[论文解读] Termination of Graph Transformation Systems Using Weighted Subgraph Counting

Roy Overbeek, Jörg Endrullis|arXiv (Cornell University)|Mar 14, 2023

Model-Driven Software Engineering Techniques参考文献 54被引用 7

一句话总结

本文提出了一种新颖的、基于加权子图计数的图变换系统终止性方法，适用于rm-黏着准拓扑范畴。该方法通过定义基于加权态射的递减度量，实现了对PBPO+、SqPO、DPO和AGREE等多种形式化系统中非线性规则的终止性证明，并实现了用于交互式相对终止性证明的REPL工具。

ABSTRACT

This is a Scala implementation of a termination method for PBPO+ graph transformation systems, described in: R. Overbeek & J. Endrullis. Termination of Graph Transformation Systems Using Weighted Subgraph Counting. arXiv, 2023. url Context The algebraic approach to graph transformation uses category theory to define graph transformation systems. PBPO+ (Overbeek et al., 2023) is a recent proposal that unifies several existing approaches in the setting of quasitoposes. Roughly speaking, a quasitopos is a category that is set-like in a variety of ways (e.g., in the way unions behave), and it contains many graph categories of interest. A fundamental question for graph transformation, and computational models in general, is when and how one can decide whether a given system terminates, i.e., whether it admits only finite computations on arbitrary input. Criteria for graph transformation are sparse, and usually defined for specific graph notions. The method proposed in (Overbeek & Endrullis, 2023) is instead defined more abstractly for more general categories; more precisely, the rm-adhesive quasitoposes. When implementing such a categorically formulated method, one may wonder how many of the categorical constructions can carry over into program code. Computational category theory tries to implement the categorical constructions as faithfully as possible. This repository follows the computational category theory philosophy. Performance considerations have been considered to a limited extent only. Code outline The source directory is structured as follows: Directory categorytheory contains all basic definitions related to general categories, as well as more specialized classes of categories such as presheaf categories and quasitoposes. Directory labeledgraph contains a definition of the category of labeled graphs, which is an instance of a topos, and thus of an rm-adhesive quasitopos. Directory rewriting contains definitions and examples of PBPO+ rules (as well as DPO rules). Directory termination implements the termination method, as well as formatting functions. Directory repl, parsing, and util speak for themselves. Usage This is a normal sbt project. You can compile code with sbt compile, and run it with sbt run. When run, the graphTT REPL starts. For an explanation of this REPL, see the research paper cited above.

研究动机与目标

• 开发一种适用于多种代数图变换框架的通用、范畴论终止性方法。
• 将终止性分析扩展至图重写系统中的非线性（复制性）规则。
• 通过在准拓扑范畴中将规则编码为PBPO+，统一PBPO+、SqPO、DPO和AGREE的终止性证明。
• 实现一个实用工具，配备REPL以支持交互式、引导式的相对终止性证明。
• 通过简化条件并包含完整证明及此前未公开的新内容，改进先前工作。

提出的方法

• 该方法通过针对每个图G的加权态射（T → G）的加权和来定义终止性度量，其中权重取自某一类态射（例如单射、rm、同构）。
• 通过拉回和上推构造递减度量，利用黏着态射在PBPO+规则中模拟重写步骤。
• 该方法在rm-黏着准拓扑范畴中运行，确保在包括图范畴和拓扑范畴在内的各类范畴中具有良好的定义性和普遍性。
• 通过将SqPO、DPO和AGREE规则范畴化编码为PBPO+，将适用范围扩展至PBPO+之外。
• 实现使用Scala库并配备REPL，以交互方式探索和验证相对终止性证明。
• 通过有限的同态集和可计算的拉回/上推来检查定理4.23的条件，确保重写关系的可判定性。

实验结果

研究问题

• RQ1 是否可以开发一种单一的、范畴论的终止性方法，统一适用于PBPO+、SqPO、DPO和AGREE等多种代数图变换框架？
• RQ2 如何通过通用的范畴论方法处理非线性（复制性）规则的终止性分析？
• RQ3 在rm-黏着准拓扑范畴中，基于加权子图计数的递减度量存在的条件是什么？
• RQ4 如何高效实现该方法，并提供交互式支持以探索证明？
• RQ5 如何改进方法以处理规则等价性和终止性证明中的反模式？

主要发现

• 该方法是首个在rm-黏着准拓扑范畴中统一适用于PBPO+、SqPO、DPO和AGREE的范畴终止性技术。
• 该方法支持非线性规则，扩展了以往多限于线性或左线性系统的终止性方法的适用范围。
• 实现包含一个REPL，可借助范畴论库交互式、引导式地探索相对终止性证明。
• 在同态集有限且构造可计算的范畴（如FinSet和函子范畴[I, FinSet]）中，该方法具有可判定性。
• 本文通过用更简单、更弱的条件（定义4.16）替代先前复杂的因式分解条件，改进了先前工作，该条件与标准因式分解系统相关（命题4.18）。
• 完整证明现已被补全，此前因篇幅限制未予包含，同时补充了第一作者博士论文中的新内容。