GNN-based approaches for learning general policies across planning domains are limited by the expressive power of $C_2$, namely; first-order logic with two variables and counting. This limitation can be overcomed by transitioning to $k$-GNNs, for $k=3$, wherein object embeddings are substituted with triplet embeddings. Yet, while $3$-GNNs have the expressive power of $C_3$, unlike $1$- and $2$-GNNs that are confined to $C_2$, they require quartic time for message exchange and cubic space for embeddings, rendering them impractical. In this work, we introduce a parameterized version of relational GNNs. When $t$ is infinity, R-GNN[$t$] approximates $3$-GNNs using only quadratic space for embeddings. For lower values of $t$, such as $t=1$ and $t=2$, R-GNN[$t$] achieves a weaker approximation by exchanging fewer messages, yet interestingly, often yield the $C_3$ features required in several planning domains. Furthermore, the new R-GNN[$t$] architecture is the original R-GNN architecture with a suitable transformation applied to the input states only. Experimental results illustrate the clear performance gains of R-GNN[$1$] and R-GNN[$2$] over plain R-GNNs, and also over edge transformers that also approximate $3$-GNNs.

翻译：基于图神经网络（GNN）的通用策略学习方法受限于$C_2$的表达能力，即带有计数功能的两变量一阶逻辑。这一限制可通过引入$k=3$的$k$-GNN（其中对象嵌入由三元组嵌入替代）加以克服。然而，尽管$3$-GNN具备$C_3$的表达能力（而限于$C_2$的$1$-GNN与$2$-GNN不具备此能力），其消息交换需四次方时间复杂度、嵌入需三次方空间复杂度，导致实际应用不可行。本文提出一种参数化的关系图神经网络（R-GNN）变体。当参数$t$取无穷大时，R-GNN[$t$]仅需二次方空间复杂度即可近似$3$-GNN；当$t$取较小值（如$t=1$或$t=2$）时，R-GNN[$t$]虽通过减少消息交换实现较弱近似，但有趣的是，其仍能提取多个规划领域所需的$C_3$特征。此外，新架构R-GNN[$t$]本质上是对原始R-GNN架构的输入状态施加适当变换的变体。实验结果表明，R-GNN[$1$]与R-GNN[$2$]在性能上显著优于普通R-GNN，也优于同样近似$3$-GNN的边缘变换网络。