Graph Transformations

Transcription

1 Graph Transformations An Introduction to the Categorical Approach Hans J. Schneider c Hans J. Schneider, Computer Science Department, University of Erlangen-Nürnberg, Germany, 2012

2 December 28, Chapter 6 Related Topics Overview of the chapter... (still missing) 6.1 Hyperedge Replacement Grammars In Chapter 2, we have defined a hypergraph as a quadruple H = (E, V, s, t) with E, V Obj Set and s, t : E V (Definition 2.2.4) and a labeled hypergraph as a sixtuple H = (E, V, s, t, l E, l V ) with set morphisms l E : E L E and l V : V L V satisfying τ s l E = lv s E τ t l E = lv t E. (Definition ). As usual, V is the set of nodes, and E denotes the set of hyperedges, i.e., the edges of the hypergraph. L E and L V are alphabets used as edge labels and node labels, respectively. The point is that the edge label determines the number and the labels of nodes the edge must visit. For this, the alphabets L E and L V are related to one another by typing functions τ s, τ t : L E L V. At that place, we have also mentioned two variations. First, it is not necessary to distinguish between s and t. Second, many applications do not need node labels; in this case, it is sufficient to define the order of the nodes. The theory of hyperedge replacement grammars follows this line. The first approaches we are aware of date from the early seventies. J.L. Pfaltz and A. Rosenfeld call them web grammars [70], J. Feder uses the term plex grammars [40]. A first study of context-free properties has been given by T. Pavlidis [68]. Lateron, A. Habel has extensively studied this approach [46], and a comprehensive summary has been presented by F. Drewes, H.- J. Kreowski, and A. Habel [15]. Definition (Multi-pointed labeled hypergraph [15]): Let L E be an alphabet and τ : L E N 0 a typing function for L E. A multi-pointed labeled hypergraph over L E is a quintuple H = (E, V, s, l E, ext) where E and V are the sets of hyperedges and nodes, respectively, s : E V assigns a sequence of pairwise distinct nodes to each hyperedge, and l E : E L E labels each hyperedge such that τ(l E (e)) = s(e). Finally, ext V is a sequence of pairwise distinct external nodes.

3 212 December 28, 2012 We say that e is a k-hyperedge if it visits k nodes, i.e., if τ(l E (e)) = k, and that H is a k-hypergraph if it has k external nodes, i.e., ext = k. The role of the external nodes becomes clear when we define a derivation step. They play the role of the interface nodes. Please note that the sequence of external nodes may be empty. Thus, the hypergraphs of Definition are 0-hypergraphs, and therefore a special case of multi-pointed hypergraphs. Example (Well-structured programs): Let us consider a 3-hyperedge and some 3-hypergraphs taken from Example describing the well-structured programs: On the left-hand side, we see a 1 s 1 s 1 cond s 2 s 3 ::= 1 s 1 stmt s 2 s 1 cond s 2 s s 1 cond s 3 s 1 s 2 s 2 2 cond s 3 3 cond s 3 s 2 s 1 3 r stmt s 2 s 1 cond s 2 2 r s 3 Figure hyperedge labeled with cond. As in the previous chapters, we represent a hyperedge by a rectangle connected to its i-th node by an arrow s i. The other graphs are 3-hypergraphs. The sequence of external nodes ext is given by the numbers. The nodes that are not external are simply represented by dots. The diagram shows three productions with identical left-hand sides: The hyperedge on the left-hand side is to be replaced by one of the right-hand sides. Please note that the left-hand side of such a production is completely determined if we know the label of e l! In our example, it would be sufficient to write cond on the left-hand side; it is not necessary to draw all the other stuff. In the later examples, we do so. Of course, we can also define the productions in this way: Definition (Hyperedge replacement production [15]): A hyperedge replacement production of type k is a pair (A, B r ) consisting of a hyperedge label A with τ(a) = k and a labeled, multi-pointed k-hypergraph B r.

4 December 28, Informally, such a production is applied in the following way: If a hyperedge e l labeled with A occurs in some hypergraph H l, (a) we remove e l from H l, (b) we add B r to the remaining graph, (c) we connect B r to the remaining graph by identifying its external nodes with the nodes that e l has visited before being removed, where the order of the nodes is obeyed. We now give a formal definition of this construction: Definition (Hyperedge replacement [46]): Let p = (A, B r ) be a hyperedge replacement production and H l, H r be hypergraphs where H l contains a hyperedge e l with l Eh l(e l ) = A. Then, H l p = H r is a derivation step if and only if the following conditions hold: (a) HE r := (HE l \ {e l }) BE r (b) HV r := (HV l \ B V l ) BV r (s B r(e)) i if e BE r (c) (s H r(e)) i := (s H l(e)) i if e HE l \ {e l } (s H l(e)) i BV l (ext B r) j if e HE l \ {e l } (s H l(e)) i = s j (e l ) { leh l(e) if e HE (d) l EH r(e) := l \ {e l } l EB r(e) if e BE r Steps (a), (b), and (d) of the construction are obvious. Step (c) implements connecting B r to the graph that remains after removing e l from H l. The first alternative defines that hyperedges coming from B r visit the same nodes as on the right-hand side of the production. The other alternatives describe the behaviour of the hyperedges coming from the remaining part of H l. As long as a hyperedge visits a node that is part of the remaining graph, too, it continues to do so. If, however, a visited node is the j-th node s j (e l ) of the removed hyperedge e l, it has been removed by step (b), and we use the j-th external node of the hypergraph on the right-hand side instead. As in Definition 3.1.2, hyperedge replacement induces the derivability relation = as its reflexive, transitive closure, and we can define a hyperedge replacement grammar and its language analogously with Definition and Definition Definition (Hyperedge replacement grammar [15]): A hyperedge replacement grammar is given by a quadruple HRG = (L, T, P, S) with P being a finite set of hyperedge replacement productions over the alphabet L. T L is called the terminal alphabet, and S is the start symbol. A hyperedge replacement grammar is said to be of order k if for all productions (A, B r ), we have τ(a) = ext B r k. Definition (Hyperedge replacement language): If HRG is a hyperedge replacement grammar, then the set L(HRG) := {H S = HRG H l EH[H E ] T is called the hyperedge replacement language generated by HRG.

