Abstract. This note presents a new formalization of graph rewritings which generalizes traditional

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1 Relational Graph Rewritings Yoshihiro MIZOGUCHI 3 and Yasuo KAWAHARA y Abstract This note presents a new ormalization o graph rewritings which generalizes traditional graph rewritings. Relational notions o graphs and their rewritings are introduced and several properties about graph rewritings are discussed using relational calculus (theory o binary relations). Single pushout approaches to graph rewritings proposed by Raoult and Kennaway are compared with our rewritings o relational (labeled) graph. Moreover a more general sucient condition or two rewritings to commute and a theorem concerning critical pairs useul to demonstrate the conuency o graph rewriting systems are also given. 1 Introduction There are many researches [1-7,9,13,14,16-18,2-22] on graph grammars and graph rewritings which have a lot o applications including sotware specication, data bases, analysis o concurrent systems, developmental biology and many others. In these one o the advantages o categorical graph rewritings is to produce a universal reduction which eases theoretical investigation considerably. Ehrig et al. [3, 4, 5] proposed algebraic graph grammars or a wide class o graphs and graph homomorphisms preserving graph structures. It is well-known that the category o graphs in [4, 7] is a topos ([8]) and so it has pushouts. In their ormalization o graph grammars with double pushouts gluing conditions or existence o pushout-complements in the category o graphs provide an essential mean o controling the semantics o rewriting rules. Gluing conditions are investigated by Ehrig and Kreowski [4] and Kawahara [9]. Using single pushouts and regarding production rules as partial unctions preserving graph structures, another ramework o graph rewritings were ormalized by Raoult [22] and Kennaway [13]. Recently Ehrig and Lowe [3, 16, 18] studied rewritings based on single pushouts in Sig-algebras and proved pushout completeness or restricted signatures with monadic operator symbols only. In this note we treat the category o (simple) graphs (with or without labeled edges) and partial unctions preserving graph structures, and present a new ormalization o graph rewritings by using a primitive pushout construction in the category. Graphs and morphisms introduced here are simple cases o relational structures and structure morphisms in the sense o [5, 16, 17]. However the notion o partial morphisms between graphs (as will be dened in Section 3) is briey dierent rom those o [5, 16, 17]. Thus graph rewritings in this note are always executed without any gluing conditions, only i a rewriting rule has a matching to a graph, and partially generalize graph derivations [4] and graph rewritings [22] in a reasonable sense. Moreover we state a more general sucient condition or two rewritings to commute and 3 Department o Control Engineering and Science, Kyushu Institute o Technology, Iizuka 82, Japan ( ym@ces.kyutech.ac.jp) y Research Institute o Fundamental Inormation Science, Kyushu University 33, Fukuoka 812, Japan ( kawahara@ris.sci.kyushu-u.ac.jp) 1

2 critical pairs useul to demonstrate the conuency o graph rewriting systems. The ramework o the note is elementary and the simplicity o discussions comes rom the usage o relational calculus (theory o binary relations). This note consists o the ollowing sections. In Section 2 we present minimum undamentals on relational calculus or the later calculations. The main subjects o this note are discussed in Section 3. We set up a new ramework o graph rewritings, that is, the notions o (simple) graphs and partial morphisms between them are dened. For a pair o partial unctions rom a common set into graphs a primitive pushout square is constructed, which indicates that the category o graphs and partial morphisms has pushouts. At the end o the section we prove a more general sucient condition or two graph rewritings to commute and a theorem on critical pairs useul to demonstrate the conuency o graph rewriting systems. In Section 4 we compare our approach with other approaches by Ehrig, Lowe, Kennaway and Okada [3, 14, 18, 21]. Some examples related to graph rewritings are listed in Section 5. In Section 6 we state how to develop our ormalization o graph rewritings or graphs with labeled edges which contains graphs in the sense o Raoult [22]. 2 Fundamentals on Relational Calculus A relation o a set A into another set B is a subset o the cartesian product A 2 B and denoted by : A + B. The inverse relation ] : B + A o is a relation such that (b; a) 2 ] i and only i (a; b) 2. The composite : A + C o : A + B ollowed by : B + C is a relation such that (a; c) 2 i and only i there exists b 2 B with (a; b) 2 and (b; c) 2. As a relation o a set A into a set B is a subset o A 2 B, the inclusion relation, union, intersection and dierence o them are available as usual and denoted by v, t, u and, respectively. The identity relation id A : A + A is a relation with id A = (a; a) 2 A2A j a 2 Ag (the diagonal set o A). The ollowings are the basic properties o relations and indicate that the totality o sets and relations orms a category Rel with involution (or shortly I-category). 2.1 Proposition (I-category) Let ; : A + B, ; : B + C and : C + D be relations. Then, (a) () = () (b) id A = id B = (associative), (identity), (c) ]] =, () ] = ] ] (involutive), (d) I v and v, then v and ] v ] (monotone). The distributive law or relations is trivial but indispensable in our relational calculus. 2.2 Proposition (Distributive Law) The distributive law ( F 23 ) = F 23 holds or relations : A + B, : B + C ( 2 3) and : C + D. A partial unction o a set A into a set B is a relation : A + B with ] v id B and it is denoted by : A! B. A (total) unction o a set A into a set B is a relation : A + B with ] v id B and id A v ], and it is also denoted by : A! B. Clearly a unction is a partial unction. Note that the identity relation id A o a set A is a unction. The denitions o partial unctions and (total) unctions here coincide with ordinary ones. A unction : A! B is injective i and only i ] = id A and surjective i and only i ] = id B. 2

