Reward machines (RMs) are a recent formalism for representing the reward function of a reinforcement learning task through a finite-state machine whose edges encode subgoals of the task using high-level events. The structure of RMs enables the decomposition of a task into simpler and independently solvable subtasks that help tackle long-horizon and/or sparse reward tasks. We propose a formalism for further abstracting the subtask structure by endowing an RM with the ability to call other RMs, thus composing a hierarchy of RMs (HRM). We exploit HRMs by treating each call to an RM as an independently solvable subtask using the options framework, and describe a curriculum-based method to learn HRMs from traces observed by the agent. Our experiments reveal that exploiting a handcrafted HRM leads to faster convergence than with a flat HRM, and that learning an HRM is feasible in cases where its equivalent flat representation is not.

Neuro-symbolic reasoning approaches proposed in recent years combine a neural perception component with a symbolic reasoning component to solve a downstream task. By doing so, these approaches can provide neural networks with symbolic reasoning capabilities, improve their interpretability and enable generalization beyond the training task. However, this often comes at the cost of poor training time, with potential scalability issues. In this paper, we propose a scalable neuro-symbolic approach, called Embed2Sym. We complement a two-stage (perception and reasoning) neural network architecture designed to solve a downstream task end-to-end with a symbolic optimisation method for extracting learned latent concepts. Specifically, the trained perception network generates clusters in embedding space that are identified and labelled using symbolic knowledge and a symbolic solver. With the latent concepts identified, a neuro-symbolic model is constructed by combining the perception network with the symbolic knowledge of the downstream task, resulting in a model that is interpretable and transferable. Our evaluation shows that Embed2Sym outperforms state-of-the-art neuro-symbolic systems on benchmark tasks in terms of training time by several orders of magnitude while providing similar if not better accuracy.

Logic Production System (LPS) is a logic-based framework for modelling reactive behaviour. Based on abductive logic programming, it combines reactive rules with logic programs, a database and a causal theory that specifies transitions between the states of the database. This paper proposes a systematic mapping of the Kernel of this framework (called KELPS) into an answer set program (ASP). For this purpose a new variant of KELPS with finite models, called n-distance KELPS, is introduced. A formal definition of the mapping from this n-distance KELPS to ASP is given and proven sound and complete. The Answer Set Programming paradigm allows to capture additional behaviours to the basic reactivity of KELPS, in particular proactive, preemptive and prospective behaviours. These are all discussed and illustrated with examples. Then a hybrid framework is proposed that integrates KELPS and ASP, allowing to combine the strengths of both paradigms.

Some probabilistic answer set programs (PASP) semantics assign probabilities to sets of answer sets and implicitly assume these answer sets to be equiprobable. While this is a common choice in probability theory, it leads to unnatural behaviours with PASPs. We argue that the user should have a level of control over what assumption is used to obtain a probability distribution when the stochastic knowledge is incomplete. To this end, we introduce the Incomplete Knowledge Semantics (IKS) for probabilistic answer set programs. We take inspiration from the field of decision making under ignorance. Given a cost function, represented by a user-defined ordering over answer sets through weak constraints, we use the notion of Ordered Weighted Averaging (OWA) operator to distribute the probability over a set of answer sets accordingly to the user’s level of optimism. The more optimistic (or pessimistic) a user is, the more (or less) probability is assigned to the more optimal answer sets. We present an implementation and showcase the behaviour of this semantics on simple examples. We also highlight the impact that different OWA operators have on weight learning, showing that the equiprobability assumption is not always the best option.