5 214 December 28, 2012 A set of hypergraphs is a hyperedge replacement language of order k if there exists a hyperedge replacement grammar of order k that generates it. To show the relationship between hyperedge replacement and the double-pushout approach, we explicitly formulate the definition of context-free hypergraph productions (Def ) without distinguishing between source and target nodes: Definition (Context-free hypergraph production): A context-free hypergraph production of order k satisfies: (a) B l E = {e l } with s l (e l ) = k (b) All the nodes of B l V occur in s l (e l ). (c) I E =, I V = {1, 2,..., k} (d) p l V is bijective and p r V is injective. Analogously, we adapt the definition of a hypergraph grammar (Def ): Definition (Context-free hypergraph grammar): A context-free hypergraph grammar of order K is a graph grammar all the productions of which are context-free hypergraph productions of order k with k K. We now discuss the relationship between hyperedge replacement and the doublepushout approach. Theorem (Hyperedge replacement as a DPO step): Let p = (A, B r ) be a k-hyperedge replacement production and H l, H r be hypergraphs with H l p = H r. Then, there exists a context-free hypergraph production p = (B l I B r ) such H l p = H r holds in the double-pushout approach. Proof: We have to construct the production p, the context graph C, and the morphisms such that the pushout property on both sides of the double-pushout diagram holds: p l B l g l H l p l p r I B r g g r C H r p r We get B l as a hypergraph consisting of one edge labeled with A and the nodes visited by this edge: BE l = {e l } l EH l(e l ) = A BV l = {(s B l(e l )) 1, (s B l(e l )) 2,..., (s B l(e l )) k } The interface graph simply consists of numbered nodes: I E = I V = {1, 2,..., k} They are mapped onto the nodes of the left-hand side and on the external nodes of the right-hand side preserving the order: p l V (i) = (s B l(e l )) i p r V (i) = (ext B r) i The context graph C is constructed from H l by removing the edge ge(e l l ), but not the nodes it visits:

6 December 28, C E = HE l \ {ge(e l l )} C V = HV l since these nodes take the role of the interface nodes: g V (i) = (s H l(ge(e l l ))) i gv l ((s B l(e l )) i ) = (s H l(ge(e l l ))) i We complete the left-hand diagram by p l V = id VC p l E(e) = e for all e C E = HE l \ {ge(e l l )} This construction results in pushout diagrams for both nodes and edges by Corollary Since this diagram is a commutative diagram in Graph, it is a pushout diagram in Graph, too (Lemma 4.5.1). On the right-hand side, the definition of hyperedge replacement (Def ) yields the set of edges and the set of nodes. We make the right-hand diagram a pushout diagram by defining: p r E(e) = e for all e C E = H l E \ {g l E(e l )} { v if H p r l V (v) = V \ BV l (ext B r) i if v = g V (i) ge(e) r = e for all e BE r gv r (v) = v for all v BV r The construction of the double-pushout production does not depend on the derivation step. We, therefore, can construct a graph grammar equivalent to the hyperedge replacement grammar. In Section 3.5, we have already considered the relationship between Chomsky languages and special hypergraph languages. That discussion has been purely based on the double-pushout approach, but of course, it is also related to hyperedge replacement. Now, we mention some results that can be easier proved in the hyperedge replacement setting. Definitions and show that Chomsky grammars can be considered as special cases of graph grammars. Those definitions represent strings by usual graphs, i.e., the edges have one source and one target. The grammar associated with a Chomsky grammar is a (1, 1)-hypergraph grammar. The definition of a contextfree Chomsky grammar assumes exactly one such edge on the left-hand side of each production. In Definition 3.5.9, we have generalized the notion of a contextfree grammar by admitting (k 1, k 2 )-hyperedges on the left-hand side. This definition allows us to show that {a n b n c n n > 0} can be generated by a contextfree hypergraph grammar of order (2, 2), although it is a typical example of a non-contextfree Chomsky language (Lemma ). It is obvious that the same result holds for a hyperedge replacement grammar of order 4. A lot of theorems well-known for contextfree Chomsky grammars and languages are valid for hyperedge replacement languages of arbitrary (finite) order, too. We mention only one of these results: the context-freeness lemma. In the Chomsky setting, it says: We have a derivation sequence consisting of n + 1 steps, i.e., A = n+1 w, if and only if there is a production p = (A, B 1 B 2... B m ) and a decomposition of w = w 1 w 2... w m such that B i = w i holds for all i = 1, 2,... m. This lemma is the basis of drawing derivation trees that represent the structure of syntactically correct strings. Extending it to contextfree hypergraph grammars, therefore, is an interesting result. 1 Since the morphisms are injective, only the first two conditions of the corollary are of interest.

