Humans have the ability to seamlessly combine low-level visual input with high-level symbolic reasoning often in the form of recognising objects, learning relations between them and applying rules. Neuro-symbolic systems aim to bring a unifying approach to connectionist and logic-based principles for visual processing and abstract reasoning respectively. This paper presents a complete neuro-symbolic method for processing images into objects, learning relations and logical rules in an end-to-end fashion. The main contribution is a differentiable layer in a deep learning architecture from which symbolic relations and rules can be extracted by pruning and thresholding. We evaluate our model using two datasets: subgraph isomorphism task for symbolic rule learning and an image classification domain with compound relations for learning objects, relations and rules. We demonstrate that our model scales beyond state-of-the-art symbolic learners and outperforms deep relational neural network architectures.

Formally verifying Deep Reinforcement Learning (DRL) systems is a challenging task due to the dynamic continuity of system behaviors and the black-box feature of embedded neural networks. In this paper, we propose a novel abstraction-based approach to train DRL systems on finite abstract domains instead of concrete system states. It yields neural networks whose input states are finite, making hosting DRL systems directly verifiable using model checking techniques. Our approach is orthogonal to existing DRL algorithms and off-the-shelf model checkers. We implement a resulting prototype training and verification framework and conduct extensive experiments on the state-of-the-art benchmark. The results show that the systems trained in our approach can be verified more efficiently while they retain comparable performance against those that are trained without abstraction.

Multi-Context Systems (MCS) model in Computational Logic distributed systems composed of heterogeneous sources, or "contexts", interacting via special rules called "bridge rules". In this paper, we consider how to enhance flexibility and generality in bridge-rules definition and application. In particular, we introduce and discuss some formal extensions of MCSs useful for a practical use in dynamic environments, and we try to provide guidelines for implementations

We introduce a new type of programming challenge called programming puzzles, as an objective and comprehensive evaluation of program synthesis, and release an open-source dataset of Python Programming Puzzles (P3). Each puzzle is defined by a short Python program $f$, and the goal is to find an input $x$ which makes $f$ output "True". The puzzles are objective in that each one is specified entirely by the source code of its verifier $f$, so evaluating $f(x)$ is all that is needed to test a candidate solution $x$. They do not require an answer key or input/output examples, nor do they depend on natural language understanding. The dataset is comprehensive in that it spans problems of a range of difficulties and domains, ranging from trivial string manipulation problems that are immediately obvious to human programmers (but not necessarily to AI), to classic programming puzzles (e.g., Towers of Hanoi), to interview/competitive-programming problems (e.g., dynamic programming), to longstanding open problems in algorithms and mathematics (e.g., factoring). The objective nature of P3 readily supports self-supervised bootstrapping. We develop baseline enumerative program synthesis and GPT-3 solvers that are capable of solving easy puzzles -- even without access to any reference solutions -- by learning from their own past solutions. Based on a small user study, we find puzzle difficulty to correlate between human programmers and the baseline AI solvers.

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.

Traditional automated theorem provers have relied on manually tuned heuristics to guide how they perform proof search. Recently, however, there has been a surge of interest in the design of learning mechanisms that can be integrated into theorem provers to improve their performance automatically. In this work, we introduce TRAIL, a deep learning-based approach to theorem proving that characterizes core elements of saturation-based theorem proving within a neural framework. TRAIL leverages (a) an effective graph neural network for representing logical formulas, (b) a novel neural representation of the state of a saturation-based theorem prover in terms of processed clauses and available actions, and (c) a novel representation of the inference selection process as an attention-based action policy. We show through a systematic analysis that these components allow TRAIL to significantly outperform previous reinforcement learning-based theorem provers on two standard benchmark datasets (up to 36% more theorems proved). In addition, to the best of our knowledge, TRAIL is the first reinforcement learning-based approach to exceed the performance of a state-of-the-art traditional theorem prover on a standard theorem proving benchmark (solving up to 17% more problems).

