We describe a framework for deriving and analyzing online optimization algorithms that incorporate adaptive, data-dependent regularization, also termed preconditioning. Such algorithms have been proven useful in stochastic optimization by reshaping the gradients according to the geometry of the data. Our framework captures and unifies much of the existing literature on adaptive online methods, ...

Gödel’s modal logic approach to analyzing provability attracted a great deal of attention and eventually led to two distinct mathematical models. The first is the modal logic GL, also known as the Provability Logic, which was shown in 1979 by Solovay to be the logic of the formal provability predicate. The second is Gödel’s original modal logic of provability S4, together with its explicit coun...

The axiomatization of arithmetical properties in theorem proving creates many straightforward inference steps. In analyzing mathematical proofs with the CERES (Cut-Elimination by Resolution) system, it is convenient to hide these inferences. The central topic of the thesis is the extension of the CERES method to allow reasoning modulo equational theories. For this, the inference systems of Sequ...