A. Corradini, F. Gadducci, A 2-Categorical Presentation of Term Graph Rewriting, Proc. CTCS'97, Springer LNCS, to appear, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of this work. While the denotational semantics of functional languages provides an elegant framework for reasoning, the semantics of the implementation of functional languages has remained relatively low level. While some attempts to model term graphs using abstract techniques have been made [6], they have yet to make a significant impact within the functional programming community. We hope the naturalness and simplicity of the coalgebraic model of term graphs, for example demonstrated by our work on recursion in coalgebraic monads, will help to bridge this gap. The paper is structured ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In Proceedings CTCS'97, volume 1290 of LNCS. Springer, 1997. 20

....of term rewrite steps. A related result is presented in [CG98b] where infinite parallel term rewriting is shown to be an adequate semantics for the rewriting of terms, i.e. terms enriched with an auto instantiation operator. A new categorical description for term graph rewriting is proposed in [CG97]. It is already known that term rewriting systems can be faithfully described by a Cartesian 2 category, where horizontal arrows represent terms, and cells represent rewriting sequences. A similar, original 2 categorical presentation is presented for term graph rewriting. In [CG98a] the main ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In Proceedings CTCS'97, volume 1290 of LNCS. Springer Verlag, 1997. 21

....on completeness for proving validity of equations and for computing term normal forms, on termination and confluence, and on term graph narrowing. The contents of the paper [AKP99] have been described in the First Annual Progress Report: In [CG98] the authors extend their previous work [CG97] on the categorical description of possibly cyclic term graph rewriting using suitable 2 categories to the rewriting of possibly cyclic term graphs. They show that this presentation is equivalent to the well accepted operational definition proposed by Barendregt et al. but for the case of ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In Proceedings CTCS'97, volume 1290 of LNCS. Springer Verlag, 1997.

....the framework presented here to weaker categories, where some of the naturality axioms for the generalized natural transformations are missing. Indeed, we are currently developing tile systems for a wide class of applications, where con gurations and observations have only a gs monoidal structure (Corradini and Gadducci 1997) rather than cartesian, i.e. where duplicators and dischargers are not natural. Indeed such structures are equipped with a notion of name sharing and have been shown useful to de ne a tile model for the (asynchronous) calculus (Milner, Parrow and Walker 1992) one of the most studied mobile ....

Corradini, A. and Gadducci, F. (1997) A 2-categorical Presentation of Term Graph Rewriting.

....two pushouts. The difference is that one considers the category of jungles instead of the category of hypergraphs. This implicitly enforces a kind of minimal collapsing in evaluation steps with non left linear term rewrite rules. A categorical treatment of garbage collection is given in [12] In [19], a description of term graph rewriting by a 2 category is presented. 23 5 Completeness In this section we consider the completeness of term graph rewriting for simulating arbitrary term rewrite derivations (Subsection 5.1) and for computing term normal forms (Subsection 5.2) We will see that ....

Andrea Corradini and Fabio Gadducci. A 2-categorical presentation of term graph rewriting. In Proc. Category Theory and Computer Science, volume 1290 of Lecture Notes in Computer Science. Springer-Verlag, 1997.

....two pushouts. The difference is that one considers the category of jungles instead of the category of hypergraphs. This implicitly enforces a kind of minimal collapsing in evaluation steps with non left linear term rewrite rules. A categorical treatment of garbage collection is given in [12] In [19], a description of term graph rewriting by a 2 category is presented. 1.5 Completeness In this section we consider the completeness of term graph rewriting for simulating arbitrary term rewrite derivations (Subsection 1.5.1) and for comput 24 CHAPTER 1. TERM GRAPH REWRITING ing term normal ....

Andrea Corradini and Fabio Gadducci. A 2-categorical presentation of term graph rewriting. In Proc. Category Theory and Computer Science, volume 1290 of Lecture Notes in Computer Science. Springer-Verlag, 1997.

.... defined interesting tile models for the (asynchronous) calculus (Milner, Parrow and Walker 1992) and for the representation of both the operational and the abstract semantics of CCS with locations (Boudol, Castellani, Hennessy and Kiehn 1993) where both configurations and effects are term graphs (Corradini and Gadducci 1997). Term graphs are a reference oriented generalization of the ordinary (value oriented) notion of term, where the sharing of subterms can be specified also for closed (i.e. without variables) term graphs (terms can share variables, but shared subterms of closed terms can be freely copied, always ....

