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 = {n1, n2, n3, n4, n5, n6} with c(ni) =            ni ; 1 ≤ i ≤ 3 {n1, n2} ; i = 4 {n3} ; i = 5 {n2, {n1, n2}, n5} ; i = 6 Atomic nodes are called vertices V = {n1, n2, n3}.

Padberg Transformation of Corecursive Graphs 10.2.2017 3

Motivation

Example

The corecursive graph G = (N, E, c, n) given by N = {n1, n2, n3, n4, n5, n6} Atomic nodes are called vertices V = {n1, n2, n3}. E = {a, b, c, d, e} with n1 : a → {n1, n3} b → {n2, b} c → {n4, a} d → {b, c, n5} e → {n1, n3}

Padberg Transformation of Corecursive Graphs 10.2.2017 3

Motivation

Example

The corecursive graph G = (N, E, c, n) given by N = {n1, n2, n3, n4, n5, n6} Atomic nodes are called vertices V = {n1, n2, n3}. 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 is an unary arc, denoted by n2

b

• .

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 Pω 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 Pω 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}}}, ..., P3(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}}}, ..., P3(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}}}, ..., P3(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}}}, ..., P3(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}}}, ..., P3(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 P(f)(x) =

• f (x)

; x ∈ M {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 M1, M2 ∈ P(M) with n + 1 nested parentheses and M1 = M2. Let x ∈ M1 ∧ x / ∈ M2. Hence P(f)(x) ∈ P(f)(M1). x / ∈ M2 implies for all m ∈ M2 that x = m. x and m have at most n nested parentheses. P(f)(x) = P(f)(m) for all m ∈ M2 as P(f) is injective for all sets with at most n nested parentheses. Thus P(f)(x) / ∈ P(f)(M2). So, P(M1) = P(M2).

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 M1, M2 ∈ P(M) with n + 1 nested parentheses and M1 = M2. Let x ∈ M1 ∧ x / ∈ M2. Hence P(f)(x) ∈ P(f)(M1). x / ∈ M2 implies for all m ∈ M2 that x = m. x and m have at most n nested parentheses. P(f)(x) = P(f)(m) for all m ∈ M2 as P(f) is injective for all sets with at most n nested parentheses. Thus P(f)(x) / ∈ P(f)(M2). So, P(M1) = P(M2).

Padberg Transformation of Corecursive Graphs 10.2.2017 8

Node- and Edge Recursion

Lemma (P preserves pullbacks along injective morphisms)

A

πB

• πC
• (PB)

B

f1

• C

g1

D

P

¯ h

• πP(B)
• πP(C)
• P(A)

P(πB)

• P(f πC )
• (1)

h

• (2)

(3)

P(B)

P(f1)

• P(C)

P(g1)

P(D)

¯ h : P → P(A) with ¯ h((X, Y )) =      (b, c) ; if X = b ∈ B, Y = c ∈ C {(x, y) | x ∈ X ∩ B, y ∈ Y ∩ C, f1(x) = g1(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 =< IdfinSets ↓ P >. G-objects: E → P(N) G-morphisms f = (fN, fE) : G1 → G2 with:

• P(fN) ◦ c1 = c2 ◦ fN
• P(fE) ◦ n1 = n2 ◦ fE

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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

Coalgebras and M-adhesive Categories

Wanted: nice categorical construct for

c : N → P(N) and n : E → P(N ⊎ E) with morphisms f : G1 → G2 based

• n mappings of nodes and mappings of edges, so that

N1

c1 fN

• P(N1)

P(fN)

• N2

c2 P(N2)

both diagrams commute E1

n1

• fE
• P(N1 ⊎ E1)

P(fE )

• E2

n2

P(N2 ⊎ E2)

Padberg Transformation of Corecursive Graphs 10.2.2017 11

Coalgebras and M-adhesive Categories

Wanted: nice categorical construct for

c : N → P(N) and n : E → P(N ⊎ E) with morphisms f : G1 → G2 based

• n mappings of nodes and mappings of edges, so that

N1

c1 fN

• P(N1)

P(fN)

• N2

c2 P(N2)

both diagrams commute E1

n1

• fE
• P(N1 ⊎ E1)