7 216 December 28, 2012 F. Drewes, H.J. Kreowski, and A. Habel have shown that some other properties typical of contextfree Chomsky grammars, e.g., the so-called Pumping Lemma, also hold for contextfree hypergraph grammars [15]. There are some interesting applications that can be modeled by contextfree hyperedge replacement. We consider the Nassi-Shneiderman diagrams well-known in structured programming [65] as an example. The set of all well-formed Nassi-Shneiderman diagrams (block diagrams) is a contextfree hyperedge replacement language. M. Minas and G. Viehstaedt have used such a grammar as an example in specifying diagram editors [58, 59]. Example : Nassi-Shneiderman diagrams are defined recursively. Simple statements as well as compound statements are drawn as rectangular blocks. A block subdivided into several blocks represents a sequence of statements. A loop is described by a block to which a vertical bar is added, indicating the range of the loop. Finally, a conditional statement is represented by two columns of blocks preceded by the condition. These basic blocks are shown in Figure a. Simple statement Sequence of statements Loop condition Range of the loop Conditional statement Alternative 1 Alternative 2 Figure a: Basic blocks of Nassi-Shneiderman diagrams Figure b shows the hyperedge replacement grammar generating well-formed Nassi-Shneiderman diagrams. 2. The start symbol nsd has only to generate the start graph that on its turn, has no external nodes, since it can not be glued together with some surrounding graph. The right-hand side of the start production is a 0-pointed labeled hypergraph consisting of one 4-hyperedge labeled with stsq and three 2-hyperedges. The stsq-hyperedge is drawn as usual; its nodes are depicted by bold dots. (We have omitted the node numbers; the order is from top left to bottom right.) Three 2-hyperedges, drawn as bold lines, connect these nodes. The bold line is an abbreviation denoting a terminal label. It means that the generated diagram contains a concrete line between these two nodes. Please note that we do not generate a line at the bottom of the sequence. The next two productions determine that a statement sequence consists of one or more statements. Since the label stsq requires a hyperedge with four nodes, the 2 The conditional statement is omitted and left to the reader as an exercise (Exercise 6.5.1)

8 December 28, nsd ::= stsq stsq ::= 1 2 stat stsq stat stat ::= bstat while st 3 4 while st ::= 1 2 cond stsq 3 4 Figure b: Hyperedge replacement grammar for Nassi-Shneiderman diagrams graphs on the right-hand sides must have four external nodes. We have explicitly numbered these external nodes to distinguish them from the other nodes. The third line contains some alternatives to expand the nonterminal edges stat, e.g., to replace a stat-hyperedge by a basic statement or by a while-statement. In the case of the basic statement, we add the terminal line at the bottom, which we have omitted in the first production to avoid duplicate lines between the blocks in a statement sequence. The last production refines the while-statement. Especially, it generates the concrete lines within the block and the part of the line at the bottom that is not generated at the end of the statement sequence. Figure c shows two derivation steps. The first step replaces the hyperedge representing a derivation sequence by a statement and a statement sequence, the second step concretizes the first statement to be a while statement. Another point must be explicitly mentioned to avoid misunderstandings. Consider the first derivation step: The nonterminal hyperedge stsq is replaced by two hyperedges: stat followed by stsq. The blocks represented by these hyperedges must be suitably connected one to another. In the production, we have

9 218 December 28, 2012 stsq = stat stsq = cond stsq stsq Figure c: Beginning of a derivation realized this connection by identifying the (not-numbered) nodes 3 and 4 of the stat-hyperedge with the nodes 1 and 2 of the stsq-hyperedge, respectively. In addition, we have put these nodes onto the lines connecting the external nodes. Of course, this is not prescribed by the production. M. Minas and G. Viehstaedt, therefore, have improved the productions by attributes constraining the geometric positions of the blocks represented by the hyperedges. 3 In this case, the coordinates of the nodes involved are identified. (In other cases, the attributes may be used to determine the size of a block such that we get good proportions.) 6.2 Inward Productions The double-pushout approach follows formal language theory: A derivation step starts with looking for a handle g l : B l G l, then g l (B l ) is removed from the host graph, and finally, a copy of B r is inserted, replacing the removed part. In term graph rewriting, however, the steps occur in the reverse order: Gluing some new structure into the graph is done first, then edges are redirected, and finally garbage is removed. In 1994, R. Banach could show that this order is also possible in the double-pushout approach [4, 5]. In Banach s productions, the arrows go inwards: q = (B l q l Q qr B r ), where B l and B r are the same graphs as in the double-pushout approach and the 3 This approach follows the lines of the attributed grammars introduced by D.E. Knuth [51].

10 December 28, morphisms q l and q r are related to p l and p r by a pushout construction. The direction of the arrows in the production leads us to use the terms inward and outward referring to Banach s approach and to the original one, respectively. Under certain assumptions, each inward derivation step unambiguously corresponds to an outward derivation step. Definition (Inward production): With each outward production p = (B l production q = (B l p l I p r B r ), we associate an inward q l Q qr B r ), where Q is the pushout object in I p r B r p l B l q r P O Q q l This definition is asymmetrical: If an outward production is given, the associated inward production can always be constructed unambiguously, but not vice versa. If the inward production is given, we need a pushout complement to get the outward production. We have examined the details of this construction in Chapter 4. The results for Set and Graph are summarized in Section 4.4 and in Construction , respectively. There may be several outward productions the given inward production is associated with, or it may happen that the inward production is not associated with any outward production at all. We often use (p l, p r ) and (q l, q r ) as abbreviations for the outward and inward production, respectively. Definition (Inward derivability): q l Let q = (B l Q qr B r ) be an inward production. We call G r derivable from G l via q: G l = q G r (with handle g l : B l G l ) if and only if there exists a morphism g r : B r G r such that in the following diagram both squares are pushouts q l q Q r B l B r g l G l P O ḡ P O D g r G r A problem of the double-pushout approach is the necessity to construct the pushout complement, i.e., after determining a handle g l in the outward version, we have to look for a g : I C such that G l is the pushout object of g and p l. There is a difference between the outward and inward approach: Using the outward one, we have to construct the pushout complement in the middle. The inward approach, however, causes us to look for a pushout complement when constructing the rightmost morphism. In 1995, R. Banach could show that in the category Graph, we can replace the outward double-pushout construction by an inward one and vice versa, if the dangling condition holds (Theorem 4.5.8) and p l is injective.