3 2.3 Proposition I ; : A + B are relations and : X! A, g : Y! B are partial unctions, then ( u )g ] = g ] u g ]. Moreover i v then ( )g ] = g ] g ]. Given a relation : A + B, the domain d() : A + A o is a relation dened by d() = ] uid A. The domain d( ] ) : B + B o ] corresponds with the image o. A partial unction : A! B is a unction i and only i d() = id A. The ollowing proposition is useul or manipulating domains o partial unctions. 2.4 Proposition Let : A + B and : B + C be relations and : A! B a partial unction. Then (a) d()d() = d() (or d() v d()), (b) d() = d(). 2.5 Proposition Let : A + A, : B + B be relations and let : A! B be a partial unction. I v ], then = ] ]. We denote the category o sets and unctions by Set and the category o sets and partial unctions by Pn. Both o Set and Pn have all small limits and colimits, so in particular, they have pushouts [19, 22, 13, 2]. Note that Pn is equivalent to the category o sets with a base point (a selected element) and base point preserving unctions. We assume that the readers are amiliar with pushout constructions [22, 16, 2, 1] in Pn. A singleton set 3g is denoted by 1 and the maximum relation rom a set A into 1 by A : A + 1, that is, A = (a; 3)ja 2 Ag. The ollowing basic properties o pushouts in Pn are indispensable or the later arguments in Section Proposition Let a square be a pushout in Pn. A gy! B C! k yh D (a) I g is an injective unction, then so is h. (b) For a unction t : X! C the composite tk : X! D is a unction i and only i d(t ] ) u d(g ] ) v d(k). 3 Rewritings or Simple Graphs 3.1 Denition A (simple) graph < A; > is a pair o a set A and a relation : A + A. A partial morphism o a graph < A; > into a graph < B; >, denoted by :< A; >!< B; >, is a partial unction : A! B satisying d() v. 3

