge Recursion 3 Coalgebras and M -adhesive Categories 4 Edge Corecursion 5 Corecursive Graphs 6 Related Work 7 Discussion Padberg Transformation of Corecursive Graphs 10.2.2017 2

Motivation Motivation various graph types with nodes within nodes hierarchies mostly what about edges between edges? define recursion on a graph’s structure so that we still obtain an M -adhesive transformation systems. Padberg Transformation of Corecursive Graphs 10.2.2017 2

Motivation Example The corecursive graph G = ( N , E , c , n ) given by N = { n 1 , n 2 , n 3 , n 4 , n 5 , n 6 } with  ; 1 ≤ i ≤ 3 n i     { n 1 , n 2 } ; i = 4  c ( n i ) = { n 3 } ; i = 5     { n 2 , { n 1 , n 2 } , n 5 } ; i = 6  Atomic nodes are called vertices V = { n 1 , n 2 , n 3 } . Padberg Transformation of Corecursive Graphs 10.2.2017 3

Motivation Example The corecursive graph G = ( N , E , c , n ) given by N = { n 1 , n 2 , n 3 , n 4 , n 5 , n 6 } Atomic nodes are called vertices V = { n 1 , n 2 , n 3 } . E = { a , b , c , d , e } with n 1 : a �→ { n 1 , n 3 } b �→ { n 2 , b } c �→ { n 4 , a } d �→ { b , c , n 5 } e �→ { n 1 , n 3 } Padberg Transformation of Corecursive Graphs 10.2.2017 3

Motivation Example The corecursive graph G = ( N , E , c , n ) given by N = { n 1 , n 2 , n 3 , n 4 , n 5 , n 6 } Atomic nodes are called vertices V = { n 1 , n 2 , n 3 } . E = { a , b , c , d , e } Edge d is an hyperedge. All other egdes are arcs, i.e.edges with one or two incident entities. Atomics arcs are A = { a , e } . The edges b and d are not node based, since n + ( b ) and n + ( d ) remain undefined. b • . b is an unary arc, denoted by n 2 Padberg Transformation of Corecursive Graphs 10.2.2017 3

Motivation Corecursive Graphs as A corecursive graph G = ( N , E , c , n ) with • only atomic nodes and all edges are atomic arcs: an undirected multi-graph • only atomic nodes and all edges are atomic: a classic hypergraph • only atomic nodes and and all edges are layered and node-based: hierarchical graphs [Drewes u. a.(2002)] • all nodes being layered and well-founded and and all edges are atomic: hierarchical graphs [Busatto u. a.(2005)] • all nodes being hierarchical and well-founded: bigraphs [Milner(2006)] Padberg Transformation of Corecursive Graphs 10.2.2017 4

Node- and Edge Recursion Table of Contents 1 Motivation 2 Node- and Edge Recursion 3 Coalgebras and M -adhesive Categories 4 Edge Corecursion 5 Corecursive Graphs 6 Related Work 7 Discussion Padberg Transformation of Corecursive Graphs 10.2.2017 5

Node- and Edge Recursion Superpower For (finite) sets M the superpower set is achieved by recursively inserting subsets of the superpower set into the superpower set. There are two possibilities: 1 P only allows sets of nodes. 2 P ω layers the nesting of nodes. 3 P allows atomic nodes as well. Padberg Transformation of Corecursive Graphs 10.2.2017 5

Node- and Edge Recursion Superpower For (finite) sets M the superpower set is achieved by recursively inserting subsets of the superpower set into the superpower set. There are two possibilities: 1 P only allows sets of nodes. 2 P ω layers the nesting of nodes. 3 P allows atomic nodes as well. we use the last one.... Padberg Transformation of Corecursive Graphs 10.2.2017 5

Node- and Edge Recursion Superpower Definition (Superpower set P ) Given a finite set M and P ( M ) the power set of M then we define the superpower set P ( M ) 1 M ⊂ P ( M ) and P ( M ) ⊂ P ( M ) 2 If M ′ ⊂ P ( M ) then M ′ ∈ P ( M ). P ( M ) is the smallest set satisfying 1. and 2. The use of the strict subset ensures that Russell’s antinomy cannot occur. Padberg Transformation of Corecursive Graphs 10.2.2017 5

Node- and Edge Recursion Example Let M = { 1 , 2 , 3 } . Then Padberg Transformation of Corecursive Graphs 10.2.2017 6

Node- and Edge Recursion Example Let M = { 1 , 2 , 3 } . Then 1 M ⊂ P ( M ) and P ( M ) ⊂ P ( M ) P ( M ) = { 1 , 2 , 3 , ∅ , { 1 } , ..., { 1 , 2 , 3 } , { 1 , { 1 , 2 }} , {{ 1 }} , {∅ , {∅}} , P ( M ) , ..., { 1 , { 1 } , {{ 1 }}} , ..., P 3 ( M ) , ... } Padberg Transformation of Corecursive Graphs 10.2.2017 6

Node- and Edge Recursion Example Let M = { 1 , 2 , 3 } . Then 2 If M ′ ⊂ P ( M ) then M ′ ∈ P ( M ). P ( M ) = { 1 , 2 , 3 , ∅ , { 1 } , ..., { 1 , 2 , 3 } , { 1 , { 1 , 2 }} , {{ 1 }} , {∅ , {∅}} , P ( M ) , ..., { 1 , { 1 } , {{ 1 }}} , ..., P 3 ( M ) , ... } Padberg Transformation of Corecursive Graphs 10.2.2017 6

