Transformation of Corecursive Graphs

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 n i ; 1 i 3 {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 n1 : 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 is an unary arc, denoted by n 2 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 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

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

Example Node- and Edge Recursion 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

Example Node- and Edge Recursion 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

Example Node- and Edge Recursion 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 ). x / M 2 implies for all m M 2 that x m. 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 π B B P π C C (PB) g 1 f 1 D h h π P(C) π P(B) (2) P(A) (3) P(π B ) P(B) P(f π C ) (1) P(f 1 ) h : P P(A) with P(C) P(g 1 ) P(D) (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)} h(x (X,Y ) (X B) (Y C), Y ) ; else Padberg Transformation of Corecursive Graphs 10.2.2017 9

Node- and Edge Recursion Corecursive F -Graph [Schneider(1999), Jäkel(2015b)] Definition (Category of corecursive graphs crfgraph) is given by a comma category crfgraph =< Id finsets P >. G-objects: E P(N) G-morphisms f = (f N, f E ) : G 1 G 2 with: 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

Coalgebras and M-adhesive Categories Wanted: nice categorical construct for c : N P(N) and n : E P(N E) with morphisms f : G 1 G 2 based on mappings of nodes and mappings of edges, so that c 1 n N 1 1 P(N 1 ) E 1 P(N 1 E 1 ) f N P(f N ) c 2 N 2 P(N 2 ) Coalgebra (see [Rutten(2000)] ) both diagrams commute f E P(f E ) n 2 E 2 P(N 2 E 2 ) A endofunctor F : Sets Sets gives rise the category of coalgebras Sets F with M α M F (M) also denoted by (M, α M ) being the objects and morphisms f : (M, α M ) (N, α N ) called F -homomorphism so that (1) commutes in Sets. M f N α M (1) F (M) F (f ) α N 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, α S : S F (S) [Rutten(2000), Adamek(2005), Jacobs(2016)]: finitely branching nondeteministic transition system with Sets Pfin, where (Q, α Q : Q P fin (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 (a 0, x 1, x 2 ) of an element a 0 A and two successor states x 1, x 2 S. Continuing the computation with both x 1 and x 2 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 Sets P(Σ ). Padberg Transformation of Corecursive Graphs 10.2.2017 12

Coalgebras and M-adhesive Categories Properties of Sets F 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 Coalg F. Lemma (Pullbacks along injections in Sets F ) Given a functor F : Sets Sets that preserves pullbacks along an injective morphism, then Sets F 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 Sets F. Obviously the class of all injective functions M F = {(A, α A ) f (B, α B ) f is injective in Sets } is PO-PB-compatible. Theorem ((Sets F, M F ) is an M-Adhesive Category) If F preserves pullbacks along injective morphisms, then (Sets F, M F ) is an M-adhesive category. Padberg Transformation of Corecursive Graphs 10.2.2017 14

Proof Idea Coalgebras and M-adhesive Categories 1 M-POs exist as Sets F is finitely cocomplete (Thm 4.2 [Rutten(2000)]) for arbitrary F : Sets Sets. 2 and are vertical weak VK squares (A, α A ) m M (B, α B ) f (1) (C, α C ) n (D, α D ) g (A, α A) (2) f m (C, α a C ) (B, α B) c (C, α C ) n f n (D, α D) d (A, α A ) g g m b (B, α B ) (D, α D ) 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 Sets Pfin, where (Q, α Q : Q P fin (Q) as finite power set functor P fin preserves pullbacks along injective morphisms M-transformation systems for infinite binary trees Sets A alphabet A with since the product functoer preserves limits M-transformation systems for labelled transition systems over a signature Σ with Sets P(Σ ), since the composition preserves pullback-preservation. over an 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 Coalg F1 over F 1 : Sets Sets Sets Sets with F 1 (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 < f V, f E >. Lemma ((Coalg F1, 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) Coalg F2 over F 2 : Sets Sets Sets Sets with F 2 (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) Coalg F3 over F 3 : Sets Sets Sets Sets with F 3 (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 ((Coalg F, M) is an M-Adhesive Category) F preserves pullbacks along monomorphisms. Padberg Transformation of Corecursive Graphs 10.2.2017 19