4 It is easily seen that a partial morphism among graphs is a partial unction preserving edges on its domain o denitions. Let :< A; >!< B; > and g :< B; >!< C; > be partial morphisms o graphs. Since d() v and d(g)g v g, we have d(g)g = d(g)d()g (by 2.4(a)) v d(g)g = d(g)g (by 2.4(b)) v g. Hence the composite o two partial morphisms o graphs is also a partial morphism o graphs. Thus we have the category Pn(Graph) o graphs and partial morphisms between them. The ollowing theorem constructs a primitive pushout or a pair o partial unctions rom a common set into graphs. 3.2 Theorem I < B; > and < C; > are graphs and the square A! B g y (1) C! D k is a pushout in Pn, then h :< B; >!< D; > and k :< C; >!< D; > are partial morphisms o graphs, where = h ] h t k ] k. Moreover, i h :< B; >!< D ; > and k :< C; >!< D ; > are partial morphisms o graphs satisying h = gk, then there exists a unique partial morphism t :< D; >!< D ; > o graphs such that h = ht and k = kt. Proo. First we see that h :< B; >!< D; > and k :< B; >!< D; > are partial morphisms o graphs. It simply ollows rom d(h)h v hh ] h (by d(h) = hh ] uid B ) v h (by = h ] h t k ] k). Next assume that h :< B; >!< D ; > and k :< C; >!< D ; > are partial morphisms o graphs satisying h = gk. Then we have d(h )h v h and d(k )k v k. As (1) is a pushout in Pn, there exists a unique partial unction t : D! D such that h = ht and k = kt. It suces to prove that d(t)t v t. But it ollows rom yh d(t)t v tt ] (h ] h t k ] k)t (d(t) = tt ] u id D ) = t(t ] h ] ht t t ] k ] kt) (by (2.2)) = t(h ] h t k ] k ) (h = ht, k = kt) = t(h ] d(h )h t k ] d(k )k ) (h = d(h )h, k = d(k )k ) = t(h ] h t k ] k ) (d(h )h v h, d(k )k v k ) v t( t ) (h ] h v id D, k ] k v id D ) = t : This completes the proo. Note that the graph < D; > in the above proo is unique up to isomorphisms. ollowing is exactly a corollary o the last theorem. The 3.3 Corollary The category Pn(Graph) o graphs and partial morphisms has pushouts. A partial morphism :< A; >!< B; > is said to be a morphism o graphs i : A! B is a unction. In other words, :< A; >!< B; > is a morphism o graphs i and only i is a unction with v. It is trivial that the composition o two morphisms o graphs is also a morphism o graphs and so one can consider the category Graph o graphs and morphisms between them. 4

5 3.4 Denition A rewriting rule p is a triple o two graphs < A; >, < B; > and a partial unction : A! B. (Note that need not to be a partial morphism o graphs.) A matching to p is a morphism g :< A; >!< G; > o graphs. Construct a pushout A gy! B G! k in Pn and dene = h ] h t k ] ( g ] g)k. Then the graph < G; > is said to be rewritten into a graph < H; > by applying a rewriting rule p along a matching g, and denoted by < G; >) p=g < H; >. More precisely < G; >) p=g < H; > is called a graph rewriting with a rewriting square < A; > g y yh H! < B; > < G; >! k < H; > : (Note that rewriting squares are not necessarily pushouts in the category o graphs and partial morphisms.) The next proposition states a sucient condition that rewriting squares are pushouts. 3.5 Proposition Let g :< A; >!< G; > be a matching to a rewriting rule p = (< A; > ; < B; >; : A! B). I :< A; >!< B; > is a partial morphism o graphs, then a rewriting square < A; > g y yh! < B; > yh < G; >! k < H; > with = h ] h t k ] ( g ] g)k is a pushout in Pn(Graph). Proo. By the virtue o 3.2 it suces to show that = h ] h t k ] k. First note that ] v ] v since d() v. Thus we have and This completes the proo. = h ] h t k ] ( g ] g)k w h ] ] h t k ] ( g ] g)k = k ] g ] gk t k ] ( g ] g)k = k ] g ] g t ( g ] g)gk = k ] k: = h ] h t k ] ( g ] g)k t k ] k = h ] h t k ] k: The last proposition suggests that our graph rewritings coincide with those o Raoult [22] i rewriting rules are partial morphisms o graphs. It is easy to understand that almost all results about the conuency and concurrency o graph rewritings in [22] are analogously valid in our case. The ollowing is a general sucient condition or two graph rewritings to commute (or to be strongly conuent). 5

