To develop rigorous knowledge about ML models -- and the systems in which they are embedded -- we need reliable measurements. But reliable measurement is fundamentally challenging, and touches on issues of reproducibility, scalability, uncertainty quantification, epistemology, and more. This dissertation addresses criteria needed to take reliability seriously: both criteria for designing meaningful metrics, and for methodologies that ensure that we can dependably and efficiently measure these metrics at scale and in practice. In doing so, this dissertation articulates a research vision for a new field of scholarship at the intersection of machine learning, law, and policy. Within this frame, we cover topics that fit under three different themes: (1) quantifying and mitigating sources of arbitrariness in ML, (2) taming randomness in uncertainty estimation and optimization algorithms, in order to achieve scalability without sacrificing reliability, and (3) providing methods for evaluating generative-AI systems, with specific focuses on quantifying memorization in language models and training latent diffusion models on open-licensed data. By making contributions in these three themes, this dissertation serves as an empirical proof by example that research on reliable measurement for machine learning is intimately and inescapably bound up with research in law and policy. These different disciplines pose similar research questions about reliable measurement in machine learning. They are, in fact, two complementary sides of the same research vision, which, broadly construed, aims to construct machine-learning systems that cohere with broader societal values.

Neural-symbolic integration aims to combine the connectionist subsymbolic with the logical symbolic approach to artificial intelligence. In this paper, we first define the answer set semantics of (boolean) neural nets and then introduce from first principles a class of neural logic programs and show that nets and programs are equivalent.

Compositional learning, mastering the ability to combine basic concepts and construct more intricate ones, is crucial for human cognition, especially in human language comprehension and visual perception. This notion is tightly connected to generalization over unobserved situations. Despite its integral role in intelligence, there is a lack of systematic theoretical and experimental research methodologies, making it difficult to analyze the compositional learning abilities of computational models. In this paper, we survey the literature on compositional learning of AI models and the connections made to cognitive studies. We identify abstract concepts of compositionality in cognitive and linguistic studies and connect these to the computational challenges faced by language and vision models in compositional reasoning. We overview the formal definitions, tasks, evaluation benchmarks, variety of computational models, and theoretical findings. We cover modern studies on large language models to provide a deeper understanding of the cutting-edge compositional capabilities exhibited by state-of-the-art AI models and pinpoint important directions for future research.

Recent studies on the Shapes Constraint Language (SHACL), a W3C specification for validating RDF graphs, rely on translating the language into first-order logic in order to provide formally-grounded solutions to the validation, containment and satisfiability decision problems. Continuing on this line of research, we introduce SHACL2FOL, the first automatic tool that (i) translates SHACL documents into FOL sentences and (ii) computes the answer to the two static analysis problems of satisfiability and containment; it also allow to test the validity of a graph with respect to a set of constraints. By integrating with existing theorem provers, such as E and Vampire, the tool computes the answer to the aforementioned decision problems and outputs the corresponding first-order logic theories in the standard TPTP format. We believe this tool can contribute to further theoretical studies of SHACL, by providing an automatic first-order logic interpretation of its semantics, while also benefiting SHACL practitioners, by supplying static analysis capabilities to help the creation and management of SHACL constraints.

While the recent Chain-of-Thought (CoT) technique enhances the reasoning ability of large language models (LLMs) with the theory of mind, it might still struggle in handling logical reasoning that relies much on symbolic expressions and rigid deducing rules. To strengthen the logical reasoning capability of LLMs, we propose a novel Symbolic Chain-of-Thought, namely SymbCoT, a fully LLM-based framework that integrates symbolic expressions and logic rules with CoT prompting. Technically, building upon an LLM, SymbCoT 1) first translates the natural language context into the symbolic format, and then 2) derives a step-by-step plan to solve the problem with symbolic logical rules, 3) followed by a verifier to check the translation and reasoning chain. Via thorough evaluations on 5 standard datasets with both First-Order Logic and Constraint Optimization symbolic expressions, SymbCoT shows striking improvements over the CoT method consistently, meanwhile refreshing the current state-of-the-art performances. We further demonstrate that our system advances in more faithful, flexible, and explainable logical reasoning. To our knowledge, this is the first to combine symbolic expressions and rules into CoT for logical reasoning with LLMs. Code is open at https://github.com/Aiden0526/SymbCoT.