....the framework presented here to weaker categories, where some of the naturality axioms for the generalized natural transformations are missing. Indeed, we are currently developing tile systems for a wide class of applications, where configurations and effects have only a gs monoidal structure (Corradini and Gadducci 1997) rather than cartesian, i.e. where duplicators and dischargers are not natural. Indeed such structures are equipped with a notion of name sharing and have been shown useful to define a tile model for the (asynchronous) calculus (Milner, Parrow and Walker 1992) one of the most studied mobile ....

Corradini, A. and Gadducci, F. (1997) A 2-categorical Presentation of Term Graph Rewriting.

....case) since it takes into account rewritings with side effects and rewriting synchronization. Tile systems can be seen as double categories [13] and tiles themselves as double cells. They can be equipped with observational equivalences and congruences. The combined used of tiles and term graphs [7,12] for modeling asynchronous calculus and CCS with locations [9] has been described in [14,15] Also coordination models equipped with flexible synchronization primitives are presented in [29,10] Ongoing work [27] aims at translating the tile model into rewriting logic, in order to take ....

A. Corradini, F. Gadducci, A 2-Categorical Presentation of Term Graph Rewriting, Proc. CTCS'97, Springer LNCS, to appear, 1997.

....also for closed (i.e. without variables) terms 3 . The distinction is made very precisely by the axiomatization of algebraic theories: terms and term graphs differ by two axioms, representing, in a categorical setting, the naturality of transformations for copying and discharging arguments [20]. Term graphs have been shown useful in [27] to define a tile model for the (asynchronous) calculus [60] one of the most studied mobile calculi) and in [28] to represent both the operational and the abstract semantics of CCS [59] with locations [9] within the tile model. In both cases, flat ....

A. Corradini and F. Gadducci. A 2-categorical Presentation of Term Graph Rewriting. In: E. Moggi, G. Rosolini, Eds., Proc. CTCS'97. Springer LNCS 1290. 1997. pp. 87--105.

....a double category whose cells directly correspond to the sequents entailed by that system. It is interesting to remark that this construction can be characterized as universal. Tiles have already been used to model several formalisms. For example, the combined used of tiles and term graphs 1 [1, 7, 8] for modeling asynchronous calculus and CCS with locations has been described in [16, 17, 18] Also, extensions of Petri nets equipped with flexible synchronization primitives and inspired by the tile model are presented in [3, 4] Finally, it is studied in [34] how to translate the tile model ....

....of gs graphs [18] which are axiomatically defined by gs monoidal theories. The latter is a logical theory similar to, but weaker then, the ordinary algebraic theory of terms and substitutions. In the Appendix we give the finitary axiomatization of (one sorted) gs monoidal theories as presented in [21, 23, 7, 8]. The models of gs monoidal theories are gs monoidal categories, a special case of symmetric monoidal categories. Here we present an algebra of open graphs which coincides with the algebra in the Appendix in the case the signature Sigma contains only edge labels. The basic open graphs are the ....

A. Corradini, F. Gadducci, A 2-Categorical Presentation of Term Graph Rewriting, in Proc. CTCS'97, LNCS 1290, 1997.

....because the only important information they offer is the sharing of common resources. A main advantage of tiles is that data structures for configurations and effects are not restricted to be syntactic terms. A small variation over ordinary terms which has proved very expressive is term graphs [15]: they are a reference oriented generalization of the ordinary (value oriented) notion of term, where the sharing of subterms can be specified also for closed (i.e. without variables) term graphs (terms can share variables, but shared subterms of closed terms can be freely copied, always yielding ....

A. Corradini and F. Gadducci. A 2-categorical Presentation of Term Graph Rewriting. In: E. Moggi, G. Rosolini, Eds., Proc. CTCS'97, LNCS 1290, 87-- 105 (1997).

.... is that of commutative monoids; other applications include Phrase Structure Grammars, for which the algebraic structure is that of monoids, Term Rewriting Systems, where states are represented as arrows of a small cartesian category, Term Graph Rewriting, where gs monoidal categories are used [CG97], and pure Logic Programming, where the models turn out to be indexed monoidal categories [CA93] Concerning our application the STS method has three main advantages: 1. Both steps, the construction of the structured transition system from the program and the construction of the model of ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In Proceedings CTCS'97, Springer LNCS 1290, 1997.