6 3.6 Theorem Let p = (< A ; >; < B ; >; : A! B ) be rewriting rules, g :< A ; >!< G; > matchings to p and < G; >) p =g < H ; > a graph rewriting induced by a rewriting square < A ; > y g! < G; > < B ; >! < H ; > h or = ; 1. I :< A ; >!< B ; > ( = ; 1) is partial morphisms o graphs and d(g ) ] u d(g 1) ] v d(k ) u d(k 1 ), then there exist a matching g :< A; >!< H 1 ; 1 > ( = ; 1) and a graph < H; > such that < H 1 ; 1 >) p =g < H; > ( = ; 1). Proo. By virtue o 3.3 we can construct the ollowing three pushouts in the category o graphs and partial morphisms : g 1 < A ; > yk g y ()! < B ; > < A 1 ; 1 >! < G; >! < H ; > k 1 y (1) yk 1 (2) yh < B 1 ; 1 >! < H 1 ; 1 >! < H; > h 1 h 1 Set g = g k 1 ( = ; 1). Then g ( = ; 1) is a unction rom the assumption and 2.6, and so g :< A ; >!< H 1 ; 1 > is a matching to p ( = ; 1). Since two squares ()+(2) and (1)+(2) are pushouts in the category o graphs and partial morphisms, we have a graph rewriting < H 1 ; 1 >) p =g < H; > ( = ; 1) by the denition 3.4, which proves the theorem. yh The rest o this section is concerned with critical pairs [15, 22, 21] that are useul to demonstrate the conuency o actual graph rewriting systems. A basic idea on critical pairs in graph rewriting systems was initiated by Raoult [22]. Our approach is an extension o his method. In what ollows we assume that rewriting rules are morphisms o graphs and matchings are injective morphisms o graphs. Thereore rewriting squares hereater are pushouts rom 3.5 and so they will be called rewriting pushouts. An essential point o the dicussion below is due to 2.6 stating that pushouts in Pn(Graph) preserve injective morphisms o graphs. 3.7 Denition A rewriting system P is simply a amily o rewriting rules (morphisms o graphs). Let < G; >) =g < H ; > be a graph rewriting induced by a rewriting pushout < A ; > y g! < G; > < B ; >! < H ; > h with 2 P or = ; 1. The pair o graph rewritings < G; >) =g < H ; > is called conuent on P i there exist rewriting rules 2 P and graph rewritings < H ; >) < =g H; > ( = ; 1) induced by rewriting pushouts yk satisying k k = k 1 k 1. < A ; > g! < H ; > y < B ; >! h yk < H; > 6

7 Let I be a set and : I + I the empty relation. Then < I; > is a discrete graph over I, that is, a graph without edges. When < A; > is a graph, every unction : I! A always induces a morphism :< I; >!< A; > o graphs. 3.8 Denition Let be a rewriting rule in a rewriting system P ( = ; 1). A critical pair ormed rom and 1 in P is a pair o morphisms t :< S; >!< T ; >( = ; 1) o graphs such that all squares in the ollowing diagram are pushouts in Graph or some pair o injective unctions i : I! A ( = ; 1). < I; > i! < A ; >! < B ; > y y i 1 y s u < A 1 ; 1 > 1 y s 1! < S; >! < T ; > t yt1 < B 1 ; 1 >! u1 < T 1 ; 1 > Note that i A and A 1 are nite sets then the set o critical pairs ormed rom and 1 is nite. 3.9 Lemma I a graph rewriting < G; >) < H ; > is induced by a rewriting pushout g < A ; >! < G; > y (1) yk < B ; >! < H ; > h or = ; 1, then there exist a critical pair t :< S; >!< T ; > ( = ; 1) and matchings s :< A ; >!< S; > ( = ; 1) and s :< S; >!< G; > such that the rewriting pushout (1) is decomposed into two pushouts in Graph through < S; > as ollows: < A ; > Proo. Construct a pullback s! < S; > s! < G; > y yk y t < B ; >! u < T ; >! v < H ; > I i! A y i 1 y g A 1! G g1 in Set and a pushout i < I; >! < A ; > i 1 y (2) ys < A 1 ; 1 >! < S; > s1 in Graph. Since (1) is a pushout there exists a unique morphism s :< S; >!< G; > such that g = s s ( = ; 1). Remark that s, s 1 and s are injective. Also construct a rewriting pushout s < A ; >! < S; > y y t < B ; >! u < T ; > 7

