Advancing Abductive Reasoning in Knowledge Graphs through
Complex Logical Hypothesis Generation

Abductive reasoning plays a vital role in generating explanatory hypotheses for observed phenomena across various research domains (Haig, 2012). It is a powerful tool with wide-ranging applications. For example, in cognitive neuroscience, reverse inference (Calzavarini and Cevolani, 2022), which is a form of abductive reasoning, is crucial for inferring the underlying cognitive processes based on observed brain activation patterns. Similarly, in clinical diagnostics, abductive reasoning is recognized as a key approach for studying cause-and-effect relationships (Martini, 2023). Moreover, abductive reasoning is fundamental to the process of hypothesis generation in humans, animals, and computational machines (Magnani, 2023). Its significance extends beyond these specific applications and encompasses diverse fields of study. In this paper, we are focused on abductive reasoning with structured knowledge, specifically, a knowledge graph.

A typical knowledge graph (KG) stores information about entities, like people, places, items, and their relations in graph structures. Meanwhile, KG reasoning is the process that leverages knowledge graphs to infer or derive new information (Zhang et al., 2021a, 2022; Ji et al., 2022). In recent years, various logical reasoning tasks are proposed over knowledge graph, for example, answering complex queries expressed in logical structure (Hamilton et al., 2018; Ren and Leskovec, 2020), or conducting logical rule mining (Galárraga et al., 2015; Ho et al., 2018; Meilicke et al., 2019).

However, the abductive perspective of KG reasoning is crucial yet unexplored. Consider the first example in Figure 1, where observation $O_{1}$ depicts five celebrities followed by a user on a social media platform. The social network service provider is interested in using structured knowledge to explain the users’ observed behavior. By leveraging a knowledge graph like Freebase (Bollacker et al., 2008), which contains basic information about these individuals, a complex logical hypothesis $H_{1}$ can be derived suggesting that they are all actors and screenwriters born in Los Angeles. In the second example from Figure 1, related to a user’s interactions on an e-commerce platform ($O_{2}$), the structured hypothesis $H_{2}$ can explain the user’s interest in $Apple$ products released in $2010$ excluding $phones$. The third example, dealing with medical diagnostics, presents three diseases ($O_{3}$). The corresponding hypothesis $H_{3}$ indicates they are diseases $V_{?}$ with symptom $V_{1}$, and $V_{1}$ can be relieved by $Panadol$. From a general perspective, these problems illustrate how abductive reasoning with knowledge graphs seeks hypotheses that best explain given observations (Josephson and Josephson, 1996; Thagard and Shelley, 1997).

A straightforward approach to tackle this reasoning task is to employ a search-based method to explore potential hypotheses based on the given observation. However, this approach faces two significant challenges. The first challenge arises from the incompleteness of knowledge graphs (KGs), as searching-based methods heavily rely on the availability of complete information. In practice, missing edges in KGs can negatively impact the performance of search-based methods (Ren and Leskovec, 2020). The second challenge stems from the complexity of logically structured hypotheses. The search space for search-based methods becomes exponentially large when dealing with combinatorial numbers of candidate hypotheses. Consequently, the search-based method struggles to effectively and efficiently handle observations that require complex hypotheses for explanation.

To overcome these challenges, we propose a solution that leverages generative models within a supervised learning framework to generate logical hypotheses for given observations. Our approach involves sampling hypothesis-observation pairs from observed knowledge graphs (Ren et al., 2020; Bai et al., 2023d) and training a transformer-based generative model (Vaswani et al., 2017) using the teacher-forcing method. However, a potential limitation of supervised training is that it primarily captures structural similarities, without necessarily guaranteeing improved explanations when applied to unseen observations. To address this, we introduce a technique called Reinforcement Learning from the Knowledge Graph (RLF-KG). It utilizes proximal policy optimization (PPO) (Schulman et al., 2017) to minimize the discrepancy between the observed evidence and the conclusion derived from the generated hypothesis. By incorporating reinforcement learning techniques, our approach aims to directly improve the explanatory capability of the generated hypotheses and ensure their effectiveness when generalized to unseen observations.

