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Proving equivalence between functional programs is a fundamental problem in program verification, which often amounts to reasoning about algebraic data types (ADTs) and compositions of structural recursions. Modern theorem provers address this problem by applying structural induction, which is insufficient for proving many equivalence theorems. In such cases, one has to invent a set of lemmas, prove these lemmas by additional induction, and use these lemmas to prove the original theorem. There is, however, a lack of systematic understanding of what lemmas are needed for inductive proofs and how these lemmas can be synthesized automatically. This paper presents directed lemma synthesis, an effective approach to automating equivalence proofs by discovering critical lemmas using program synthesis techniques. We first identify two induction-friendly forms of propositions that give formal guarantees to the progress of the proof. We then propose two tactics that synthesize and apply lemmas, thereby transforming the proof goal into induction-friendly forms. Both tactics reduce lemma synthesis to a specialized class of program synthesis problems with efficient algorithms. Experimental results demonstrate the effectiveness of our approach: Compared to state-of-the-art equivalence checkers employing heuristic-based lemma enumeration, directed lemma synthesis saves 95.47% runtime on average and solves 38 more tasks over an extended version of the standard benchmark set.

Counterfactuals, or modified inputs that lead to a different outcome, are an important tool for understanding the logic used by machine learning classifiers and how to change an undesirable classification. Even if a counterfactual changes a classifier's decision, however, it may not affect the true underlying class probabilities, i.e. the counterfactual may act like an adversarial attack and ``fool'' the classifier. We propose a new framework for creating modified inputs that change the true underlying probabilities in a beneficial way which we call Trustworthy Actionable Perturbations (TAP). This includes a novel verification procedure to ensure that TAP change the true class probabilities instead of acting adversarially. Our framework also includes new cost, reward, and goal definitions that are better suited to effectuating change in the real world. We present PAC-learnability results for our verification procedure and theoretically analyze our new method for measuring reward. We also develop a methodology for creating TAP and compare our results to those achieved by previous counterfactual methods.

The Hawkes process is a model for counting the number of arrivals to a system which exhibits the self-exciting property - that one arrival creates a heightened chance of further arrivals in the near future. The model, and its generalizations, have been applied in a plethora of disparate domains, though two particularly developed applications are in seismology and in finance. As the original model is elegantly simple, generalizations have been proposed which: track marks for each arrival, are multivariate, have a spatial component, are driven by renewal processes, treat time as discrete, and so on. This paper creates a cohesive review of the traditional Hawkes model and the modern generalizations, providing details on their construction, simulation algorithms, and giving key references to the appropriate literature for a detailed treatment.

Information pooling has been extensively formalised across various logical frameworks in distributed systems, characterized by diverse information-sharing patterns. These approaches generally adopt an intersection perspective, aggregating all possible information, regardless of whether it is known or unknown to the agents. In contrast, this work adopts a unique stance, emphasising that sharing knowledge means distributing what is known, rather than what remains uncertain. This paper introduces new modal logics for knowledge pooling and sharing, ranging from a novel language of knowledge pooling to a dynamic mechanism for knowledge sharing. It also outlines their axiomatizations and discusses a potential framework for permissible knowledge pooling.

The contextual integrity model is a widely accepted way of analyzing the plurality of norms that are colloquially called "privacy norms". Contextual integrity systematically describes such norms by distinguishing the type of data concerned, the three social agents involved (subject, sender, and recipient) and the transmission principle governing the transfer of information. It allows analyzing privacy norms in terms of their impact on the interaction of those agents with one another. This paper places contextual integrity in a strict game theoretic framework. When such description is possible it has three key advantages: Firstly, it allows indisputable utilitarian justification of some privacy norms. Secondly, it better relates privacy to topics which are well understood by stakeholders whose education is predominantly quantitative, such as engineers and economists. Thirdly, it is an absolute necessity when describing ethical constraints to machines such as AI agents. In addition to describing games which capture paradigmatic informational norms, the paper also analyzes cases in which the game, per se, does not encourage normative behavior. The paper discusses two main forms of mechanisms which can be applied to the game in such cases, and shows that they reflect accepted privacy regulation and technologies.