Over the last two decades there has been a growing interest in logic-based machine learning, where the goal is to learn a logic program, called a hypothesis, that together with a given background knowledge explains a set of examples. Although logic-based machine learning has traditionally addressed the task of learning definite logic programs (with no negation), our logic-based machine learning approaches have extended this field to a wider class of formalisms for knowledge representation, captured by the answer set programming (ASP) semantics. The ASP formalism is truly declarative and due to its non-monotonicity it is particularly well suited to commonsense reasoning. It allows constructs such as choice rules, hard and weak constraints, and support for default inference and default assumptions. Choice rules and weak constraints are particularly useful for modelling human preferences, as the choice rules can represent the choices available to the user, and the weak constraints can specify which choices a human prefers. In the recent years we have made fundamental contributions to the field of logic-based machine learning by extending it to the learning of the full class of ASP programs and the first part of this talk provides an introduction to these results and to the general field of learning under the answer set semantics, referred here as learning from answer sets (LAS). To be applicable to real-world problems, LAS has to be tolerant to noise in the data, scalable over large search spaces, amenable to user-defined domain-specific optimisation criteria and capable of learning interpretable knowledge from structured and unstructured data. The second part of this talk shows how these problems are addressed by our recently proposed FastLAS approach for learning Answer Set Programs, which is targeted at solving restricted versions of observational and non-observational predicate learning from answer sets tasks. The advanced features of our family of LAS systems have made it possible to solve a variety of real-world problems in a manner that is data efficient, scalable and robust to noise. LAS can be combined with statistical learning methods to realise neuro-symbolic solutions that perform both fast, low-level prediction from unstructured data, and high-level logic-based learning of interpretable knowledge. The talk concludes with presenting two such neuro-symbolic solutions for respectively solving image classification problems in the presence of distribution shifts, and discovering sub-goal structures for reinforcement learning agents.

In the past decade, several systems for learning Answer Set Programs (ASP) have been proposed, including the recent FastLAS system. Compared to other state-of-the-art approaches to learning ASP, FastLAS is more scalable, as rather than computing the hypothesis space in full, it computes a much smaller subset relative to a given set of examples that is nonetheless guaranteed to contain an optimal solution to the task (called an OPT-sufficient subset). On the other hand, like many other Inductive Logic Programming (ILP) systems, FastLAS is designed to be run on a fixed learning task meaning that if new examples are discovered after learning, the whole process must be run again. In many real applications, data arrives in a stream. Rerunning an ILP system from scratch each time new examples arrive is inefficient. In this paper we address this problem by presenting IncrementalLAS, a system that uses a new technique, called hypothesis space expansion, to enable a FastLAS-like OPT-sufficient subset to be expanded each time new examples are discovered. We prove that this preserves FastLAS's guarantee of finding an optimal solution to the full task (including the new examples), while removing the need to repeat previous computations. Through our evaluation, we demonstrate that running IncrementalLAS on tasks updated with sequences of new examples is significantly faster than re-running FastLAS from scratch on each updated task.

The Credal semantics is a probabilistic extension of the answer set semantics which can be applied to programs that may or may not be stratified. It assigns to atoms a set of acceptable probability distributions characterised by its lower and upper bounds. Performing exact probabilistic inference in the Credal semantics is computationally intractable. This paper presents a first solver, based on sampling, for probabilistic inference under the Credal semantics called PASOCS (Parallel Approximate SOlver for the Credal Semantics). PASOCS performs both exact and approximate inference for queries given evidence. Approximate solutions can be generated using any of the following sampling methods: naive sampling, Metropolis-Hastings and Gibbs Markov Chain Monte-Carlo. We evaluate the fidelity and performance of our system when applied to both stratified and non-stratified programs. We perform a sanity check by comparing PASOCS to available systems for stratified programs, where the semantics agree, and show that our system is competitive on unstratified programs.

In this paper we present ISA, an approach for learning and exploiting subgoals in episodic reinforcement learning (RL) tasks. ISA interleaves reinforcement learning with the induction of a subgoal automaton, an automaton whose edges are labeled by the task's subgoals expressed as propositional logic formulas over a set of high-level events. A subgoal automaton also consists of two special states: a state indicating the successful completion of the task, and a state indicating that the task has finished without succeeding. A state-of-the-art inductive logic programming system is used to learn a subgoal automaton that covers the traces of high-level events observed by the RL agent. When the currently exploited automaton does not correctly recognize a trace, the automaton learner induces a new automaton that covers that trace. The interleaving process guarantees the induction of automata with the minimum number of states, and applies a symmetry breaking mechanism to shrink the search space whilst remaining complete. We evaluate ISA in several gridworld and continuous state space problems using different RL algorithms that leverage the automaton structures. We provide an in-depth empirical analysis of the automaton learning performance in terms of the traces, the symmetry breaking and specific restrictions imposed on the final learnable automaton. For each class of RL problem, we show that the learned automata can be successfully exploited to learn policies that reach the goal, achieving an average reward comparable to the case where automata are not learned but handcrafted and given beforehand.

