Compositionality of Rewriting Rules with Conditions

Abstract

We extend the notion of compositional associative rewriting as recently studied in the rule algebra framework literature to the setting of rewriting rules with conditions. Our methodology is category-theoretical in nature, where the definition of rule composition operations encodes the non-deterministic sequential concurrent application of rules in Double-Pushout (DPO) and Sesqui-Pushout (SqPO) rewriting with application conditions based upon M-adhesive categories. We uncover an intricate interplay between the category-theoretical concepts of conditions on rules and morphisms, the compositionality and compatibility of certain shift and transport constructions for conditions, and thirdly the property of associativity of the composition of rules.

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

“…Together with a corresponding notion of constraint-preserving application conditions on rewriting rules, this approach yields a versatile extension of rewriting theory. In the DPO setting, this modification had been well-known [8,10,15,16], while it has been only very recently introduced for the SqPO setting [9]. For the rule algebra constructions presented in the main part of this contribution, we require in addition a certain compositionality property of our rewriting theories (established for the DPO case in [6,3], for the SqPO case in [7], and for both settings augmented with conditions in [9]).…”

“…We refer the interested readers to Definition 6 for the key application of the two operations (i.e., in defining the operation of sequential composition of rewriting rules with conditions), as well as to Example 4 for a detailed illustration of the uses of the two operations in computations of application conditions for rewriting rules. Definition 3 ( [8,9]). In an M-adhesive category satisfying the version of Assumption 1 appropriating for the type T ∈ {DP O, SqP O} of rewriting, we define the shift operation, denoted Shift, and the transport operation, denoted Trans, via their effect on conditions:…”

“…Crucially, there exist algorithmic implementations for both of these constructions in DPO-rewriting [8] as well as in SqPO-rewriting [9]: Theorem 2 ( [8,9]). With notations as in Definition 3, the action of the operation Shift on an application condition c X ∈ cond(C) over some object X along an M-morphism (y : X → Y ) is implemented concretely via the following recursive algorithm:…”

“…As a reference for notational conventions and in order to recall some of the standard definitions necessary in the main text, we collect here some of the materials contained in our recent paper [9] for the readers' convenience. 1.…”

“…The class of monomorphisms M of (C, M) is said to be stable under FPCs[9] if for every pair (b, a) of composable M-morphisms the FPC (c, d) (if it exists) is a pair of M-morphisms.…”