P(fE )

• E2

n2

P(N2 ⊎ E2)

Coalgebra (see [Rutten(2000)] )

A endofunctor F : Sets → Sets gives rise the category of coalgebras SetsF with M

αM

− → F(M) – also denoted by (M, αM) – being the objects and morphisms f : (M, αM) → (N, αN) – called F-homomorphism – so that (1) commutes in Sets. M

αM f

• (1)

F(M)

F(f )

• N

αN F(N)

Padberg Transformation of Corecursive Graphs 10.2.2017 11

Coalgebras and M-adhesive Categories

Examples of Coalgebras

Let F : Sets → Sets an F-coalgebra is a pair (S, αS : S → F(S) [Rutten(2000), Adamek(2005), Jacobs(2016)]:

• finitely branching nondeteministic transition system with SetsPfin,

where (Q, αQ : Q → Pfin(Q): assigns each state q a finite collection of successor states.

• infinite binary trees over an alphabet A with F(S) = A × S × S:

given a state x ∈ S, a one-step computation yields a triple (a0, x1, x2)

• f an element a0 ∈ A and two successor states x1, x2 ∈ S. Continuing

the computation with both x1 and x2 yields two more elements in A, and four successor states, etc. This yields for each x ∈ S an infinite binary tree with one label from A at each node.

• Labelled transition systems over a signature Σ with SetsP(Σ× ).

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Coalgebras and M-adhesive Categories

Properties of SetsF

4.2.5. Proposition [Jacobs(2016)]

Assume a functor F : C → C that preserves (ordinary) pullbacks. If the category C has pullbacks, then so has the category of coalgebras CoalgF.

Lemma (Pullbacks along injections in SetsF)

Given a functor F : Sets → Sets that preserves pullbacks along an injective morphism, then SetsF has pullbacks along an injective F-homomorphism.

Padberg Transformation of Corecursive Graphs 10.2.2017 13

Coalgebras and M-adhesive Categories

Transformation System for Coalgebras

According to Prop. 4.7 in [Rutten(2000)] if f : M → N is injective in Sets then f is an F-monomorphism in SetsF. Obviously the class of all injective functions MF = {(A, αA)

f

֒ → (B, αB) | f is injective in Sets } is PO-PB-compatible.

Theorem ((SetsF, MF) is an M-Adhesive Category)

If F preserves pullbacks along injective morphisms, then (SetsF, MF) is an M-adhesive category.

Padberg Transformation of Corecursive Graphs 10.2.2017 14

Coalgebras and M-adhesive Categories

Proof Idea

1 M-POs exist as SetsF is finitely cocomplete (Thm 4.2

[Rutten(2000)]) for arbitrary F : Sets → Sets.

2 and are vertical weak VK squares

(A, αA)

m∈M f

• (B, αB)

g

• (1)

(C, αC)

n

(D, αD)

(A′, α′

A) a

• f ′
• m′

(2) (C ′, α′

C) c

• n′

(B′, α′

B) b

• g′
• (D′, α′

D) d

• (A, αA)

f

• m

(C, αC)

n

• (B, αB)

g

• (D, αD)

Since (finite) colimits and pullbacks along M-morphisms are constructed on the underlying set, square (1) and the VK-cube are given for the underlying sets in Sets as well.

Padberg Transformation of Corecursive Graphs 10.2.2017 15

Coalgebras and M-adhesive Categories

M-Transformation Systems for F-Coalgebras

• M-transformation systems for finitely branching non-deterministic

transition systems SetsPfin, where (Q, αQ : Q → Pfin(Q) as finite power set functor Pfinpreserves pullbacks along injective morphisms

• M-transformation systems for infinite binary trees SetsA× × over an

alphabet A with since the product functoer preserves limits

• M-transformation systems for labelled transition systems over a

signature Σ with SetsP(Σ× ), since the composition preserves pullback-preservation.

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Coalgebras and M-adhesive Categories

M-Transformation Systems for F-Coalgebras

• M-transformation systems for finitely branching non-deterministic

transition systems SetsPfin, where (Q, αQ : Q → Pfin(Q) as finite power set functor Pfinpreserves pullbacks along injective morphisms

• M-transformation systems for infinite binary trees SetsA× × over an

alphabet A with since the product functoer preserves limits

• M-transformation systems for labelled transition systems over a

signature Σ with SetsP(Σ× ), since the composition preserves pullback-preservation.

