# grounding_methods_for_neuralsymbolic_ai__fecac2ef.pdf

Grounding Methods for Neural-Symbolic AI

Rodrigo Castellano Ontiveros1 , Francesco Giannini2 , Marco Gori1 , Giuseppe Marra3 and Michelangelo Diligenti1

1University of Siena, Italy 2Scuola Normale Superiore, Italy 3KU Leuven, Belgium {rodrigo.castellano,marco.gori,michelangelo.diligenti}@unisi.it, francesco.giannini@sns.it, giuseppe.marra@kuleuven.be

A large class of Neural-Symbolic (Ne Sy) 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 Ne Sy 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 Ne Sy method itself.

1 Introduction Neural-Symbolic (Ne Sy) methods [d Avila Garcez et al., 2009; Marra et al., 2024] integrate the strengths of neural networks and symbolic reasoning, and heavily rely on the process of logic grounding, which bridges the semantic gap between neural representations and symbolic logic. However, classical grounding techniques in Ne Sy are impractical, and they do not take advantage of the generalization capabilities of Ne Sy approaches, which can deduce true facts using statistical methods without the need for a classical proof. In fact, the selection of appropriate grounding represents a significant challenge for Ne Sy methodologies. Existing approaches exhibit a dichotomy between exhaustive derivations that main-

tain full expressiveness of logical knowledge but suffer from scalability issues, and heuristic-based selective derivations that sacrifice logical coherence for computational efficiency. Indeed, the employed grounding process is often completely neglected in the Ne Sy literature, or relegated to a single sentence in the appendix. This hinders the possibility to reproduce, compare, and even fully understand the strengths and weaknesses of the different methods. This paper aims to fill this gap by exploring diverse grounding methods in Ne Sy AI. We contend that the selection of a suitable grounding criterion is as pivotal as the design of the Ne Sy method itself. To support this claim, we propose a parameterized generalization of the classic Backward Chaining algorithm, which leverages the well-studied grounding approaches employed in logic inference, but it can be relaxed to take advantage of the statistical nature of Ne Sy approaches. This allows the control of the trade-off between scalability and preservation of the logical context. Experimentally, we show the efficacy of the proposed grounding methodologies across various Ne Sy methods on the link prediction task in Knowledge Graphs (KGs).

Contributions. The main contributions of this paper are: (i) we formalize a variety of parametrized grounding methods for Ne Sy models. The parametrization controls the trade-off between scalability, ensuring the applicability of Ne Sy methods to large datasets, and expressiveness, hence allowing the model to take full advantage of the knowledge; (ii) we show that our results are general and not specific to a single Ne Sy method, and we provide an extensive experimental evaluation to support this claim, (iii) we redefine a large class of popular Ne Sy methods within a common message passing schema, allowing us to share a common implementation and reuse the same grounding and scaling techniques.

2 Preliminaries

First-Order Logic (FOL). A function-free FOL language [Barwise, 1977] is defined by a set of constants (entities) C, variables V and predicates (relations) R. An atomic formula, or atom, is r(a1, ..., ak) with a1, ..., ak C V, r R and k = arity(r). A literal is an atom or its negation, and each compound FOL formula α is obtained by recursively combining atomic formulas with connectives ( , , , , ) and quantifiers ( and ). Examples of FOL

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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 Located In (Loc In) and Neighbour Of (Neigh Of), and the rule x y z Loc In(x, z) Neigh Of(x, y) Loc In(y, z).

formulas are x Nation(x) y Capital Of(y, x), expressing that All nations have a capital or x y z Loc In(x, y) Loc In(y, z) Loc In(x, z), expressing the transitiveness of being located in a place. Given a formula α, we define vars(α) as the set of variables contained in α. For simplicity, in the following sections we will focus on FOL theories, i.e. sets of formulas, containing only specific formulas called Horn clauses, which play a fundamental role in computational logic. A Horn clause is a disjunction of literals with a single positive literal, which is equivalent to formulas in the form b1 . . . bn h, where the atoms b1, . . . , bn are called body atoms and the single atom h is called head.

