Detecting and Exploiting Symmetry in Discrete-State Markov Models

Abstract

Dependable systems are usually designed with multiple instances of components or logical processes, and often possess symmetries that may be exploited in model-based evaluation. The problem of how best to exploit symmetry in models has received much attention from the modeling community, but no solution has garnered widespread support, primarily because each solution is limited in terms of either the types of symmetry that can be exploited, or the difficulty of translating from the system description to the model formalism. We propose a new method for detecting and exploiting model symmetry in which 1) models retain the structure of the system, and 2) all symmetry inherent in the structure of the model can be detected and exploited for the purposes of state-space reduction. Composed models are constructed from models through specification of connections between models that correspond to shared state fragments. The composed model is interpreted as an undirected graph, and results from group theory, and graph theory are used to develop procedures for automatically detecting, and exploiting all symmetries in the composed model. We discuss the necessary algorithms to detect and exploit model symmetry, and provide a proof that the theory generates an equivalent model. After a thorough analysis of the added complexity, a state-space generator which implements these algorithms within Mobius is then presented.