生命科学重写理论：CTMC语义学统一理论
Rewriting Theory for the Life Sciences: A Unifying Theory of CTMC Semantics (Long version)
摘要：
Kappa生物化学和MPERD有机化学框架是迄今为止生命科学中基于重写的方法的最密集开发的应用之一。这些类型的重写理论的一个典型特征是必须对要重写的对象实施某些结构约束（根据经验发现蛋白质具有特定的位点特征，一个碳原子最多可以形成四个键，......）。在本文中，我们提供了一些原始的发展，允许实现一个普遍的理论连续时间马尔可夫链（CTMCs）的随机重写系统。我们的核心数学概念是一个新的规则代数结构的相关设置重写规则的条件，无论是在双和Sesqui-Pushout语义，增强了一个合适的随机力学形式主义扩展，允许获得动态演化方程的模式计数统计。我们的论文的第二个主要贡献是一个新的框架限制重写理论，其中包括一个规则代数演算的限制下，所谓的约束保持完成的应用条件（规则被认为是只对对象的基础类别满足一个全球固定的结构约束）。这种新的框架，反过来又呈现一个忠实的编码的生物和有机化学重写的意义上的Kappa和MPERD可能的，这使我们能够得到一个重写为基础的配方的反应系统，包括一个完整的CTMC语义的实例，我们的通用CTMC框架。虽然提供了一个有趣的新的视角和概念简化的语义设置的Kappa，无论是正式的编码和CTMC语义的有机化学反应系统的动机MPERD框架是第一个这样的结果。
英文摘要：
The Kappa biochemistry and the MØD organic chemistry frameworks are amongst the most intensely developed applications of rewriting-based methods in the life sciences to date. A typical feature of these types of rewriting theories is the necessity to implement certain structural constraints on the objects to be rewritten (a protein is empirically found to have a certain signature of sites, a carbon atom can form at most four bonds, ...). In this paper, we contribute a number of original developments that permit to implement a universal theory of continuous-time Markov chains (CTMCs) for stochastic rewriting systems. Our core mathematical concepts are a novel rule algebra construction for the relevant setting of rewriting rules with conditions, both in Double- and in Sesqui-Pushout semantics, augmented by a suitable stochastic mechanics formalism extension that permits to derive dynamical evolution equations for pattern-counting statistics. A second main contribution of our paper is a novel framework of restricted rewriting theories, which comprises a rule-algebra calculus under the restriction to so-called constraint-preserving completions of application conditions (for rules considered to act only upon objects of the underlying category satisfying a globally fixed set of structural constraints). This novel framework in turn renders a faithful encoding of bio- and organo-chemical rewriting in the sense of Kappa and MØD possible, which allows us to derive a rewriting-based formulation of reaction systems including a full-fledged CTMC semantics as instances of our universal CTMC framework. While offering an interesting new perspective and conceptual simplification of this semantics in the setting of Kappa, both the formal encoding and the CTMC semantics of organo-chemical reaction systems as motivated by the MØD framework are the first such results of their kind.
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