Grounding FOL Theories. Let us fix a FOL language. A substitution θ is an assignment from variables to domain constants, like θ = {x : Italy, y : Europe}. A ground formula is a FOL formula where all the variables are constants. For example, given α = Loc In(x, y), we can ground it with θ and obtain θα = Loc In(Italy, Europe). The Herbrand Base HB = {r(c1, ..ck)|r R, k = arity(r), (c1, ..ck) Ck} is the set of all possible ground atoms formed using the predicate and constant symbols. Given a FOL theory Γ, its Herbrand Universe is defined as HUΓ = {θα : α Γ, θ is a substitution}, i.e. HUΓ is the set of all possible ground formulas that can be formed from Γ. The process of constructing the entire Herbrand Universe is called (full) grounding. In general, we refer to grounding as the process of constructing a subset of the Herbrand Universe for a given FOL Theory. Some methods like Markov Logic Networks (MLNs) [Richardson and Domingos, 2006], use a graphical representation of the entire Herbrand Universe (or a subset) called a Grounded Markov Network (GMN). In the GMN, each node represents a ground atom, and there is an edge between two nodes only if the corresponding ground atoms appear together in at least one grounding of a formula (Figure 1).

Knowledge Graph Embeddings. A Knowledge Graph (KG) represents knowledge as a graph of entities (nodes) and relations (edges), with facts as (subject, relation, object) triples. Knowledge Graph Embeddings (KGEs) [Wang et al., 2017] map KG entities and relations to low-dimensional vectors, encoding semantic meaning and statistical dependencies. KGEs enable effective reasoning even with incom-

plete/noisy graphs, for tasks like link prediction and query answering. Prominent examples include Dist Mult [Yang et al., 2015] and Compl Ex [Trouillon et al., 2016].

3 Neural-Symbolic Models

Neural networks, while effective at feature processing, are limited by opaque decision-making, large data requirements, and poor generalization. Symbolic AI, though interpretable, struggles with real-world complexity. Neurosymbolic (Ne Sy) models combine their strengths. This paper focuses on Ne Sy methods using a neural network to process FOL constants/predicates, using the GMN topology to define the reasoner. The overall structure of these methods, inspired by [Barbiero et al., 2024], is shown in Figure 2, and it is composed of two high-level components:

1. Atom Processing Layer mapping each ground atom into an embedding and computing its initial prediction.

2. Reasoning Layers updating the atoms embedding and score according to the structure imposed by the GMN associated to the logic theory, and computing a final updated version of the predictions for each ground atom.

3.1 Atom Processing Layer

The class of considered neural-symbolic methods assigns a numeric representation and an initial output prediction to all the ground atoms in the HB. This can be formalized as: given a ground atom α = P(c1, . . . , cn) and Emb(ci) the embedding of ci C, the atom processing layer outputs the embedding Emb0(α) and the initial output prediction o0(α) as:

Emb0(α) = Enc P (Emb(c1), ..., Emb(cn))

o0(α) = Score(Emb0(α))

where Enc P is a generic encoding function associated with the predicate P R, and Score is a generic scoring function. Many different representation learning approaches can be employed to implement and co-train the functions Enc, Score. While this paper focuses on KGEs and their scoring functions, the approach can be trivially applied to pattern recognition problems using neural networks.

3.2 Reasoning Layers

The reasoning layers take the outputs of the atom processing layer and provide an updated representation (both as embedding and prediction) for each ground atom in the head of the rules. Finally, it outputs the predictions for all atoms. In the following, we indicate the set of rules of a FOL theory having an atom h as head in the considered FOL grounding as R(h). The reasoner can be seen as the unfolding network of a message passing process [Gilmer et al., 2017] over the GMN, where messages flow from atoms appearing in the body of a ground formula to the atom in the head of the same ground formula. Below, we provide the message passing equations for some of the methods tested in this paper. The messages flow from an atom node b in the body of the j-th formula rj to the atom node h in the head of the same

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Figure 2: In the considered Ne Sy 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.