8 or = ; 1. Thus we have a critical pair (t ; t 1 ) ormed rom and 1 and there exists a unique morphism v o graphs such that sk = t v and h = u v ( = ; 1). s < A ; >! < S; >! < G; > y y t (3) yk < B ; >! < T ; >! < H ; > u v By the basic property o pushouts the square (3) is a pushout. This completes the proo. A rewriting sysytem P is conuent i every pair o rewritings < G; >) < H ; >( = ; 1) in P is conuent on P. The ollowing is a main theorem o the note, which asserts that the conuency o rewriting systems are reduced to that o critical pairs. 3.1 Theorem A graph rewriting system P is conuent i and only i every critical pair in P is conuent. Proo. The only-i part is trivial. So we will show the i part. Assume that :< A ; >!< B ; > is a rewriting rule in P and < G; >) =g < H ; > is a graph rewriting induced by a rewriting pushout g < A ; >! < G; > y (1) yk < B ; >! < H ; > h or = ; 1. By 3.9 there exists a critical pair t :< S; >!< T ; > ( = ; 1) such that the rewriting pushout (1) is decomposed into two pushouts in Graph as ollows: < A ; > s! < S; > s s! < G; > y yk y t < B ; >! u < T ; >! v < H ; > From the assumption that every critical pair in P is conuent there exists a pair o rewriting pushouts g < A ; >! < T ; > y yk < B ; >! < T; > h ( = ; 1) such that t k = t 1 k1. Construct a pushout < S; > t k =t 1k1! < T; > sy yk < G; >! h < H ; > : Then there exists a unique morphism w :< H ; >!< H ; > ( = ; 1) o graphs making the square (3) below a pushout. < A ; > y g < B ; >! h! < T ; >! < H ; > yk (3) y w < T; >! < H ; > k The above diagram induces a graph rewriting < H ; >) =g v < H ; >( = ; 1) in P. Hence the given pair o graph rewritings is conuent on P. v 8

9 g Figure 1: 4 Observation We rst compare our category o graphs with that o Lowe and Ehrig [18]. Let < A; > be a graph in our sense. We have two unctions i p :! A and i q :! A, where i :! A2A is an inclusion unction and p : A 2 A! A and q : A 2 A! A are projections. Then (; A; i p; i q) is naturally considered as a Sig-algebra with Sig = s; t : E! V g in the sense o [18]. Thus a graph in our sense exactly corresponds to a Sig-algebra (G E ; G V ; s G ; t G ) such that a unction (s G ; t G ) : G E! G V 2G V is injective. A partial Sig-algebra morphism rom < A; > to < B; > is a tuple (< A ; >; i ; t ) o a subgraph < A ; > o < A; >, an inclusion unction i :< A ; >!< A; >, and a (total) graph morphism t :< A ; >!< B; >. It corresponds to a notion o partial morphisms [23, 14] over Graph. But Graph has pushouts which are not hereditary in the sense o Kennaway [14], so the category o partial morphisms constructed rom Graph is not pushout complete [14]. Figure 1 illustrates a pushout which is not hereditary. Let :< A; >!< B; > be a partial morphism o graphs in our sense. We have the domain A o partial unction : A! B, an inclusion unction i : A! A and a unction t : A! B such that = i ] t. Dene by constructing a pullback! A 2 A PB A 2 A i y! i yi 2i in Set. Since d() v it ollows by assumptions that i :< A ; >!< A; > and t :< A ; >!< B; > are morphisms o graphs. But there may be many subgraphs < A ; 3 > o < A; > such that t :< A ; 3 >!< B; > is a morphism o graphs. This is a dierence between our partial morphisms o graphs and those in [18]. Figure 1 indicates that Lowe's pushout construction is not closed under the subclass o our graphs and so it is meaningul to prove the pushout completeness o the category Pn(Graph) in our sense(c. 3.3). Ehrig and Lowe [16, 18] proved the pushout completeness o the category o Sig-algebras whose signature contains monadic operator symbols only. In this case the category o Sigalgebras is equivalent to a unctor category over Set which is a topos [8]. Relations and partial unctions can be similarly considered in topoi [9, 1]. We present a new pushout completeness [1] in the ollowing 4.1 Theorem I a topos E has the ollowing properties: 9

