## An Inductive View of Graph Transformation (1998)

Venue: | In Workshop on Algebraic Development Techniques |

Citations: | 30 - 12 self |

### BibTeX

@INPROCEEDINGS{Gadducci98aninductive,

author = {F. Gadducci and R. Heckel},

title = {An Inductive View of Graph Transformation},

booktitle = {In Workshop on Algebraic Development Techniques},

year = {1998},

pages = {223--237},

publisher = {Springer Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

. The dynamic behavior of rule-based systems (like term rewriting systems [24], process algebras [27], and so on) can be traditionally determined in two orthogonal ways. Either operationally, in the sense that a way of embedding a rule into a state is devised, stating explicitly how the result is built: This is the role played by (the application of) a substitution in term rewriting. Or inductively, showing how to build the class of all possible reductions from a set of basic ones: For term rewriting, this is the usual definition of the rewrite relation as the minimal closure of the rewrite rules. As far as graph transformation is concerned, the operational view is by far more popular: In this paper we lay the basis for the orthogonal view. We first provide an inductive description for graphs as arrows of a freely generated dgs-monoidal category. We then apply 2-categorical techniques, already known for term and term graph rewriting [29, 7], recasting in this framework the...

### Citations

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Communication and Concurrency
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Citation Context ...di Pisa, Dipartimento di Informatica, Corso Italia 40, 56124 Pisa, Italy, (reiko@di.unipi.it). Abstract. The dynamic behavior of rule-based systems (like term rewriting systems [24], process algebras =-=[27]-=-, and so on) can be traditionally determined in two orthogonal ways. Either operationally, in the sense that a way of embedding a rule into a state is devised, stating explicitly how the result is bui... |

566 | Term rewriting systems
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Citation Context ...lin.de). 2 Universit`a di Pisa, Dipartimento di Informatica, Corso Italia 40, 56124 Pisa, Italy, (reiko@di.unipi.it). Abstract. The dynamic behavior of rule-based systems (like term rewriting systems =-=[24]-=-, process algebras [27], and so on) can be traditionally determined in two orthogonal ways. Either operationally, in the sense that a way of embedding a rule into a state is devised, stating explicitl... |

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Handbook of Graph Grammars and Computing by Graph Transformation
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Citation Context ...y known for term and term graph rewriting [29, 7], recasting in this framework the usual description of graph transformation via double-pushout [13]. 1 Introduction The theory of graph transformation =-=[30]-=- basically studies a variety of formalisms which extend the theories of formal languages and term rewriting, respectively, in order to deal with structures more general than strings and terms. In both... |

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Functorial semantics of algebraic theories
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Citation Context ...respondence between pointers and underlying structures (so that e.g. t; ! b =! a : deleting a pointer is the same as deleting the structure), and allows to recast the usual notion of algebraic theory =-=[26]-=-. Definition 8 (dgs-monoidal categories). A dgs-monoidal category C is a eight-tuple hC 0 ;\Omega ; e; ae; r; !; \Delta; ?i, such that both the six-tuples hC 0 ;\Omega ; e; ae; r; !i and h(C 0 ) op ;\... |

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Citation Context ...ed pointer, without destroying the new name, then ? a ; ! a is in general different from id e . 3.2 On the notion of feedback Traced monoidal categories have been studied per se as a categorical tool =-=[20]-=-. However, they already surfaced in the literature related to algebraic theories. In fact, there is a strong connection between traced categories and iteration theories (that is, algebraic theories wi... |

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Citation Context ... this section we briefly present some notions about 2-categories needed in the rest of the paper. We first recall the basic definitions: For an introduction, we refer the reader to the classical work =-=[23]-=-. Definition 15 (2-categories). A 2-category C is a four-tuple hOb c ; C; ; idi such that Ob c is a set of 2-objects and, indexed by elements in Ob c , C is a family of categories C[a; b] (the hom-cat... |

96 |
Graph-grammars: An algebraic approach
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Citation Context ...R Network GETGRATS (General Theory of Graph Transformation Systems) through the Technical University of Berlin and the University of Pisa. In the double-pushout (DPO) approach to graph transformation =-=[15, 13]-=- (and in most of the other approaches) the operational definition is by far more popular. Inductive definitions of DPO graph transformation have been given in [1, 12], but they do not have the same ro... |

75 |
Cartesian Bicategories I
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- 1987
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Citation Context ...t. ut Equivalent notions of dgs-monoidal categories have surfaced quite frequently in recent years. A (bicategorical) presentation is used as a description of the (bi)category of relations already in =-=[5]-=-, which forms the basis for some recent work on the categorical description of circuits [21, 17]: Arrows are processes, and the bicategorical structure allows to relate specifications which are equiva... |

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Graph expressions and graph rewritings
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Citation Context ...approach to graph transformation [15, 13] (and in most of the other approaches) the operational definition is by far more popular. Inductive definitions of DPO graph transformation have been given in =-=[1, 12]-=-, but they do not have the same role as in the theory of term rewriting. One reason may be that, unlike for strings and terms, there is no straightforward inductive definition of graphs. Rather, each ... |

