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.

Recent inductive logic programming (ILP) approaches learn optimal hypotheses. An optimal hypothesis minimises a given cost function on the training data. There are many cost functions, such as minimising training error, textual complexity, or the description length of hypotheses. However, selecting an appropriate cost function remains a key question. To address this gap, we extend a constraint-based ILP system to learn optimal hypotheses for seven standard cost functions. We then empirically compare the generalisation error of optimal hypotheses induced under these standard cost functions. Our results on over 20 domains and 1000 tasks, including game playing, program synthesis, and image reasoning, show that, while no cost function consistently outperforms the others, minimising training error or description length has the best overall performance. Notably, our results indicate that minimising the size of hypotheses does not always reduce generalisation error.

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.

The goal of inductive logic programming (ILP) is to find a set of logical rules that generalises training examples and background knowledge. We introduce an ILP approach that identifies pointless rules. A rule is pointless if it contains a redundant literal or cannot discriminate against negative examples. We show that ignoring pointless rules allows an ILP system to soundly prune the hypothesis space. Our experiments on multiple domains, including visual reasoning and game playing, show that our approach can reduce learning times by 99% whilst maintaining predictive accuracies.

We describe a datalog query evaluation approach based on efficient and composable boolean matrix manipulation modules. We first define an overarching problem, Boolean Matrix Logic Programming (BMLP), which uses boolean matrices as an alternative computation to evaluate datalog programs. We develop two novel BMLP modules for bottom-up inferences on linear dyadic recursive datalog programs, and show how additional modules can extend this capability to compute both linear and non-linear recursive datalog programs of arity two. Our empirical results demonstrate that these modules outperform general-purpose and specialised systems by factors of 30x and 9x, respectively, when evaluating large programs with millions of facts. This boolean matrix approach significantly enhances the efficiency of datalog querying to support logic programming techniques.

Many inductive logic programming (ILP) methods are incapable of learning programs from probabilistic background knowledge, e.g. coming from sensory data or neural networks with probabilities. We propose Propper, which handles flawed and probabilistic background knowledge by extending ILP with a combination of neurosymbolic inference, a continuous criterion for hypothesis selection (BCE) and a relaxation of the hypothesis constrainer (NoisyCombo). For relational patterns in noisy images, Propper can learn programs from as few as 8 examples. It outperforms binary ILP and statistical models such as a Graph Neural Network.

Reinforcement learning (RL) has proven to be a powerful tool for training agents that excel in various games. However, the black-box nature of neural network models often hinders our ability to understand the reasoning behind the agent's actions. Recent research has attempted to address this issue by using the guidance of pretrained neural agents to encode logic-based policies, allowing for interpretable decisions. A drawback of such approaches is the requirement of large amounts of predefined background knowledge in the form of predicates, limiting its applicability and scalability. In this work, we propose a novel approach, Explanatory Predicate Invention for Learning in Games (EXPIL), that identifies and extracts predicates from a pretrained neural agent, later used in the logic-based agents, reducing the dependency on predefined background knowledge. Our experimental evaluation on various games demonstrate the effectiveness of EXPIL in achieving explainable behavior in logic agents while requiring less background knowledge.

The goal of inductive logic programming (ILP) is to search for a logic program that generalises training examples and background knowledge. We introduce an ILP approach that identifies minimal unsatisfiable subprograms (MUSPs). We show that finding MUSPs allows us to efficiently and soundly prune the search space. Our experiments on multiple domains, including program synthesis and game playing, show that our approach can reduce learning times by 99%.

Symbolic rule learners generate interpretable solutions, however they require the input to be encoded symbolically. Neuro-symbolic approaches overcome this issue by mapping raw data to latent symbolic concepts using a neural network. Training the neural and symbolic components jointly is difficult, due to slow and unstable learning, hence many existing systems rely on hand-engineered rules to train the network. We introduce NeuralFastLAS, a scalable and fast end-to-end approach that trains a neural network jointly with a symbolic learner. For a given task, NeuralFastLAS computes a relevant set of rules, proved to contain an optimal symbolic solution, trains a neural network using these rules, and finally finds an optimal symbolic solution to the task while taking network predictions into account. A key novelty of our approach is learning a posterior distribution on rules while training the neural network to improve stability during training. We provide theoretical results for a sufficient condition on network training to guarantee correctness of the final solution. Experimental results demonstrate that NeuralFastLAS is able to achieve state-of-the-art accuracy in arithmetic and logical tasks, with a training time that is up to two orders of magnitude faster than other jointly trained neuro-symbolic methods.

Synthesizing large logic programs through symbolic Inductive Logic Programming (ILP) typically requires intermediate definitions. However, cluttering the hypothesis space with intensional predicates typically degrades performance. In contrast, gradient descent provides an efficient way to find solutions within such high-dimensional spaces. Neuro-symbolic ILP approaches have not fully exploited this so far. We propose extending the {\delta}ILP approach to inductive synthesis with large-scale predicate invention, thus allowing us to exploit the efficacy of high-dimensional gradient descent. We show that large-scale predicate invention benefits differentiable inductive synthesis through gradient descent and allows one to learn solutions for tasks beyond the capabilities of existing neuro-symbolic ILP systems. Furthermore, we achieve these results without specifying the precise structure of the solution within the language bias.