ground formula as: Embj(h) = Fj(Mne(h,1) h, ..., Mne(h,dj) h)

oj(h) = Gj(Mne(h,1) h, ..., Mne(h,dj) h)

where M j b h is the message defining the different neuralsymbolic methods, Fj, Gj are generic functions aggregating back the encoded representation and the output of a node, ne(h, 1), . . . , ne(h, dj) are the neighbours of node h for the j-th rule, i.e. the set of body atoms associated to atom h for the rule rj, with dj = |ne(h)| the number of atoms occurring in the body of the j-th rule rj. The embedding of an atom node h is updated by aggregating the representation for all the rules having atom h as head: Emb(h) = Ae

rj R(h) Embj(h), where Ae is an ag-

gregation operator like sum, mean or max. Other methods instead aggregate directly over the outputs of a node: o(h) = Ao

rj R(h) oj(h), where Ao can be any aggregation op-

erator, even if most methods set it as a disjunctive operator, like a max, as shown below. Relational Reasoning Networks (R2N). R2N [Marra et al., 2025] propagates the embeddings as representations of the atoms, using the following message passing schema: Mb h = Emb(b) Embj(h) = MLP e j (Mne(h,1) h, ..., Mne(h,dj) h)

rj R(h) Embj(h)

o(h) = MLP o(Emb(h))

where the embeddings are initialized using Emb0, and MLP e j , MLP o are multi-layer perceptrons (MLP) processing the embeddings and computing the outputs, respectively. Logic Tensor Networks (LTN). LTN [Badreddine et al., 2022] and Semantic Based Regularization (SBR) [Diligenti et al., 2017] aggregate the predictions of the body atoms of the j-th rule using a selected t-norm and enforce the consistency of the predictions with the logic rule. These methods were limited to unary predicates in the original papers, but here we propose a relational extension based on message passing:

Mb h = o(b) oj(h) = t norm(Mne(h,1) h, . . . , Mne(h,dj) h)

o(h) = max rj R(h) oj(h)

where o(α) is initialized using o0(α) and the original formulation was running a single propagation step. Deep Concept Reasoners (DCR). DCR [Barbiero et al., 2023] builds a formula from a set of candidate atoms before computing the output of a formula using a t-norm:

Mb h = Ψj(Emb0(b), Φj(Emb0(b), o(b)))

oj(h) = t norm Mne(h,1) h, . . . , Mne(h,dj) h

o(h) = max rj R(h) oj(h)

where oj(α) is initialized using o0 j(α), and the functions Φj : Rn+1 [0, 1] and Ψj : Rn+1 [0, 1], assuming Emb0(α) Rn, process the embedded representation of

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Figure 3: Illustration of some accepted/rejected proofs for different queries with respect to w, d (BCw,d) parameters.

each atom in each rule j, to compute its polarity and relevance values, respectively, to get a learned Horn Clause. The process can possibly be iterated, even if the original formulation is defined for a single step of propagation.

4 Grounding Methods Logical reasoning is often used as a form of theorem proving [Loveland, 2016] in computational logic. An example is query answering in Prolog, where the answer to a query is computed by searching for supporting facts given the set of rules of the program. Hence, proving can be cast to a search over the GMN, looking for subtrees (i.e., proofs) that connect the query to a set of supporting known facts. The known facts in a logic theory are typically much smaller than the HB. Classic Backward Chaining in Prolog considers unknown facts as possibly provable, and recursively builds a sub-proof for them if they are the head of a logic rule. While this process is effective in finding a formal proof for a query given the available known facts, it can be very time-consuming. A commonly used strategy consists in fixing a maximum depth for the backward search, thus admitting to find only proofs within a maximum number of reasoning steps. In Ne Sy, the logic facts can be neither true nor false, as they are assigned a score (e.g. a probability of being true) by the corresponding input layer. Therefore, it would be limiting to only consider known true facts in the proving process. Many other facts can provide uncertain information (i.e. context) to predict a query and should be considered. Unfortunately,

the size of the portion of the ground network relevant to the query may grow exponentially. However, such a limitation also introduces a new opportunity. Since Ne Sy models can score any fact in the knowledge base, one does not necessarily need to search the supporting known facts, as the scored facts at shallow depths in a proof can provide enough context for the learning to happen at a statistical level. Nevertheless, the majority of existing Ne Sy models completely neglect such opportunity, requiring very deep proving processes and resulting in intractable methods for inference.

