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Learning More Expressive General Policies for Classical Planning Domains

Simon Ståhlberg, Blai Bonet, Hector Geffner

TL;DR

This work addresses the expressive bottleneck of learning general planning policies under $C_2$ by introducing a parametric Relational GNN, R-GNN[$t$], which transforms inputs to emulate higher-order interactions. By adjusting $t$, the model balances expressive power and computational cost, with $t=1$ providing strong $C_3$-level capabilities while maintaining quadratic space and time. Empirical results across numerous planning domains show that R-GNN[$1$] often yields higher coverage and better-quality plans than plain R-GNN, 2-GNNs, and Edge Transformers, particularly in $C_3$-level domains. Theoretical discussion links R-GNN[$t$] to $C_3$-relational joins via derived predicates, highlighting a practical middle ground between $C_2$ and $C_3$ expressivity and pointing to future work on intermediate representations that further reduce overhead.

Abstract

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 overcame 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 to store embeddings, rendering them infeasible in practice. In this work, we introduce a parameterized version R-GNN[$t$] (with parameter $t$) of Relational GNNs. Unlike GNNs, that are designed to perform computation on graphs, Relational GNNs are designed to do computation on relational structures. When $t=\infty$, R-GNN[$t$] approximates $3$-GNNs over graphs, but 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 expressivity 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 inputs only. Experimental results illustrate the clear performance gains of R-GNN[$1$] over the plain R-GNNs, and also over Edge Transformers that also approximate $3$-GNNs.

Learning More Expressive General Policies for Classical Planning Domains

TL;DR

This work addresses the expressive bottleneck of learning general planning policies under by introducing a parametric Relational GNN, R-GNN[], which transforms inputs to emulate higher-order interactions. By adjusting , the model balances expressive power and computational cost, with providing strong -level capabilities while maintaining quadratic space and time. Empirical results across numerous planning domains show that R-GNN[] often yields higher coverage and better-quality plans than plain R-GNN, 2-GNNs, and Edge Transformers, particularly in -level domains. Theoretical discussion links R-GNN[] to -relational joins via derived predicates, highlighting a practical middle ground between and expressivity and pointing to future work on intermediate representations that further reduce overhead.

Abstract

GNN-based approaches for learning general policies across planning domains are limited by the expressive power of , namely; first-order logic with two variables and counting. This limitation can be overcame by transitioning to -GNNs, for , wherein object embeddings are substituted with triplet embeddings. Yet, while -GNNs have the expressive power of , unlike - and -GNNs that are confined to , they require quartic time for message exchange and cubic space to store embeddings, rendering them infeasible in practice. In this work, we introduce a parameterized version R-GNN[] (with parameter ) of Relational GNNs. Unlike GNNs, that are designed to perform computation on graphs, Relational GNNs are designed to do computation on relational structures. When , R-GNN[] approximates -GNNs over graphs, but using only quadratic space for embeddings. For lower values of , such as and , R-GNN[] achieves a weaker approximation by exchanging fewer messages, yet interestingly, often yield the expressivity required in several planning domains. Furthermore, the new R-GNN[] architecture is the original R-GNN architecture with a suitable transformation applied to the inputs only. Experimental results illustrate the clear performance gains of R-GNN[] over the plain R-GNNs, and also over Edge Transformers that also approximate -GNNs.
Paper Structure (34 sections, 1 theorem, 24 equations, 1 figure, 2 tables, 1 algorithm)

This paper contains 34 sections, 1 theorem, 24 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. General policies from logic.
  4. General policies from neural nets.
  5. GNNs, R-GNNs, and $C_k$ logics.
  6. Background
  7. Planning and Generalized Planning
  8. Graph Neural Networks (GNNs)
  9. Relational GNNs (R-GNNs)
  10. Weisfeiler-Leman Coloring Algorithms
  11. 1-Dimensional WL (1-WL).
  12. Folklore $k$-Dimensional WL ($k$-FWL).
  13. Oblivious $k$-Dimensional WL ($k$-OWL).
  14. 2-FWL vs. 2-OWL.
  15. Expressive power.
  16. ...and 19 more sections

Key Result

Theorem 2

Let $\sigma$ be a relational language, and let $\mathcal{D}\xspace$ be a finite collection of $C_3$-relational joins. Then, there is a tuple of parameters $\langle t,k,L \rangle\xspace$ and network $N$ in $\text{R-GNN}[\sigma,t,k,L]\xspace$ that computes $\mathcal{D}\xspace$.

Figures (1)

  • Figure 1: Two $8{\times}4$ test instances of the Navig-xy domain where the robot has to reach the green cell in a grid with obstacles, and where the objects are the values of each one of the two coordinates, and not the cells themselves. The problem is not in $C_2$ in this representation, and indeed, after training, the baseline R-GNN solves the instance on the left but not the one on the right, while R-GNN[$t$], for $t\,{=}\,1$, solves all the instances in the test set.

Theorems & Definitions (2)

  • Definition 1: $C_3$-Relational Joins
  • Theorem 2: Computation of $C_3$-Relational Joins