Verifying the interactive convergence clock synchronization algorithm using the Boyer-Moore theorem prover

The application of formal methods to the analysis of computing systems promises to provide higher and higher levels of assurance as the sophistication of our tools and techniques increases. Improvements in tools and techniques come about as we pit the current state of the art against new and challenging problems. A promising area for the application of formal methods is in real-time and distributed computing. Some of the algorithms in this area are both subtle and important. In response to this challenge and as part of an ongoing attempt to verify an implementation of the Interactive Convergence Clock Synchronization Algorithm (ICCSA), we decided to undertake a proof of the correctness of the algorithm using the Boyer-Moore theorem prover. This paper describes our approach to proving the ICCSA using the Boyer-Moore prover

Adaptive Discontinuous Galerkin Methods for Variational Inequalities with Applications to Phase Field Models

Solutions of variational inequalities often have limited regularity. In particular, the nonsmooth parts are local, while other parts of the solution have higher regularity. To overcome this limitation, we apply hp-adaptivity, which uses a combination of locally finer meshes and varying polynomial degrees to separate the different features of the the solution. For this, we employ Discontinuous Galerkin (DG) methods and show some novel error estimates for the obstacle problem which emphasize the use in hp-adaptive algorithms. Besides this analysis, we present how to efficiently compute numerical solutions using error estimators, fast algebraic solvers which can also be employed in a parallel setup, and discuss implementation details. Finally, some numerical examples and applications to phase field models are presented

Decidability and coincidence of equivalences for concurrency

There are two fundamental problems concerning equivalence relations in con-currency. One is: for which system classes is a given equivalence decidable? The second is: when do two equivalences coincide? Two well-known equivalences are history preserving bisimilarity (hpb) and hereditary history preserving bisimi-larity (hhpb). These are both ‘independence ’ equivalences: they reflect causal dependencies between events. Hhpb is obtained from hpb by adding a ‘back-tracking ’ requirement. This seemingly small change makes hhpb computationally far harder: hpb is well-known to be decidable for finite-state systems, whereas the decidability of hhpb has been a renowned open problem for several years; only recently it has been shown undecidable. The main aim of this thesis is to gain insights into the decidability problem for hhpb, and to analyse when it coincides with hpb; less technically, we might say, to analyse the power of the interplay between concurrency, causality, and conflict. We first examine the backtracking condition, and see that it has two dimen

Glue TAG semantics for binary branching syntactic structures

This thesis presents Gl-TAG, a new semantics for a fragment of natural language including simple in/transitive sentences with quantifiers. Gl-TAG utilises glue semantics, a proof-theoretic semantics based on linear logic, and TAG, a tree-based syntactic theory. We demonstrate that Gl-TAG is compositional, and bears interesting similarities to other approaches to the semantics of quantifiers. Chapter 1, rather than discussing the arguments of the thesis as a whole, outlines the global picture of language and semantic theory we adopt, introducing different semantics for quantification, so that Gl-TAG is understood in the proper context. Chapter 2, the heart of the thesis, introduces Gl-TAG, illustrating its application to quantifier scope ambiguity (Qscope ambiguity) and binding. Ways of constricting quantifier scope where necessary are suggested, but their full development is a topic of future research. Chapter 3 demonstrates that our semantics is compositional in certain formal senses there distinguished. Our account of quantification bears striking similarities to that proposed in Heim and Kratzer (1998), and also to Cooper storage (Cooper ((1983))); in fact, we can set up a form of Cooper storage within Gl-TAG. We suggest in conclusion that the features in common between frameworks highlight the possible formal similarities between the approaches. One philosophically interesting aspect of our semantics left aside is that it depends on proof theoretic methods; glue semantics combines semantic values both by harnessing the inferential power of linear logic and by exploiting the Curry-Howard isomorphism (CHI) familiar from proof theory (see chapter 2 for a brief explanation of the CHI). The semantic value of a proposition is thus a proof, as some proof theorists have desired (see Martin-Lof (1996). This raises a question for future research; namely, whether Gl-TAG is an inferential semantics in the sense that some philosophers have discussed (Murzi and Steinberger (2015))

Computational surface partial differential equations

Surface partial differential equations model several natural phenomena; for example in uid mechanics, cell biology and material science. The domain of the equations can often have complex and changing morphology. This implies analytic techniques are unavailable, hence numerical methods are required. The aim of this thesis is to design and analyse three methods for solving different problems with surface partial differential equations at their core. First, we define a new finite element method for numerically approximating solutions of partial differential equations in a bulk region coupled to surface partial differential equations posed on the boundary of this domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and to use piecewise polynomial boundary faces as an approximation of the surface and solve using isoparametric finite element spaces. We study this method in the context of a model elliptic problem. The main result in this chapter is an optimal order error estimate which is confirmed in numerical experiments. Second, we use the evolving surface finite element method to solve a Cahn- Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport formulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subsequence, of the finite element scheme. We conclude the chapter by deriving error estimates and present various numerical examples. Finally, we stray from surface finite element method to consider new unfitted finite element methods for surface partial differential equations. The idea is to use a fixed bulk triangulation and approximate the surface using a discrete approximation of the distance function. We describe and analyse two methods using a sharp interface and narrow band approximation of the surface for a Poisson equation. Error estimates are described and numerical computations indicate very good convergence and stability properties