Padberg Transformation of Corecursive Graphs 10.2.2017 16

Edge Corecursion

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 17

Edge Corecursion

Corecursive Hyperedges

graphs with undirected edges as many sorted coalgebras using the functor F : Sets × Sets → Sets × Sets with F(V , E) = (V , E)

(!,<s,t>)

− − − − − − → (1, V × V ) where 1 is the final object and ! the corresponding final morphism.[Rutten(2000)]

Definition (Corecursive Hyperedges)

Given a set of vertices V and a set of edge names E and a function yielding the neighbouring entities n : E → P(V ⊎ E). Then the category of coalgebras CoalgF1 over F1 : Sets × Sets → Sets × Sets with F1(V , E) = (1, P(V ⊎ E) yields the category of graphs with corecursive hyperedges. The class M is given by the class of pairs of injective morphisms < fV , fE >.

Lemma ((CoalgF1, M) is an M-Adhesive Category)

Padberg Transformation of Corecursive Graphs 10.2.2017 17

Edge Corecursion

Corecursive Hyperedges

graphs with undirected edges as many sorted coalgebras using the functor F : Sets × Sets → Sets × Sets with F(V , E) = (V , E)

(!,<s,t>)

− − − − − − → (1, V × V ) where 1 is the final object and ! the corresponding final morphism.[Rutten(2000)]

Definition (Corecursive Hyperedges)

Given a set of vertices V and a set of edge names E and a function yielding the neighbouring entities n : E → P(V ⊎ E). Then the category of coalgebras CoalgF1 over F1 : Sets × Sets → Sets × Sets with F1(V , E) = (1, P(V ⊎ E) yields the category of graphs with corecursive hyperedges. The class M is given by the class of pairs of injective morphisms < fV , fE >.

Lemma ((CoalgF1, M) is an M-Adhesive Category)

Padberg Transformation of Corecursive Graphs 10.2.2017 17

Edge Corecursion

Corecursive Hyperedges

graphs with undirected edges as many sorted coalgebras using the functor F : Sets × Sets → Sets × Sets with F(V , E) = (V , E)

(!,<s,t>)

− − − − − − → (1, V × V ) where 1 is the final object and ! the corresponding final morphism.[Rutten(2000)]

Definition (Corecursive Hyperedges)

Given a set of vertices V and a set of edge names E and a function yielding the neighbouring entities n : E → P(V ⊎ E). Then the category of coalgebras CoalgF1 over F1 : Sets × Sets → Sets × Sets with F1(V , E) = (1, P(V ⊎ E) yields the category of graphs with corecursive hyperedges. The class M is given by the class of pairs of injective morphisms < fV , fE >.

Lemma ((CoalgF1, M) is an M-Adhesive Category)

Padberg Transformation of Corecursive Graphs 10.2.2017 17

Edge Corecursion

More Corecursive Edges

This can be extended to various types of corecursive edges:

• corecursive undirected edges:

vertices V , edges E and a function n : E → P(1,2)(V ⊎ E) CoalgF2 over F2 : Sets × Sets → Sets × Sets with F2(V , E) = (1, P(1,2)(V ⊎ E) together with class M is an M-adhesive category.

• corecursive directed edges:

vertices V , edges E and a function n : E → (V ⊎ E) × (V ⊎ E) CoalgF3 over F3 : Sets × Sets → Sets × Sets with F3(V , E) = (1, (V ⊎ E) × (V ⊎ E) together with class M is an M-adhesive category.

Padberg Transformation of Corecursive Graphs 10.2.2017 18

Edge Corecursion

More Corecursive Edges

This can be extended to various types of corecursive edges:

• corecursive undirected edges:

vertices V , edges E and a function n : E → P(1,2)(V ⊎ E) CoalgF2 over F2 : Sets × Sets → Sets × Sets with F2(V , E) = (1, P(1,2)(V ⊎ E) together with class M is an M-adhesive category.