4.1 Parameterized Backward Chaining Grounder The discussed limitations of classical Backward Chaining for Ne Sy motivate a novel definition of a flexible parameterized class of Backward Chaining grounders, which may take advantage of the generalization capabilities of Ne Sy methods over partially proved trees. A grounder in this parametrized class is defined as BCw,d, where w, d N are referred to as the width and depth of the grounder, respectively. The width w determines the maximum number of atoms in the body of each ground rule that is allowed to not be in the training data and, in turn, to be proved as sub-goals. The depth d controls the maximum depth that can be achieved by any proof for any query passing through its sub-goals. If the number of unknown ground atoms is bigger than w in any ground rule, the proof is immediately discarded even if more depth is allowed. Moreover, once the proof has been built by the grounder, we allow two options: i) to discard the proof if it contains unknown atoms (as done by Backward Chaining), ii) to accept the proof by using the scores of the unknown ground atom. We denote these grounders allowing uncertain atoms as BCu w,d. For simplicity, if not stated differently, we will assume option i). When using option i), BC0,1 BC1,1, so we will use BC0,1 in the rest of the paper. To illustrate the effects of depths and widths, Figure 3 compares the acceptance of diverse proofs across different parameter values. Section 5 discusses the experimental results for different versions of BCw,d. Here below, we show how BCw,d subsumes as special cases grounding techniques commonly used in the literature.

Backward Chaining Grounder (w = , d = ). Backward Chaining is an inference strategy used in logic programming and automated theorem provers to efficiently prove unknown facts. Given a query to prove, it searches for a rule, whose head atom matches the query. If some atom in the body of the rule is not in the training data, it becomes a new sub-goal that needs to be proven. The process is repeated recursively, until the query is proved with an associated proof tree. Notice that, for w = d = we mean w = B, being B the maximum amount of atoms in the body of all the rules. This is the grounding strategy employed by Deep Prob Log [Manhaeve et al., 2018], but its application is unfeasible at the scale of the learning tasks considered in this paper.

Backward Chaining Grounder with limited depth (w = , d = n). In practical settings, the growth of the admissible proofs is limited, as often done in logic programming. This can be accomplished by setting d = n, where n > 0 determines the maximum depth of the proofs.

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Known Body Grounder (w = 0, d = 1). Several popular Ne Sy models, like e.g. Express GNN [Zhang et al., 2020] and RNNLOGIC [Qu et al., 2020], accept a grounding only if all body atoms are known facts, and recursion is limited to depth 1. Therefore, these approaches are equivalent to BC0,1. Full Grounder (w = , d = 1). Full grounding of a FOL theory Γ involves generating the HUΓ, and it is used by models like LTN, SBR or MLNs. This grounder assumes option ii) for proofs, i.e. proofs are accepted even with unknown ground atoms, and thus it is denoted as BCu ,1. Roughly, the Full Grounder considers each atom in the HB as a possible query. Hence, all possible ground atoms are represented in the graph, often including irrelevant information, but at the same time ensuring that all the required information needed to perform a correct inference could be recovered. Complexity Analysis. The performances of BCw,d for Ne Sy applications are bound by two conflicting effects. On one hand, a grounder should be able to find the proofs needed to prove the queries. In this regard, increasing the width and the depth will result in extending the number of admissible proofs, thus increasing the number of provable queries. Proposition 1. The set of facts that are provable via BCw,d is a subset of the provable facts via BCw,d+1 and of the provable facts via BCw+1,d.

Proof. The proof trivially follows by observing that the set of buildable proofs in BCw,d is monotone w.r.t. w and d. Indeed, any proof with maximum width w (or depth d) is also a proof with maximum width w + 1 (or depth d + 1).

However, a grounder should not instantiate more nodes than needed, as this may affect the generalization capabilities, as stated by the following theorem. Theorem 1. [Scarselli et al., 2018] The Vapnik Chervonenkis dimension (VCD) of the function ϕ computed by a GNN with sigmoidal outputs on the domain of graphs with up to N nodes satisfies: V CD(ϕ) = O(p2 N 2) , where p is the number of parameters of the GNN. The Vapnik Chervonenkis theory states that the generalization error grows as the square root of the V CD, therefore the error bound grows linearly with the graph size. The number of nodes of the underlying reasoning graph built by BCw,d increases as O(Gdwd 1m), where G is the number of groundings with at most w atoms not in the known KG, and m is the maximum number of body atoms within any rule. For instance, if the rule with the longest body is a chain in the form of p(x1, x2) . . . p(xn 1, xn) p(x1, xn) with n representing the number of variables, the number of groundings grows as: G Cmin(w,n 2) for a single reasoning step, where C is the number of constants, and the overall ground network size grows as O(Cmin(w,n 2) dwd 1m). Theorem 1 shows that the generalization error is bound to grow linearly with the size of the grounded network. Higher values for w or d lead to an increase in the size of the GMN, therefore it is fundamental for the success of the Ne Sy approach to select values for w, d such that the logical context is preserved, while the generalization capabilities of the GNN are not hindered.

