An Implementation of Mermera: A Shared Memory System that Mixes Coherence with Non-coherence

Coherent shared memory is a convenient, but inefficient, method of inter-process communication for parallel programs. By contrast, message passing can be less convenient, but more efficient. To get the benefits of both models, several non-coherent memory behaviors have recently been proposed in the literature. We present an implementation of Mermera, a shared memory system that supports both coherent and non-coherent behaviors in a manner that enables programmers to mix multiple behaviors in the same program[HS93]. A programmer can debug a Mermera program using coherent memory, and then improve its performance by selectively reducing the level of coherence in the parts that are critical to performance. Mermera permits a trade-off of coherence for performance. We analyze this trade-off through measurements of our implementation, and by an example that illustrates the style of programming needed to exploit non-coherence. We find that, even on a small network of workstations, the performance advantage of non-coherence is compelling. Raw non-coherent memory operations perform 20-40~times better than non-coherent memory operations. An example application program is shown to run 5-11~times faster when permitted to exploit non-coherence. We conclude by commenting on our use of the Isis Toolkit of multicast protocols in implementing Mermera.

Fixed Point vs. First-Order Logic on Finite Ordered Structures with Unary Relations

We prove that first order logic is strictly weaker than fixed point logic over every infinite classes of finite ordered structures with unary relations: Over these classes there is always an inductive unary relation which cannot be defined by a first-order formula, even when every inductive sentence (i.e., closed formula) can be expressed in first-order over this particular class. Our proof first establishes a property valid for every unary relation definable by first-order logic over these classes which is peculiar to classes of ordered structures with unary relations. In a second step we show that this property itself can be expressed in fixed point logic and can be used to construct a non-elementary unary relation.

Efficient Support for Common Relations in Lightweight Formal Reasoning Systems

In work that involves mathematical rigor, there are numerous benefits to adopting a representation of models and arguments that can be supplied to a formal reasoning or verification system: reusability, automatic evaluation of examples, and verification of consistency and correctness. However, accessibility has not been a priority in the design of formal verification tools that can provide these benefits. In earlier work [Lap09a], we attempt to address this broad problem by proposing several specific design criteria organized around the notion of a natural context: the sphere of awareness a working human user maintains of the relevant constructs, arguments, experiences, and background materials necessary to accomplish the task at hand. This work expands one aspect of the earlier work by considering more extensively an essential capability for any formal reasoning system whose design is oriented around simulating the natural context: native support for a collection of mathematical relations that deal with common constructs in arithmetic and set theory. We provide a formal definition for a context of relations that can be used to both validate and assist formal reasoning activities. We provide a proof that any algorithm that implements this formal structure faithfully will necessary converge. Finally, we consider the efficiency of an implementation of this formal structure that leverages modular implementations of well-known data structures: balanced search trees and transitive closures of hypergraphs.