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....only conditions (2) and 14 (3) of Definition 3.1, so they can have several roots and need not be connected. Moreover, function symbols are equipped with a list of argument sorts and a result sort, and sorts are represented as node labels. For results about graph rewriting on jungles, we refer to [75, 43, 51, 61, 23, 89]. In the literature there exists a variety of definitions of term graphs. Besides hypergraphs, directed graphs, terms with labels, and recursion equations have been used as underlying structures. Acyclic graphs have been dealt with in [34, 95, 96, 97] while [83, 92, 59, 15, 37, 63, 32] also ....

Richard Kennaway. Graph rewriting in some categories of partial morphisms. In Proc. Graph Grammars and Their Application to Computer Science, volume 532 of Lecture Notes in Computer Science, pages 490--504. SpringerVerlag, 1991.

....in a completely uniform way. 1 Introduction The classical theory of Term Graph Rewriting studies the issue of representing finite terms with directed, acyclic graphs, and of modeling term rewriting via graph rewriting (among the many contributions to this theory, we cite [Sta80, BvEG 87, Ken91, HP91, CR93] The main advantage of using graphs is that the sharing of common subterms can be represented explicitly in a graph. Therefore the rewriting process is speeded up, because the rewriting steps do not have to be repeated for each copy of an identical subterm. For example, the rewrite ....

....: g . 2. g is the result of the simultaneous application of R to an infinite number of redexes in f : in a single step all the occurrences of f in f are replaced by g. 3 A variation of the algebraic approach making use of a single pushout construction is used for example in [Ken91] Other differences among the various proposal for TGR concern the way terms are represented as graphs, but these are a mere syntactical issue (see [CMR 91] 2 f ffl ABCED oo R gffl ABCED oo H H 0 Fig. 2. An example of cyclic term graph rewriting The interesting fact is ....

J.R. Kennaway. Graph rewriting in some categories of partial morphisms. In Graph Grammars and their Application in Computer Science, volume 532 of LNCS, pages 490--504. Springer Verlag, 1991.

....only conditions (2) and (3) of Definition 1.3.1, so they can have several roots and need not be connected. Moreover, function symbols are equipped with a list of argument sorts and a result sort, and sorts are represented as node labels. For results about graph rewriting on jungles, we refer to [76,43,51,62,23,91]. In the literature there exists a variety of definitions of term graphs. Besides hypergraphs, directed graphs, terms with labels, and recursion equations have been used as underlying structures. Acyclic graphs have been dealt with in [34,97,98,99] while [85,94,60,15,37,64, 32] also consider ....

Richard Kennaway. Graph rewriting in some categories of partial morphisms. In Proc. Graph Grammars and Their Application to Computer Science, volume 532 of Lecture Notes in Computer Science, pages 490--

....compared with our rewritings of relational (labeled) graph. Moreover a more general sufficient condition for two rewritings to commute and a theorem concerning critical pairs useful to demonstrate the confluency of graph rewriting systems are also given. 1 Introduction There are many researches [1 7,9,13,14,16 18,20 22] on graph grammars and graph rewritings which have a lot of applications including software specification, data bases, analysis of concurrent systems, developmental biology and many others. In these one of the advantages of categorical graph rewritings is to produce a universal reduction which ....

....At the end of the section we prove a more general sufficient condition for two graph rewritings to commute and a theorem on critical pairs useful to demonstrate the confluency of 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 formalization of graph rewritings for graphs with labeled edges which contains graphs in the sense of Raoult [22] 2 Fundamentals on Relational Calculus A relation ff of a set A into ....

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R. Kennaway, Graph rewriting in some categories of partial morphisms, Lecture Notes in Computer Science 532 (Springer, Berlin, 1990), 490--504.

....pushout construction of total morphisms [7] or the construction of a single pushout of partial morphisms [10] As soon as rewriting is studied in an arbitrary category, however, some problems arise. A partial morphism in an arbitrary category that has all binary pullbacks has been defined in [9, 15] as an equivalence class (up to isomorphism of the central object D) of pairs of morphisms A D F NaN F NaN oo i F NaN F NaN h C with i a monomorphism. Although partial morphisms can be defined in this way for any category having all binary pullbacks, and this kind of definitions up to ....

.... (S0) and (S1) which are needed to define the category of spans S(C; M;H) and to understand M and H as subcategories of C, and two further conditions: S2) which asks that H has all pushouts; and (S3) a technical condition closely related to the existence of hereditary pushouts condition in [9], which relates pushouts in H with distinguished pullbacks. Then: We establish a necessary and sufficient condition for the existence of the pushout of two given spans in S(C; M;H) when (C; M;H) satisfies conditions (S0) to (S3) We prove that if (C; M;H) satisfies conditions (S0) to ....

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R. Kennaway, "Graph rewriting in some categories of partial morphisms." Proc. 4th Int. Workshop on Graph Grammars and their Applications to Computer Science, Springer-Verlag, Lect. Notes in Comp. Sc. 532 (1991), pp. 490--504.

....bound variables used relation algebraic notation from the beginning, we have been able to complete the proofs in a way that is still readable and at the same time su#ciently formal to support confidence in the correctness. On the other hand, the algebraic approach to term graph rewriting itself [35, 27, 28, 31, 5] had to be extended, but that is outside the scope of the present article. This paper provides a more gentle introduction to the fundamental rationale behind the relational formalisation of term graphs developed in [21, 22] and goes on to extend this formalisation even more into the relational ....