....of functions (i.e. source and target) between them. 3 In fact, just by varying the environment category, structured transition systems exactly describe such diverse models of computation as P T Petri nets in the sense of [21] concurrent grammars, concurrent term rewriting, term graph rewriting [4], graph rewriting [6,17] and Horn Clause Logic [3] More interestingly, the free functor (which exists under mild conditions on C) mapping the category of structured transition systems on C to the category of internal categories in C actually corresponds to de ning the operational semantics of ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In Proc. CTCS'97, LNCS 1290. Springer Verlag, 1997.

.... a fundamental property of correspondence can be shown between gs monoidal theories (over ordinary signatures) and term graphs: Each arrow t : n m identifies a term graph over Sigma with a specified m tuple of roots and a specified n tuple of variable nodes, and composition is graph replacement [19]. The acronym gs stands indeed for graph substitution. Example 2.1 (terms and theories) Let us consider the signature Sigma e = S 3 i=0 Sigma i , where Sigma 0 = fa; bg, Sigma 1 = ffg, Sigma 2 = fgg and Sigma 3 = fhg. Some of the arrows in GS( Sigma e ) are a; f : 0 1, a Omega f ) g : ....

.... concrete, than the one allowed by the ordinary description as elements of a term algebra, separating in a better way the Sigma structure from the additional algebraic structure that the meta operators used in the set theoretic presentation of term algebras (like substitution) implicitly enjoy [19,21]. 3 Rewriting logic 3.1 The unconditional, one sorted rewriting logic We assume the reader to be familiar with the usual, set theoretic presentation of algebraic specifications. In particular, given a signature Sigma and a set of variables X, then T Sigma (X) denotes the free algebra over X. ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In E. Moggi and G. Rosolini, editors, Category Theory and Computer Science, volume 1290 of Lect. Notes in Comp. Science, pages 87--105. Springer Verlag, 1997.

....studied by many authors, but only in operational style. In fact, term graphs have been represented as directed graphs satisfying a number of constraints [3] as suitable labelled hyper graphs called jungles [36] or as sets of recursive equations [1] among others. Only recently we have shown in [13] that the rewriting of acyclic term graphs can also be presented in a categorical way similar to the 2 categorical presentation of term rewriting. In fact, ranked, acyclic) term graphs over a signature Sigma are in one to one correspondence with the arrows of the free gs monoidal category ....

.... shown in [15] the non naturality of r is related to the fact that term graphs with different degree of sharing are distinct, while that of allows for the presence of garbage in a term graph (i.e. nodes not reachable from the roots) The main contribution of this paper is the generalisation of [13] to the categorical representation of possibly cyclic term graph rewriting. This is not a minor point, since it is shown in [30] that in the presence of suitable, quite natural, sharing strategies, cyclic term graphs can be generated during rewriting even if starting from an acyclic graph and ....

[Article contains additional citation context not shown here]

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In E. Moggi and G. Rosolini, editors, Category Theory and Computer Science, volume 1290 of LNCS, pages 87--105. Springer Verlag, 1997.

....arrows, called duplicators. They are reminiscent of arrows of the same name which are obtained in cartesian categories as pairings of two instances of an identity. However they do not form a natural transformation as duplicators in cartesian categories. Besides duplicators, gs monoidal categories [8] are equipped with dischargers and they differ from cartesian categories just for missing the naturality axioms on duplicators and dischargers. The arrows of the gs monoidal category freely generated by a signature Sigma represent the term graphs [8] on Sigma . Symmetric strict monoidal ....

....Besides duplicators, gs monoidal categories [8] are equipped with dischargers and they differ from cartesian categories just for missing the naturality axioms on duplicators and dischargers. The arrows of the gs monoidal category freely generated by a signature Sigma represent the term graphs [8] on Sigma . Symmetric strict monoidal categories equipped with duplicators (without naturality) are called share categories; match share, if both them and their opposite are share categories. Those are the categories where processes of contextual nets live: The main result of the paper states ....

[Article contains additional citation context not shown here]

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In Proceedings CTCS'97, volume 1290 of LNCS. Springer Verlag, 1997.

.... the bicategories of processes of Walters [22,23] to the pre monoidal categories of Power and Robinson [34] to the action structures of Milner [30] to the interaction categories of Abramsky [1] to the sharing graphs of Hasegawa [20] and to the gs monoidal categories of Corradini and Gadducci [10,11], just to mention a few (see also [12,16,18,35] It is noteworthy that all these structures can be seen as enrichments of symmetric monoidal categories, which give the basis for the description of a distributed environment in terms of a wire and box diagram. We propose a schema for describing ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In E. Moggi and G. Rosolini, editors, Proc. of CTCS'97, Category Theory and Computer Science, volume 1290 of Lect. Notes in Comp. Science, pages 87-105. Springer Verlag, 1997.