10 (a) the set Sub(A) o subobjects o an object A is a complete lattice by inclusion, (b) the distributive law (c. 2.2) o relations holds, then the category o partial unctions in E is nitely cocomplete. Kennaway [14] introduced the notion o hereditary pushouts and showed that i E satisying the condition (a) o 4.1 has hereditary pushouts, then P (E) has pushouts. When 4.1 holds, every pushout square in E is also a pushout in the category o partial unctions in E), that is, it is hereditary [14]. Next we consider Ehrig's double pushout approach [3] in our category Graph, that is, assume that the ollowing two squares are pushouts in Graph and that m is an injective unction. < A; > g y m < D; > ys! < B; > < G; > < E; " >! < H; > n k Then m v m, s v s", v, = g ] g t n ] "n and = h ] h t k ] "k by 3.2. Since nn ] = id E by the pushout property it is easy to see that s ] s v " and yh n( g ] g)n ] = (ng ] gn ] t nn ] "nn ] ) ng ] gn ] (by 2.3) = (ng ] gn ] t ") ng ] gn ] (nn ] = id E ) = " ng ] gn ] v ": Hence n( g ] g)n ] t s ] s v ". Now put ^" = n( g ] g)n ] t s ] s. From n ] "n g ] g = n ] n(n ] "n g ] g)n ] n (by 2.5) = n ] (" ng ] gn ] )n (by 2.2) and nn ] = id E, we have g ] g t n ]^"n = g ] g t n ] n( g ] g)n ] n t n ] s ] sn (by 2.3) = g ] g t n ] (" ng ] gn ] )n t n ] s ] sn = g ] g t (n ] "n g ] g) t n ] s ] sn = g ] g t n ] "n t n ] s ] sn = g ] g t n ] "n (s ] s v ") = : Thus ^" : E + E is the least relation such that s ] s v ^" and = g ] g t n ]^"n. Hence it is reasonable to assume that " = ^". In this case we have = h ] h t k ] n( g ] g)n ] k t k ] s ] sk = h ] h t k ] n( g ] g)n ] k t h ] ] h (h = sk) = h ] h t k ] n( g ] g)n ] k ( ] v ): This shows that ^" : E + E is the least relation " which makes the above squares pushouts. In our category o graphs Graph the pushout complement is not always exist and not unique (c. 5.1). I there exists a pushout complement, our rewriting using single pushout coincides with the double pushout rewriting which uses the least pushout complement. Finally we consider the boundary graphs (or B-graphs) due to Okada and Hayashi[21] in Pn(Graph). I a matching g :< A; >!< G; > to p = (< A; >; < B; >; : A! B) is an injective morphism o graphs such that deg(g(a)) = deg(a) or each a 2 A on which is undened, then the rewritings coincides with those o B-graphs. 1

11 1 2 (1) (2) 1 2 (1) 3 3 g g g(1) g(2) 1 2 g(3) 4 4 g(1)=g(2) Figure 2: Figure 3: 5 Examples o Graph Rewritings In this section a ew examples related to graph rewritings are listed. The rst example shows that pushout-complements are not unique in Graph. 5.1 Let ; ; : A + A be relations with v v. Then because o 3.2 the square < A; > id A! < A; > id A y yid A < A; >! < A; > ida is a pushout in the category o graphs and morphisms between them. Thereore the square is a pushout or any choice o satisying v v. The choice o ^" in 4 means the most economical way to have pushout-complements. Next we present two simple examples o graph rewritings to which conventional graph rewritings cannot be applied. 5.2 In Figure 2 g is a neat morphism o graphs with respect to theories o Ehrig [3], Raoult [22] and ours. But is not a morphism o graphs and it is not worth to be a rewriting rule in the sense o Raoult [22]. On the other hand means a ast production in the double pushout aproach o [3] but unortunately the necessary pushout-complement does not exist since the gluing condition is not satised. However we have the bottom right resultant graph by applying our ormalization. 5.3 In Figure 3 g is a morphism o graphs and is a partial morphism o graphs in all theories o Ehrig [3], Raoult [22] and ours. However graph rewritings o Ehrig [3] and Raoult [22] does not work again because the gluing conditions are not valid. In this case the resultant graph given by our graph rewritings is one point graph without edges. The nal two examples indicate reasons why rewriting rules are not restricted to morphisms o graphs and why matchings must be morphisms o graphs in the denition 3.4 o graph rewritings. 11