• corecursive directed edges:

vertices V , edges E and a function n : E → (V ⊎ E) × (V ⊎ E) CoalgF3 over F3 : Sets × Sets → Sets × Sets with F3(V , E) = (1, (V ⊎ E) × (V ⊎ E) together with class M is an M-adhesive category.

Padberg Transformation of Corecursive Graphs 10.2.2017 18

Edge Corecursion

More Corecursive Edges

This can be extended to various types of corecursive edges:

• corecursive undirected edges:

vertices V , edges E and a function n : E → P(1,2)(V ⊎ E) CoalgF2 over F2 : Sets × Sets → Sets × Sets with F2(V , E) = (1, P(1,2)(V ⊎ E) together with class M is an M-adhesive category.

• corecursive directed edges:

vertices V , edges E and a function n : E → (V ⊎ E) × (V ⊎ E) CoalgF3 over F3 : Sets × Sets → Sets × Sets with F3(V , E) = (1, (V ⊎ E) × (V ⊎ E) together with class M is an M-adhesive category.

Padberg Transformation of Corecursive Graphs 10.2.2017 18

Edge Corecursion

More Corecursive Edges

This can be extended to various types of corecursive edges:

• corecursive undirected edges:

vertices V , edges E and a function n : E → P(1,2)(V ⊎ E) CoalgF2 over F2 : Sets × Sets → Sets × Sets with F2(V , E) = (1, P(1,2)(V ⊎ E) together with class M is an M-adhesive category.

• corecursive directed edges:

vertices V , edges E and a function n : E → (V ⊎ E) × (V ⊎ E) CoalgF3 over F3 : Sets × Sets → Sets × Sets with F3(V , E) = (1, (V ⊎ E) × (V ⊎ E) together with class M is an M-adhesive category.

Padberg Transformation of Corecursive Graphs 10.2.2017 18

Edge Corecursion

More Corecursive Edges

This can be extended to various types of corecursive edges:

• corecursive undirected edges:

vertices V , edges E and a function n : E → P(1,2)(V ⊎ E) CoalgF2 over F2 : Sets × Sets → Sets × Sets with F2(V , E) = (1, P(1,2)(V ⊎ E) together with class M is an M-adhesive category.

• corecursive directed edges:

vertices V , edges E and a function n : E → (V ⊎ E) × (V ⊎ E) CoalgF3 over F3 : Sets × Sets → Sets × Sets with F3(V , E) = (1, (V ⊎ E) × (V ⊎ E) together with class M is an M-adhesive category.

Padberg Transformation of Corecursive Graphs 10.2.2017 18

Edge Corecursion

More Corecursive Edges

This can be extended to various types of corecursive edges:

• corecursive undirected edges:

vertices V , edges E and a function n : E → P(1,2)(V ⊎ E) CoalgF2 over F2 : Sets × Sets → Sets × Sets with F2(V , E) = (1, P(1,2)(V ⊎ E) together with class M is an M-adhesive category.

• corecursive directed edges:

vertices V , edges E and a function n : E → (V ⊎ E) × (V ⊎ E) CoalgF3 over F3 : Sets × Sets → Sets × Sets with F3(V , E) = (1, (V ⊎ E) × (V ⊎ E) together with class M is an M-adhesive category.

Padberg Transformation of Corecursive Graphs 10.2.2017 18

Edge Corecursion

More Corecursive Edges

This can be extended to various types of corecursive edges:

• corecursive undirected edges:

vertices V , edges E and a function n : E → P(1,2)(V ⊎ E) CoalgF2 over F2 : Sets × Sets → Sets × Sets with F2(V , E) = (1, P(1,2)(V ⊎ E) together with class M is an M-adhesive category.

• corecursive directed edges:

vertices V , edges E and a function n : E → (V ⊎ E) × (V ⊎ E) CoalgF3 over F3 : Sets × Sets → Sets × Sets with F3(V , E) = (1, (V ⊎ E) × (V ⊎ E) together with class M is an M-adhesive category.