11 220 December 28, 2012 Example 6.2.3: We illustrate the equivalence of outward and inward derivation steps by an example in Graph. With Banach s result in mind, we choose an outward production with injective morphisms p l and p r. Figure 6.2.3(a) shows a simple derivation step using this production. B l I B r p l p r g l g G l C G r p l p r g r Figure 6.2.3(a): Derivation step using an outward production In Figure 6.2.3(b), we construct the inward production q = (B l Q qr B r ) associated with p = (p l, p r ) as the pushout of p l and p r. Of course, the resulting morphisms q l and q r are also injective. Finally, Figure 6.2.3(c) shows the inward derivation step based on the production q = (q l, q r ). Since the morphisms are injective, the pushout complement on the right-hand side can be constructed unambiguously. We omit the details of the construction, especially checking the dangling condition. Of course, it is satisfied in the outward version as well as in the inward one. Banach has proved that under certain assumptions, outward derivation steps are equivalent to inward derivation steps. One part of his result, however, can be obtained without any restrictions as long as the category under consideration has pushouts: Theorem (From outward to inward derivations [5, 94, 95]): If we have an outward derivation step G l = p G r in a category that has pushouts, then G l = q G r does also hold with q being the inward production associated with the outward production p. Proof: We consider an outward production p = (p l, p r ) and the inward production q = (q l, q r ) associated with it, i.e., q l p l = q r p r is a pushout diagram. This is the upper part q l

12 December 28, I 1 2 B r 1 2 p r 5 6 B l p l q l Q 3 1 q r Figure 6.2.3(b): Construction if the inward production (q l, q r ) of the figure. We assume that G r is derivable from G l by applying the outward production p with the handle g l, i.e., the squares g l p l = p l g and g r p r = p r g are pushout diagrams, too: We complete the lower part of the cube by constructing D as the pushout object in the square q l p l = q r p r. Now, the outermost edges establish a commutative diagram: ( q l g l ) p l = q l p l g = q r p r g = ( q r g r ) p r and therefore, the pushout property of Q yields a unique ḡ making q l g l = ḡ q l and q r g r = ḡ q r commutative. To show the pushout property of these squares, we use the composition lemma (Lemma ): With g r p r = p r g and q r p r = q l p l being pushouts, ( q r g r ) p r = q l ( p l g) also is a pushout. Up to isomorphism, (ḡ q r ) p r = q l (g l p l ) is the same pushout. Its second half, namely ḡ q l = q l g l, is a pushout, too, due to part (b) of the composition lemma. Analogously, we see that q r g r = ḡ q r is a pushout. Putting these pushouts together, we get the inward derivability via q. Please note that this proof does not make use of any special properties of the category under consideration as long as it has pushouts. If the outward derivation step exists, the associated inward production can also be applied. In Graph, this means that if the dangling condition holds in the outward derivation step, then the dangling condition needed in constructing the inward derivation step is satisfied, too! Furthermore, we have not assumed any special properties of the morphisms, especially, they need not be injective. In the case of a non-injective p l, we can apply the outward production if the pushout complement of g l p l exists, but C and G r may be ambiguous. Theorem says that all the graphs G r derivable by the outward production can also be derived by the associated inward production, i.e., the existence of the pushout complement of ḡ q r implicitly follows from the existence of the pushout complement of g l p l! Conversely, it is easy to prove that every inward derivation step corresponds to an

13 222 December 28, 2012 B l Q B r q l q r g l ḡ G l D G r q l q r g r Figure 6.2.3(c): Derivation step using an inward production I pl B l q l g Q p r B r q r g r g l ḡ p r C p l q r D q l G l G r Figure 6.2.4: Relationship between outward and inward derivation step outward one under more or less strong assumptions. In general, we need only that existence of pushout complements is inherited from the inward part of the cube in Figure to the outward part (Exercise 6.5.4). In Set and Graph, it is sufficient to assume the top square of the cube to be a bicartesian square (Exercise 6.5.5). This idea can also be used to prove the equivalence in an adhesive grammar [19]. Authors preferring HLR1-categories usually assume both p l and p r to be in the distinguished class M. (The same holds true for the cospan approach discussed by H. Ehrig, F. Her-

14 December 28, mann, and U. Prange [28]). But we can prove the equivalence assuming only p l to be in the distinguished class of morphisms. As in Chapter 5, we consider the class of PIT-categories (see Def ): Theorem (Equivalence of inward and outward derivations): If in a PIT-category, p is an outward production with p l M, q is the associated inward production and if we have a handle g l : B l G l, then G l = p G r holds if and only if G l = q G r. Proof: The direction from left to right is a consequence of Theorem Conversely, we assume G l = q G r, i.e., three of the pushouts in the cube of Figure exist: q r p r = q l p l due to the definition of associated productions and ḡ q l = q l g l as well as ḡ q r = q r g r because of the assumption (Figure 6.2.5a). p r I pl P O q r B l Q g l G l q l P O q l ḡ D Figure 6.2.5a P O q r B r g r G r p r B r q r I pl B l Q q l g = = ḡ p r C p l G l P B q r D q l g l Figure 6.2.5b g r G r Please keep in mind that q r and q r are in M since p l is by definition of a PIT-category. Now, we construct C as the pullback object of q l and q r (Figure 6.2.5b), and we get a unique morphism g : I C such that g r p r = p r g and g l p l = p l g commute. We have to show that these two diagrams are pushouts. By the composition lemma, the diagram q r (g r p r ) = (ḡ q l ) p l is a pushout diagram, but q r ( p r g) = ( q l g l ) p l is the same diagram. Analogously, q l ( p l g) = ( q r g r ) p r is a pushout diagram. In Figure 6.2.5c, therefore, the large rectangles are pushouts, the right-hand corner I B r I C G r B l = = P B q r G l D Figure 6.2.5(c) is a pullback and q r is in the class of distinguished morphisms. Condition (c) in the definition of a PIT-category completes the proof.

