Automated theorem provers and formal proof assistants are general reasoning systems that are in theory capable of proving arbitrarily hard theorems, thus solving arbitrary problems reducible to mathematics and logical reasoning. In practice, such systems however face large combinatorial explosion, and therefore include many heuristics and choice points that considerably influence their performance. This is an opportunity for trained machine learning predictors, which can guide the work of such reasoning systems. Conversely, deductive search supported by the notion of logically valid proof allows one to train machine learning systems on large reasoning corpora. Such bodies of proof are usually correct by construction and when combined with more and more precise trained guidance they can be boostrapped into very large corpora, with increasingly long reasoning chains and possibly novel proof ideas. In this paper we provide an overview of several automated reasoning and theorem proving domains and the learning and AI methods that have been so far developed for them. These include premise selection, proof guidance in several settings, AI systems and feedback loops iterating between reasoning and learning, and symbolic classification problems.

Artificial Intelligence for Theorem Proving has given rise to a plethora of benchmarks and methodologies, particularly in Interactive Theorem Proving (ITP). Research in the area is fragmented, with a diverse set of approaches being spread across several ITP systems. This presents a significant challenge to the comparison of methods, which are often complex and difficult to replicate. Addressing this, we present BAIT, a framework for fair and streamlined comparison of learning approaches in ITP. We demonstrate BAIT's capabilities with an in-depth comparison, across several ITP benchmarks, of state-of-the-art architectures applied to the problem of formula embedding. We find that Structure Aware Transformers perform particularly well, improving on techniques associated with the original problem sets. BAIT also allows us to assess the end-to-end proving performance of systems built on interactive environments. This unified perspective reveals a novel end-to-end system that improves on prior work. We also provide a qualitative analysis, illustrating that improved performance is associated with more semantically-aware embeddings. By streamlining the implementation and comparison of Machine Learning algorithms in the ITP context, we anticipate BAIT will be a springboard for future research.

One of the main challenges in the area of Neuro-Symbolic AI is to perform logical reasoning in the presence of both neural and symbolic data. This requires combining heterogeneous data sources such as knowledge graphs, neural model predictions, structured databases, crowd-sourced data, and many more. To allow for such reasoning, we generalise the standard rule-based language Datalog with existential rules (commonly referred to as tuple-generating dependencies) to the fuzzy setting, by allowing for arbitrary t-norms in the place of classical conjunctions in rule bodies. The resulting formalism allows us to perform reasoning about data associated with degrees of uncertainty while preserving computational complexity results and the applicability of reasoning techniques established for the standard Datalog setting. In particular, we provide fuzzy extensions of Datalog chases which produce fuzzy universal models and we exploit them to show that in important fragments of the language, reasoning has the same complexity as in the classical setting.

Defeasible reasoning is a kind of reasoning where some generalisations may not be valid in all circumstances, that is general conclusions may fail in some cases. Various formalisms have been developed to model this kind of reasoning, which is characteristic of common-sense contexts. However, it is not easy for a modeller to choose among these systems the one that better fits its domain from an ontological point of view. In this paper we first propose a framework based on the notions of exceptionality and defeasibility in order to be able to compare formalisms and reveal their ontological commitments. Then, we apply this framework to compare four systems, showing the differences that may occur from an ontological perspective.

We present the first algorithm for query learning of a general class of Mealy machines with timers (MMTs) in a black-box context. Our algorithm is an extension of the L# algorithm of Vaandrager et al. to a timed setting. Like the algorithm for learning timed automata proposed by Waga, our algorithm is inspired by ideas of Maler & Pnueli. Based on the elementary languages of, both Waga's and our algorithm use symbolic queries, which are then implemented using finitely many concrete queries. However, whereas Waga needs exponentially many concrete queries to implement a single symbolic query, we only need a polynomial number. This is because in order to learn a timed automaton, a learner needs to determine the exact guard and reset for each transition (out of exponentially many possibilities), whereas for learning an MMT a learner only needs to figure out which of the preceding transitions caused a timeout. As shown in our previous work, this can be done efficiently for a subclass of MMTs that are race-avoiding: if a timeout is caused by a preceding input then a slight change in the timing of this input will induce a corresponding change in the timing of the timeout ("wiggling"). Experiments with a prototype implementation, written in Rust, show that our algorithm is able to efficiently learn realistic benchmarks.

