Before revision:  At the heart of our algorithm is a reduction to the problem of estimating the number of strings of length\neqP n accepted by a succinct non-deterministic finite automaton(NFA), which is an NFA where the transitions are labeled by succinctly encoded sets of symbols, whose sizes can be exponential in the encoding length.
After revision:  At the heart of our algorithm is a reduction to the problem of estimating the number of strings of length\neqP). In fact, our FPRAS is more general, and also applies to conjunctive queries with bounded hypertree width, as well as unions of such queries. The key ingredient in our proof is the resolution of a fundamental counting problem from automata theory. Specifically, we demonstrate the first FPRAS and polynomial time sampler for the set of trees of size n accepted by a succinct non-deterministic finite automaton(NFA), which is an NFA where the transitions are labeled by succinctly encoded sets of symbols, whose sizes can be exponential in the encoding length.