Inductive Logic Programming (ILP) approaches like Meta \-/ Interpretive Learning (MIL) can learn, from few examples, recursive logic programs with invented predicates that generalise well to unseen instances. This ability relies on a background theory and negative examples, both carefully selected with expert knowledge of a learning problem and its solutions. But what if such a problem-specific background theory or negative examples are not available? We formalise this question as a new setting for Self-Supervised ILP and present a new MIL algorithm that learns in the new setting from some positive labelled, and zero or more unlabelled examples, and automatically generates, and labels, new positive and negative examples during learning. We implement this algorithm in Prolog in a new MIL system, called Poker. We compare Poker to state-of-the-art MIL system Louise on experiments learning grammars for Context-Free and L-System languages from labelled, positive example strings, no negative examples, and just the terminal vocabulary of a language, seen in examples, as a first-order background theory. We introduce a new approach for the principled selection of a second-order background theory as a Second Order Definite Normal Form (SONF), sufficiently general to learn all programs in a class, thus removing the need for a backgound theory tailored to a learning task. We find that Poker's performance improves with increasing numbers of automatically generated examples while Louise, bereft of negative examples, over-generalises.

Visual generative abductive learning studies jointly training symbol-grounded neural visual generator and inducing logic rules from data, such that after learning, the visual generation process is guided by the induced logic rules. A major challenge for this task is to reduce the time cost of logic abduction during learning, an essential step when the logic symbol set is large and the logic rule to induce is complicated. To address this challenge, we propose a pre-training method for obtaining meta-rule selection policy for the recently proposed visual generative learning approach AbdGen [Peng et al., 2023], aiming at significantly reducing the candidate meta-rule set and pruning the search space. The selection model is built based on the embedding representation of both symbol grounding of cases and meta-rules, which can be effectively integrated with both neural model and logic reasoning system. The pre-training process is done on pure symbol data, not involving symbol grounding learning of raw visual inputs, making the entire learning process low-cost. An additional interesting observation is that the selection policy can rectify symbol grounding errors unseen during pre-training, which is resulted from the memorization ability of attention mechanism and the relative stability of symbolic patterns. Experimental results show that our method is able to effectively address the meta-rule selection problem for visual abduction, boosting the efficiency of visual generative abductive learning.

We introduce T-CREx, a novel model-agnostic method for local and global counterfactual explanation (CE), which summarises recourse options for both individuals and groups in the form of human-readable rules. It leverages tree-based surrogate models to learn the counterfactual rules, alongside 'metarules' denoting their regions of optimality, providing both a global analysis of model behaviour and diverse recourse options for users. Experiments indicate that T-CREx achieves superior aggregate performance over existing rule-based baselines on a range of CE desiderata, while being orders of magnitude faster to run.

Despite recent advances in modern machine learning algorithms, the opaqueness of their underlying mechanisms continues to be an obstacle in adoption. To instill confidence and trust in artificial intelligence systems, Explainable Artificial Intelligence has emerged as a response to improving modern machine learning algorithms' explainability. Inductive Logic Programming (ILP), a subfield of symbolic artificial intelligence, plays a promising role in generating interpretable explanations because of its intuitive logic-driven framework. ILP effectively leverages abductive reasoning to generate explainable first-order clausal theories from examples and background knowledge. However, several challenges in developing methods inspired by ILP need to be addressed for their successful application in practice. For example, existing ILP systems often have a vast solution space, and the induced solutions are very sensitive to noises and disturbances. This survey paper summarizes the recent advances in ILP and a discussion of statistical relational learning and neural-symbolic algorithms, which offer synergistic views to ILP. Following a critical review of the recent advances, we delineate observed challenges and highlight potential avenues of further ILP-motivated research toward developing self-explanatory artificial intelligence systems.

