Bounded Refinement Types

We present a notion of bounded quantification for refinement types and show how it expands the expressiveness of refinement typing by using it to develop typed combinators for: (1) relational algebra and safe database access, (2) Floyd-Hoare logic within a state transformer monad equipped with combinators for branching and looping, and (3) using the above to implement a refined IO monad that tracks capabilities and resource usage. This leap in expressiveness comes via a translation to "ghost" functions, which lets us retain the automated and decidable SMT based checking and inference that makes refinement typing effective in practice.Comment: 14 pages, International Conference on Functional Programming, ICFP 201

Bounded Refinement Types

Abstract We present a notion of bounded quantification for refinement types. We show how bounded quantification expands the expressiveness of refinement typing by (1) developing typed combinators for relational algebra and safe database access, (2) encoding Floyd-Hoare logic in a state transformer monad equipped with combinators for branching and looping, and (3) using the above to implement a refined IO monad that tracks capabilities and resource usage. Fortunately, we show that by translating bounds into "ghost" functions, the increased expressiveness comes while preserving the automated and decidable SMT based checking and inference that makes refinement typing effective in practice

An odyssey into local refinement and multilevel preconditioning III: Implementation and numerical experiments

In this paper, we examine a number of additive and multiplicative multilevel iterative methods and preconditioners in the setting of two-dimensional local mesh refinement. While standard multilevel methods are effective for uniform refinement-based discretizations of elliptic equations, they tend to be less effective for algebraic systems, which arise from discretizations on locally refined meshes, losing their optimal behavior in both storage and computational complexity. Our primary focus here is on Bramble, Pasciak, and Xu (BPX)-style additive and multiplicative multilevel preconditioners, and on various stabilizations of the additive and multiplicative hierarchical basis (HB) method, and their use in the local mesh refinement setting. In parts I and II of this trilogy, it was shown that both BPX and wavelet stabilizations of HB have uniformly bounded condition numbers on several classes of locally refined two- and three-dimensional meshes based on fairly standard (and easily implementable) red and red-green mesh refinement algorithms. In this third part of the trilogy, we describe in detail the implementation of these types of algorithms, including detailed discussions of the data structures and traversal algorithms we employ for obtaining optimal storage and computational complexity in our implementations. We show how each of the algorithms can be implemented using standard data types, available in languages such as C and FORTRAN, so that the resulting algorithms have optimal (linear) storage requirements, and so that the resulting multilevel method or preconditioner can be applied with optimal (linear) computational costs. We have successfully used these data structure ideas for both MATLAB and C implementations using the FEtk, an open source finite element software package. We finish the paper with a sequence of numerical experiments illustrating the effectiveness of a number of BPX and stabilized HB variants for several examples requiring local refinement

Operations in Hilbert space

This thesis reviews some of the major topics in elementary Hilbert space theory. The theory of operators is developed by providing details regarding several types of operators, in particular compact operators. This study of compact operators is the start of the refinement of bounded linear operators to those which are also members of the Schatten p-class operators

On the beliefs off the path: equilibrium refinement due to quantal response and level-k

This paper studies the relevance of equilibrium and nonequilibrium explanations of behavior, with respects to equilibrium refinement, as players gain experience. We investigate this experimentally using an incomplete information sequential move game with heterogeneous preferences and multiple perfect equilibria. Only the limit point of quantal response (the limiting logit equilibrium), and alternatively that of level-k reasoning (extensive form rationalizability), restricts beliefs off the equilibrium path. Both concepts converge to the same unique equilibrium, but the predictions differ prior to convergence. We show that with experience of repeated play in relatively constant environments, subjects approach equilibrium via the quantal response learning path. With experience spanning also across relatively novel environments, though, level-k reasoning tends to dominate

MeGARA: Menu-based Game Abstraction and Abstraction Refinement of Markov Automata

Markov automata combine continuous time, probabilistic transitions, and nondeterminism in a single model. They represent an important and powerful way to model a wide range of complex real-life systems. However, such models tend to be large and difficult to handle, making abstraction and abstraction refinement necessary. In this paper we present an abstraction and abstraction refinement technique for Markov automata, based on the game-based and menu-based abstraction of probabilistic automata. First experiments show that a significant reduction in size is possible using abstraction.Comment: In Proceedings QAPL 2014, arXiv:1406.156

Combinatorial modulus and type of graphs

Let a $A$ be the 1-skeleton of a triangulated topological annulus. We establish bounds on the combinatorial modulus of a refinement $A'$, formed by attaching new vertices and edges to $A$, that depend only on the refinement and not on the structure of $A$ itself. This immediately applies to showing that a disk triangulation graph may be refined without changing its combinatorial type, provided the refinement is not too wild. We also explore the type problem in terms of disk growth, proving a parabolicity condition based on a superlinear growth rate, which we also prove optimal. We prove our results with no degree restrictions in both the EEL and VEL settings and examine type problems for more general complexes and dual graphs.Comment: 24 pages, 12 figure