Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Grounding Methods for Neural-Symbolic AI

Rodrigo Castellano Ontiveros, Francesco Giannini, Marco Gori, Giuseppe Marra, Michelangelo Diligenti

TL;DR

This work addresses the grounding bottleneck in Neural-Symbolic AI by introducing a parametrized framework that generalizes Backward Chaining to balance expressiveness and scalability. The authors formalize a family of BC_{w,d} grounders, connect them to established approaches, and show how grounding strategy critically influences NeSy performance across multiple KG link-prediction tasks. Through extensive experiments with R2N, SBR, and DCR models, they demonstrate that modest grounding depths can capture essential multi-hop reasoning with substantial gains, while deeper grounding or full grounding often yields diminishing returns or worsens generalization on large graphs. The findings underscore the practical importance of selecting appropriate grounding criteria and pave the way for dynamic, end-to-end trainable grounders that adapt to the learner’s predictions.

Abstract

A large class of Neural-Symbolic (NeSy) methods employs a machine learner to process the input entities, while relying on a reasoner based on First-Order Logic to represent and process more complex relationships among the entities. A fundamental role for these methods is played by the process of logic grounding, which determines the relevant substitutions for the logic rules using a (sub)set of entities. Some NeSy methods use an exhaustive derivation of all possible substitutions, preserving the full expressive power of the logic knowledge. This leads to a combinatorial explosion in the number of ground formulas to consider and, therefore, strongly limits their scalability. Other methods rely on heuristic-based selective derivations, which are generally more computationally efficient, but lack a justification and provide no guarantees of preserving the information provided to and returned by the reasoner. Taking inspiration from multi-hop symbolic reasoning, this paper proposes a parametrized family of grounding methods generalizing classic Backward Chaining. Different selections within this family allow us to obtain commonly employed grounding methods as special cases, and to control the trade-off between expressiveness and scalability of the reasoner. The experimental results show that the selection of the grounding criterion is often as important as the NeSy method itself.

Grounding Methods for Neural-Symbolic AI

TL;DR

This work addresses the grounding bottleneck in Neural-Symbolic AI by introducing a parametrized framework that generalizes Backward Chaining to balance expressiveness and scalability. The authors formalize a family of BC_{w,d} grounders, connect them to established approaches, and show how grounding strategy critically influences NeSy performance across multiple KG link-prediction tasks. Through extensive experiments with R2N, SBR, and DCR models, they demonstrate that modest grounding depths can capture essential multi-hop reasoning with substantial gains, while deeper grounding or full grounding often yields diminishing returns or worsens generalization on large graphs. The findings underscore the practical importance of selecting appropriate grounding criteria and pave the way for dynamic, end-to-end trainable grounders that adapt to the learner’s predictions.

Abstract

A large class of Neural-Symbolic (NeSy) methods employs a machine learner to process the input entities, while relying on a reasoner based on First-Order Logic to represent and process more complex relationships among the entities. A fundamental role for these methods is played by the process of logic grounding, which determines the relevant substitutions for the logic rules using a (sub)set of entities. Some NeSy methods use an exhaustive derivation of all possible substitutions, preserving the full expressive power of the logic knowledge. This leads to a combinatorial explosion in the number of ground formulas to consider and, therefore, strongly limits their scalability. Other methods rely on heuristic-based selective derivations, which are generally more computationally efficient, but lack a justification and provide no guarantees of preserving the information provided to and returned by the reasoner. Taking inspiration from multi-hop symbolic reasoning, this paper proposes a parametrized family of grounding methods generalizing classic Backward Chaining. Different selections within this family allow us to obtain commonly employed grounding methods as special cases, and to control the trade-off between expressiveness and scalability of the reasoner. The experimental results show that the selection of the grounding criterion is often as important as the NeSy method itself.

Paper Structure

This paper contains 31 sections, 2 theorems, 5 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Preliminaries
  4. First-Order Logic (FOL).
  5. Grounding FOL Theories.
  6. Knowledge Graph Embeddings.
  7. Neural-Symbolic Models
  8. Atom Processing Layer
  9. Reasoning Layers
  10. Relational Reasoning Networks (R2N).
  11. Logic Tensor Networks (LTN).
  12. Deep Concept Reasoners (DCR).
  13. Grounding Methods
  14. Parameterized Backward Chaining Grounder
  15. Backward Chaining Grounder $(w=\infty, d=\infty)$.
  16. ...and 16 more sections

Key Result

Proposition 1

The set of facts that are provable via $BC_{w,d}$ is a subset of the provable facts via $BC_{w,d+1}$ and of the provable facts via $BC_{w+1,d}$.

Figures (6)

  • Figure 1: Portion of a GMN. We used the same color for atoms occurring in the same ground formula. The FOL theory includes the constants Italy (IT), France (FR), Spain (ES) and Europe (EU), the predicates LocatedIn (LocIn) and NeighbourOf (NeighOf), and the rule $\forall x\forall y\forall z~ LocIn(x, z) \land NeighOf(x, y) \rightarrow LocIn(y, z)$.
  • Figure 2: In the considered NeSy models an atom processor takes as input the representation of the constants and compute an embedding and an initial prediction of the atoms. The representations and predictions are refined via message passing, thus performing the reasoning process.
  • Figure 3: Illustration of some accepted/rejected proofs for different queries with respect to $w,d$ ($BC_{w,d}$) parameters.
  • Figure 4: MRR results for different methods using $BC_{1,d}$ with different depths for the Countries_abl datasets $AS_1, AS_2, AS_3$.
  • Figure 5: Results for the studied NeSy models and $BC_{w,d}$ grounders for Countries S2,S3 (S1 omitted as it is perfectly solved by all methods).
  • ...and 1 more figures

Theorems & Definitions (3)

  • Proposition 1
  • proof
  • Theorem 1