Inductive logic programming (ILP) is a form of machine learning. The goal of ILP is to induce a hypothesis (a set of logical rules) that generalises training examples. As ILP turns 30, we provide a new introduction to the field. We introduce the necessary logical notation and the main learning settings; describe the building blocks of an ILP system; compare several systems on several dimensions; describe four systems (Aleph, TILDE, ASPAL, and Metagol); highlight key application areas; and, finally, summarise current limitations and directions for future research.

How, he asks, do we complete this pattern? Now a student might say that the next term is 12. When the teacher asks him why, he says, "I looked out the window and saw the number 12 bus go by." One thing you might say is that there's a metarule, a rule about rules, and the metarule is: The only valid rules are ones that don't involve anything specific about the classroom in which the question is asked. So then the student says, fine, the next number in the series is 5. And this time, when you ask him why, he says it's because it's the fifth term in the series.

In Meta-Interpretive Learning (MIL) the metarules, second-order datalog clauses acting as inductive bias, are manually defined by the user. In this work we show that second-order metarules for MIL can be learned by MIL. We define a generality ordering of metarules by $\theta$-subsumption and show that user-defined sort metarules are derivable by specialisation of the most-general matrix metarules in a language class; and that these matrix metarules are in turn derivable by specialisation of third-order punch metarules with variables that range over the set of second-order literals and for which only an upper bound on their number of literals need be user-defined. We show that the cardinality of a metarule language is polynomial in the number of literals in punch metarules. We re-frame MIL as metarule specialisation by resolution. We modify the MIL metarule specialisation operator to return new metarules rather than first-order clauses and prove the correctness of the new operator. We implement the new operator as TOIL, a sub-system of the MIL system Louise. Our experiments show that as user-defined sort metarules are progressively replaced by sort metarules learned by TOIL, Louise's predictive accuracy is maintained at the cost of a small increase in training times. We conclude that automatically derived metarules can replace user-defined metarules.

Discovering novel high-level concepts is one of the most important steps needed for human-level AI. In inductive logic programming (ILP), discovering novel high-level concepts is known as predicate invention (PI). Although seen as crucial since the founding of ILP, PI is notoriously difficult and most ILP systems do not support it. In this paper, we introduce POPPI, an ILP system that formulates the PI problem as an answer set programming problem. Our experiments show that (i) PI can drastically improve learning performance when useful, (ii) PI is not too costly when unnecessary, and (iii) POPPI can substantially outperform existing ILP systems.

The program synthesis problem within the Inductive Logic Programming (ILP) community has typically been seen as untyped. We consider the benefits of user provided types on background knowledge. Building on the Meta-Interpretive Learning (MIL) framework, we show that type checking is able to prune large parts of the hypothesis space of programs. The introduction of polymorphic type checking to the MIL approach to logic program synthesis is validated by strong theoretical and experimental results, showing a cubic reduction in the size of the search space and synthesis time, in terms of the number of typed background predicates. Additionally we are able to infer polymorphic types of synthesized clauses and of entire programs. The other advancement is in developing an approach to leveraging refinement types in ILP. Here we show that further pruning of the search space can be achieved, though the SMT solving used for refinement type checking comes at a significant cost timewise.

Meta-Interpretive Learners, like most ILP systems, learn by searching for a correct hypothesis in the hypothesis space, the powerset of all constructible clauses. We show how this exponentially-growing search can be replaced by the construction of a Top program: the set of clauses in all correct hypotheses that is itself a correct hypothesis. We give an algorithm for Top program construction and show that it constructs a correct Top program in polynomial time and from a finite number of examples. We implement our algorithm in Prolog as the basis of a new MIL system, Louise, that constructs a Top program and then reduces it by removing redundant clauses. We compare Louise to the state-of-the-art search-based MIL system Metagol in experiments on grid world navigation, graph connectedness and grammar learning datasets and find that Louise improves on Metagol's predictive accuracy when the hypothesis space and the target theory are both large, or when the hypothesis space does not include a correct hypothesis because of "classification noise" in the form of mislabelled examples. When the hypothesis space or the target theory are small, Louise and Metagol perform equally well.