Indirect Applications

Abstract

In this chapter, we return to the general framework of Chapter 1 and re-examine the notions of approximate counting and almost uniform generation in the light of our work on Markov chains. Our main result is a dramatic improvement of the reduction from generation to counting for selfreducible relations presented in Theorem 1.10, which allows much larger errors in the counter to be handled. The reduction is achieved by constructing an ergodic Markov chain based on the tree of derivations. As always, the crucial feature of the chain from our point of view is that it converges rapidly to its stationary distribution. The machinery developed in Chapter 2 will enable us to establish this property painlessly.