We show in this article that some concepts from homotopy theory, in algebraic topology, are relevant for studying concurrent programs. We exhibit a natural semantics of semaphore programs, based on partially ordered topological spaces, which are studied up to "elastic deformation" or homotopy, giving information about important properties of the program, such as deadlocks, unreachables, serializability, essential schedules, etc. In fact, it is not quite ordinary homotopy that has to be used, but rather a "directed homotopy" that does not reverse the flow of time. We show some of the essential differences between ordinary and directed homotopy through examples. We also relate the topological view to a combinatorial view of concurrent programs closer to transition systems, through the notion of a cubical set. Finally we apply some of these concepts to the proof of the safeness of a two-phase protocol, well-known and used in concurrent database theory. We end up with a list of problems from both a mathematical and a computer-scientific point of view.
We use a geometric description for deadlocks occuring in scheduling problems for concurrent systems to construct a partial order and hence a directed graph, in which the local maxima correspond to deadlocks. Algorithms finding deadlocks are described and assessed.
Despite the numerous technological applications of amorphous materials, such as glasses, the understanding of their medium-range order (MRO) structure—and particularly the origin of the first sharp diffraction peak (FSDP) in the structure factor—remains elusive. Here, we use persistent homology, an emergent type of topological data analysis, to understand MRO structure in sodium silicate glasses. To enable this analysis, we introduce a self-consistent categorization of rings with rigorous geometrical definitions of the structural entities. Furthermore, we enable quantitative comparison of the persistence diagrams by computing the cumulative sum of all points weighted by their lifetime. On the basis of these analysis methods, we show that the approach can be used to deconvolute the contributions of various MRO features to the FSDP. More generally, the developed methodology can be applied to analyze and categorize molecular dynamics data and understand MRO structure in any class of amorphous solids.
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