Padberg Transformation of Corecursive Graphs 10.2.2017 18

Edge Corecursion

More Corecursive Edges

This can be extended to various types of corecursive edges:

• corecursive undirected edges:

vertices V , edges E and a function n : E → P(1,2)(V ⊎ E) CoalgF2 over F2 : Sets × Sets → Sets × Sets with F2(V , E) = (1, P(1,2)(V ⊎ E) together with class M is an M-adhesive category.

• corecursive directed edges:

vertices V , edges E and a function n : E → (V ⊎ E) × (V ⊎ E) CoalgF3 over F3 : Sets × Sets → Sets × Sets with F3(V , E) = (1, (V ⊎ E) × (V ⊎ E) together with class M is an M-adhesive category.

Padberg Transformation of Corecursive Graphs 10.2.2017 18

Corecursive Graphs

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 19

Corecursive Graphs

Corecursive Graphs

Definition (Corecursive graphs)

G = (N, E, c : N → P(V ), n : E → P(P(N) ⊎ E)) can be considered to be an coalgebra over F : Sets × Sets → Sets × Sets with F(N, E) = (P(N), P(P(N) ⊎ E)).

Lemma ((CoalgF, M) is an M-Adhesive Category)

F preserves pullbacks along monomorphisms.

Padberg Transformation of Corecursive Graphs 10.2.2017 19

Corecursive Graphs

Overview

definition nodes edges n : E → P(N) containers have no names atomic nodes may exist hyperedges without order n : E → P(N) containers have no names every node is a container hyperedges without order c : N → P(N) n : E → P(N) containers have a name atomic nodes may exist hyperedges without order c : N → P(N) n : E → P(N)(1,2) containers have a name atomic nodes may exist undirected edges c : N → P(N) s, t : E → N containers have a name atomic nodes may exist directed edges c : N → P(N) s, t : E → (N)∗ containers have a name atomic nodes may exist directed hyperedges with an

• rder

Padberg Transformation of Corecursive Graphs 10.2.2017 20

Corecursive Graphs

Overview

definition nodes edges ! : V → 1 n : E → P(V ⊎ E)

• nly vertices

corecursive hyperedges ! : V → 1 n : E → P(1,2)(V ⊎ E)

• nly vertices

corecursive undirected edges ! : V → 1 n : E → (V ⊎ E) × (V ⊎ E)

• nly vertices

corecursive directed edges c : N → P(N) n : E → P(N ⊎ E) containers have a name atomic nodes may exist corecursive hyperedges without order

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Corecursive Graphs

Properties of Corecursive Nodes

1 Nodes are unique if c is injective. 2 Vertices are the atomic nodes that refer to themselves:

V = {n | c(n) = n}

3 Nodes are containers if c(n) ∈ P(N) − N 4 The set of nodes is well-founded if and only if

• X ∈ N ∧ Y ∈ c(X) implies, that Y ∈ c(N)
• X ∈ c(N) ∧ Y ∈ (X − N) implies, that Y ∈ c(N)

5 The set of nodes is hierarchical if and only if

• c(n) ∩ c(n′) = ∅ implies n = n′

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Corecursive Graphs

Properties of Corecursive Hyperedges

1 The set of atomic hyperedges E := {e ∈ E | n(e) ∈ P(N)}. 2 Edges are noded based if the function n+ : E → P(N) defined by

n+(e) = {n ∈ N | n ∈ n(e)} ∪

x∈n(e) n+(x) is well-defined. 3 Edges are atomic if they are noded-based and if the function

n+(E) ⊆ V only yields vertices. Analogously the properties for (un-)directed edges.

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Related Work

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

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Related Work

Related Work

• recursive graphs

A recursive graph G = (V, E) is recursive, if V , the set of vertices is a recursive subset of the natural numbers N and E, the set of edges is a recursive subset of N(2), the set of unordered pairs from N [Bean(1976), Remmel(1986)].

