We propose the Neural Logic Machine (NLM), a neural-symbolic architecture for both inductive learning and logic reasoning. NLMs exploit the power of both neural networks---as function approximators, and logic programming---as a symbolic processor for objects with properties, relations, logic connectives, and quantifiers. After being trained on small-scale tasks (such as sorting short arrays), NLMs can recover lifted rules, and generalize to large-scale tasks (such as sorting longer arrays). In our experiments, NLMs achieve perfect generalization in a number of tasks, from relational reasoning tasks on the family tree and general graphs, to decision making tasks including sorting arrays, finding shortest paths, and playing the blocks world. Most of these tasks are hard to accomplish for neural networks or inductive logic programming alone.

In this paper, we consider the problem of learning a (first-order) theorem prover where we use a representation of beliefs in mathematical claims instead of a proof system to search for proofs. The inspiration for doing so comes from the practices of human mathematicians where a proof system is typically used after the fact to justify a sequence of intuitive steps obtained by "plausible reasoning" rather than to discover them. Towards this end, we introduce a probabilistic representation of beliefs in first-order statements based on first-order distributive normal forms (dnfs) devised by the philosopher Jaakko Hintikka. Notably, the representation supports Bayesian update and does not enforce that logically equivalent statements are assigned the same probability---otherwise, we would end up in a circular situation where we require a prover in order to assign beliefs. We then examine (1) conjecturing as (statistical) model selection and (2) an alternating-turn proving game amenable (in principle) to self-play training to learn a prover that is both complete in the limit and sound provided that players maintain "reasonable" beliefs. Dnfs have super-exponential space requirements so the ideas in this paper should be taken as conducting a thought experiment on "learning to prove". As a step towards making the ideas practical, we will comment on how abstractions can be used to control the space requirements at the cost of completeness.

Over the last decades, a class of important mathematical results have required an ever increasing amount of human effort to carry out. For some, the help of computers is now indispensable. We analyze the implications of this trend towards "big mathematics", its relation to human cognition, and how machine support for big math can be organized. The central contribution of this position paper is an information model for "doing mathematics", which posits that humans very efficiently integrate four aspects: inference, computation, tabulation, and narration around a well-organized core of mathematical knowledge. The challenge for mathematical software systems is that these four aspects need to be integrated as well. We briefly survey the state of the art.

Answer Set Programming (ASP) is a prominent knowledge representation language with roots in logic programming and non-monotonic reasoning. Biennial ASP competitions are organized in order to furnish challenging benchmark collections and assess the advancement of the state of the art in ASP solving. In this paper, we report on the design and results of the Seventh ASP Competition, jointly organized by the University of Calabria (Italy), the University of Genova (Italy), and the University of Potsdam (Germany), in affiliation with the 14th International Conference on Logic Programming and Non-Monotonic Reasoning (LPNMR 2017). (Under consideration for acceptance in TPLP).

Children learn though play. We introduce the analogous idea of learning programs through play. In this approach, a program induction system (the learner) is given a set of tasks and initial background knowledge. Before solving the tasks, the learner enters an unsupervised playing stage where it creates its own tasks to solve, tries to solve them, and saves any solutions (programs) to the background knowledge. After the playing stage is finished, the learner enters the supervised building stage where it tries to solve the user-supplied tasks and can reuse solutions learnt whilst playing. The idea is that playing allows the learner to discover reusable general programs on its own which can then help solve the user-supplied tasks. We claim that playing can improve learning performance. We show that playing can reduce the textual complexity of target concepts which in turn reduces the sample complexity of a learner. We implement our idea in Playgol, a new inductive logic programming system. We experimentally test our claim on two domains: robot planning and real-world string transformations. Our experimental results suggest that playing can substantially improve learning performance. We think that the idea of playing (or, more verbosely, unsupervised bootstrapping for supervised program induction) is an important contribution to the problem of developing program induction approaches that self-discover BK.

By their nature, the composition of black box models is opaque. This makes the ability to generate explanations for the response to stimuli challenging. The importance of explaining black box models has become increasingly important given the prevalence of AI and ML systems and the need to build legal and regulatory frameworks around them. Such explanations can also increase trust in these uncertain systems. In our paper we present RICE, a method for generating explanations of the behaviour of black box models by (1) probing a model to extract model output examples using sensitivity analysis; (2) applying CNPInduce, a method for inductive logic program synthesis, to generate logic programs based on critical input-output pairs; and (3) interpreting the target program as a human-readable explanation. We demonstrate the application of our method by generating explanations of an artificial neural network trained to follow simple traffic rules in a hypothetical self-driving car simulation. We conclude with a discussion on the scalability and usability of our approach and its potential applications to explanation-critical scenarios.

Good parameter settings are crucial to achieve high performance in many areas of artificial intelligence (AI), such as propositional satisfiability solving, AI planning, scheduling, and machine learning (in particular deep learning). Automated algorithm configuration methods have recently received much attention in the AI community since they replace tedious, irreproducible and error-prone manual parameter tuning and can lead to new state-of-the-art performance. However, practical applications of algorithm configuration are prone to several (often subtle) pitfalls in the experimental design that can render the procedure ineffective. We identify several common issues and propose best practices for avoiding them. As one possibility for automatically handling as many of these as possible, we also propose a tool called GenericWrapper4AC.

We extend description logics (DLs) with non-monotonic reasoning features. We start by investigating a notion of defeasible subsumption in the spirit of defeasible conditionals as studied by Kraus, Lehmann and Magidor in the propositional case. In particular, we consider a natural and intuitive semantics for defeasible subsumption, and investigate KLM-style syntactic properties for both preferential and rational subsumption. Our contribution includes two representation results linking our semantic constructions to the set of preferential and rational properties considered. Besides showing that our semantics is appropriate, these results pave the way for more effective decision procedures for defeasible reasoning in DLs. Indeed, we also analyse the problem of non-monotonic reasoning in DLs at the level of entailment and present an algorithm for the computation of rational closure of a defeasible ontology. Importantly, our algorithm relies completely on classical entailment and shows that the computational complexity of reasoning over defeasible ontologies is no worse than that of reasoning in the underlying classical DL ALC.

We present an environment, benchmark, and deep learning driven automated theorem prover for higher-order logic. Higher-order interactive theorem provers enable the formalization of arbitrary mathematical theories and thereby present an interesting, open-ended challenge for deep learning. We provide an open-source framework based on the HOL Light theorem prover that can be used as a reinforcement learning environment. HOL Light comes with a broad coverage of basic mathematical theorems on calculus and the formal proof of the Kepler conjecture, from which we derive a challenging benchmark for automated reasoning. We also present a deep reinforcement learning driven automated theorem prover, DeepHOL, with strong initial results on this benchmark.

We propose a novel learning paradigm for Deep Neural Networks (DNN) by using Boolean logic algebra. We first present the basic differentiable operators of a Boolean system such as conjunction, disjunction and exclusive-OR and show how these elementary operators can be combined in a simple and meaningful way to form Neural Logic Networks (NLNs). We examine the effectiveness of the proposed NLN framework in learning Boolean functions and discrete-algorithmic tasks. We demonstrate that, in contrast to the implicit learning in MLP approach, the proposed neural logic networks can learn the logical functions explicitly that can be verified and interpreted by human. In particular, we propose a new framework for learning the inductive logic programming (ILP) problems by exploiting the explicit representational power of NLN. We show the proposed neural ILP solver is capable of feats such as predicate invention and recursion and can outperform the current state of the art neural ILP solvers using a variety of benchmark tasks such as decimal addition and multiplication, and sorting on ordered list.