Reversible Sesqui-Pushout Rewriting

The paper proposes a variant of sesqui-pushout rewriting (SqPO) that allows one to develop the theory of nested application conditions (NACs) for arbitrary rule spans; this is a considerable generalisation compared with existing results for NACs, which only hold for linear rules (w.r.t. a suitable class of monos). Besides this main contribution, namely an adapted shifting construction for NACs, the paper presents a uniform commutativity result for a revised notion of independence that applies to arbitrary rules; these theorems hold in any category with (enough) stable pushouts and a class of monos rendering it weak adhesive HLR. To illustrate results and concepts, we use simple graphs, i.e. the category of binary endorelations and relation preserving functions, as it is a paradigmatic example of a category with stable pushouts; moreover, using regular monos to give semantics to NACs, we can shift NACs over arbitrary rule spans. Introduction Nested application conditions (nacs) for rules of graph transformation systems (gtss) are a popular and intuitive means to increase the versatility of graph transformation. Tools such as agg3 and Groove4 support a weakened form of nacs, namely negative application conditions. So far, the theory of nacs is fully developed only for double pushout (dpo) rewriting with so-called linear rules, which means that transformation operations are restricted to deletion and addition of nodes and edges; for the ubiquitous example of simple graphs, linearity is even more restrictive, namely, it is not allowed to add or delete edges between pairs of unchanged nodes. We show that none of these restriction are necessary if we use a suitable combination of dpo and sesqui-pushout (sqpo) rewriting, which coincides with dpo for the case of linear rules (in adhesive categories [16]). More precisely, we shall extend the theory of nacs, notably the Translation Theorem [13, Theorem 6], to arbitrary spans as rules, which means that we accommodate not only for the deletion of edges between preserved nodes in simple graphs but we can ? This research was sponsored by the European Research Council (ERC) under grants 587327 “DOPPLER” and 320823 “RULE”. 3 http://user.cs.tu-berlin.de/~gragra/agg/ 4 http://groove.sourceforge.net/groove-index.html also handle the operations of merging and cloning of nodes – at least, if rule applications are free of side-effects. Absence of side-effects will be made formal by the definition of reversible sqpo (sqpor) rewriting, which is the new approach that we shall propose in this paper; its definition is quite natural: it merely amounts to restricting to those sqpo-diagrams that are also sqpo-diagrams “backwards” for the reversed rule. In the end, we obtain a variation of dpo rewriting, avoiding complications involving uniqueness of pushout complements (by use of final pullback complements [8]) and thus we do not need any restriction on rules or matches any more – having the best of both worlds. Besides the extension of the theory of nacs to arbitrary rules, we provide a suitable notion of independence for sqpor rewriting and give the corresponding commutativity result, which specialises to the existing theory for the dpo approach (with linear rules). The only categorical requirements are pullbacks and (enough) stable pushouts since we do not rely on uniqueness of pushout complements any more, which allows to drop the restriction to (left-)linear rules. Roughly, reversible sqpo-rewriting combines the controlled rewriting mechanism of dpo rewriting with the expressive power of sqpo-rewriting while being in line with the usual notions and results about independence of adjacent rule applications. We plan to put to use our stronger version of nacs in the context of formal modeling languages for systems biology, in particular the Kappa language [14]. The rewrite semantics of Kappa has been formalized as a gts over a particular category of structured graphs [6,14]. However, it seems natural to reformulate these semantics using an adhesive category [16] and nacs: some of the extra structure present in the patterns of Kappa rules intuitively specifies (positive) application conditions; moreover, matches are required to preserve so-called free sites, which amounts to a simple family of nacs. Kappa also supports a quantitative analysis that approximates the evolution of the expected number of occurrences of a given set of observable graphs over time using a system of differential equations [5]; if we want to formalise observables as graphs with a nac, we also need to keep track of the change in their occurrence counts for this quantitative analysis. This is the point where the shifting constructions for nacs as formulated in the literature [13,10] are the tool of choice. One might also consider extending this type of quantitative analysis to process calculi (via graphical encodings), in which case merging rules, as supported by the sqpo approach, become relevant, e.g. for encoding substitution rules. However, as we will see in Example 6, nac translation may break down when using the sqpo approach; this is why we consider the sqpor approach. The final contribution of this paper, is a first tentative solution to the failure of nac translation for the sqpo approach (see Example 6): we first construct for each sqpo-diagram the “best approximation” by a sqpor-diagram using a suitable “minimally extended” rule instance, which intuitively just contains enough additional context to make the side-effects of sqpo-rewriting (notably deletion of dangling edges) explicit; we then translate nacs to all possible sqpo-rule-instances, for which we finally can use sqpor-rewriting. Even though this solution is not effective, we conjecture that it will be viable for graph transformation systems with bounded node degree. We illustrate the new rewriting approach and our results through examples in the category of simple graphs; in fact, that our results apply to the category of simple graphs is interesting in itself. In summary, it should become clear that the proposal of the sqpor approach to rewriting is not merely triggered by the recent interest in reversible computation, but that it is of interest for the core theory of graph transformation and contributes to versatility in applications. Structure of the Paper We first recall the notion of (final) pullback complements [8] and sesqui-pushout (sqpo) rewriting [4] in Section 1, where we also state the relevant composition and decomposition results for the corresponding pullback squares, and recall related results on stable pushouts, which all together will be the technical backbone of our main theorems. We define reversible sesquipushout rewriting (sqpor) in Section 2 together with a notion of independence, for which we derive a uniform commutativity result (Theorem 1), assuming that (enough) stable pushouts exist. Then we recall the syntax and semantics of nested application conditions (nacs) in Section 3 and present our main result about nacs in Theorem 2, after describing the required categorical assumptions. In Section 4, we discuss how this result might be applied even to sqpo-rewriting. Related and future work is discussed in Section 5 where we also quickly discuss suitable categorical frameworks, before we conclude with a summary of our results in Section 6. 1 Preliminaries A secondary theme of the present paper is the use of an algebraic approach to perform rewriting on simple graphs, which are ubiquitous in computer science and beyond, but are not as well-behaved w.r.t. algebraic graph rewriting as for example (multi-)hypergraphs; we use the following definition. Definition 1 (Category of Simple Graphs). A simple graph is a pair of sets G = (VG, EG) where EG ⊆ VG×VG is an endorelation over VG; the elements of VG are nodes or vertices and EG contains all edges of the graph G. A graph morphism f : G H is a function f : VG VH such that EH ⊇ (f × f)(EG) = {( f(e), f(e′) ) | (e, e′) ∈ EG } . The category of simple graphs, denoted by G, has simple graphs as objects and graph morphisms as morphisms; composition and identities are given by (f ◦ g)(v) = f(g(v)) and idK(v) = v for all morphisms f : G H, g : K G and nodes v ∈ VK . The category of simple graphs will serve as running example to illustrate the concepts and results of the paper; we have chosen to keep the list of preliminary categorical concepts as short as possible. A short discussion of suitable categories of graph-like structures is given later in Section 5. 1.1 Final Pullback Complements and Stable Pushouts The crucial concept of sesqui-pushout rewriting that goes beyond standard textbooks on category theory are (final) pullback complements [8]; we use the original definition in terms of the universal property illustrated on the right in Figure 1.