Traversals of Infinite Graphs with Random Local Orientations

We introduce the notion of a "random basic walk" on an infinite graph, give numerous examples, list potential applications, and provide detailed comparisons between the random basic walk and existing generalizations of simple random walks. We define analogues in the setting of random basic walks of the notions of recurrence and transience in the theory of simple random walks, and we study the question of which graphs have a cycling random basic walk and which a transient random basic walk. We prove that cycles of arbitrary length are possible in any regular graph, but that they are unlikely. We give upper bounds on the expected number of vertices a random basic walk will visit on the infinite graphs studied and on their finite analogues of sufficiently large size. We then study random basic walks on complete graphs, and prove that the class of complete graphs has random basic walks asymptotically visit a constant fraction of the nodes. We end with numerous conjectures and problems for future study, as well as ideas for how to approach these problems.Comment: This is my masters thesis from Wesleyan University. Currently my advisor and I are selecting a journal where we will submit a shorter version. We plan to split this work into two papers: one for the case of infinite graphs and one for the finite case (which is not fully treated here

Static Analysis of a Concurrent Programming Language by Abstract Interpretation

Static analysis is an approach to determine information about the program without actually executing it. There has been much research in the static analysis of concurrent programs. However, very little academic research has been done on the formal analysis of message passing or process-oriented languages. We currently miss formal analysis tools and techniques for concurrent process-oriented languages such as Erasmus . In this dissertation, we focus on the problem of static analysis of large Erasmus programs. This can help us toward building more reliable Erasmus software systems. Reasoning about non-deterministic large Erasmus program using static analyzer is hard. These kinds of programs can quickly exhaust the computational and memory resources of the static analyzer tool. We use Abstract Interpretation to reason about Erasmus programs. To use the Abstract Interpretation theory, we introduce a lattice for Erasmus communications and an Event Order Predictor algorithm to statically determine the order that events happen in an Erasmus program. By using fixed-point theory of lattice, we compute a safe approximation of reachable states of an Erasmus programs. We also offer a Resettable Event order Vector for Erasmus processes to realistically implement our vector for large Erasmus programs using bounded space. We believe that our formal approach is also applicable to other types of process-oriented programs and MPI programs