Since the advent of Byzantine replicated systems such as PBFT, numerous follow-up variants can maintain consistent replications in the presence of malicious nodes. However, Byzantine replication systems have been traditionally monolithic and homogeneous: the quorums nodes trust are fixed and uniform. This dissertation considers multiple aspects of heterogeneity for Byzantine replicated systems.

First, it considers heterogeneous replicated systems that ensure the trustworthiness specifications for a given computation. In the face of Byzantine attacks, the ultimate goal is to assure end-to-end policies for confidentiality, integrity, and availability. Given a computation and its policies, this dissertation synthesizes a distributed computation that flows through a sequence of heterogeneous replicated systems. It presents a security-typed object-based language, a partitioning transformation, an operational semantics, an information flow type inference system, and a synthesis tool called Hamraz. Hamraz automatically places and replicates fields and methods of the class. The experiments show the resulting systems can gracefully tolerate attacks that are as strong as the specified policies.

Second, it investigates heterogeneous quorum systems (HQS) where each process can declare its own quorums. In traditional replication systems, the set of trusted quorums is homogeneous across processes. However, trust is a subjective matter. This dissertation presents a general model of HQS. When quorum systems are not uniform, the properties that they should maintain to support reliable broadcast and consensus are not well understood. It was shown that quorum intersection and availability are necessary. This dissertation proves that they are not sufficient. It then defines the notion of quorum subsumption, and shows that the three conditions together are sufficient: it presents reliable broadcast and consensus protocols and proves their correctness.

Traditionally, the trusted set of quorums was not only homogeneous but fixed. However, trust evolves; processes might need to change their quorums. This dissertation presents reconfiguration protocols for HQS including joining and leaving of a process, and adding and removing of a quorum, and further, proves their correctness. The design of the protocols is informed by the trade-offs for the properties that reconfigurations can preserve. The dissertation further presents a graph characterization of HQS and its application for reconfiguration optimization.

Hydrogen transfer reactions are among the most fundamental organic reactions. They are involved in nearly all enzyme-catalyzed reactions necessary for the sustenance of life, and they are often involved in key steps in synthetically useful reductions. Notable among synthetically useful reactions are the class of reactions known as C–H bond functionalization. It is surprising, then, that this most fundamental chemical event is so poorly understood. To address this knowledge gap, we have developed new experimental and computational approaches that allow interrogation of the core physical nature of cyclic hydrogen transfer reactions. In general, our laboratory utilizes kinetic isotope effects (KIEs) to probe chemical reactions. This technique involves accurately determining how replacement of atoms with their heavier isotopes affects the reaction rate (or product distribution). In order to extract the most insightful information from KIE studies, suitable experiments and measurement methods need to be well-designed and carefully executed.

This dissertation is composed of four chapters. The first chapter serves as a general introduction that lays out the background of the mechanistic study of cyclic hydrogen transfer, the concepts of kinetic isotope effect and a concise review of traditional and new techniques that utilize KIEs to elucidate reaction mechanisms. The remaining chapters present three unique studies that culminate in a unified view of cyclic hydrogen transfers. Each study is designed to interrogate the key physical aspects of a synthetically useful and mechanistically interesting hydrogen transfer. The first study explores the Chugaev elimination, which is used to transform alcohols into alkenes in a typically regioselective manner. In this study, we have used a combination of intermolecular and intramolecular 2H KIE measurements to demonstrate that this reaction is dominated by quantum mechanical behavior even though it occurs at elevated temperatures. Comparison between the experimental and computational results imply that the observed KIEs are influenced by the location of the bottleneck (or transition state) which is found to be dependent on the isotopic substitution. Chapter 3 explores the Cope elimination, which can be used to install alkenes adjacent to an amine, demonstrated that seemingly simple cyclic hydrogen transfer reactions can proceed simultaneously via multiple mechanisms depending upon driving force and temperatures. Our approaches in the Cope and Chugaev eliminations were then utilized to explore an exciting new C–H functionalization reaction developed by DuBois. This study demonstrates that cyclic H-transfers mediated by a Ru-nitrene species are fundamentally similar to relatively simpler syn- eliminations in terms of the involvement of quantum mechanical phenomena. The reaction models built with prototype reactions can be generalized to more complicated metal catalyzed reactions to facilitate a better understanding of the reaction in terms of both reactivity and selectivity.

Statistics and machine learning have achieved remarkable successes in solving data problems including driving new biomedical discoveries. In particular, prediction and hypothesis testing are two important applications of statistics and machine learning to biomedical data. In this dissertation, we will investigate how appropriate interpretations of prediction algorithms, and scrutiny of efficiency of hypothesis testing techniques can help us extend the capability of statistical and machine learning approaches in biomedical science.

Chapter 1 of this dissertation provides an overview of the topics covered, as well as the background information for the rest of the dissertation. Chapters 2 and 3 introduce the applications of an interpretable machine learning prediction pipeline for two biomedical problems: drug response prediction, and molecular partner prediction in clathrin-mediated endocytosis.In the drug response prediction task, our predictive and stability-driven pipeline achieves the state-of-the-art performance in identifying stable, predictive -omics features for drug response. In the molecular partner prediction task, we developed a interpretable deep learning model that achieves state-of-the-art accuracy in predicting whether a clathrin-coated pit is abortive or valid.

Chapter 4 focuses on the interpretation of a specific algorithm: random forest. Random forest has witnessed numerous applications in biomedical sciences, and its interpretation has become an important topic of research. We derived the first finite sample bound on the bias of Mean Decrease Impurity, one of the most widely used measure of feature importance.To reduce this bias, we proposed a new feature importance measure, called MDI-oob. MDI-oob achieves state-of-the-art performance in feature selection from random forest in biology inspired simulations.

Chapters 5 and 6 aim to provide a more comprehensive understanding of some of the most popular high dimensional tests for biomedical data. Of particular interest is the comparison between special-purpose tests with the Bonferroni correction (or the closely associated max test in global testing), a simple and transparent test whose Type-I error (false positive) is robust to arbitrary dependence between $p$-values of univariate null hypotheses. In the context of global testing, we showed that the max test is optimal for detecting sparse signals, provided that the distribution of the signals has Gaussian or heavier tails. We also derived the first general negative results for knockoff methods. We give realistic conditions on the covariance matrix of the design matrix under which the true positive rate of the best achievable knockoff method must converge to zero, even when the true positive rate of Bonferroni correction converges to 1.