• bigraphs
• hierarchical graphs
• abstraction for graphs

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Related Work

Bigraph: Application Example

from [Milner(2006)]

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Related Work

Bigraph: Abstract Example

from [Milner(2006)]

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Related Work

Bigraph as a Corecursive Graph

hierarchical nodes c : N → P(N), N = {0, 1, v0, v1, v2, v3, 0, 1, 2} directed nested hyperedges s, t : E → P(N ⊎ E), E = {e1, e2, e3, e4, e5} c : 0→{v0, v2} 1→{v3, 1} v0→{v1} v1→{0} v2→v2 v3→{2} i→i;for 0 ≤ i ≤ 2 s : e1→{v1, v2, v3} y0→{v2} y1→{v2, v3} x0→{x0} x1→{x1} t : e1→{v1, v2, v3} y0→{v2} y1→{v2, v3} x0→{x0} x1→{x1}

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Related Work

Hierarchical Hypergraphs [Drewes u. a.(2002)]

Hypergraphs with order, so att : E → V ∗ Hierarchy in layers, edges within one layer < G, F, cts : F → H >∈ H with special edges F that contain subgraphs c = !, n : E → N∗ × Pω(N) so that

• edges are node-based

n : a→< xyz, ∅ > b→< nm, ∅ > c→< v2v4, ∅ > e1→< v1v2v3, {x, y, z} > e2→< v4, {n, m} >

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Related Work

Hierarchical Graphs [Busatto u. a.(2005)]

graphs are grouped into packages via a coupling graph c : N → Pω(N) being well-founded packages are the nodes that are not atomic n : E → F(N) where F determines the type of the underlying graphs N = {n, m, x, y, z, p1, p2, p3} c(v) =            v ; if v ∈ {n, m, x, y, z} {x, y, z} ; if v = p1 {n, m} ; if v = p2 {p1, p2} ; if v = p3 completeness condition: ∀n ∈ N : c(n) = n ⇒ ∃p ∈ N : n ∈ c(p)

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Discussion

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

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Discussion

Open Questions

• transformation of coalgebras??
• nomenclature??
• recursive vs corecursive
• atomic nodes vs. vertices
• edges vs. arcs/ atomic

hyperedge — edge — atomic edge = arc??

• abstraction for graphs
• F-Graphs
• M-adhesive transformation systems
• ??
• P preserving PBs along injections for arbitrary sets

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Discussion Padberg Transformation of Corecursive Graphs 10.2.2017 31

Discussion

Adamek, Jiri: Introduction to coalgebra. In: Theory and Applications of Categories 14 (2005), 157–199. http://www.tac.mta.ca/tac/volumes/14/8/14-08abs.html Bean, Dwight R.: Effective coloration. In: Journal of Symbolic Logic 41 (1976), Nr. 2, S. 469–480 Busatto, Giorgio ; Kreowski, Hans-J¨

• rg ; Kuske, Sabine:

Abstract hierarchical graph transformation. In: Mathematical Structures in Computer Science 15 (2005), Nr. 4, 773–819. Drewes, Frank ; Hoffmann, Berthold ; Plump, Detlef: Hierarchical Graph Transformation. In: J. Comput. Syst. Sci. 64 (2002), Nr. 2, 249–283. http://dx.doi.org/10.1006/jcss.2001.1790. – DOI 10.1006/jcss.2001.1790. – ISSN 0022–0000 Ehrig, H. ; Ehrig, K. ; Prange, U. ; Taentzer, G.: Fundamentals of Algebraic Graph Transformation. Springer, 2006 (EATCS Monographs in TCS) Jacobs, Bart: Introduction to Coalgebra: Towards Mathematics of States and Observation. Bd. 59. Cambridge University Press, 2016

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Discussion

J¨ akel, C.: A coalgebraic model of graphs. 2015 J¨ akel, C.: A unified categorical approach to graphs. 2015 Milner, Robin: Pure bigraphs: Structure and dynamics. In: Inf. Comput. 204 (2006), Nr. 1, 60–122. Remmel, J.B.: Graph colorings and recursively bounded 0

1-classes.

In: Annals of Pure and Applied Logic 32 (1986), S. 185 – 194. Rutten, J.J.M.M.: Universal coalgebra: a theory of systems. In: Theoretical Computer Science 249 (2000), Nr. 1, 3 - 80. Schneider, H. J.: Describing systems of processes by means of high-level replacement. In: Handbook of Graph Grammars and Computing by Graph Transformation, Volume 3. World Scientific, 1999, S. 401–450 in Computer Science), 494–501

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