Overview Corecursive Graphs definition nodes edges n : E P(N) n : E P(N) c : N P(N) n : E P(N) c : N P(N) n : E P(N) (1,2) c : N P(N) s, t : E N containers have no names atomic nodes may exist containers have no names every node is a container containers have a name atomic nodes may exist containers have a name atomic nodes may exist containers have a name atomic nodes may exist c : N P(N) containers have a name s, t : E (N) atomic nodes may exist hyperedges without order hyperedges without order hyperedges without order undirected edges directed edges directed hyperedges with an order Padberg Transformation of Corecursive Graphs 10.2.2017 20

Overview Corecursive Graphs definition nodes edges! : V 1 n : E P(V E)! : V 1 n : E P (1,2) (V E)! : V 1 n : E (V E) (V E) c : N P(N) n : E P(N E) only vertices only vertices only vertices containers have a name atomic nodes may exist corecursive hyperedges corecursive undirected edges corecursive directed edges corecursive hyperedges without order Padberg Transformation of Corecursive Graphs 10.2.2017 21

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 Padberg Transformation of Corecursive Graphs 10.2.2017 22

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. Padberg Transformation of Corecursive Graphs 10.2.2017 23

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 Padberg Transformation of Corecursive Graphs 10.2.2017 24

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 Padberg Transformation of Corecursive Graphs 10.2.2017 24

Related Work Bigraph: Application Example from [Milner(2006)] Padberg Transformation of Corecursive Graphs 10.2.2017 25

Related Work Bigraph: Abstract Example from [Milner(2006)] Padberg Transformation of Corecursive Graphs 10.2.2017 26

Related Work Bigraph as a Corecursive Graph hierarchical nodes c : N P(N), N = {0, 1, v 0, v 1, v 2, v 3, 0, 1, 2} directed nested hyperedges s, t : E P(N E), E = {e 1, e 2, e 3, e 4, e 5 } c : 0 {v 0, v 2 } 1 {v 3, 1} v 0 {v 1 } v 1 {0} v 2 v 2 v 3 {2} i i;for 0 i 2 s : e 1 {v 1, v 2, v 3 } y 0 {v 2 } y 1 {v 2, v 3 } x 0 {x 0 } x 1 {x 1 } t : e 1 {v 1, v 2, v 3 } y 0 {v 2 } y 1 {v 2, v 3 } x 0 {x 0 } x 1 {x 1 } Padberg Transformation of Corecursive Graphs 10.2.2017 27

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 P ω (N) so that edges are node-based n : a < xyz, > b < nm, > c < v2v4, > e 1 < v1v2v3, {x, y, z} > e 2 < v4, {n, m} > Padberg Transformation of Corecursive Graphs 10.2.2017 28

Related Work Hierarchical Graphs [Busatto u. a.(2005)] graphs are grouped into packages via a coupling graph c : N P ω (N) being N = {n, well-founded m, x, y, z, p1, p2, p3} packages are the nodes that v ; if v {n, m, x, y, z} are not atomic {x, y, z} ; if v = p1 c(v) = n : E F (N) where F {n, m} ; if v = p2 determines the type of the {p1, p2} ; if v = p3 underlying graphs completeness condition: n N : c(n) = n p N : n c(p) Padberg Transformation of Corecursive Graphs 10.2.2017 29

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 Padberg Transformation of Corecursive Graphs 10.2.2017 30

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 Padberg Transformation of Corecursive Graphs 10.2.2017 30

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-Jörg ; 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 Padberg Transformation of Corecursive Graphs 10.2.2017 31

Discussion Jäkel, C.: A coalgebraic model of graphs. 2015 Jäkel, 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 Padberg Transformation of Corecursive Graphs 10.2.2017 31