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25
Term Graph Rewriting
, 1998
"... Term graph rewriting is concerned with the representation of functional expressions as graphs, and the evaluation of these expressions by rulebased graph transformation. Representing expressions as graphs allows to share common subexpressions, improving the efficiency of term rewriting in space ..."
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Cited by 72 (5 self)
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Term graph rewriting is concerned with the representation of functional expressions as graphs, and the evaluation of these expressions by rulebased graph transformation. Representing expressions as graphs allows to share common subexpressions, improving the efficiency of term rewriting in space and time. Besides efficiency, term graph rewriting differs from term rewriting in properties like termination and confluence. This paper is a survey of (acyclic) term graph rewriting, where emphasis is given to the relations between term and term graph rewriting. We focus on soundness of term graph rewriting with respect to term rewriting, on completeness for proving validity of equations and for computing term normal forms, on termination and confluence, and on term graph narrowing. Keywords: term graph rewriting, termination, confluence, term rewriting, narrowing Classification: 68Q05, 68Q40, 68Q42 (AMS '91); D.1.1, F.1.1, F.4.2, I.1.1 (CR '98) Note: This paper will appear in H...
The Tile Model
 PROOF, LANGUAGE AND INTERACTION: ESSAYS IN HONOUR OF ROBIN MILNER
, 1996
"... In this paper we introduce a model for a wide class of computational systems, whose behaviour can be described by certain rewriting rules. We gathered our inspiration both from the world of term rewriting, in particular from the rewriting logic framework [Mes92], and of concurrency theory: among the ..."
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Cited by 65 (24 self)
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In this paper we introduce a model for a wide class of computational systems, whose behaviour can be described by certain rewriting rules. We gathered our inspiration both from the world of term rewriting, in particular from the rewriting logic framework [Mes92], and of concurrency theory: among the others, the structured operational semantics [Plo81], the context systems [LX90] and the structured transition systems [CM92] approaches. Our model recollects many properties of these sources: first, it provides a compositional way to describe both the states and the sequences of transitions performed by a given system, stressing their distributed nature. Second, a suitable notion of typed proof allows to take into account also those formalisms relying on the notions of synchronization and sideeffects to determine the actual behaviour of a system. Finally, an equivalence relation over sequences of transitions is defined, equipping the system under analysis with a concurrent semantics, ...
An Algebraic Presentation of Term Graphs, via GSMonoidal Categories
 Applied Categorical Structures
, 1999
"... . We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the wellknown characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particula ..."
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Cited by 37 (24 self)
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. We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the wellknown characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particular, we show that term graphs over a signature \Sigma are onetoone with the arrows of the free gsmonoidal category generated by \Sigma. Such a category satisfies all the axioms for Cartesian categories but for the naturality of two transformations (the discharger ! and the duplicator r), providing in this way an abstract and clear relationship between terms and term graphs. In particular, the absence of the naturality of r and ! has a precise interpretation in terms of explicit sharing and of loss of implicit garbage collection, respectively. Keywords: algebraic theories, directed acyclic graphs, gsmonoidal categories, symmetric monoidal categories, term graphs. Mathematical Subject Clas...
Process and Term Tile Logic
, 1998
"... In a similar way as 2categories can be regarded as a special case of double categories, rewriting logic (in the unconditional case) can be embedded into the more general tile logic, where also sideeffects and rewriting synchronization are considered. Since rewriting logic is the semantic basis o ..."
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Cited by 33 (25 self)
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In a similar way as 2categories can be regarded as a special case of double categories, rewriting logic (in the unconditional case) can be embedded into the more general tile logic, where also sideeffects and rewriting synchronization are considered. Since rewriting logic is the semantic basis of several language implementation efforts, it is useful to map tile logic back into rewriting logic in a conservative way, to obtain executable specifications of tile systems. We extend the results of earlier work by two of the authors, focusing on some interesting cases where the mathematical structures representing configurations (i.e., states) and effects (i.e., observable actions) are very similar, in the sense that they have in common some auxiliary structure (e.g., for tupling, projecting, etc.). In particular, we give in full detail the descriptions of two such cases where (net) processlike and usual term structures are employed. Corresponding to these two cases, we introduce two ca...
An Inductive View of Graph Transformation
 In Workshop on Algebraic Development Techniques
, 1998
"... . The dynamic behavior of rulebased 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 i ..."
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Cited by 30 (12 self)
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. The dynamic behavior of rulebased 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 dgsmonoidal category. We then apply 2categorical techniques, already known for term and term graph rewriting [29, 7], recasting in this framework the...