The ModelWriter platform provides a generic framework for automated traceability analysis. In this paper, we demonstrate how this framework can be used to trace the consistency and completeness of technical documents that consist of a set of System Installation Design Principles used by Airbus to ensure the correctness of aircraft system installation. We show in particular, how the platform allows the integration of two types of reasoning: reasoning about the meaning of text using semantic parsing and description logic theorem proving; and reasoning about document structure using first-order relational logic and finite model finding for traceability analysis.

Classically, in saturation-based proof systems, unification has been considered atomic. However, it is also possible to move unification to the calculus level, turning the steps of the unification algorithm into inferences. For calculi that rely on unification procedures returning large or even infinite sets of unifiers, integrating unification into the calculus is an attractive method of dovetailing unification and inference. This applies, for example, to AC-superposition and higher-order superposition. We show that first-order superposition remains complete when moving unification rules to the calculus level. We discuss some of the benefits this has even for standard first-order superposition and provide an experimental evaluation.

Partially Observable Markov Decision Processes (POMDPs) are a powerful framework for planning under uncertainty. They allow to model state uncertainty as a belief probability distribution. Approximate solvers based on Monte Carlo sampling show great success to relax the computational demand and perform online planning. However, scaling to complex realistic domains with many actions and long planning horizons is still a major challenge, and a key point to achieve good performance is guiding the action-selection process with domain-dependent policy heuristics which are tailored for the specific application domain. We propose to learn high-quality heuristics from POMDP traces of executions generated by any solver. We convert the belief-action pairs to a logical semantics, and exploit data- and time-efficient Inductive Logic Programming (ILP) to generate interpretable belief-based policy specifications, which are then used as online heuristics. We evaluate thoroughly our methodology on two notoriously challenging POMDP problems, involving large action spaces and long planning horizons, namely, rocksample and pocman. Considering different state-of-the-art online POMDP solvers, including POMCP, DESPOT and AdaOPS, we show that learned heuristics expressed in Answer Set Programming (ASP) yield performance superior to neural networks and similar to optimal handcrafted task-specific heuristics within lower computational time. Moreover, they well generalize to more challenging scenarios not experienced in the training phase (e.g., increasing rocks and grid size in rocksample, incrementing the size of the map and the aggressivity of ghosts in pocman).

Human mathematicians are often good at recognizing modular and reusable theorems that make complex mathematical results within reach. In this paper, we propose a novel method called theoREm-from-prooF extrACTOR (REFACTOR) for training neural networks to mimic this ability in formal mathematical theorem proving. We show on a set of unseen proofs, REFACTOR is able to extract 19.6% of the theorems that humans would use to write the proofs. When applying the model to the existing Metamath library, REFACTOR extracted 16 new theorems. With newly extracted theorems, we show that the existing proofs in the MetaMath database can be refactored. The new theorems are used very frequently after refactoring, with an average usage of 733.5 times, and help shorten the proof lengths. Lastly, we demonstrate that the prover trained on the new-theorem refactored dataset proves more test theorems and outperforms state-of-the-art baselines by frequently leveraging a diverse set of newly extracted theorems. Code can be found at https://github.com/jinpz/refactor.

Databases are often assumed to have definite content. The reality, though, is the database at hand may be deficient due to missing, invalid, or uncertain information. As a simple illustration, the primary address of a person may be missing, or it may conflict with another primary address, or it may be improbable given the presence of nearby businesses. A common practice to address this challenge is to rectify the database by fixing the gaps, as done in data imputation, entity resolution, and data cleaning. The process of rectifying the database, however, may involve arbitrary choices due to computational limitations, such as errors in statistical or machine-learning models, or mere lack of information that even humans cannot cope with in full confidence. In turn, answers to queries over the deficient database may depend on the choices made to rectify it; thus, the answers to queries may vary from one choice to choice, even though both choices may be equally legitimate. In the pursuit of principled solutions, there has been a continuous research effort to develop fundamental approaches for handling database deficiency with no (or with less) arbitrariness. The purpose of this review article is to highlight some of the ways in which the possible world semantics has been deployed as a principled approach to overcome database deficiency in different contexts. In this approach, we acknowledge that we need to rectify the deficiency: fill in missing information, delete wrong records (hereafter tuples or facts), correct erroneous values, and so on. Yet, since many rectifications may exist and since we do not know which is the correct one, we do not commit to a specific one. Instead, we view our deficient database as a representation of the results of all conceivable rectifications, each such rectification giving rise to a legitimate candidate of a valid database that we call a possible world. Since the possible worlds differ from each other, a query may produce different collections of answers (which are also tuples) when applied to different possible worlds. Therefore, query answering requires the use of an aggregation method to combine the query results over the possible worlds.