15 224 December 28, 2012 This theorem essentially coincides with Banach s result [4, 5]. Trivially, it also holds in the case that both sides of the given outward production are in the distinguished class of morphisms. If we require the right-hand side to be in M, but allow the left-hand side to be an arbitrary morphism, the symmetry of the cube in Figure yields: Corollary 6.2.6: If in a PIT-category, p is an outward production with p r M, q is the associated inward production, and if we have a handle g l : B l G l, then G l = p G r holds if and only if G l = q G r. In this case, the outward derivation step allows ambiguous solutions, but all the solutions can be derived by inward derivation steps, too. The advantage of our proof in a purely categorical setting is that we can apply it to more general structures such as hierarchical graphs [92], or structurally labeled graphs [99, 100]. Lemma gives a criterion which restrictions a structured alphabet must obey such that the structurally labeled graphs constitute a PIT-category and the equivalence theorem holds. At this point, we switch over to an interesting operational approach to graph transformations. These approaches describe replacing a subgraph with another one by set operations that can be easily implemented. Such an approach is the -grammar approach, which has been introduced by S.M. Kaplan, J.P. Loyall, and S.K. Goering [50]. In this approach, all relevant items of a production are incorporated into one tripartite graph: Definition ( -production [50]): A -production is a tripartite graph = ( l E c E r E, l V c V r V, s, t) with s, t : l E c E r E l V c V r V where the following conditions hold: (a) e l E s(e) r V t(e) r V (b) e r E s(e) l V t(e) l V (c) e c E s(e) c V t(e) c V The definition excludes edges from l to r or vice versa. 4 Usually, c is depicted within a triangle, and l and r are drawn on the left and on the right of this triangle, respectively. Example 6.2.8: We consider a simple example. Figure depicts the -version of the production we have already used in Figure 6.2.3(a). c consists of the nodes within the triangle and the edges not leaving it. In our example, the edge from node 1 to node 2 belongs to c. All the edges a part of which is drawn outside the triangle belong to l or to r, respectively, i.e., the edge from 2 to 1 is part of l and the loop at node 1 belongs to r. The edges connecting nodes one of which is in l or in r belong to l or to r, respectively, e.g., the edge from 1 to 3 is part of l. 4 x is shorthand for the edges x E and the nodes x V.

16 December 28, l c r 2 6 Figure 6.2.8: Example of a -production By definition c is a graph. In general, however, l and r are not graphs, but Def ensures that l c and c r are graphs if we restrict s, t to l E c E and r E c E, respectively. l is the fragment of the graph removed during applying the production, r is the graph fragment replacing l. c denotes a subgraph that is identified, but not changed during rewriting. It is common to both the left-hand side and the right-hand side of the production. 5 Definition ( -derivation [50]): If a -production is given and if we have an injective graph morphism g l : ( l E c E, l V c V, s p, t p ) G l where s p, t p are the restrictions of s, t to l E c E, then G r is called derivable from G l with : G l = G r if G r is constructed as follows: G r E := (G l E \ g l [ l E]) r E G r V := (G l V \ g l [ l V ]) r V s G l(e) if e G l E \ g l [ l E] s G r(e) := s(e) if e r E s(e) r V g l (s(e)) if e r E s(e) c V t G r(e) analogously Constructions of s G r and t G r ensure that G r is a graph. Theorem (Equivalence of -derivability with double-pushout approach): In the case of injective handles -derivability and double-pushout derivability are equivalent. Proof: Since in Set and Graph, the coproduct is the disjoint union, we can translate a - production into an outward production straightforwardly 6 : p = ( l c p l c p r c r ) 5 Kaplan s productions have two further components, a negative application condition and a textually expressed guard; both are omitted here. 6 l c denotes the coproduct of l and c. Please remember that this operation is associative.

17 226 December 28, 2012 with p l and p r being the natural coproduct morphisms. Since in a -production, p l and p r are injective, l and r are the coproduct complements of c with respect to l c and c r, respectively. Figure illustrates the situation. Since - derivability assumes g l to be injective, g l : l c G l has a coproduct complement R G l, as we have shown in Example We can, therefore, rewrite G l as a coproduct: G l = ( l c ) R. By Lemma 4.2.5, the subdiagram (1) is a pushout if l c p l c p r c r g l (1) (2) ( l c ) R C ( c r ) R R Figure g r we choose C = c R. By the same argument, subdiagram (2) is a pushout. On the other hand, we can rewrite the pushout object of (2) as ( c r ) R = ( c R) r. In a set-theoretical formulation, c R is G l \ l, and ( c R) r is (G l \ l ) r. This completes the proof. (Definitions of s G r and t G r are straightforward.) We have embedded the -approach into the double-pushout approach. Why we have discussed this equivalence in the context of inward productions? The answer is simple: Corollary : Consider a -production, and translate it into an outward production. Then, the graph Q in the associated inward production (B l Q qr B r ) is the tripartite graph representing the given -production: Q = l c r. q l c p r r c p r q r q l l c l c r Figure Figure refers to the canonical construction of pushouts (Theorem 2.5.6): After the coproduct ( l c ) ( r c ) has been constructed, the coequalizer identifies the elements of c. This means that we can implement the double-pushout approach (at least in the case of injective productions) storing only one graph, namely Q indicating the partitioning by suitable flags. The original definition of -derivability allows only injective handles g l. The categorical treatment, however, shows that this assumption is too strong:

18 December 28, Lemma : A -production can also be applied using a noninjective handle g l identification condition. satisfying the If we do so, we have to observe the identification condition (Lemma 4.1.2): Only elements of c may be put together. Proof: We replace Figure by Figure (a) taking advantage of Corollary instead of Lemma By the identification condition, g l p is injective, and we can p l l c p l g l p g l p l l C c Figure (a) g C p r c r g r G r construct the pushout complement p l : C l C as the coproduct complement of g l p, i.e., p l is the natural injection. Furthermore, the identification condition ensures g l [ c ] C. The morphism g l, therefore, can be written as the coproduct of two morphisms (cf. Def ): g l = g l g l l c, and we get g = gl as shown in c Figure (b). On the right-hand side, we symmetrically get G r = C r. In the in l l g l l l c in c c g l l g l c l C g l c in C C Figure (b) p r c r g r G r set-theoretic notation, this means that G r = C r = (G l \ g l [ l ]) r in accordance with the definition of -derivability. Finally, we have to show that the definition of s G r in Definition does not depend on the choice of edges that are put together. We consider e, e c E r C with g r (e) = g r (e ). Both e and e must come from c and put together by g l c, i.e., gl E(e) = ge(e l ), implying gv l (s(e)) = s G l(ge(e)) l = s G l(ge(e l )) = gv l (s(e )). Although the -grammar approach is explixitly based on the outward version of the double-pushout approach (cf. [50]), it may be better understood considering the inward version. Kaplan s -approach is very similar to Göttlers Y-model [43, 44], which has been extended to the X-model lateron [45].