Node- and Edge Recursion Example Let M = { 1 , 2 , 3 } . Then 2 If M ′ ⊂ P ( M ) then M ′ ∈ P ( M ). P ( M ) = { 1 , 2 , 3 , ∅ , { 1 } , ..., { 1 , 2 , 3 } , { 1 , { 1 , 2 }} , {{ 1 }} , {∅ , {∅}} , P ( M ) , ..., { 1 , { 1 } , {{ 1 }}} , ..., P 3 ( M ) , ... } Padberg Transformation of Corecursive Graphs 10.2.2017 6

Node- and Edge Recursion Example Let M = { 1 , 2 , 3 } . Then 2 If M ′ ⊂ P ( M ) then M ′ ∈ P ( M ). P ( M ) = { 1 , 2 , 3 , ∅ , { 1 } , ..., { 1 , 2 , 3 } , { 1 , { 1 , 2 }} , {{ 1 }} , {∅ , {∅}} , P ( M ) , ..., { 1 , { 1 } , {{ 1 }}} , ..., P 3 ( M ) , ... } Padberg Transformation of Corecursive Graphs 10.2.2017 6

Node- and Edge Recursion Example Let M = { 1 , 2 , 3 } . Then P ( M ) = { 1 , 2 , 3 , ∅ , { 1 } , ..., { 1 , 2 , 3 } , { 1 , { 1 , 2 }} , {{ 1 }} , {∅ , {∅}} , P ( M ) , ..., { 1 , { 1 } , {{ 1 }}} , ..., P 3 ( M ) , ... } P ( M ) can be inductively enumerated by the depth of the nested parentheses provided M is finite. Padberg Transformation of Corecursive Graphs 10.2.2017 6

Node- and Edge Recursion Superpower Set P Lemma ( P is a functor) P : finSets → finSets is defined for finite sets as above and for functions f : M → N by P ( f ) : P ( M ) → P ( N ) with � f ( x ) ; x ∈ M P ( f )( x ) = { P ( f )( x ′ ) | x ′ ∈ x } ; else Lemma ( P preserves injections) Given injective function f : M → N then P ( f ) : P ( M ) → P ( N ) is injective. Padberg Transformation of Corecursive Graphs 10.2.2017 7

Node- and Edge Recursion Proof Sketch Induction over the number of nested parentheses: • P ( f ) is injective on the elements of M since f is injective. Padberg Transformation of Corecursive Graphs 10.2.2017 8

Node- and Edge Recursion Proof Sketch Induction over the number of nested parentheses: • P ( f ) is injective on the elements of M since f is injective. • Let P ( f ) be injective on the sets of P ( M ) with at most n nested parentheses. Given M 1 , M 2 ∈ P ( M ) with n + 1 nested parentheses and M 1 � = M 2 . Let x ∈ M 1 ∧ x / ∈ M 2 . Hence P ( f )( x ) ∈ P ( f )( M 1 ). ∈ M 2 implies for all m ∈ M 2 that x � = m . x / x and m have at most n nested parentheses. P ( f )( x ) � = P ( f )( m ) for all m ∈ M 2 as P ( f ) is injective for all sets with at most n nested parentheses. Thus P ( f )( x ) / ∈ P ( f )( M 2 ). So, P ( M 1 ) � = P ( M 2 ). Padberg Transformation of Corecursive Graphs 10.2.2017 8

Node- and Edge Recursion Proof Sketch Induction over the number of nested parentheses: • P ( f ) is injective on the elements of M since f is injective. • Let P ( f ) be injective on the sets of P ( M ) with at most n nested parentheses. Given M 1 , M 2 ∈ P ( M ) with n + 1 nested parentheses and M 1 � = M 2 . Let x ∈ M 1 ∧ x / ∈ M 2 . Hence P ( f )( x ) ∈ P ( f )( M 1 ). ∈ M 2 implies for all m ∈ M 2 that x � = m . x / x and m have at most n nested parentheses. P ( f )( x ) � = P ( f )( m ) for all m ∈ M 2 as P ( f ) is injective for all sets with at most n nested parentheses. Thus P ( f )( x ) / ∈ P ( f )( M 2 ). So, P ( M 1 ) � = P ( M 2 ). Padberg Transformation of Corecursive Graphs 10.2.2017 8

� � � � � � � � � � Node- and Edge Recursion Lemma ( P preserves pullbacks along injective morphisms) A � � B P π B � � π P ( B ) ¯ h π C ( PB ) f 1 h (2) � D C � � g 1 P ( A ) � � P ( B ) P ( π B ) π P ( C ) (3) P ( f π C ) (1) P ( f 1 ) � P ( D ) P ( C ) � � P ( g 1 ) ¯ h : P → P ( A ) with  ( b , c ) ; if X = b ∈ B , Y = c ∈ C   ¯ h (( X , Y )) = { ( x , y ) | x ∈ X ∩ B , y ∈ Y ∩ C , f 1 ( x ) = g 1 ( y ) } ( X ′ , Y ′ ) ∈ ( X − B ) × ( Y − C ) ¯  h ( X ′ , Y ′ ) ; else ∪ �  Padberg Transformation of Corecursive Graphs 10.2.2017 9

Node- and Edge Recursion Corecursive F -Graph [Schneider(1999), J¨ akel(2015b)] Definition (Category of corecursive graphs crFGraph ) is given by a comma category crFGraph = < Id finSets ↓ P > . G -objects: G -morphisms f = ( f N , f E ) : G 1 → G 2 with: E → P ( N ) • P ( f N ) ◦ c 1 = c 2 ◦ f N • P ( f E ) ◦ n 1 = n 2 ◦ f E Padberg Transformation of Corecursive Graphs 10.2.2017 10

Coalgebras and M -adhesive Categories Table of Contents 1 Motivation 2 Node- and Edge Recursion 3 Coalgebras and M -adhesive Categories 4 Edge Corecursion 5 Corecursive Graphs 6 Related Work 7 Discussion Padberg Transformation of Corecursive Graphs 10.2.2017 11