Artificial intelligence assisted mathematical proof has become a highly focused area nowadays. One key problem in this field is to generate formal mathematical proofs from natural language proofs. Due to historical reasons, the formal proof languages adopted by traditional theorem provers were not intended to represent natural language proofs. Therefore, they are not well-suited for the aforementioned tasks and proof-checking work for educational purposes. In this paper, we design a proof language and its corresponding abstract syntax tree and implement a proof checking tool for it. This language can be easily converted from natural language, thus providing a rich corpus of formal proof. Additionally, it supports the handling of issues in informal proofs through static analysis, and enhances the expressive power of the language by introducing the structure of partial proofs. This design combines the expressiveness of natural language and the accuracy of formal language, resulting in an improved mathematical proof language.

Recent advancements in imitation learning have been largely fueled by the integration of sequence models, which provide a structured flow of information to effectively mimic task behaviours. Currently, Decision Transformer (DT) and subsequently, the Hierarchical Decision Transformer (HDT), presented Transformer-based approaches to learn task policies. Recently, the Mamba architecture has shown to outperform Transformers across various task domains. In this work, we introduce two novel methods, Decision Mamba (DM) and Hierarchical Decision Mamba (HDM), aimed at enhancing the performance of the Transformer models. Through extensive experimentation across diverse environments such as OpenAI Gym and D4RL, leveraging varying demonstration data sets, we demonstrate the superiority of Mamba models over their Transformer counterparts in a majority of tasks. Results show that HDM outperforms other methods in most settings. The code can be found at //github.com/meowatthemoon/HierarchicalDecisionMamba.

In traditional semantics for classical logic and its extensions, such as modal logic, propositions are interpreted as subsets of a set, as in discrete duality, or as clopen sets of a Stone space, as in topological duality. A point in such a set can be viewed as a "possible world," with the key property of a world being primeness--a world makes a disjunction true only if it makes one of the disjuncts true--which classically implies totality--for each proposition, a world either makes the proposition true or makes its negation true. This chapter surveys a more general approach to logical semantics, known as possibility semantics, which replaces possible worlds with possibly partial "possibilities." In classical possibility semantics, propositions are interpreted as regular open sets of a poset, as in set-theoretic forcing, or as compact regular open sets of an upper Vietoris space, as in the recent theory of "choice-free Stone duality." The elements of these sets, viewed as possibilities, may be partial in the sense of making a disjunction true without settling which disjunct is true. We explain how possibilities may be used in semantics for classical logic and modal logics and generalized to semantics for intuitionistic logics. The goals are to overcome or deepen incompleteness results for traditional semantics, to avoid the nonconstructivity of traditional semantics, and to provide richer structures for the interpretation of new languages.

The adaptive processing of structured data is a long-standing research topic in machine learning that investigates how to automatically learn a mapping from a structured input to outputs of various nature. Recently, there has been an increasing interest in the adaptive processing of graphs, which led to the development of different neural network-based methodologies. In this thesis, we take a different route and develop a Bayesian Deep Learning framework for graph learning. The dissertation begins with a review of the principles over which most of the methods in the field are built, followed by a study on graph classification reproducibility issues. We then proceed to bridge the basic ideas of deep learning for graphs with the Bayesian world, by building our deep architectures in an incremental fashion. This framework allows us to consider graphs with discrete and continuous edge features, producing unsupervised embeddings rich enough to reach the state of the art on several classification tasks. Our approach is also amenable to a Bayesian nonparametric extension that automatizes the choice of almost all model's hyper-parameters. Two real-world applications demonstrate the efficacy of deep learning for graphs. The first concerns the prediction of information-theoretic quantities for molecular simulations with supervised neural models. After that, we exploit our Bayesian models to solve a malware-classification task while being robust to intra-procedural code obfuscation techniques. We conclude the dissertation with an attempt to blend the best of the neural and Bayesian worlds together. The resulting hybrid model is able to predict multimodal distributions conditioned on input graphs, with the consequent ability to model stochasticity and uncertainty better than most works. Overall, we aim to provide a Bayesian perspective into the articulated research field of deep learning for graphs.

We introduce an approach for deep reinforcement learning (RL) that improves upon the efficiency, generalization capacity, and interpretability of conventional approaches through structured perception and relational reasoning. It uses self-attention to iteratively reason about the relations between entities in a scene and to guide a model-free policy. Our results show that in a novel navigation and planning task called Box-World, our agent finds interpretable solutions that improve upon baselines in terms of sample complexity, ability to generalize to more complex scenes than experienced during training, and overall performance. In the StarCraft II Learning Environment, our agent achieves state-of-the-art performance on six mini-games -- surpassing human grandmaster performance on four. By considering architectural inductive biases, our work opens new directions for overcoming important, but stubborn, challenges in deep RL.