19 228 December 28, Comma Category Concept In the previous chapters, we have studied a large variety of graphs and different labeling mechanisms. In each case, we had to prove that the variation under consideration constitutes a category satisfying the properties we need. Now, we show that it is sufficient to prove interesting properties only for some basic categories, since we can lift them to other categories by the comma-category technique. 7 For this, we need functors, i.e., maps between categories that preserve the category structure. Definition (Functor [48]): A functor F : A B from category A to category B is a pair of functions F O : Obj A Obj B and F M : Mor A Mor B preserving identities and composition: (a) F O (id a ) = idf(a) (b) Whenever g f is defined, the equality F M (g f) = F M (g) F M (f) holds. The functions F M and F O cannot be confused with one another; therefore, we usually omit the indices. Remark: Since there is a one-to-correspondence between objects and identities, it would be sufficient to define a functor by its F M. From this, we get F O by F O (a) = dom(f M (id a )). Some examples, however, are easier to comprehend if F O is given explicitly. A comma category is constructed from two functors with a common target: Definition (Comma category [42, 81]): Let A, B, C be categories and let L : A C and R : B C be two functors. Then, the comma category (L, R) is defined as follows: (a) The objects of (L, R) are of the form (a, b, c) with a Obj A, b Obj B and c : L(a) R(b) being a morphism of C. (b) A morphism f : (a, b, c) (a, b, c ) of (L, R) is a pair of morphisms f = (f l : a a, f r : b b ) with f l Mor A, f r Mor B such that the following diagram commutes: L(a) c R(b) L(f l ) = R(f r ) L(a ) R(b ) c (c) Composition and identities are defined componentwise. An object of the comma category (L, R) takes an object of A and an object of B and establishes a relationship between their images in C by a suitable morphism. We illustrate this by defining the category of graphs as a comma category. Informally, A represents the edges, B the nodes, and the morphisms of C define source and target nodes. The example is a little bit too simple because all three categories are the same category, namely Set: 7 From a systematic point of view, it seems better to discuss this approach in Chapter 2. Then, we can take advantage of it in the following chapters. For didactic reasons, however, that chapter has been kept as short as possible.

20 December 28, Example (Graph): Consider the identity functor Id : Set Set and the diagonal functor X : Set Set defined by Id(S) := S X(S) := S S Id(f) := f X(f)((a, b)) := (f(a), f(b)) The comma category (Id, X) is the category of graphs. The comma-category definition considers a graph to be a triple G = (E, V, c) with c : E V V, where E is the set of edges and V is the set of nodes. Function c connects each edge with a source and a target node: c(e) = (s(e), t(e)). As usual, a graph morphism is a pair (g E : E E, g V : V V ). On the left-hand side Id(E) c X(V ) Id(g E ) = X(g V ) g E = (g V, g V ) Id(E ) X(V ) c E V V c Figure 6.3.3: Comma category interpretation of graph morphisms E c V V of the figure, we show the diagram according to the definition of a comma category, and on the right-hand side, you see the interpretation with Id and X replaced by their definitions. In the previous chapters, we have discussed different versions of hypergraphs. Here, we use the version well-suited to represent term graphs. For this, we modify Definition such that each edge visits at least one target: Example (Hypergraphs): Let C : Set Set be the functor defined by C(S) := S S + C(f)((w, w )) := (f (w), f + (w )) Then, comma category (Id, C) is the category of hypergraphs with (a possibly empty sequence of) sources and at least one target. Hence, a hypergraph is a triple H = (E, V, c) with c : E V V +, and a hypergraph Id(E) c C(V ) c E V V + Id(g E ) = C(g V ) g E = (gv, g+ V ) Id(E ) C(V ) c E V V + c Figure 6.3.4: Interpretation of hypergraph morphisms morphism is pair (g e : E E, g V commutative. : V V ) making the diagram in the figure

21 230 December 28, 2012 Petri nets are based on bipartite graphs. Defining this variant of graphs, we need the product A B of two categories. The set of objects is Obj A Obj B, and the morphisms are constructed componentwise. Example (Bipartite graphs): Let B : Set Set Set be the functor defined by B(S, T ) := S T T S { (f(u), g(v)) if u S, v T B(f, g)(u, v) := (g(u), f(v)) if u T, v S Then, comma category Bgraph = (Id, B) is the category of bipartite graphs. From this, a bipartite graph is a graph (E, (S, T ), c) with c : E S T T S. A Id(E) c B(S, T ) Id(g E ) = B(g S, g T ) Id(E ) B(S, T ) c E c S T T S g E = B(g S, g T ) E S T T S c Figure 6.3.5: Interpretation of morphisms in Bgraph morphism is a pair of morphisms the second component is a pair again: (g e : E 1 E 2, (g S : S 1 S 2, g T : T 1 T 2 )) mapping places to places and transitions to transitions. The comma category concept unifies not only treating different graph structures, but also the different labeling mechanisms. On the one hand, we can distinguish between labeling only nodes, only edges or both nodes and edges. On the other hand, we can use a simple labeling alphabet assuming morphisms to preserve labels (Def ) or we may impose a structure on the alphabet requiring morphisms to be compatible with this structure (Def ) or we may even use graphs as labels again [92]. First, we introduce selecting functors that decide on whether nodes or edges are to be labeled. We slightly modify Schied s definition [82]. For this, we define the category the objects of which are the Cartesian products of sets: Definition (Category of pairs): The category CartSet has Obj Set Obj Set as objects, and Mor Set Mor Set as morphisms. Composition and identities are defined componentwise. Of course, limits and colimits can be also constructed component by component. Definition (Selected labeling): Let G be any category of graphs and W E, W V : G Set, W EV : G CartSet selecting functors defined as follows: W E ((E, V, c)) := E W E ((f E, f V )) := f E W V ((E, V, c)) := V W V ((f E, f V )) := f V W EV ((E, V, c)) := (E, V ) W EV ((f E, f V )) := (f E, f V )

