Before revision:  We demonstrate that, by applying our FPRAS for tree automata, we can obtain an FPRAS for many hitherto open problems in the fields of constraint satisfaction problems ( CSPs) , database systems, software verification, and knowledge compilation. Specifically, we obtain an FPRAS for counting solutions for CSPs that are acyclic or, more generally, that have bounded hypertree-width, which results in an FPRAS for counting the number of answers to conjunctive queries that are acyclic or which have bounded hypertree-width. Moreover, these results can be extended to unions of acyclic conjunctive queries, and to the more general class of unions of conjunctive queries with bounded hypertree-width . Finally, we also obtain FPRAS for the problems of counting the number of error threads in programs with nested call subroutines, and counting valid assignments to structured DNNF circuits.
After revision:  We demonstrate that, by applying our FPRAS for tree automata, we can obtain an FPRAS for many hitherto open problems in the fields of constraint satisfaction problems ( CSP) with bounded hypertree-width . Finally, we also obtain FPRAS for the problems of counting the number of error threads in programs with nested call subroutines, and counting valid assignments to structured DNNF circuits.