12 Figure 4: 5.4 In order to treat with more general graph rewritings we do not restrict rewriting rules to (partial) morphisms o graphs (C.3.4). Figure 4 illustrates an important example o graph rewriting rules being not morphisms o graphs, because the edge denoted by a bold arrow is not preserved. The rule expresses usual associative laws. 5.5 Recall that matchings to rewriting rules are dened to be morphisms o graphs but not partial morphisms (C. 3.4). We now observe what happens when matchings are allowed to be partial morphisms o graphs. First we note that any couple o rewriting rules being partial morphisms o graphs commute, because rewriting squares are pushouts in the category o graphs and partial morphisms by 3.5. Hence every set o rewriting rules consisting o partial morphisms o graphs is strongly conuent, which seems to exceed. Let p = (< A; > ; < B; >; : A! B) and assume that (A) = B and there exists a 2 A such that is undened on a and a has no loops. (This rewriting rule p is not so special.) For any vertex v o an arbitrary graph < G; >, dene a matching g :< A; >!< G; > such that g(a) = v and undened otherwise. Then g is in act a partial morphism o graphs. The resultant graph H ater applying p along g is a graph obtained by subtracting rom G the vertex v and all edges connected with v. Thus this claims that any nite graph is reduced into the empty graph by iterating applications o p. 6 Rewritings or Graphs with Labeled Edges In this section we rst dene graphs with labeled edges and partial morphisms between them, and a primitive pushout construction similar to 3.2 is stated or graphs with labeled edges. The readers may easily understand analogies with results in the section 3 are also valid in this case. Let 6 be a set o labels. A graph < A; > with 6-labeled edges is a pair o a set A and a collection = : A + A j 2 6g o relations indexed by 6. A partial morphism o a graph < A; > with 6-labeled edges into a graph < B; > with 6-labeled edges, denoted by :< A; >!< B; >, is a partial unction : A! B satisying d() v or all 2 6. Similarly we have the category o graphs with 6-labeled edges and partial morphisms between them. The ollowing theorem constructs a primitive pushout or a pair o partial unctions rom a common set into graphs with labeled edges. 6.1 Theorem I < B; > and < C; > are graphs with 6-labeled edges and i the square A! B g y (1) C! D k 12 yh

13 is a pushout in Pn, then h :< B; >!< D; > and k :< C; >!< D; > are partial morphisms o graphs with 6-labeled edges, where = h ] htk ] k or each 2 6. Moreover, i h :< B; >!< D ; > and k :< C; >!< D ; > are partial morphisms o graphs with 6-labeled edges satisying h = gk, then there exists a unique partial morphism t :< D; >!< D ; > o graphs with 6-labeled edges such that h = ht and k = kt. Similarly we have the ollowing corollary rom the last theorem. 6.2 Corollary The category o graphs Pn(6-Graph) with 6-labeled edges and partial morphisms between them has pushouts. Remark. A graph < A; > with ] = is just an undirected graph. Hence almost all results in this note are also valid or undirected graphs. 6.3 Example Let N be the set o natural numbers. A graph < A; > with N-labeled edges satisying the ollowing conditions: (a) i is a partial unction or any i 2 N, i.e. ] i i v id A, (b) d( j ) v d( i ) or any i j (i; j 2 N), is equivalent to a graph < A; A : A! A 3 > in the sense o Raoult [22]. Since graph morphisms in [22] are identical with morphisms o graphs with 6-labeled edges, the category o graphs in [22] is a subcategory o Pn(N-Graph). Though the category o graphs in [22] does not have pushouts Raoult [22] showed a sucient condition or existence o pushouts. 7 Concluding Remark Ehrig and Lowe [16, 17, 18] have extensively developed the theory o graph rewritings using partial unctions and single pushouts rom an algebraic viewpoint. They reexamined that several properties o graph grammars can be simply proved within single pushout approaches and demonstrated the eciency o the single pushout ormalization. Kennaway [14] investigated the pushout completeness o abstract categories o partial morphisms. But their categories o graphs are dierent rom Pn(Graph). We proved the pushout completeness o the category o simple graphs and partial morphisms using the relational calculus. We claim two points. First our notions and proos are simple and clear. The relational calculus is convenient to deal with partial unctions. Second our ramework can be extended to more general relational categories which have many applications. For example, a relational structure < A; >, a pair o a set A and a relation : SA + T A, is considered as a generalization o graphs, where S; T : Set! Set are two unctors. We can construct a category o relational structures in which similar properties to the case o simple graphs also hold [11]. Acknowledgement The authors are much indebted to the reeree or his careul reading the manuscript and many valuable comments to improve this note. 13

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