Dataset #Entities #Relations #Facts #Degree #Rules Countries S1 272 3 1,110 4.28 1 Countries S2 272 4 1,062 4.35 2 Countries S3 272 4 978 4.35 3 Kinship 104 25 28,356 272.7 48 WN18RR 40,943 18 93,003 2.3 17 FB15k-237 14505 237 310079 21.4 500

Table 1: Basic statistics for the employed datasets.

5 Experiments

Several experiments have been performed on KG link prediction to compare the effectiveness of the grounders.1

Datasets. The Countries [Bouchard et al., 2015], Kinship [Kok and Domingos, 2007], WN18RR [Dettmers et al., 2018] and FB15k-237 [Dou et al., 2021] datasets are used to capture diverse sizes and complexities. Countries is split into three tasks S{1,2,3} of increasing complexity, predicting country location based on regions and neighborhoods. Kinship represents familial relationships. Dataset statistics are in Section 5.

Rules. The rules associated with the Countries dataset are R1 = Loc In(x, w) Loc In(w, z) Loc In(x, z), R2 = Neigh Of(x, y) Loc In(y, z) Loc In(x, z) and R3 = Neigh Of(x, y) Neigh Of(y, k) Loc In(k, z) Loc In(x, z), where Loc In refers to the predicate Located In and Neigh Of refers to Neighbour Of. The task S1 takes R1, S2 takes R1, R2, and S3 all of them. The Countries_abl dataset is based only on R2. Rules for the remaining datasets were extracted using AMIE [Galárraga et al., 2013], selecting the top rules based on confidence. The number of selected rules per dataset is shown in Section 5.

Hyperparameters. The KGE uses a fixed embedding size of 100, with overall 1000 head and tail corruptions for WN18RR. Adam optimizer (10 2 learning rate) and binary cross-entropy loss were used over 100 epochs.

Models. Compl Ex is used as the KGE baseline and as the input layer for Ne Sy methods to enable a reasoning ablation study. The KGE baseline is compared with Ne Sy models of varying characteristics: R2N, which is highly expressive but less interpretable as it learns rules over groups of atoms, excels in complex, loosely formalized reasoning but risks overfitting when flexibility is unnecessary. SBR, less expressive but highly interpretable, performs best in simpler, well-formalized tasks as it strictly applies logic rules. DCR serves as a middle ground between these two approaches.

Metrics. For the evaluation, we use Mean Reciprocal Rank (MRR) and Hits@N metrics. All metric evaluations have been averaged over five runs for the countries dataset.

5.1 Experimental Results The first experiment serves as a proof of concept using the Countries_abl dataset. Three splits (AS1, AS2, AS3) are generated by ablating facts from the original dataset. The splits are designed with queries requiring one, two, and three reasoning steps, respectively, based on consecutive applications

1https://github.com/rodrigo-castellano/Grounding-Methods

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Figure 4: MRR results for different methods using BC1,d with different depths for the Countries_abl datasets AS1, AS2, AS3.

Figure 5: Results for the studied Ne Sy models and BCw,d grounders for Countries S2,S3 (S1 omitted as it is perfectly solved by all methods).

of the rule Loc In(x, w) Loc In(w, z) Loc In(x, z). The BC0,1 grounder is sufficient to solve the simple one-hop reasoning task defined by AS1, whereas BC1,2 or BC1,3 are required to provide good MRR performances in AS2, as shown in Figure 4. Finally, BC1,3 is the only grounder capable of solving AS3 with high MRR with all methods. This is consistent with our expectations, as the grounding depth defines the number of hops preserved in the context of a query.

