Satisfiability of algebraic circuits over sets of natural numbers

We investigate the complexity of satisfiability problems   for {∪,∩,−,+,×}{∪,∩,−,+,×}-circuits computing sets of natural numbers. These problems are a natural generalization of membership problems for expressions and circuits studied by Stockmeyer and Meyer (1973) [10] and McKenzie and Wagner (2003) [8].Our work shows that satisfiability problems capture a wide range of complexity classes such as NL, P, NP, PSPACE, and beyond. We show that in several cases, satisfiability problems are harder than membership problems. In particular, we prove that testing satisfiability for {∩,+,×}{∩,+,×}-circuits is already undecidable. In contrast to this, the satisfiability for {∪,+,×}{∪,+,×}-circuits is decidable in PSPACE.