.... the bicategories of processes of Walters [18, 19] to the pre monoidal categories of Power and Robinson [28] to the action structures of Milner [24] to the interaction categories of Abramsky [1] to the sharing graphs of Hasegawa [16] and to the gs monoidal categories of Corradini and Gadducci [7, 8], just to mention a few (see also [9, 11, 15, 29] All these structures can be seen as enrichments of symmetric monoidal categories, which give the basis for the description of a distributed environment in terms of a wire and box diagram. We propose a schema for describing normal forms for this ....

....offer a concrete mathematical structure corresponding to a normalized representation for gs graphs. 2 The main result of [8] states that the free gs monoidal category over a (ordinary) signature is isomorphic to the class of (ranked) term graphs labeled over it. Such a property is exploited in [7] to give an inductive, algebraic account of term graph rewriting. It is in this setting that we recover the intuitive interpretation of copying and discharging as suitable operations over graphical structures. Also the open graphs of [26] form a free gs monoidal category: The one generated by the ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In Category Theory and Computer Science, volume 1290 of LNCS. Springer Verlag, 1997.

....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 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 ....

....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 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 slightly from the standard definition (see for example [13] because our main concern is the algebraic structure of graphs. Definition 1 (directed concrete graphs) A ....

[Article contains additional citation context not shown here]

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In Proceedings CTCS'97, volume 1290 of LNCS. Springer Verlag, 1997.

.... term graphs (roughly, graphs whose nodes are labeled by operators, as defined e.g. in the introductory chapter of [44] Each arrow n m identifies a term graph t over Sigma with a specified m tuple of roots and a specified n tuple of variables nodes, and arrow composition is graph replacement [7]. This correspondence motivates the acronym, where gs stands for graph substitution. Example 1 (terms and theories) Let us consider the signature Sigma ffl = S 2 i=0 Sigma i , where Sigma 0 = fa; bg, Sigma 1 = ff; gg and Sigma 2 = fhg (that same signature is often used in the next ....

....(monoidal, graph) if, in defining R, l and r are elements of GS( Sigma oe ) M( Sigma oe ) and G( Sigma oe ) respectively) The choice depends on how expressive we need our system to be. For example, a context system, 1 In this view, and r describe respectively garbage collection and sharing [7, 8]. 7 introduced in [30] in order to generalise sos rules to deal with process contexts, is just a graph system, where R : N Sigma oe Theta G( Sigma ) Theta Sigma Theta Sigma oe , with the further restriction that a : 1 1 for all a 2 Sigma (hence, for all d 2 N , if R(d) hl; a; b; ....

[Article contains additional citation context not shown here]

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In E. Moggi and G. Rosolini, editors, Category Theory and Computer Science, volume 1290 of LNCS, pages 87--105. Springer Verlag, 1997.

....important role in the theory since it allows for definitions and proofs by structural induction. From a categorical view point, such inductive definitions have been given for various formalisms via the construction of free categories equipped with an orthogonal (algebraic or categorical) structure [3,7,8,10,16,25 27,29,30]. Often such categorical models of rewriting do not only axiomatise the rewrite relation but impose an equivalence on rewriting sequences which captures the basic concurrency properties of the system. In the double pushout (DPO) approach to graph transformation [11,15] the operational definition ....

....of graph derivations. The restriction to iso cells is necessary since cells are meant to represent rewrite steps which have to be explicitly specified by productions. 5 From DPO rewrites to bi categories We already mentioned how most of the categorical descriptions of productionbased systems [7,8,16,18,20,25,27,29,30] simulate computations via cells. To a certain extent, they all share the same view, representing such a system as a computad [31] namely, a category (in our case, a discrete bi category) augmented with a graph structure over hom sets (informally, a set S of cells 4 Actually, in [5] the authors ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In E. Moggi and G. Rosolini, editors, Category Theory and Computer Science, volume 1290 of LNCS, pages 87--105. Springer Verlag, 1997.

.... a fundamental property of correspondence can be shown between gs monoidal theories (over ordinary signatures) and term graphs: Each arrow t : n m identifies a term graph over Sigma with a specified m tuple of roots and a specified n tuple of variable nodes, and composition is graph replacement [19]. The acronym gs stands indeed for graph substitution. Example 2.1 (terms and theories) Let us consider the signature Sigma e = S 3 i=0 Sigma i , where Sigma 0 = fa; bg, Sigma 1 = ffg, Sigma 2 = fgg and Sigma 3 = fhg. Some of the arrows in GS( Sigma e ) are a; f : 0 1, a Omega f ) g : ....