Glue TAG semantics for binary branching syntactic structures

This thesis presents Gl-TAG, a new semantics for a fragment of natural language including simple in/transitive sentences with quantifiers. Gl-TAG utilises glue semantics, a proof-theoretic semantics based on linear logic, and TAG, a tree-based syntactic theory. We demonstrate that Gl-TAG is compositional, and bears interesting similarities to other approaches to the semantics of quantifiers. Chapter 1, rather than discussing the arguments of the thesis as a whole, outlines the global picture of language and semantic theory we adopt, introducing different semantics for quantification, so that Gl-TAG is understood in the proper context. Chapter 2, the heart of the thesis, introduces Gl-TAG, illustrating its application to quantifier scope ambiguity (Qscope ambiguity) and binding. Ways of constricting quantifier scope where necessary are suggested, but their full development is a topic of future research. Chapter 3 demonstrates that our semantics is compositional in certain formal senses there distinguished. Our account of quantification bears striking similarities to that proposed in Heim and Kratzer (1998), and also to Cooper storage (Cooper ((1983))); in fact, we can set up a form of Cooper storage within Gl-TAG. We suggest in conclusion that the features in common between frameworks highlight the possible formal similarities between the approaches. One philosophically interesting aspect of our semantics left aside is that it depends on proof theoretic methods; glue semantics combines semantic values both by harnessing the inferential power of linear logic and by exploiting the Curry-Howard isomorphism (CHI) familiar from proof theory (see chapter 2 for a brief explanation of the CHI). The semantic value of a proposition is thus a proof, as some proof theorists have desired (see Martin-Lof (1996). This raises a question for future research; namely, whether Gl-TAG is an inferential semantics in the sense that some philosophers have discussed (Murzi and Steinberger (2015))

Metascientific aspects of topoi of spaces

This thesis presents a study of the importance of topoi for Science. It is argued that whenever the concept of space enters the practice of Science then formal (mathematical) theories should be interpreted in a topos of spaces. It is claimed that these topoi encode knowledge of space arising directly out of the needs of Science, in that the constitutive questions of the Sciences can be traced back to their leading knowledge interests and these determine the role of mathematics as a methodical device. In the Natural Sciences the constitutive questions involve the study of non-intentional objects, in terms of a causal nexus to be explained geometrically, and this facilitates the introduction of geometric objects as the methodical device for posing questions to Nature. Although the study of intentional subjects in the Human Sciences requires ordinary language, not mathematics, to pose questions to each other, secondary methodological objectifications permit a conception of geometric objects analogous to that of the Natural Sciences. Lawvere*s axioms for the gros and petit topoi illustrate attempts to formalise the idea of topoi of spaces, as a rational reconstruction of categories in which geometric objects satisfying the formal theories of Science can be found. The catalysing function of this knowledge of topoi of spaces can lead to a diagnosis of mathematical difficulties caused by a failure to align mathematical conceptions with these topoi. This is illustrated through Varela's use of self-reference in Biology and Atkin's use of algebraic topology in Social Studies

Contraction of Hamiltonian K-spaces

In the spirit of recent work of Harada-Kaveh and Nishinou-Nohara-Ueda, we study the symplectic geometry of Popov's horospherical degenerations of complex algebraic varieties with the action of a complex linearly reductive group. We formulate an intrinsic symplectic contraction of a Hamiltonian space, which is a surjective, continuous map onto a new Hamiltonian space that is a symplectomorphism on an explicitly defined dense open subspace. This map is given by a precise formula, using techniques from the theory of symplectic reduction and symplectic implosion. We then show, using the Vinberg monoid, that the gradient-Hamiltonian flow for a horospherical degeneration of an algebraic variety gives rise to this contraction from a general fiber to the special fiber. We apply this construction to branching problems in representation theory, and finally we show how the Gel'fand-Tsetlin integrable system can be understood to arise this way.Comment: v3: 42 pages, various minor edits, to appear in IMR

On observational variance learning for multivariate Bayesian time series and related models

This thesis is concerned with variance learning in multivariate dynamic linear models (DLMs). Three new models are developed in this thesis. The first one is a dynamic regression model with no distributional assumption of the unknown variance matrix. The second is an extension of a known model that enables comprehensive treatment of any missing observations. For this purpose new distributions that replace the inverse Wishart and matrix T and that allow conjugacy are introduced. The third model is the general multivariate DLM without any precise assumptions of the error sequences and of the unknown variance matrix. We find analytic updatings of the first two moments based on weak assumptions that are satisfied for the usual models. Missing observations and time varying variances are considered in detail for every model. For the first time, deterministic and stochastic variance laws for the general multivariate DLM are presented. Also, by introducing a new distribution that replaces the matrix-beta of a previous work, we prove results on stochastic changes in variance that are in line with missing observation analysis and variance intervention