A Heterogeneous Pushout Approach to Term-Graph Transformation

Abstract

Abstract. We address the problem of cyclic termgraph rewriting. We propose a new framework where rewrite rules are tuples of the form (L, R, τ, σ) such that L and R are termgraphs representing the left-hand and the right-hand sides of the rule, τ is a mapping from the nodes of L to those of R and σ is a partial function from nodes of R to nodes of L. The mapping τ describes how incident edges of the nodes in L are connected in R, it is not required to be a graph morphism as in classical algebraic approaches of graph transformation. The role of σ is to indicate the parts of L to be cloned (copied). Furthermore, we introduce a notion of heterogeneous pushout and define rewrite steps as heterogeneous pushouts in a given category. Among the features of the proposed rewrite systems, we quote the ability to perform local and global redirection of pointers, addition and deletion of nodes as well as cloning and collapsing substructures.

Graph Transformation with Focus on Incident Edges

“…In [7] an algebraic approach of termgraph transformation, based on heterogeneous pushouts (HPO), has been proposed. With respect to cloning abilities, the HPO approach offers the possibility to make one or more copies of a node together with its outgoing edges.…”

“…Therefore, this way of cloning nodes is limited to the outgoing edges only and contrasts with the flexible possibilities of cloning edges proposed in the present paper. In fact, whenever a graph G rewrites into H according to the HPO approach using a rule (L, R, τ, σ) [7,Definition 5], the graph G can also be rewritten into H according to a rule L l ← K r → R where morphisms l and r encode the functions τ and σ.…”

“…The classical DPO and SPO approaches of graph transformation are clearly not well suited to perform cloning of nodes. As far as we are aware of, there are two algebraic attempts to deal with node cloning : the sesqui-pushout approach (SqPO) [4] and the heterogeneous pushout approach (HPO) [7]. The sesquipushout approach has the ability to clone nodes with all their incident edges whereas the HPO clones a node only with its outgoing edges.…”

Semantics-Preserving DPO-Based Term Graph Rewriting

“…For example, Banach [Ban93] defines "DACTL" term graph rewriting using a modified opfibration, and Kahl [Kah96,Kah97] uses both fibrations and opfibrations to define rewriting of term graphs with variable binding. A different approach to using separate classes of horizontal and vertical morphisms for term graph rewriting has been proposed by Duval et al [DEP09], who are using a specific rule concept as morphisms in the horizontal direction in their "heterogeneous pushout approach". More recently, motivated by attributed graphs, which share some characteristics with term graphs, Habel and Plump [HP12] propose "M,N -adhesive transformation systems" as one general framework to accommodate different classes of morphisms in the horizontal and vertical directions of the double-pushout setting.…”

Graph rewriting with polarized cloning

“…The classical DPO and SPO approaches of graph transformation are clearly not well suited to perform cloning of nodes. As far as we are aware of, there are two algebraic attempts to deal with cloning: the sesqui-pushout approach (SqPO) [4] and the heterogeneous pushout approach (HPO) [7]. They can be illustrated as follows:…”