Rule Algebras for Adhesive Categories

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

“…Proof. The proof is entirely analogous to the one for the case without application conditions [9,4] (cf. Appendix B.2).…”

“…Rewriting theories of DPO-and SqPO-type for rules with conditions over Madhesive categories are poised to provide a rich theoretical and algorithmic framework for modeling stochastic dynamical systems in the life sciences. The main result of the present paper consists in the introduction of a rule algebra framework that extends the pre-existing constructions [9,4,6] precisely via incorporating the notion of conditions. The sophisticated Kappa [11] and MØD [1] bio-/organo-chemistry platforms and related developments have posed one of the main motivations for this work.…”

“…In the present paper, we provide the necessary technical constructions in order to consider the CTMCs and analysis methods of relevance for more general types of compositional rewriting theories with conditions, with key examples provided in the form of biochemical graph rewriting in the sense of the Kappa framework (https://kappalanguage.org) [11], 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 [9,10,4] by the rewriting theoretical design feature of incorporating rules with conditions as well as constraints on objects (Section 3). Based upon these technical developments, we then investigate to which extent it is possible to utilize the rule-algebraic stochastic mechanics frameworks of the relevant types (Section 4) in order to compute evolution equations for the moments of pattern-count observables within the Kappa and MØD frameworks (Section 5 and 6).…”

“…In the DPO setting, this modification had been well-known [26,20,22,21], while it has been only very recently introduced for the SqPO setting [8]. 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 [9,10], for the SqPO case in [4], and for both settings augmented with conditions in [8]).…”

“…isomorphism class of objects) to a basis vector |X of a vector space Ĉ; then a rule r is lifted to a linear operator acting on Ĉ, 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 [6,9,4] to the present setting of rewriting rules with conditions.…”