The results for the Country dataset are presented in Figure 5, where the baseline is placed as d = 0 (this is consistent with the model definitions as the baseline is the model output if no message passing of the reasoner is performed on top of the input atom processing layer). The task S1 is quite simple so that all methods including the baseline are able to fully solve it with MRR = 1, therefore it is omitted in the plots. For S2, all Ne Sy methods outperform the baseline, with depth 1 (BCw,1) sufficient for strong performance, and higher depths plateauing. Regarding the more complex reasoning required by the S3 task, we observed that R2N tends to overfit for this dataset in the tested configuration. All other Ne Sy methods provide a large improvement over the baseline. Optimal results are achieved at either depth 2 or 3, which is indeed the number of reasoning hops required to solve the task. Figure 6 compares the Ne Sy methods against the baseline when using the Full grounder for the S1,S2,S3 tasks. The performances of the Full grounder are, in general, not improving over the BCw,d grounders in Figure 5, with actually smaller average gains over the baseline. The Full grounder tends to ground many unnecessary atoms with respect to the focused BCw,d grounders, and this negatively affects model generalization (as stated by Theorem 1). The results for the WN18RR and Kinship are presented in Table 2, where we report only the grounders that could be applied in less than 10h

Figure 6: MRR results for the Country datasets and the Full grounder using different Ne Sy methods and the baseline.

of computation on our workstation (12 core i7 CPU, 64GB RAM, OS Linux). For example, the application of the Full grounder is unfeasible for WN18RR or Kinship. MRR gains over the baseline of approximately 9%, and 3% for Kinship, and WN18RR, respectively. As shown by the inference times in Table 2, increasing the depth of the grounder in the larger datasets strongly impacts the required inference times. In practice, any grounder with a depth d > 2 would be very impractical to use for large KGs. Therefore we limited the experiments to BC0,1 and BC1,2. We observed generally improved results are obtained as the depth increases from d = 1 to d = 2 to retain enough information to improve the classification. However, the statistical nature of Ne Sy tasks makes it possible to get competitive performances even when limited to BC0,1. For instance, in the Kinship dataset DCR works very well already at depth 1, even if R2N and SBR can provide slightly improved results at depth 2. In the WN18RR dataset, BC0,1 already provides significant gains over the baseline, even if BC1,2 provides the best results both for DCR and SBR. These differences are also found in larger

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Model Grounder MRR Hits@1 Hits@3 Hits@10 Train Time(s) Test Time(s)

Kinship DCR BC0,1 90.1 84.1 95.9 97.0 16479.6 7658.7 BC1,2 90.1 84.1 95.9 97.0 16295.2 7517.4 R2N BC0,1 94.0 92.1 95.6 96.5 9573.3 6616.2 BC1,2 91.8 87.1 96.4 97.4 48808.8 28249.2 SBR BC0,1 86.9 78.0 95.6 97.1 9066.8 6208.7 BC1,2 87.7 79.1 96.0 97.3 43354.7 27447.9 Compl Ex - 85.9 79.2 92.2 94.5 773.3 284.9 WN18RR DCR BC0,1 44.2 42.2 44.8 47.6 26133.3 2337.9 BC1,2 45.6 42.9 47.1 50.2 74626.7 6944.2 R2N BC0,1 44.2 42.3 44.6 47.3 20613.7 2183.4 BC1,2 44.1 41.4 45.4 48.1 72213.3 7353.4 SBR BC0,1 44.0 42.3 44.2 46.6 21940.5 1909.8 BC1,2 44.7 42.5 45.2 48.2 67851.6 6666.0 Compl Ex - 42.7 40.8 42.9 45.9 1079.2 138.7

Table 2: Results for different Ne Sy methods and grounders for the Kinship and WN18RR datasets compared against the baseline.

Grounder MRR Hits@1 Hits@3 Hits@10 Compl Ex - 25.9 16.3 29.2 45.2 RNNLogic(emb.) BC0,1 34.4 25.2 38.0 53.0 R2N BC0,1 34.7 25.4 38.2 53.1

Table 3: MRR, Hits@N metrics on the FB15k-237 dataset.

datasets. For WN18RR, moving from BC0,1 to BC1,2 with the DCR model improved the MRR metric by 1.6 points, but increased training and inference times by a 3x factor. This observation underscores the critical importance of efficient and effective grounding techniques for scaling Ne Sy methods to larger real-world KGs. Results for the larger FB15k-237 dataset are provided in Section 5.1, where the computational constraints limited experiments to BC0,1, which still showed substantial improvements over the baseline for R2N. We did not add SBR and DCR to the table as the available rules were not covering enough atoms to provide a fair comparison.

6 Related Work

Full Groundings Approaches. A large class of Ne Sy models was defined over a full grounding approach, this includes SBR [Diligenti et al., 2017], LTN [Badreddine et al., 2022], and Relational Neural Machines [Marra et al., 2020]. The main problem with this grounding method is its lack of scalability, as shown in the experimental section.