67 |
Semantics of weakening and contraction
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Citation Context ... duplication of the pointer b, so that t; r b can be seen as a shared instance of t. These structures fill the gap between monoidal and cartesian categories: It can be considered categorical folklore =-=[18, 25, 19, 7]-=- that, equipping a monoidal category with suitable natural transformations, we obtain a cartesian category: See e.g. [9] for a recollection. In our case, an instance of the theorem is obtained simply ... |

60 |
Varieties of iteration theories
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- 1988
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Citation Context ...ted to algebraic theories. In fact, there is a strong connection between traced categories and iteration theories (that is, algebraic theories with an equational characterization of (least) fix-point =-=[2, 3]-=-) as it is pointed out in the works on flownomial calculus [32, 33]. Definition 9 (traced monoidal categories). A traced monoidal category C is a five-tuple hC 0 ;\Omega ; e; ae; tri, where hC 0 ;\Ome... |

49 |
M.: Algebraic approaches to graph transformation I: Basic concepts and double pushout approach
- Corradini, Montanari, et al.
- 1997
(Show Context)
Citation Context ...category. We then apply 2-categorical techniques, already known for term and term graph rewriting [29, 7], recasting in this framework the usual description of graph transformation via double-pushout =-=[13]-=-. 1 Introduction The theory of graph transformation [30] basically studies a variety of formalisms which extend the theories of formal languages and term rewriting, respectively, in order to deal with... |

42 | Bicategories of processes
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- 1997
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Citation Context ...es; we then show a finitary encoding of traced monoidal categories into the dgs-monoidal structures, from which the completeness result can be inferred via a folklore characterization of hyper-graphs =-=[33, 21]-=-. 3.1 On (d)gs-monoidal categories In this section we introduce dgs-monoidal categories, an extension of gs-monoidal ones [9, 7], which are used for our equational presentation of graphs. Definition 7... |

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- 1999
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Citation Context ... only atomic graphs as constants and composition and union as operators. ut Space limitations force us to omit the proof. Nevertheless, a similar one for (acyclic) term graph can be actually found in =-=[9]-=-, carried out by induction on G id 1 ffl 1 G ae 1 2 ffl ffl 1 2 Ge 1 ffl ffl 1 G ; ; ; Gr 1 ffl 1 2 G ! 1 ffl ; G \Delta 1 2 ffl 1 G ? ; ffl 1 Fig. 2. Atomic graphs. the number of nodes of a given gra... |

34 | A 2-categorical presentation of term graph rewriting
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- 1997
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Citation Context ...ew. We first provide an inductive description for graphs as arrows of a freely generated dgs-monoidal category. We then apply 2-categorical techniques, already known for term and term graph rewriting =-=[29, 7]-=-, recasting in this framework the usual description of graph transformation via double-pushout [13]. 1 Introduction The theory of graph transformation [30] basically studies a variety of formalisms wh... |

33 |
Categorical structures
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- 1996
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Citation Context ...g term rewriting (see e.g. [31, 28], and the more recent [34, 11]), while in [7, 8] similar categorical models have been proposed for term graph rewriting, (traced) gs-monoidal 2-theories. A computad =-=[35]-=- is a category equipped with a graph structure over homsets (informally, a set of cells not closed under composition), which intuitively represent a rule based system, the states being the arrows and ... |

32 |
Hyperedge replacement grammars and languages, volume 643 of LNCS
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Citation Context ...all often write G i j to recall that G has rank (i; j). ut The idea of equipping graphs with lists of distinguished nodes in order to define composition operations on them is not new (see for example =-=[1, 16]-=-): Roughly, for a graph G i j , the components i; j represent discrete interfaces, through which graphs can be equipped with a compositional structure. And, dealing with isomorphism classes of concret... |

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Span(Graph): A categorical algebra of transition systems
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Citation Context ... and let DGS(d) be the associated free dgs-monoidal category. Then DGS(d) is isomorphic to DG via a dgs-monoidal functor. ut 4 We just discovered an equivalent description, provided without proof, in =-=[22]-=-. 4 A 2-category for Graph Rewriting We open this section recalling a few definitions about the double-pushout approach to graph transformation. We will then present the basic notions regarding 2-cate... |

22 | Sesqui-Pushout Rewriting
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Citation Context ...lution should take into account three different aspects of the problem. First, the dgs-monoidal structure should be lifted to the 2-categorical level, as for similar results with iteration 2-theories =-=[10, 4]-=-, also obtaining a generalized version of Theorem 11. This would provide us with the syntax of parallel productions and derivations. Second, the structure of the interfaces should be enriched, at leas... |

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Citation Context ... duplication of the pointer b, so that t; r b can be seen as a shared instance of t. These structures fill the gap between monoidal and cartesian categories: It can be considered categorical folklore =-=[18, 25, 19, 7]-=- that, equipping a monoidal category with suitable natural transformations, we obtain a cartesian category: See e.g. [9] for a recollection. In our case, an instance of the theorem is obtained simply ... |

