The geometry of semaphore programs

Abstract

Synchronization errors in concurrent programs are notoriously difficult to find and correct. Deadlock, partial deadlock, and unsafeness are conditions that constitute such errors.A model of concurrent semaphore programs based on multidimensional, solid geometry is presented. While previously reported geometric models are restricted to two-process mutual exclusion problems, the model described here applies to a broader class of synchronization problems. The model is shown to be exact for systems composed of an arbitrary, yet fixed number of concurrent processes, each consisting of a straight line sequence of arbitrarily ordered semaphore operations.

State spaces and dipaths up to dihomotopy

“…1. For so-called semaphore programs (explained below), these progress graphs have been exploited for an algorithmic determination of deadlocs and unreachable states [23,5,9]. A systematic framework for studying schedules of actions of distributed computations by means of geometric properties was proposed by V. Pratt [25] and subsequently R. van Glabbeek [30].…”

“…Even if the po-space X does not have any deadlock point x (i.e., π 1 (X)(x, X 1 ) = ∅ for all x ∈ X, cf. [23,5,9]), the mapping spaces very often have lots of them. If X is the po-space from the left part of Fig.…”

Classification of di-embeddings of the $n$-cube into $\mathbb{R}^n$

“…Progress graphs were used in [1] to prove the existence of deadlocks in certain types of concurrent systems. More recently, progress graphs appear in the work of Fajstrup, Goubault, Raussen and others [3,5,6] where various tools from algebraic topology are adapted to their study.…”

Detecting Deadlocks in Concurrent Systems

“…A description of deadlocks in terms of the geometry of the progress graph had been given earlier by Carson and Reynolds [1], and we stick to their terminology.…”

“…The main idea in [1] is to model a discrete concurrency problem in a continuous geometric set-up: A system of n concurrent processes will be represented as a subset of Euclidean space R n . Each coordinate axis corresponds to one of the processes.…”