.... concrete, than the one allowed by the ordinary description as elements of a term algebra, separating in a better way the Sigma structure from the additional algebraic structure that the meta operators used in the set theoretical presentation of term algebras (like substitution) implicitly enjoy [19,21]. 3 Rewriting logic 3.1 The unconditional, one sorted rewriting logic We assume the reader to be familiar with the usual, set theoretic presentation of algebraic specifications. In particular, given a signature Sigma and a set of variables X, then T Sigma (X) denotes the free algebra over X. ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In E. Moggi and G. Rosolini, editors, Category Theory and Computer Science, volume 1290 of LNCS, pages 87--105. Springer Verlag, 1997.

....composition. Furthermore, the generated rewriting sequences are subject to an equivalence that coincides with the so called permutation equivalence [7] due to the axioms of 2 categories [21, 42] A similar construction can be followed for term graph rewriting as well, as it is shown in [10]. The idea is to add to the gs monoidal category generated by Sigma cells representing the rules of a term graph rewriting system, and to consider the (gs monoidal) 2 category freely generated by such cells. The main result is that the cells in the resulting 2 category represent term graph ....

....can delete a sub term of unbounded size) This motivated our study of structures related to terms, but more concrete, i.e. term graphs. After the characterisation of term graphs using gs monoidal theories presented here and the 2 categorical presentation of their rewriting proposed in [10], we intend to study to which extent the resulting equivalence on rewriting sequences is satisfactory for a concurrent semantics. Other authors have considered categories similar to ours with different motivations. The most related to ours, in our opinion, is the study of the algebra of flow ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In E. Moggi and G. Rosolini, editors, Category Theory and Computer Science, volume 1290 of LNCS, pages 87--105. Springer Verlag, 1997.

.... defined in the literature in many yet operational in flavour ways, representing term graphs for example as directed acyclic graphs satisfying a number of constraints [3] as suitable labelled hypergraphs called jungles [12] or as sets of recursive equations [1] Only recently we have shown in [7] that the rewriting of acyclic term graphs can also be presented in a categorical way similar to the 2 categorical presentation of term rewriting. In fact, ranked, acyclic) term graphs over a signature Sigma are in one to one correspondence with the arrows of the free gsmonoidal category ....

.... the non naturality of r is related to the fact that term graphs with different degree of sharing are distinct, while that of allows for the presence of garbage in a term graph (i.e. nodes not reachable from the roots) The main contribution of this paper is the generalization of the results of [7] to the categorical representation of possibly cyclic term graph rewriting. In Section 2 we first introduce (possibly cyclic) ranked term graphs , and three operations on them: Composition (a counterpart of term substitution) disjoint) union and feedback ; next we introduce our definition of ....

[Article contains additional citation context not shown here]

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In Proceedings CTCS'97, volume 1290 of LNCS. Springer Verlag, 1997.

....plan to investigate if, and how, our taxonomy can be extended and generalized to categories other than Set. We are thinking in particular of Graph, given the importance of the resulting categories in the modeling of the operational behaviour of rewriting systems and of automata, as pointed out in [9,11,20] and [30,31] respectively. In this sense, we consider interesting the results on monads and pseudo algebras presented in [25] ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In E. Moggi and G. Rosolini, editors, Category Theory and Computer Science, volume 1290 of LNCS, pages 87-105. Springer Verlag, 1997.

.... the bicategories of processes of Walters [22,23] to the pre monoidal categories of Power and Robinson [34] to the action structures of Milner [30] to the interaction categories of Abramsky [1] to the sharing graphs of Hasegawa [20] and to the gs monoidal categories of Corradini and Gadducci [10,11], just to mention a few (see also [12,16,18,35] It is noteworthy that all these structures can be seen as enrichments of symmetric monoidal categories, which give the basis for the description of distributed environments in terms of wire and box diagrams. We propose a schema for describing ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In E. Moggi and G. Rosolini, editors, Proc. of CTCS'97, Category Theory and Computer Science, volume 1290 of Lect. Notes in Comp. Science, pages 87-105. Springer Verlag, 1997.