20 |
Foundations of equational deduction: A categorical treatment of equational proofs and unification algorithms
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- 1987
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Citation Context ...transformations are given by the cells of a 2-category freely generated from basic cells which represent the rules of the system. Such 2-categorical models are well-known for term rewriting: See e.g. =-=[28, 31]-=-. More recently, they have been applied to term graph rewriting [7]. 2 Graphs This section introduces (ranked) graphs as isomorphism classes of (ranked) concrete graphs. This presentation departs slig... |

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- Corradini, Gadducci, et al.
- 1995
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Citation Context ...ells. As far as rewriting is concerned, it is well-known that a suitable class of 2-categories, algebraic 2-theories, can be used for describing term rewriting (see e.g. [31, 28], and the more recent =-=[34, 11]-=-), while in [7, 8] similar categorical models have been proposed for term graph rewriting, (traced) gs-monoidal 2-theories. A computad [35] is a category equipped with a graph structure over homsets (... |

16 |
An abstract formulation for rewrite systems
- Power
- 1989
(Show Context)
Citation Context ...transformations are given by the cells of a 2-category freely generated from basic cells which represent the rules of the system. Such 2-categorical models are well-known for term rewriting: See e.g. =-=[28, 31]-=-. More recently, they have been applied to term graph rewriting [7]. 2 Graphs This section introduces (ranked) graphs as isomorphism classes of (ranked) concrete graphs. This presentation departs slig... |

15 |
An algebra of graphs and graph rewriting
- CORRADINI, MONTANARI
- 1991
(Show Context)
Citation Context ...approach to graph transformation [15, 13] (and in most of the other approaches) the operational definition is by far more popular. Inductive definitions of DPO graph transformation have been given in =-=[1, 12]-=-, but they do not have the same role as in the theory of term rewriting. One reason may be that, unlike for strings and terms, there is no straightforward inductive definition of graphs. Rather, each ... |

11 |
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- 1987
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Citation Context ...between traced categories and iteration theories (that is, algebraic theories with an equational characterization of (least) fix-point [2, 3]) as it is pointed out in the works on flownomial calculus =-=[32, 33]-=-. Definition 9 (traced monoidal categories). A traced monoidal category C is a five-tuple hC 0 ;\Omega ; e; ae; tri, where hC 0 ;\Omega ; e; aei is a symmetric strict monoidal category, 3 which is equ... |

9 |
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Citation Context ...at is, transformations of complex graphs are build from transformations of simpler components. In fact, this notion of rewriting is closely related to distributed graph transformation in the sense of =-=[14]-=-. Technically speaking, the transformations are given by the cells of a 2-category freely generated from basic cells which represent the rules of the system. Such 2-categorical models are well-known f... |

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Abstract Graph Derivations in the Double-Pushout Approach
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- 1994
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Citation Context ...ations. Finally, the most delicate point is to ensure that the equivalence induced on cells by the coherence axioms of 2-categories resembles the usual shift equivalence on abstract graph derivations =-=[6]-=-, like it happens for permutation equivalence in categorical models of term rewriting (see e.g. [11]). Here, particular attention should be paid to the isomorphism question between graphs: A problem t... |

8 |
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- 1994
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Citation Context ...es; we then show a finitary encoding of traced monoidal categories into the dgs-monoidal structures, from which the completeness result can be inferred via a folklore characterization of hyper-graphs =-=[33, 21]-=-. 3.1 On (d)gs-monoidal categories In this section we introduce dgs-monoidal categories, an extension of gs-monoidal ones [9, 7], which are used for our equational presentation of graphs. Definition 7... |

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Citation Context ...t years. A (bicategorical) presentation is used as a description of the (bi)category of relations already in [5], which forms the basis for some recent work on the categorical description of circuits =-=[21, 17]-=-: Arrows are processes, and the bicategorical structure allows to relate specifications which are equivalent but show a different internal structure (a different implementation, so to say). Intuitivel... |

6 |
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Citation Context ...ells. As far as rewriting is concerned, it is well-known that a suitable class of 2-categories, algebraic 2-theories, can be used for describing term rewriting (see e.g. [31, 28], and the more recent =-=[34, 11]-=-), while in [7, 8] similar categorical models have been proposed for term graph rewriting, (traced) gs-monoidal 2-theories. A computad [35] is a category equipped with a graph structure over homsets (... |

5 |
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- 1997
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Citation Context ...lution should take into account three different aspects of the problem. First, the dgs-monoidal structure should be lifted to the 2-categorical level, as for similar results with iteration 2-theories =-=[10, 4]-=-, also obtaining a generalized version of Theorem 11. This would provide us with the syntax of parallel productions and derivations. Second, the structure of the interfaces should be enriched, at leas... |

5 |
On partial recursive definitions and programs
- Hoenke
- 1977
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Citation Context ... duplication of the pointer b, so that t; r b can be seen as a shared instance of t. These structures fill the gap between monoidal and cartesian categories: It can be considered categorical folklore =-=[18, 25, 19, 7]-=- that, equipping a monoidal category with suitable natural transformations, we obtain a cartesian category: See e.g. [9] for a recollection. In our case, an instance of the theorem is obtained simply ... |

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- 1990
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