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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.
Some Algebraic Laws for Spans (and Their Connections With MultiRelations)
 Proceedings of RelMiS 2001, Workshop on Relational Methods in Software. Electronic Notes in Theoretical Computer Science, n.44 v.3, Elsevier Science (2001
, 2001
"... This paper investigates some basic algebraic properties of the categories of spans and cospans (up to isomorphic supports) over the category Set of (small) sets and functions, analyzing the monoidal structures induced over both spans and cospans by the cartesian product and disjoint union of sets. O ..."
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Cited by 9 (3 self)
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This paper investigates some basic algebraic properties of the categories of spans and cospans (up to isomorphic supports) over the category Set of (small) sets and functions, analyzing the monoidal structures induced over both spans and cospans by the cartesian product and disjoint union of sets. Our results nd analogous counterparts in (and are partly inspired by) the theory of relational algebras, thus our paper also shed some light on the relationship between (co)spans and the categories of (multi)relations and of equivalence relations. And, since (co)spans yields an intuitive presentation in terms of dynamical system with input and output interfaces, our results introduce an expressive, twofold algebra that can serve as a specication formalism for rewriting systems and for composing software modules and open programs. Key words: Spans, multirelations, monoidal categories, system specications. Introduction The use of spans [1,6] (and of the dual notion of cospans) have been...
Some Algebraic Properties of (Co)Spans
, 2000
"... The paper investigates the algebraic properties of the categories of spans and cospans (up to isomorphic supports) over the category Set of (small) sets and functions. We analyze the monoidal structures induced over both spans and cospans by the cartesian product and disjoint union of sets, show ..."
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The paper investigates the algebraic properties of the categories of spans and cospans (up to isomorphic supports) over the category Set of (small) sets and functions. We analyze the monoidal structures induced over both spans and cospans by the cartesian product and disjoint union of sets, showing that they characterize dierent selfdual algebraic structures. Furthermore, our results shed some light on their relationship with the categories of (multi)relations and of equivalence relations. Introduction Recent years have seen a large diusion of equational presentations for dierent graphlike structures. The motivations underlying these works are quite varied, and dicult to recast in a unitary thread, even if they could be roughly divided into two halves, albeit tightly intertwined. On the one hand, there have been the studies regarding the syntax for diagrammatic presentations of, e.g., nets and circuits. The initial works can be considered those introducing the ownomial c...