22 December 28, Consider a category L of labels and a labeling functor Lab : L Set. Then, (W E, Lab) yields a category of edge-labeled graphs, (W V, Lab) yields a category of node-labeled graphs. With Lab : L CartSet, comma category (W EV, Lab) yields a category of graphs both the nodes and edges of which are labeled. The domain of W selects a kind of graphs such as usual graphs, hypergraphs, or bipartite graphs. The codomain defines the component to be labeled, e.g., nodes, edges, or both. In detail, a Lab-labeled graph is a triple (G, L, l), where G = (E, V, c) is a graph, L Obj L is an object of the labeling category and l is a labeling function. The figure depicts the case of node labeling. We see the definition on the left-hand W V (G) l Lab(L) W V ((f E, f V )) = Lab(...) f V = Lab(...) W V (G ) Lab(L ) l V Lab(L ) l Figure 6.3.7: Interpretation of a selecting functor V l Lab(L) side and a draft of the interpretation on the right-hand side. We hesitate to make the morphism Lab(...) concrete, since it depends on the way we label the graphs. We first consider the case of morphisms preserving the labels: Example (Node-labeled graphs): Let 1 be the category consisting of one object and one morphism id. Consider functor Lab : 1 Set defined by Lab( ) := L V Lab(id ) := id LV Then, (W V, Lab) is the category of node-labeled graphs with the labeling alphabet L V. Replacing Lab in Figure by this definition, we get the well-known diagram of Definition W V (G) W V ((f E, f V )) W V (G ) l V = l V Lab( ) Lab(id ) Lab( ) V l V f V = L V V l V Figure 6.3.8: Interpretation of node-labeled graph morphisms Analogously, you can define the category of edge-labeled graphs by simply replacing W V and L V by W E and L E, respectively. Labeling both nodes and edges, we must take into consideration that the labeling alphabets for nodes and edges usually are different.

23 232 December 28, 2012 Example (Node- and edge-labeled graphs): Let 1 as before and consider the functor Lab : 1 CartSet defined by Lab( ) := (L E, L V ) Lab(id ) := id (LE,L V ) Then, (W EV, Lab) is isomorphic to the category Lgraph with L E and L V labeling alphabets. as the Again, we illustrate the construction by two diagrams (Fig (a)). By definition, W EV (G) W EV ((f E, f V )) W EV (G ) l V = l V Lab( ) Lab(id ) Lab( ) (l E, l V ) (E, V ) (L E, L V ) (f E, f V ) = (id LE, id LV ) (E, V ) (l E, l V ) (E, V ) Figure 6.3.9(a): Comma-category interpretation of Lgraph V l V f V = L V V l V Figure 6.3.9(b) E l E f E = E l E L E the objects of (W, Lab) are of the form (G, (L E, L V ), (l E, l V )). Figure 6.3.9(b) shows the right-hand diagram decomposed into its components. We get the same diagrams as in Def In G. Schied s version of Definition 6.3.7, all the functors have the codomain Set. He has defined W EV ((E, V, c)) := E V and W EV ((f E, f V )) := f E f V. 8 As a consequence of this, he gets Lab( ) := L E L V, and he must ensure that l also is of the form l E l V. In [97], we have proposed another approach. We introduce a two-step comma-category construction. Without loss of generality, we start with labeling edges using W E and Lab E : L E Set. Objects of (W E, Lab E ) are of the form (G, L E, l E ) = ((E, V, c), L E, l E ) and morphisms are of the form ((f E, f V ), f LE ), where the first components (f E, f V ) are graph morphisms. Although (W E, Lab E ) is a category of graphs, we must slightly modify W V to access the nodes: Definition (Two-step labeling): Let Lab E : L E Set and Lab V : L V Set be two labeling functors. With W V : (W E, Lab E ) Set defined by W V (G, L E, l E ) = W V (G) W V (f, f LE ) = W V (f), we construct the comma category (W V, Lab V ) and denote the result by (W E, W V, Lab E, Lab V ), explicitly mentioning the four morphisms. 8 f E f V : E V f E f V is the function f E on E and is the function f V on V.