Richard Kennaway. Graph rewriting in some categories of partial morphisms. In Ehrig et al.

....[13, 2] making use of a notion of partial morphisms of term graphs, and discussed about the Church Rosser property and critical pairs of production rules by a categorical setting. So far the single pushout rewritings has been extensively developed from various view points, for example by [2, 4, 5, 11, 12, 16]. The aim of this paper is to generalize relational graph rewritings [14] of (crisp or ordinary) graphs and to formalize a fuzzy graph rewriting with single pushout approach from a viewpoint of fuzzy relational calculus [9, 10] A fuzzy graph here means a pair of a set of nodes and a fuzzy ....

R. Kennaway, Graph rewriting in some categories of partial morphisms, Lecture Notes in Computer Science 532(1991), 490--504.

....for edges. We define a functor (L20) 3 : Pfn Pfn as follows. For a set A, L2A) 3 is the set of finite strings of pairs of a label and an element of A. Other definition of the functor is similar to Example 3.2. An object of G( L 2 0) 3 ) is similar to the closed term hypergraph of Kennaway [10]. Example 3.6 (L labeled powerset functor) We similarly define a functor P (L20) Pfn Pfn like Example 3.3 and Example 3.5. Theorem 3.7 Let f : A; a) B; b) and g : A; a) C; c) be morphisms in G(T ) If the square A f 000 B y g (1) yh C 000 k D is a pushout in ....

R. Kennaway. Graph rewriting in some categories of partial morphisms. In Proc. 4th Intern. Workshop on graph grammars and their application to computer science, pages 490--504, 1990. (LNCS 532).

....condition on a pair of quomorphisms to have a pushout; we recall it in x2.3. In this paper we give, for conformisms, c quomorphisms and cdc quomorphisms, a detailed description of the pushout of two such partial morphisms as the pushout of two total closed homomorphisms, following the spirit of [11] or [12] Our construction of pushouts in CF Alg Gamma , CQ Alg Gamma and CDCQ Alg Gamma , Gamma = S; Omega ; j) a graph structure, runs roughly as follows. Given two partial mappings of S sets f : K A and g : K B, let K 0 be the greatest subset of K such that f Gamma1 (f(K 0 ) ....

R. Kennaway, Graph Rewriting in Some Categories of Partial Morphisms. In: Proc. Fourth Int. Workshop on Graph Grammars and their Application to Computer Science, Lect. Notes in Comp. Sc. 532 (1991), 490--504.

....7.1 INTRODUCTION In this chapter, we re examine the problem of providing a categorical semantics for the core of the general term graph rewriting language DACTL. Partial success in this area has been obtained by describing graph rewrites as certain kinds of pushout. See [Ken87, HP88, HKP88, Ken91] Nevertheless, none of these constructions successfully describe the whole of the operational models of [BvEG 87] where term graph rewriting was introduced, or of its generalization in the language DACTL itself [GKSS88, GHK 88, GKS91, Ken90] The main stumbling blocks for all of these ....

J.R. Kennaway. Graph rewriting in some categories of partial morphisms. Graph Grammars and their Application to Computer Science, LNCS 532, pp. 490-504, Springer-Verlag, 1991.

....from G to L is usually required to be a monomorphism (i.e. injective) An important result of the double pushout approach is that existence of a pushout complement H is equivalent to the validity of the gluing condition. L . R A . B . Raoult[1984] and Kennaway [1987; 1990] and independently Lowe [1990] developed the approach of the single pushout by moving to the corresponding category of graphs with partial morphisms . The only application condition is that the embedding of the left hand rule side into the application graph is a total morphism; no gluing ....

Richard Kennaway. Graph rewriting in some categories of partial morphisms. In Ehrig et al.

.... fi Theta fi Theta fi A 6 Gamma Gamma Gamma B Theta Gamma Gamma Gamma Theta ff A fi According to the definition of rules as single homomorphisms, rewriting will be defined by a single pushout construction 2 the difference to the single pushout approach of [Ken90, Low90] is that here we still consider total homomorphisms: Definition 3.5 A rewrite step for a rule (L r Gamma R) and a relational diagram G together with a homomorphism f from L to G is the pushout L r R f g G s H of r and f ; the result diagram is the pushout object H . A derivation ....

Richard Kennaway. Graph rewriting in some categories of partial morphisms. In Ehrig et al. [EKR90], pages 490--504.

....Establishing confluence was not altogether trivial since it faced the need for solving the problem of cyclic collapsing terms [Ari92, Ari93, FW91] a g a Y b g Figure 1. Previous definitions of term graph rewriting tend to use one of two ways: 1) categorytheory oriented [Ken87, Ken88, Ken90, Rao84] 2) implementation oriented [PvE93] The first describes graph rewrite steps as push outs in a category, and some papers have been devoted to analyzing whether this can be done by single or double push out constructions [Lo93] The second uses notions like pointers, redirections, ....

J. R. Kennaway. Graph rewriting on some categories of partial morphisms. In Proc. 4th International Workshop on Graph Grammars and their Application to Computer Science, Bremen, Germany, Springer-Verlag LNCS 532, pages 490-- 504, 1990.

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J.R. Kennaway. Graph rewriting in some categories of partial morphisms, 1991.

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