Recently, novel ILP systems under the answer set semantics have been proposed, some of which are robust to noise and scalable over large hypothesis spaces. One such system is FastLAS, which is significantly faster than other state-of-the-art ASP-based ILP systems. FastLAS is, however, only capable of Observational Predicate Learning (OPL), where the learned hypothesis defines predicates that are directly observed in the examples. It cannot learn knowledge that is indirectly observable, such as learning causes of observed events. This class of problems, known as non-OPL, is known to be difficult to handle in the context of non-monotonic semantics. Solving non-OPL learning tasks whilst preserving scalability is a challenging open problem. We address this problem with a new abductive method for translating examples of a non-OPL task to a set of examples, called possibilities, such that the original example is covered iff at least one of the possibilities is covered. This new method allows an ILP system capable of performing OPL tasks to be "upgraded" to solve non-OPL tasks. In particular, we present our new FastNonOPL system, which upgrades FastLAS with the new possibility generation. We compare it to other state-of-the-art ASP-based ILP systems capable of solving non-OPL tasks, showing that FastNonOPL is significantly faster, and in many cases more accurate, than these other systems.

Explainability in AI is gaining attention in the computer science community in response to the increasing success of deep learning and the important need of justifying how such systems make predictions in life-critical applications. The focus of explainability in AI has predominantly been on trying to gain insights into how machine learning systems function by exploring relationships between input data and predicted outcomes or by extracting simpler interpretable models. Through literature surveys of philosophy and social science, authors have highlighted the sharp difference between these generated explanations and human-made explanations and claimed that current explanations in AI do not take into account the complexity of human interaction to allow for effective information passing to not-expert users. In this paper we instantiate the concept of structure of scientific explanation as the theoretical underpinning for a general framework in which explanations for AI systems can be implemented. This framework aims to provide the tools to build a "mental-model" of any AI system so that the interaction with the user can provide information on demand and be closer to the nature of human-made explanations. We illustrate how we can utilize this framework through two very different examples: an artificial neural network and a Prolog solver and we provide a possible implementation for both examples.

In this work we present ISA, a novel approach for learning and exploiting subgoals in reinforcement learning (RL). Our method relies on inducing an automaton whose transitions are subgoals expressed as propositional formulas over a set of observable events. A state-of-the-art inductive logic programming system is used to learn the automaton from observation traces perceived by the RL agent. The reinforcement learning and automaton learning processes are interleaved: a new refined automaton is learned whenever the RL agent generates a trace not recognized by the current automaton. We evaluate ISA in several gridworld problems and show that it performs similarly to a method for which automata are given in advance. We also show that the learned automata can be exploited to speed up convergence through reward shaping and transfer learning across multiple tasks. Finally, we analyze the running time and the number of traces that ISA needs to learn an automata, and the impact that the number of observable events have on the learner's performance.

The Winograd Schema Challenge (Levesque et al., 2012) is a set of 273 expert-crafted pronoun resolution problems which form a benchmark for commonsense reasoning. In recent years, several approaches to the WSC using neural language models have signi cantly pushed forward the state-of-the-art accuracy on the dataset. However, the performance of these approaches on certain subsets of problems with strong statistical hints, as well as their reliance on extensive ne-tuning procedures, has brought into question whether these models are truly demonstrating commonsense reasoning, or whether they are instead exploiting spurious biases present in the dataset. One frequently proposed method to test whether an approach to the WSC demonstrates true commonsense capabilities is to require it to explain its answers, but there is currently no published approach that does this.In this project, we propose an approach to the WSC that uses inductive logic programming to learn explanations from similar examples that are sourced automatically. In particular, we make use of answer set grammars to automatically generate structured representations for natural language texts, and propose a fully-automated approach to generating induction tasks which learn commonsense rules directly within these grammars. We are then able to determine the answers to Winograd schemas by applying these learned rules. Our work makes two main contributions. Firstly, we show that answer set grammars can be used to elegantly encode the syntax and semantics of natural language, producing structured representations for texts that are richer than dependency …

We introduce a novel approach for the computation of stable and supported models of normal logic programs in continuous vector spaces by a gradient-based search method. Specifically, the application of the immediate consequence operator of a program reduct can be computed in a vector space. To do this, Herbrand interpretations of a propositional program are embedded as 0-1 vectors in $\mathbb{R}^N$ and program reducts are represented as matrices in $\mathbb{R}^{N \times N}$. Using these representations we prove that the underlying semantics of a normal logic program is captured through matrix multiplication and a differentiable operation. As supported and stable models of a normal logic program can now be seen as fixed points in a continuous space, non-monotonic deduction can be performed using an optimisation process such as Newton's method. We report the results of several experiments using synthetically generated programs that demonstrate the feasibility of the approach and highlight how different parameter values can affect the behaviour of the system.

