Learning General Policies for Classical Planning Domains: Getting Beyond C₂

GNN-based approaches for learning general policies across planning domains are limited by the expressive power of C₂, 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 C₃, unlike $1$- and $2$-GNNs that are confined to C₂, 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 C₃ 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.