We evaluate the proposed methods for effectiveness and efficiency on three knowledge graphs: FB15k-237 (Toutanova and Chen, 2015), WN18RR (Toutanova and Chen, 2015), and DBpedia50 (Shi and Weninger, 2018). The results consistently demonstrate the superiority of our approach over supervised generation baselines and search-based methods, as measured by two evaluation metrics across all three datasets. Our contributions can be summarized as follows:

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  We introduce the task of complex logical hypothesis generation, which aims to identify logical hypotheses that best explain a given set of observations. This task can be seen as a form of abductive reasoning with KGs.

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  To address the challenges posed by the incompleteness of knowledge graphs and the complexity of logical hypotheses, we propose a generation-based method. This approach effectively handles these difficulties and enhances the quality of generated hypotheses.

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  Additionally, we developed the Reinforcement Learning from the Knowledge Graph (RLF-KG) technique. By incorporating feedback from the knowledge graph, RLF-KG further improves the hypothesis generation model. It minimizes the discrepancies between the observations and the conclusions of the generated hypotheses, leading to more accurate and reliable results.

In this task, a knowledge graph is denoted as ${\mathcal{G}}=(\mathcal{V},\mathcal{R})$, where $\mathcal{V}$ is the set of vertices and $\mathcal{R}$ is the set of relation types. Each relation type $r\in\mathcal{R}$ is a function $:\mathcal{V}\times\mathcal{V}\to\{\mathtt{true},\mathtt{false}\}$, with $r(u,v)=\mathtt{true}$ indicating the existence of a directed $(u,r,v)$ from vertex $u$ to vertex $v$ of type $r$ in the graph , and $\mathtt{false}$ otherwise.

We adopt the open-world assumption of knowledge graphs (Drummond and Shearer, 2006), treating missing edges as unknown rather than false. The reasoning model can only access the observed KG ${\mathcal{G}}$, while the true KG $\mathcal{\bar{G}}$ is hidden and encompasses the observed graph ${\mathcal{G}}$.

Abductive reasoning is a type of logical reasoning that involves making educated guesses to infer the most likely reasons for the observations (Josephson and Josephson, 1996; Thagard and Shelley, 1997). For further details on the distinctions between abductive, deductive, and inductive reasoning, refer to Appendix A. In this work, we focus on a specific abductive reasoning type in the context of knowledge graphs. We first introduce some concepts in this context.

An observation is a set $O$ of entities in ${\mathcal{V}}$. A logical hypothesis $H$ on a graph ${\mathcal{G}}=(\mathcal{V},\mathcal{R})$ is defined as a predicate of a variable vertex $V_{?}$ in first-order logical form, including existential quantifiers, AND ($\land$), OR ($\lor$), and NOT ($\lnot$). The hypothesis can always be written in disjunctive normal form,

where each $r_{ij}$ can take the forms: $r_{ij}=r(v,V)$, $r_{ij}=\neg r(v,V)$, $r_{ij}=r(V,V^{\prime})$, $r_{ij}=\neg r(V,V^{\prime})$, where $v$ represents a fixed vertex, the $V,V^{\prime}$ represent variable vertices in $\{V_{?},V_{1},V_{2},\dots,V_{k}\}$, and $r$ is a relation type.

The subscript $|_{{\mathcal{G}}}$ denotes that the hypothesis is formulated based on the given graph ${\mathcal{G}}$. This means that all entities and relations in the hypothesis must exist in ${\mathcal{G}}$, and the domain for variable vertices is the entity set of ${\mathcal{G}}$. For example, please refer to Appendix  B. The same hypothesis $H$ can be applied to a different knowledge graph, ${\mathcal{G}}^{\prime}$, provided that ${\mathcal{G}}^{\prime}$ includes the entities and edges present in $H$. When the context is clear or the hypothesis pertains to a general statement applicable to multiple knowledge graphs (e.g., observed and hidden graphs), the symbol $H$ is used without the subscript.

The conclusion of the hypothesis $H$ on a graph ${\mathcal{G}}$, denoted by $[\![H]\!]_{{\mathcal{G}}}$, is the set of entities for which $H$ holds true on ${\mathcal{G}}$:

Suppose $O=\{v_{1},v_{2},...,v_{k}\}$ represents an observation, ${\mathcal{G}}$ is the observed graph, and $\bar{{\mathcal{G}}}$ is the hidden graph. Then abductive reasoning in knowledge graphs aims to find the hypothesis $H$ on ${\mathcal{G}}$ whose conclusion on the hidden graph $\bar{{\mathcal{G}}}$, $[\![H]\!]_{\bar{{\mathcal{G}}}}$, is most similar to $O$. Formally, the similarity is quantified using the Jaccard index, defined as:

In other words, the goal is to find a hypothesis $H$ that maximizes $\mathtt{Jaccard}([\![H]\!]_{\bar{{\mathcal{G}}}},O)$.