Inductive Logic Programming (ILP) systems aim to find a set of logical rules, called a hypothesis, that explain a set of examples. In cases where many such hypotheses exist, ILP systems often bias towards shorter solutions, leading to highly general rules being learned. In some application domains like security and access control policies, this bias may not be desirable, as when data is sparse more specific rules that guarantee tighter security should be preferred. This paper presents a new general notion of a scoring function over hypotheses that allows a user to express domain-specific optimisation criteria. This is incorporated into a new ILP system, called FastLAS, that takes as input a learning task and a customised scoring function, and computes an optimal solution with respect to the given scoring function. We evaluate the accuracy of FastLAS over real-world datasets for access control policies and show that varying the scoring function allows a user to target domain-specific performance metrics. We also compare FastLAS to state-of-the-art ILP systems, using the standard ILP bias for shorter solutions, and demonstrate that FastLAS is significantly faster and more scalable.

Various sub-symbolic approaches for reasoning and learning have been proposed in the literature. Among these approaches, the neural theorem prover (NTP) approach uses a backward chaining reasoning mechanism to guide a machine learning architecture to learn vector embedding representations of predicates and to induce first-order clauses from a given knowledge base. NTP is however known for being not scalable, as the computation trees generated by the backward chaining process can grow exponentially with the size of the given knowledge base. In this paper we address this limitation by extending the NTP approach with a topic-based method for controlling the induction of first-order clauses. Our proposed approach, called TNTP for Topical NTP, identifies topic-based clusters over a large knowledge-base and uses these clusters to control the soft unification of predicates during the learning process with the effect of reducing the size of the computation tree needed to induce first-order clauses. Our TNTP framework is capable of learning a diverse set of induced rules that have improved predictive accuracy, whilst reducing the computational time by several orders of magnitude. We demonstrated this by evaluating our approach on three different datasets (UMLS, Kinship and Nations) and comparing our results with that of the NTP method, chosen here as our baseline.

The goal of Inductive Logic Programming (ILP) is to learn a program that explains a set of examples in the context of some pre-existing background knowledge. Until recently, most research on ILP targeted learning Prolog programs. Our own ILASP system instead learns Answer Set Programs, including normal rules, choice rules and hard and weak constraints. Learning such expressive programs widens the applicability of ILP considerably; for example, enabling preference learning, learning common-sense knowledge, including defaults and exceptions, and learning non-deterministic theories. In this paper, we first give a general overview of ILASP's learning framework and its capabilities. This is followed by a comprehensive summary of the evolution of the ILASP system, presenting the strengths and weaknesses of each version, with a particular emphasis on scalability.

In teaching mathematics, theorem provers have been used only seldomly due to their technical nature. Theorem provers like Elfe can bridge the gap between informal and formal reasoning by using automated theorem provers to verify intermediate steps in a proof that are passed over when reasoning intuitively. In this paper we present the inner workings of Elfe and how it can be used to prove lemmas in synthetic geometry. We compare the system to other approaches to formalized mathematics and give an outlook where the development may lead.

Saliency map generation techniques are at the forefront of explainable AI literature for a broad range of machine learning applications. Our goal is to question the limits of these approaches on more complex tasks. In this paper we apply Layer-Wise Relevance Propagation (LRP) to a sequence-to-sequence attention model trained on a text summarization dataset. We obtain unexpected saliency maps and discuss the rightfulness of these explanations. We argue that we need a quantitative way of testing the counterfactual case to judge the truthfulness of the saliency maps. We suggest a protocol to check the validity of the importance attributed to the input and show that the saliency maps obtained sometimes capture the real use of the input features by the network, and sometimes do not. We use this example to discuss how careful we need to be when accepting them as explanation.