Robustness Verification of Support Vector Machines

We study the problem of formally verifying the robustness to adversarial examples of support vector machines (SVMs), a major machine learning model for classification and regression tasks. Following a recent stream of works on formal robustness verification of (deep) neural networks, our approach relies on a sound abstract version of a given SVM classifier to be used for checking its robustness. This methodology is parametric on a given numerical abstraction of real values and, analogously to the case of neural networks, needs neither abstract least upper bounds nor widening operators on this abstraction. The standard interval domain provides a simple instantiation of our abstraction technique, which is enhanced with the domain of reduced affine forms, which is an efficient abstraction of the zonotope abstract domain. This robustness verification technique has been fully implemented and experimentally evaluated on SVMs based on linear and nonlinear (polynomial and radial basis function) kernels, which have been trained on the popular MNIST dataset of images and on the recent and more challenging Fashion-MNIST dataset. The experimental results of our prototype SVM robustness verifier appear to be encouraging: this automated verification is fast, scalable and shows significantly high percentages of provable robustness on the test set of MNIST, in particular compared to the analogous provable robustness of neural networks

Logical presentations of domains

Bibliography: pages 168-174.This thesis combines a fairly general overview of domain theory with a detailed examination of recent work which establishes a connection between domain theory and logic. To start with, the theory of domains is developed with such issues as the semantics of recursion and iteration; the solution of recursive domain equations; and non-determinism in mind. In this way, a reasonably comprehensive account of domains, as ordered sets, is given. The topological dimension of domain theory is then revealed, and the logical insights gained by regarding domains as topological spaces are emphasised. These logical insights are further reinforced by an examination of pointless topology and Stone duality. A few of the more prominent categories of domains are surveyed, and Stone-type dualities for the objects of some of these categories are presented. The above dualities are then applied to the task of presenting domains as logical theories. Two types of logical theory are considered, namely axiomatic systems, and Gentzen-style deductive systems. The way in which these theories describe domains is by capturing the relationships between the open subsets of domains

Ranking for Objects and Attribute Reductions in Intuitionistic Fuzzy Ordered Information Systems

We aim to investigate intuitionistic fuzzy ordered information systems. The concept of intuitionistic fuzzy ordered information systems is proposed firstly by introducing an intuitionistic fuzzy relation to ordered information systems. And a ranking approach for all objects is constructed in this system. In order to simplify knowledge representation, it is necessary to reduce some dispensable attributes in the system. Theories of rough set are investigated in intuitionistic fuzzy ordered information systems by defining two approximation operators. Moreover, judgement theorems and methods of attribute reduction are discussed based on discernibility matrix in the systems, and an illustrative example is employed to show its validity. These results will be helpful for decisionmaking analysis in intuitionistic fuzzy ordered information systems

Abstract Interpretation of Supermodular Games

Supermodular games find significant applications in a variety of models, especially in operations research and economic applications of noncooperative game theory, and feature pure strategy Nash equilibria characterized as fixed points of multivalued functions on complete lattices. Pure strategy Nash equilibria of supermodular games are here approximated by resorting to the theory of abstract interpretation, a well established and known framework used for designing static analyses of programming languages. This is obtained by extending the theory of abstract interpretation in order to handle approximations of multivalued functions and by providing some methods for abstracting supermodular games, in order to obtain approximate Nash equilibria which are shown to be correct within the abstract interpretation framework

Decision Rules Acquisition for Inconsistent Disjunctive Set-Valued Ordered Decision Information Systems

Set-valued information system is an important formal framework for the development of decision support systems. We focus on the decision rules acquisition for the inconsistent disjunctive set-valued ordered decision information system in this paper. In order to derive optimal decision rules for an inconsistent disjunctive set-valued ordered decision information system, we define the concept of reduct of an object. By constructing the dominance discernibility function for an object, we compute reducts of the object via utilizing Boolean reasoning techniques, and then the corresponding optimal decision rules are induced. Finally, we discuss the certain reduct of the inconsistent disjunctive set-valued ordered decision information system, which can be used to simplify all certain decision rules as much as possible

Resource Theories as Quantale Modules

We aim to counter the tendency for specialization in science by advancing a language that can facilitate the translation of ideas and methods between disparate contexts. The methods we address relate to questions of "resource-theoretic nature". In a resource theory, one identifies resources and allowed manipulations that can be used to transform them. Some of the main questions are: How to optimize resources? What are the trade-offs between them? Can a given resource be converted to another one via the allowed manipulations? Because of the ubiquity of such questions, methods for answering them in one context can be used to tackle corresponding questions in new contexts. The translation occurs in two stages. Firstly, concrete methods are generalized to the abstract language to find under what conditions they are applicable. Then, one can determine whether potentially novel contexts satisfy these conditions. Here, we mainly focus on the first part of this two-stage process. The thesis starts with a more thorough introduction to resource theories and our perspective on them in chapter 1. Chapter 2 then provides a selection of mathematical ideas that we make heavy use of in the rest of the manuscript. In chapter 3, we present two variants of the abstract framework, whose relations to existing ones are summarized in table 1.1. The first one, universally combinable resource theories, offers a structure in which resources, desired tasks, and resource manipulations may all be viewed as "generalized resources". Blurring these distinctions, whenever appropriate, is a simplification that lets us understand the abstract results in elementary terms. It offers a slightly distinct point of view on resource theories from the traditional one, in which resources and their manipulations are considered independently. In this sense, the second framework in terms of quantale modules follows the traditional conception. Using these, we make contributions towards the task of generalizing concrete methods in chapter 4 by studying the ways in which meaningful measures of resources may be constructed. One construction expresses a notion of cost (or yield) of a resource, summarized in its generalized form in theorems 4.21 and 4.22. Among other applications, this construction may be used to extend measures from a subset of resources to a larger domain—such as from states to channels and other processes. Another construction allows the translation of resource measures between resource theories. A particularly useful version thereof is the translation of measures of distinguishability to other resource theories, which we study in detail. Special cases include resource robustness and weight measures as well as relative entropy based measures quantifying minimal distinguishability from freely available resources. We instantiate some of these ideas in a resource theory of distinguishability in chapter 5. It describes the utility of systems with probabilistic behavior for the task of distinguishing between hypotheses, which said behavior may depend on