Vector sequence analysis and full weak safety for concurrent systems

Abstract

In this paper we analyze a particular sequence of points in the poset of nonnegative lattice points in n-dimensions. For this sequence, in 2-dimensions, we obtain optimal upper bounds on the number of points to be selected in order to ensure desired chain lengths among selected points; for higher dimensions, the upper bounds obtained are not optimal [6]. We use these bounds to decide full weak safety for a simple resource-oriented model for concurrent systems ([5, 12]). In this model, processes execute a sequence of operations (a program) which are controlled by a finite state device called the synchronizer. Concurrency among processes is modeled as an interleaving of their operations. A concurrent system consisting of a synchronizer and infinitely many identical processes is full weak safe if and only if all valid interleavings (of any finite number of processes) are accepted by the synchronizer. To decide full weak safety, we use our chain length result to give an effective procedure to obtain a threshold τ, such that the model is full weak safe if and only if all interleavings of operations length τ or less are accepted by the synchronizer.