24 December 28, The objects of this category are of the form (((E, V, c), L E, l E ), L V, l V ) and morphisms look like (((f E, f V ), f LE ), f LV ). We can easily establish a close relation to the previous version substituting 1 for both L E and L V : ((E, V, c), L E, l E ), L V, l V ) ((E, V, c), (L E, L V ), (l E, l V )) ((f E, f V ), f LE ), f LV ) ((f E, f V ), (f LE, f LV )) Up to now, we have considered only the case of label-preserving morphisms. Example (Substituting labels): Let (L, ) be a structured alphabet and CL L the category with L as the only object and all nondecreasing functions h as morphisms, i.e., a h(a) for all a L. Consider the functor CL : CL L Set defined by CL(L) := L CL(h) := h Then, CLgraph = (W V, CL) is a category of node-labeled graphs the morphisms of which are allowed to change labels observing the structure of the alphabet. We illustrate this example by labeling the nodes with elements of an alphabet L V. The right-hand diagram shows that each label a of a node is replaced by h(a) on l W V (G) V CL(L V ) W V (f) = SL(h) f V = W V (G ) CL(L V ) l V V l V Figure : Morphisms substituting labels V l V L V h L V the image of the node. Contrary to Definition 3.6.5, however, all occurrences of a label must be replaced by the same label! This property meets the requirements in constructing derivation steps on term graphs (Definition 3.7.3), even if the terms are not normalized. Some other examples such as the problem of the dining philosophers can not be modeled in this way. Example (Dining Philosophers): The figure recapitulates one of the productions we have considered in Example to illustrate derivability in SLgraph, with the subset relation as the strucure B l I B r p 2 ps satisf ied b pt p l Figure p r ps satisf ied b pt f 2, f 3 p 2 of the alphabet. A morphism in this category allows a node with label to be

25 234 December 28, 2012 mapped onto a node with label {p 2 } and another node with the same label onto a node with label {f 2, f 3 }, as you can see on the right-hand side of the production. We use a trick due to Hess and Mayoh [49] and extend the labels by adding indices from an (arbitrary) index set I. We define h : I M(S) I M(S) to be nondecreasing if the second component is: Example (Structurally labeled graphs [97]): Let (L, ) be a structured alphabet, I a set of indices, and SL IL the category with I L as the only object and all functions h with h(i, l) = (i, l ) implying l l as morphisms. Consider the functor SL : SL IL Set defined by SL(I L) := I L SL(h) := h Then, SLgraph = (W V, SL) is the category of structurally labeled graphs. If two nodes are labeled with the same index and the same label, then the image nodes must have identical labels. If the indices are different, then the image nodes may (but need not) have different labels. What about the index set? A simple choice is the set of nodes itself. This choice covers all the examples which allow the labels on different nodes to change independently. If you, however, want to allow a label to be replaced arbitrarily on some nodes (obeying the structure, of course), but to be replaced with the same label on some other nodes, you must choose a special set of indices. This approach gives the designer of an application a lot of freedom. What are the benefits of this constructions? We can infer interesting properties of the constructed categories such as existence of pushouts from the properties of the constituent categories. Evidently, every commutative diagram qf = pg in (L, R) can be divided into two commutative diagrams q l f l = p l g l in A and q r f r = p r g r in B, since the morphisms of (L, R) are pairs of morphisms. Hence, pushouts and other colimits in (L, R) induce the corresponding constructions in the constituent categories. The proof is simple (cf. Example 3.8.1). This gives us a necessary condition for the existence of colimits in the comma category: A commutative diagram in (L, R) is a pushout diagram only if the component diagrams are pushout diagrams in A and B, respectively. But the converse requires an additional condition. Definition (Cocontinuous functor): A functor L : A C is called (finitely) cocontinuous if it preserves all (finite) colimits, i.e., if a commutative diagram composed of the morphisms f 1, f 2,... is a colimit in A, then the diagram composed of the morphisms L(f 1 ), L(f 2 ),... is a colimit of the same type in C. A functor is continuous if it preserves limits. Cocontinuous functors allow us to use the idea of Theorem in order to get a general way to construct colimits in comma categories. By way of example, we explicitly prove the case of pushouts: Lemma (Pushouts in a comma category): Let L : A C and R : B C be two functors. The comma category (L, R) has pushouts if A and B have pushouts and L is cocontinuous.

26 December 28, Please note that we assume only L to be cocontinuous. The dual lemma states that (L, R) has pullbacks if A and B have pullbacks and if R is continuous. Proof: We follow the lines of the proof of Theorem 3.1.4, but we have to pay attention to some details arising from working in different categories. We start with two morphisms p = (p A, p B ) and g = (g A, g B ) in the category (L, R) as shown in Figure (a), and we I p B I A p A B A I B p B B B g ḡ g A P O ḡ A g B P O ḡ B C p G C A pa G A C B pb G B Fig (a) Fig (b) Fig (c) remember that objects of (L, R) are of the form I = (I A, I B, f I ) where I A and I B are objects of A and of B, respectively, and f I is a morphism f I : L(I A ) R(I B ) in C. The first step is to construct the pushout diagrams of the components in the categories A and B, respectively, as depicted in Figures (b) and (c). These diagrams exist, since A and B have pushouts by assumption, and they can be carried over to L(p A ) L(I A ) L(B A ) R(p B ) R(I B ) R(B B ) L(g A ) P O L(ḡ A ) R(g B ) = R(ḡ B ) L(C A ) L( p A ) L(G A ) R(C B ) R( p B ) R(G B ) Fig (b ) Fig (c ) category C by the functors L and R, respectively. We get the diagrams (b ) and (c ). Since L is cocontinuous, the left-hand diagram is a pushout again, whereas the righthand diagram need not, but the definition of functors ensures its commutativity. In Figure (d), we connect these two diagrams by the morphisms f I, f B, and f C. We must prove that ḡ = (ḡ A, ḡ B ) and p = ( p A, p B ) complete the diagram (a) such that we get a commutative one. First, we have to make G A and G B an object of (L, R) by adding a suitable morphism f G : L(G A ) R(G B ) factorizing R(ḡ B ) f B and R( p B ) f C such that (ḡ A, ḡ B ) and ( p A, p B ) become morphisms in (L, R). Existence and uniqueness of f G follow from the pushout property of L(ḡ A ) L(p A ) = L( p A ) L(g A ) and the fact that the outer paths of the diagram commute: R(ḡ B ) f B L(p A ) = R( p B ) f C L(g A ). This part of the proof also makes clear why R need not be cocontinous; commutativity of diagram (c ) is sufficient. The final step is proving the universal property of the diagram in Figure (a). Consider the left-hand diagram of Figure (e) that is a diagram in (L, R). We have to show