We formally verify an algorithm for approximate policy iteration on Factored Markov Decision Processes using the interactive theorem prover Isabelle/HOL. Next, we show how the formalized algorithm can be refined to an executable, verified implementation. The implementation is evaluated on benchmark problems to show its practicability. As part of the refinement, we develop verified software to certify Linear Programming solutions. The algorithm builds on a diverse library of formalized mathematics and pushes existing methodologies for interactive theorem provers to the limits. We discuss the process of the verification project and the modifications to the algorithm needed for formal verification.

Reinforcement learning (RL) has proven to be a powerful tool for training agents that excel in various games. However, the black-box nature of neural network models often hinders our ability to understand the reasoning behind the agent's actions. Recent research has attempted to address this issue by using the guidance of pretrained neural agents to encode logic-based policies, allowing for interpretable decisions. A drawback of such approaches is the requirement of large amounts of predefined background knowledge in the form of predicates, limiting its applicability and scalability. In this work, we propose a novel approach, Explanatory Predicate Invention for Learning in Games (EXPIL), that identifies and extracts predicates from a pretrained neural agent, later used in the logic-based agents, reducing the dependency on predefined background knowledge. Our experimental evaluation on various games demonstrate the effectiveness of EXPIL in achieving explainable behavior in logic agents while requiring less background knowledge.

We investigate the out-of-domain generalization of random feature (RF) models and Transformers. We first prove that in the `generalization on the unseen (GOTU)' setting, where training data is fully seen in some part of the domain but testing is made on another part, and for RF models in the small feature regime, the convergence takes place to interpolators of minimal degree as in the Boolean case (Abbe et al., 2023). We then consider the sparse target regime and explain how this regime relates to the small feature regime, but with a different regularization term that can alter the picture in the non-Boolean case. We show two different outcomes for the sparse regime with q-ary data tokens: (1) if the data is embedded with roots of unities, then a min-degree interpolator is learned like in the Boolean case for RF models, (2) if the data is not embedded as such, e.g., simply as integers, then RF models and Transformers may not learn minimal degree interpolators. This shows that the Boolean setting and its roots of unities generalization are special cases where the minimal degree interpolator offers a rare characterization of how learning takes place. For more general integer and real-valued settings, a more nuanced picture remains to be fully characterized.

We introduce a constructive method applicable to a large number of description logics (DLs) for establishing the concept-based Beth definability property (CBP) based on sequent systems. Using the highly expressive DL RIQ as a case study, we introduce novel sequent calculi for RIQ-ontologies and show how certain interpolants can be computed from sequent calculus proofs, which permit the extraction of explicit definitions of implicitly definable concepts. To the best of our knowledge, this is the first sequent-based approach to computing interpolants and definitions within the context of DLs, as well as the first proof that RIQ enjoys the CBP. Moreover, due to the modularity of our sequent systems, our results hold for any restriction of RIQ, and are applicable to other DLs by suitable modifications.

Today's distributed and pervasive computing addresses large-scale cyber-physical ecosystems, characterised by dense and large networks of devices capable of computation, communication and interaction with the environment and people. While most research focusses on treating these systems as "composites" (i.e., heterogeneous functional complexes), recent developments in fields such as self-organising systems and swarm robotics have opened up a complementary perspective: treating systems as "collectives" (i.e., uniform, collaborative, and self-organising groups of entities). This article explores the motivations, state of the art, and implications of this "collective computing paradigm" in software engineering, discusses its peculiar challenges, and outlines a path for future research, touching on aspects such as macroprogramming, collective intelligence, self-adaptive middleware, learning, synthesis, and experimentation of collective behaviour.