Automated Inference of Graph Transformation Rules

TL;DR

The paper introduces a fully automated, data-driven method for inferring graph transformation rules from observed transitions, framing the problem as compressing a transition system into a small set of rules. It defines a formal pipeline built on the DPO (double-pushout) graph-transformation framework, introducing maximum rules, candidate subrules, and a set-cover optimization to obtain minimal exact or lossy generating rule sets. The approach yields a Kolmogorov-like complexity measure for graph transformation models and demonstrates its utility through experiments on formal languages, Tic-Tac-Toe strategies, and the Formose reaction, showing both exact rule recovery and model completion via lossy compression. The method offers a general, adaptable framework for reverse-engineering rule-based dynamics in domains like chemistry and biology, with potential extensions to abstract interpretation and more expressive rule formalisms.

Abstract

The explosion of data available in life sciences is fueling an increasing demand for expressive models and computational methods. Graph transformation is a model for dynamic systems with a large variety of applications. We introduce a novel method of the graph transformation model construction, combining generative and dynamical viewpoints to give a fully automated data-driven model inference method. The method takes the input dynamical properties, given as a "snapshot" of the dynamics encoded by explicit transitions, and constructs a compatible model. The obtained model is guaranteed to be minimal, thus framing the approach as model compression (from a set of transitions into a set of rules). The compression is permissive to a lossy case, where the constructed model is allowed to exhibit behavior outside of the input transitions, thus suggesting a completion of the input dynamics. The task of graph transformation model inference is naturally highly challenging due to the combinatorics involved. We tackle the exponential explosion by proposing a heuristically minimal translation of the task into a well-established problem, set cover, for which highly optimized solutions exist. We further showcase how our results relate to Kolmogorov complexity expressed in terms of graph transformation.

Figures (12)

• Figure 1: Commuting diagram of a rule $p = (L\xleftarrow{l}K\xrightarrow{r}R)$ applied to a graph $G$ using the match $m$, yielding a direct derivation $\delta = G \xRightarrow{m, p} H$. By definition, $p(\delta) = (G\xleftarrow{l'}D\xrightarrow{r'}H)$ is also a rule. The partial functions $f_p$ and $f_{p(\delta)}$ represent the element maps of the two rules.
• Figure 2: Commuting diagram of a subrule relation between rules $p_1 = (L_1\xleftarrow{l_1}K_1\xrightarrow{r_1}R_1)$ and $p_2 = (L_2\xleftarrow{l_2}K_2\xrightarrow{r_2}R_2)$.
• Figure 3: A commuting diagram showing a rule $p$ and three other rules, $p_0, p_1$ and $p_2$ with matches into the left side $L$. The relative position of the vertices is used to indicate the graph morphisms. $p_1$ is shown to be a subrule of $p$ and $p_0$ a subrule of both $p$ and $p_1$. $p_2$ is not a subrule of $p$ as $\varphi_{R_2}$ is just a monomorphism, not an isomorphism.
• Figure 4: An example transition system (\ref{['fig:transition_system']}) and the input transition system expressing equivalent behaviour (\ref{['fig:input_transitions']}). The relative position of the vertices is used to indicate the element maps. The element maps $f_0'$ and $f_0"$ are variations on the element map $f_0$ adapted to larger graphs. Same for $f_1$ and $f_2$.
• Figure 5: Example of three generating rule sets for the input transition system from Example \ref{['ex:input_transition_system']}. Unlike the rule set in (\ref{['fig:minimal_not_exact']}), both of the rule sets in (\ref{['fig:generating_rule_set']}) and (\ref{['fig:minimal_semantics']}) are exact. (\ref{['fig:minimal_semantics']}) is additionally a smallest exact rule set generating the input transition system.

Theorems & Definitions (11)

• Definition 2.1: Graph
• Definition 2.2: Rule
• Definition 2.3: Subrule
• Example 2.4: Subrules
• Definition 2.5: Input Transition System
• Example 2.6: Input Transition System
• Example 3.1: Generating Rule Sets
• Definition 3.2: Maximum Rule
• Example 3.3: Maximum Rules
• Example 3.4: Candidate Rules