Neural models and symbolic algorithms have recently been combined for tasks requiring both perception and reasoning. Neural models ground perceptual input into a conceptual vocabulary, on which a classical reasoning algorithm is applied to generate output. A key limitation is that such neural-to-symbolic models can only be trained end-to-end for tasks where the output space is symbolic. In this paper, we study neural-symbolic-neural models for reasoning tasks that require a conversion from an image input (e.g., a partially filled sudoku) to an image output (e.g., the image of the completed sudoku). While designing such a three-step hybrid architecture may be straightforward, the key technical challenge is end-to-end training -- how to backpropagate without intermediate supervision through the symbolic component. We propose NSNnet, an architecture that combines an image reconstruction loss with a novel output encoder to generate a supervisory signal, develops update algorithms that leverage policy gradient methods for supervision, and optimizes loss using a novel subsampling heuristic. We experiment on problem settings where symbolic algorithms are easily specified: a visual maze solving task and a visual Sudoku solver where the supervision is in image form. Experiments show high accuracy with significantly less data compared to purely neural approaches.

Despite recent successes of large pre-trained language models in solving reasoning tasks, their inference capabilities remain opaque. We posit that such models can be made more interpretable by explicitly generating interim inference rules, and using them to guide the generation of task-specific textual outputs. In this paper we present COINS, a recursive inference framework that i) iteratively reads context sentences, ii) dynamically generates contextualized inference rules, encodes them, and iii) uses them to guide task-specific output generation. We apply COINS to a Narrative Story Completion task that asks a model to complete a story with missing sentences, to produce a coherent story with plausible logical connections, causal relationships, and temporal dependencies. By modularizing inference and sentence generation steps in a recurrent model, we aim to make reasoning steps and their effects on next sentence generation transparent. Our automatic and manual evaluations show that the model generates better story sentences than SOTA baselines, especially in terms of coherence. We further demonstrate improved performance over strong pre-trained LMs in generating commonsense inference rules. The recursive nature of COINS holds the potential for controlled generation of longer sequences.

In this work we study how to learn good algorithms for selecting reasoning steps in theorem proving. We explore this in the connection tableau calculus implemented by leanCoP where the partial tableau provides a clean and compact notion of a state to which a limited number of inferences can be applied. We start by incorporating a state-of-the-art learning algorithm -- a graph neural network (GNN) -- into the plCoP theorem prover. Then we use it to observe the system's behaviour in a reinforcement learning setting, i.e., when learning inference guidance from successful Monte-Carlo tree searches on many problems. Despite its better pattern matching capability, the GNN initially performs worse than a simpler previously used learning algorithm. We observe that the simpler algorithm is less confident, i.e., its recommendations have higher entropy. This leads us to explore how the entropy of the inference selection implemented via the neural network influences the proof search. This is related to research in human decision-making under uncertainty, and in particular the probability matching theory. Our main result shows that a proper entropy regularisation, i.e., training the GNN not to be overconfident, greatly improves plCoP's performance on a large mathematical corpus.

Decision-making policies for agents are often synthesized with the constraint that a formal specification of behaviour is satisfied. Here we focus on infinite-horizon properties. On the one hand, Linear Temporal Logic (LTL) is a popular example of a formalism for qualitative specifications. On the other hand, Steady-State Policy Synthesis (SSPS) has recently received considerable attention as it provides a more quantitative and more behavioural perspective on specifications, in terms of the frequency with which states are visited. Finally, rewards provide a classic framework for quantitative properties. In this paper, we study Markov decision processes (MDP) with the specification combining all these three types. The derived policy maximizes the reward among all policies ensuring the LTL specification with the given probability and adhering to the steady-state constraints. To this end, we provide a unified solution reducing the multi-type specification to a multi-dimensional long-run average reward. This is enabled by Limit-Deterministic B\"uchi Automata (LDBA), recently studied in the context of LTL model checking on MDP, and allows for an elegant solution through a simple linear programme. The algorithm also extends to the general $\omega$-regular properties and runs in time polynomial in the sizes of the MDP as well as the LDBA.