....state, and the algebra on transitions ensures that basic transitions can be fired in any context and also in parallel. It has been shown that programs of many computational formalisms (including, among others, P T Petri nets in the sense of [25] term rewriting systems, term graph rewriting [7], graph rewriting [15, 9, 18] Horn Clause Logic [6] can be encoded as heterogeneous graphs having as collection of nodes algebras with respect to a suitable algebraic specification, and usually a poorer structure on arcs (often they are just a set) Structured transition systems are defined ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In E. Moggi and G. Rosolini, editors, Category Theory and Computer Science, volume 1290 of LNCS, pages 87--105. Springer Verlag, 1997.

.... the bicategories of processes of Walters [24,25] to the pre monoidal categories of Power and Robinson [36] to the action structures of Milner [31] to the interaction categories of Abramsky [1] to the sharing graphs of Hasegawa [22] and to the gs monoidal categories of Corradini and Gadducci [8,9], just to mention a few (see also [10,18,20,38] All these structures can be seen as enrichments of symmetric monoidal categories, which give the basis for the description of a distributed environment in terms of a wire and box diagram. 1 Terms may share variables, but shared sub terms of a ....

....sense we take as normal forms those terms with maximal parallelism and minimal duplication. 2 The main result of [9] states that the free gs monoidal category over a (onesorted, ordinary) signature is isomorphic to the class of (ranked) term graphs labeled over it. Such a property is exploited in [8] to give an inductive, algebraic account of term graph rewriting, together with the resulting functorial semantics. It is in this setting that we recover the intuitive interpretation of copying and discharging as suitable operations over graphical structures. We recall that term graphs are ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In E. Moggi and G. Rosolini, editors, Category Theory and Computer Science, volume 1290 of LNCS, pages 87--105. Springer Verlag, 1997.

.... term graphs (roughly, graphs whose nodes are labeled by operators, as defined e.g. in the introductory chapter of [42] Each arrow n m identifies a term graph t over Sigma with a specified m tuple of roots and a specified n tuple of variables nodes, and arrow composition is graph replacement [8]. This correspondence motivates the acronym, where gs stands for graph substitution. Example 1 (terms and theories) Let us consider the signature Sigma ffl = S 2 i=0 Sigma i , where Sigma 0 = fa; bg, Sigma 1 = ff; gg and Sigma 2 = fhg (that same signature is often used in the next ....

.... as a module (kind of basic component of a system) carrying information (equivalently, expressing a few restrictions) on the possible behaviour of its sub components (that is, of the terms to which it can be applied) 2 In this view, and r describe respectively garbage collection and sharing [9, 8]. Definition7 (algebraic rewriting systems) An algebraic rewriting system (ars) R is a tuple h Sigma oe ; Sigma ; N;Ri, where Sigma oe ; Sigma are signatures, N is a set of (rule) names and R is a function R : N A( Sigma oe ) Theta G( Sigma ) Theta G( Sigma ) Theta A( Sigma oe ....

[Article contains additional citation context not shown here]

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In Proceedings CTCS'97, volume 1290 of LNCS. Springer Verlag, 1997.

.... the bicategories of processes of Walters [18, 19] to the pre monoidal categories of Power and Robinson [28] to the action structures of Milner [24] to the interaction categories of Abramsky [1] to the sharing graphs of Hasegawa [16] and to the gs monoidal categories of Corradini and Gadducci [7, 8], just to mention a few (see also [9, 11, 15, 29] All these structures can be seen as enrichments of symmetric monoidal categories, which give the basis for the description of a distributed environment in terms of a wire and box diagram. We propose a schema for describing normal forms for this ....

....offer a concrete mathematical structure corresponding to a normalized representation for gs graphs. 2 The main result of [8] states that the free gs monoidal category over a (ordinary) signature is isomorphic to the class of (ranked) term graphs labeled over it. Such a property is exploited in [7] to give an inductive, algebraic account of term graph rewriting. It is in this setting that we recover the intuitive interpretation of copying and discharging as suitable operations over graphical structures. Also the open graphs of [26] form a free gs monoidal category: The one generated by the ....

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In Category Theory and Computer Science, volume 1290 of LNCS. Springer Verlag, 1997.

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A. Corradini, F. Gadducci, A 2-Categorical Presentation of Term Graph Rewriting, Proc. CTCS'97, Springer LNCS, to appear, 1997.

No context found.

A. Corradini and F. Gadducci. A 2-categorical presentation of term graph rewriting. In Proc. CTCS'97, volume 1290 of Lecture Notes in Computer Science. Springer, 1997.

No context found.

A. Corradini and F. Gadducci. A 2-categorical Presentation of Term Graph Rewriting. In: E. Moggi, G. Rosolini, Eds., Proc. CTCS'97. LNCS 1290, 87-- 105 (1997).

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