Markovian analysis of large finite state machines

Regarding nite state machines as Markov chains facilitates the application of probabilistic methods to very large logic synthesis and formal veriication problems. In this paper we present symbolic algorithms to compute the steady-state probabilities for very large nite state machines (up to 10 27 states). These algorithms, based on Algebraic Decision Diagrams (ADDs) | an extension of BDDs that allows arbitrary values to be associated with the terminal nodes of the diagrams | determine the steady-state probabilities by regarding nite state machines as homogeneous, discrete-parameter Markov chains with nite state spaces, and by solving the corresponding Chapman-Kolmogorov equations. We rst consider nite state machines with state graphs composed of a single terminal strongly connected component; for this type of systems we have implemented two solution techniques: One is based on the Gauss-Jacobi iteration, the other one is based on simple matrix multiplication. Then we extend our treatment to the most general case of systems which can be modeled as nite state machines with arbitrary transition structures; here our approach exploits structural information to decompose and simplify the state graph of the machine. We report experimental results obtained for problems on which traditional methods fail.