Sesqui-Pushout Rewriting: Concurrency, Associativity and Rule Algebra Framework

Abstract

Sesqui-pushout (SqPO) rewriting is a variant of transformations of graph-like and other types of structures that fit into the framework of adhesive categories where deletion in unknown context may be implemented. We provide the first account of a concurrency theorem for this important type of rewriting, and we demonstrate the additional mathematical property of a form of associativity for these theories. Associativity may then be exploited to construct so-called rule algebras (of SqPO type), based upon which in particular a universal framework of continuous-time Markov chains for stochastic SqPO rewriting systems may be realized. * This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 753750.1 While non-linear rules in SqPO rewriting have interesting applications in their own right (permitting e.g. the cloning and fusing of vertices in graphs), this most general case is left for future work.SqPO Rewriting: Concurrency, Associativity and Rule Algebra Framework adhesive categories (see Assumption 1) in order to ensure certain technical properties necessary for our concurrency and associativity theorems to hold. To the best of our knowledge, apart from some partial results in the direction of developing a concurrency theorem for SqPO-type rewriting in [16,36,15], prior to this work neither of the aforementioned theorems had been available in the SqPO framework.Associativity of SqPO rewriting theories plays a pivotal role in our development of a novel form of concurrent semantics for these theories, the so-called SqPO-type rule algebras. Previous work on associative DPO-type rewriting theories [3, 5, 7] (see also [8]) has led to a category-theoretical understanding of associativity that may be suitably extended to the SqPO setting. In contrast to the traditional and well-established formalisms of concurrency theory for rewriting systems (see e.g. [42,25,23,15] for DPO-type semantics and [16,15] for a notion of parallel independence and a Local Church-Rosser theorem for SqPO-rewriting of graphs), wherein the focus of the analysis is mostly on derivation traces and their sequential independence and parallelism properties, the focus of our rule-algebraic approach differs significantly: we propose instead to put sequential compositions of linear rules at the center of the analysis (rather than the derivation traces), and moreover to employ a vector-space based semantics in order to encode the non-determinism of such rule compositions. It is for this reason that the concurrency theorem plays a quintessential role in our rule algebra framework, in that it encodes the relationship between sequential compositions of linear rules and derivation traces, which in turn gives rise to the socalled canonical representations of the rule algebras (see Section 4). This approach in particular permits to uncover certain combinatorial properties of rewriting systems that would otherwise not be accessible. While undoubtedly not a stand...

Rewriting Theory for the Life Sciences: A Unifying Theory of CTMC Semantics

“…Proof. The proof that we provide in [9, Appendix B.1] is entirely analogous to the one for the case without application conditions [4,10].…”

“…org) [12], and (organo-) chemical graph rewriting in the sense of the MØD framework (https://cheminf.imada.sdu.dk/mod/) [1]. The present paper aims to serve two main purposes: the first consists in providing an extension of the existing category-theoretical rule-algebra frameworks [4,10,11] by the rewriting theoretical design feature of incorporating rules with conditions as well as constraints on objects (Sect. 3).…”

“…isomorphism class of objects) to a basis vector |X of a vector spaceĈ; then a rule r is lifted to a linear operator acting onĈ, with the idea that this operator acting upon a basis vector |X should evaluate to the "sum over all possibilities to act with r on X". We will extend here the general rule algebra framework [4,6,10] to the present setting of rewriting rules with conditions. We will first lift the notion of rule composition into the setting of a composition operation on a certain abstract vector space over rules, thus realizing the heuristic concept of "summing over all possibilities to compose rules".…”

Rate Equations for Graphs

“…Partial differential equations describing the generating function of such observables can be derived systematically [7]. As the theory can handle adhesive categories in general and sesqui-pushout rewriting [5], it offers an treatment of irreversible rewrites alternative to the one presented in this paper. (The rule-algebraic approach can also handle application conditions [9]).…”

“…Let f 1 , f 2 , g, g 1 be matches, and α, β, γ rules, such that the diagrams (5) and (6) are derivations. Then there is a unique match g 2 , such that diagram 7 Proof.…”