We study the computational complexity of emptiness problems for circuits over sets of natural numbers with the operations union, intersection, complement, addition, and multiplication. For most settings of allowed operations we precisely characterize the complexity in terms of completeness for classes like NL, NP, and PSPACE. The case where intersection, addition, and multiplication is allowed turns out to be equivalent to the complement of polynomial identity testing (PIT).

Our results imply the following improvements and insights on problems studied in earlier papers. We improve the bounds for the membership problem MC($\cup,\cap,\overline{\phantom{x}},+,\times$) studied by McKenzie and Wagner 2007 and for the equivalence problem EQ($\cup,\cap,\overline{\phantom{x}},+,\times$) studied by Glaßer et al. 2010. Moreover, it turns out that the following problems are equivalent to PIT, which shows that the challenge to improve their bounds is just a reformulation of a well-studied, major open problem in algebraic computing complexity:

1. membership problem MC($\cap,+,\times$) studied by McKenzie and Wagner 2007

2. integer membership problems MC$_{\mathbb{Z}}$($+,\times$), MC$_{\mathbb{Z}}$($\cap,+,\times$) studied by Travers 2006

3. equivalence problem EQ($+,\times$) studied by Glaßer et al. 2010