A BiCategorical Axiomatisation of Concurrent Graph Rewriting
, 1999
"... In this paper the concurrent semantics of doublepushout (DPO) graph rewriting, which is classically defined in terms of shiftequivalence classes of graph derivations, is axiomatised via the construction of a free monoidal bicategory. In contrast to a previous attempt based on 2categories, the us ..."
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Cited by 18 (10 self)
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In this paper the concurrent semantics of doublepushout (DPO) graph rewriting, which is classically defined in terms of shiftequivalence classes of graph derivations, is axiomatised via the construction of a free monoidal bicategory. In contrast to a previous attempt based on 2categories, the use of bicategories allows to define rewriting on concrete graphs. Thus, the problem of composition of isomorphism classes of rewriting sequences is avoided. Moreover, as a first step towards the recovery of the full expressive power of the formalism via a purely algebraic description, the concept of disconnected rules is introduced, i.e., rules whose interface graphs are made of disconnected nodes and edges only. It is proved that, under reasonable assumptions, rewriting via disconnected rules enjoys similar concurrency properties like in the classical approach.
Graph Rewriting, Constraint Solving and Tiles for Coordinating Distributed Systems
 Applied Categorical Structures
, 1999
"... . In this paper we describe an approach to model the dynamics of distributed systems. For distributed systems we mean systems consisting of concurrent processes communicating via shared ports and posing certain synchronization requirements, via the ports, to the adjacent processes. The basic idea is ..."
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Cited by 17 (14 self)
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. In this paper we describe an approach to model the dynamics of distributed systems. For distributed systems we mean systems consisting of concurrent processes communicating via shared ports and posing certain synchronization requirements, via the ports, to the adjacent processes. The basic idea is to use graphs to represent states of such systems, and graph rewriting to represent their evolution. The kind of graph rewriting we use is based on simple contextfree productions which are however combined by means of a synchronization mechanism. This allows for a good level of expressivity in the system without sacrifying full distribution. To formally model this kind of graph rewriting, however, we do not adopt the classical graph rewriting style but a more general framework, called the tile model, which allows for a clear separation between sequential rewriting and synchronization. Then, since the problem of satisfying the synchronization requirements may be a complex combinatorial pro...
Internal Strategies in a Rewriting Implementation of Tile Systems
 Rewriting Logic and its Applications, volume 15 of Electronic Notes in Theoretical Computer Science. Elsevier Sciences
, 1998
"... Tile logic extends rewriting logic, taking into account rewriting with sideeffects and rewriting synchronization. Since rewriting logic is the semantic basis of several language implementation efforts, it is interesting to map tile logic back into rewriting logic in a conservative way, to obtain ex ..."
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Cited by 15 (11 self)
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Tile logic extends rewriting logic, taking into account rewriting with sideeffects and rewriting synchronization. Since rewriting logic is the semantic basis of several language implementation efforts, it is interesting to map tile logic back into rewriting logic in a conservative way, to obtain executable specifications of tile systems. The resulting implementation requires a metalayer to control the rewritings, so that only tile proofs are accepted. However, by exploiting the reflective capabilities of the Maude language, such metalayer can be specified as a kernel of internal strategies. It turns out that the required strategies are very general and can be reformulated in terms of search algorithms for nonconfluent systems equipped with a notion of success. We formalize such strategies, giving their detailed description in Maude, and showing their application to modeling uniform tile systems. 1 Introduction The evolution of a process in a concurrent system often depends on the ...
Axioms for Contextual Net Processes
 In Automata, Languages and Programming, volume 1443 of LNCS
, 1998
"... . In the classical theory of Petri nets, a process is an operational description of the behaviour of a net, which takes into account the causal links between transitions in a sequence of firing steps. In the categorical framework developed in [19, 11], processes of a P/T net are modeled as arrows of ..."
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Cited by 14 (9 self)
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. In the classical theory of Petri nets, a process is an operational description of the behaviour of a net, which takes into account the causal links between transitions in a sequence of firing steps. In the categorical framework developed in [19, 11], processes of a P/T net are modeled as arrows of a suitable monoidal category: In this paper we lay the basis of a similar characterization for contextual P/T nets, that is, P/T nets extended with read arcs, which allows a transition to check for the presence of a token in a place, without consuming it. 1 Introduction Petri nets [24] are probably the best studied and most used model for concurrent systems: Their range of applications covers a wide spectrum, from their use as a specification tool to their analysis as a suitable semantical domain. A recent extension to the classical model concerns a class of nets where transitions are able to check for the presence of a token in a place without actually consuming it. While the possibility ...