Our methodology, referred to as reinforcement learning from the knowledge graph (RLF-KG), is depicted in Fig. 4 and involves the following steps: (1) Randomly sample observation-hypothesis pairs from the knowledge graph. (2) Train a generative model to generate hypotheses from observations using the pairs. (3) Enhance the generative model using RLF-KG, leveraging reinforcement learning to minimize discrepancies between observations and generated hypotheses.

We utilize three distinct knowledge graphs, namely FB15k-237 (Toutanova and Chen, 2015), DBpedia50 (Shi and Weninger, 2018), and WN18RR (Toutanova and Chen, 2015), for our experiments. Table 1 provides an overview of the number of training, evaluation, and testing edges, as well as the total number of nodes in each knowledge graph. To ensure consistency, we randomly partition the edges of these knowledge graphs into three sets - training, validation, and testing - using an 8:1:1 ratio. Consequently, we construct the training graph ${\mathcal{G}}_{\text{train}}$, validation graph ${\mathcal{G}}_{\text{valid}}$, and testing graph ${\mathcal{G}}_{\text{test}}$ by including the corresponding edges: training edges only, training + validation edges, and training + validation + testing edges, respectively.

Following the methodology outlined in Section 3.2, we proceed to sample pairs of observations and hypotheses. To ensure the quality and diversity of the samples, we impose certain constraints during the sampling process. Firstly, we restrict the size of the observation sets to a maximum of thirty-two elements. This limitation is enforced ensuring that the observations remain manageable. Additionally, specific constraints are applied to the validation and testing hypotheses. Each validation hypothesis must incorporate additional entities in the conclusion compared to the training graph, while each testing hypothesis must have additional entities in the conclusion compared to the validation graph. This progressive increase in entity complexity ensures a challenging evaluation setting. The statistics of queries sampled for datasets are detailed in Appendix C.

In line with previous work on KG reasoning (Ren and Leskovec, 2020; Ren et al., 2020), we utilize thirteen pre-defined logical patterns to sample the hypotheses. Eight of these patterns, known as existential positive first-order (EPFO) hypotheses (1p/2p/2u/3i/ip/up/2i/pi), do not involve negations. The remaining five patterns are negation hypotheses (2in/3in/inp/pni/pin), which incorporate negations. It is important to note that the generated hypotheses may or may not match the type of the reference hypothesis. The structures of the hypotheses are visually presented in Figure 5, and the corresponding numbers of samples drawn for each hypothesis type can be found in Table 5.

The problem of abductive knowledge graph reasoning shares connections with various other knowledge graph reasoning tasks, including knowledge graph completion, complex logical query answering, and rule mining.

In summary, this paper has introduced the task of abductive logical knowledge graph reasoning. Meanwhile, this paper has proposed a generation-based method to address knowledge graph incompleteness and reasoning efficiency by generating logical hypotheses. Furthermore, this paper demonstrates the effectiveness of our proposed reinforcement learning from knowledge graphs (RLF-KG) to enhance our hypothesis generation model by leveraging feedback from knowledge graphs.

Our proposed methods and techniques in the paper are evaluated on a specific set of knowledge graphs, namely FB15k-237, WN18RR, and DBpedia50. It is unclear how well these approaches would perform on other KGs with different characteristics or domains. Meanwhile, knowledge graphs can be massive and continuously evolving, our method is not yet able to address the dynamic nature of knowledge evolutions, like conducting knowledge editing automatically. It is important to note that these limitations should not undermine the significance of the work but rather serve as areas for future research and improvement.

We thank the anonymous reviewers and the area chair for their constructive comments. The authors of this paper were supported by the NSFC Fund (U20B2053) from the NSFC of China, the RIF (R6020-19 and R6021-20), and the GRF (16211520 and 16205322) from RGC of Hong Kong. We also thank the support from the UGC Research Matching Grants (RMGS20EG01-D, RMGS20CR11, RMGS20CR12, RMGS20EG19, RMGS20EG21, RMGS23CR05, RMGS23EG08).