Theorem proving approaches. Automatically answering a query (i.e. proving a theorem) given a set of definite clauses is at the core of logic programming [Clocksin and Mellish, 2012]. When proving a fact, only a portion of the GMN is relevant, thus a full grounding is usually not necessary. Standard techniques to avoid the full grounding include Backward Chaining [Kapoor and Bahl, 2016] and Forward Chaining [Al-Ajlan, 2015] or a mix of the two [Mumick and Pirahesh, 1994]. Since known facts are in general sparse over the HB, the relevant portion of the Markov network is much smaller than the full GMN obtained by grounding the rules over the entire domain. A large class of Ne Sy systems is based on theorem proving, this class includes systems exploiting a Prolog engine like Deep Prob Log [Manhaeve et al., 2018], Deep Stoch Log [Winters et al., 2022], an ASP engine [Shakarian et al., 2023] or Datalog (e.g. LRNN [Sourek

et al., 2018]). However, the exact proving approach does not allow the methods to scale beyond small inference problems. Therefore, some methods try to further limit the graph size. For example, DPLA* [Manhaeve et al., 2021] uses an A* approach to select the most promising portions of the GMN.

Heuristics-based approaches. The most common approach used in the literature is to rely on some ad-hoc heuristics to make grounding tractable. However, it did not emerge in the literature a clear understanding of the trade-off between performance and computing time. RNNLogic [Qu et al., 2020] generates logic rules and then scores the query triplets using the grounding paths in the training knowledge. This makes the model account for the grounded rules where all body atoms are true in the training facts, which is the special case where Backward Chaining is limited to depth 1. Discriminative Gaifman Models [Niepert, 2016] considers local neighbourhoods of a KG as features to predict queries. The local neighbourhoods can be seen as a generalization of a Horn clause, representing a chain of facts in the knowledge that imply the query. Similarly to RNN Logic, this considers only the grounded rules where all body atoms are true in the training facts. Express GNN [Zhang et al., 2020] operates directly on the knowledge graph rather than the extensive GMN grounding graph. This is similar to the approach of Uni KER [Cheng et al., 2021], which focuses on instantiating rules only with the training triples in the knowledge graph but iteratively enriching the training data with new facts derived using forward chaining. KALE [Guo et al., 2016] includes groundings where at least one atom is true in the training data, and uses a full grounding approach for the other variables. RUGE [Guo et al., 2018] considers a grounding when the triples in the rule s body are observed in the training data, and the head triple is not.

7 Conclusions and Future Work

This paper studies the effect of different grounding schemas on the predictive accuracy of Ne Sy methods. The main contributions of the work are in the definition of new parametrized grounding approaches, tailored to Ne Sy methods, and in the extensive experimental comparison of these grounders, offering insights into the trade-offs between grounding depth, model performance, and computational efficiency. The results on link prediction on KGs show that increasing the depth of the parameterized grounder enhances performance in complex reasoning tasks, but the statistical nature of Ne Sy allows preserving a good prediction accuracy even using shallow and efficient grounders. This contribution is not only methodological but also practical, highlighting the critical role of the grounding schema in bridging the gap between symbolic reasoning and neural learning. This study is limited to static grounders without any feedback from the Ne Sy method, which is the most used approach. In future work, we plan to explore dynamic grounders, which iteratively exploit the predictions of the learner to dynamically incorporate new atoms into the set of known facts. This could lead to the definition of parametric grounders which can be trained end-to-end with the learner.

Proceedings of the Thirty-Fourth International Joint Conference on Artificial Intelligence (IJCAI-25)

Ethical Statement There are no ethical issues.

Acknowledgments This work has been partially supported by the project PNRR M4C2 FAIR - Future Artificial Intelligence Research - Spoke 1 Human-centered AI , code PE0000013, CUP E53C22001610006. This work was also supported by the EU Framework Program for Research and Innovation Horizon under the Grant Agreement No 101073307 (MSCA-DN Le Mu R). This work has been partially supported by the project CONSTR: a COllectionless-based Neuro Symbolic Theory for learning and Reasoning , PARTENARIATO ESTESO Future Artificial Intelligence Research - FAIR , SPOKE 1 Human-Centered AI Università di Pisa, Next Generation EU , CUP I53C22001380006.

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