Equivalence Problems for Circuits over Sets of Natural Numbers

We investigate the complexity of equivalence problems for {∪,∩,⁻,+,×}-circuits computing sets of natural numbers. These problems were first introduced by Stockmeyer and Meyer (1973). We continue this line of research and give a systematic characterization of the complexity of equivalence problems over sets of natural numbers. Our work shows that equivalence problems capture a wide range of complexity classes like NL, C ₌ L, P,Π₂^P, PSPACE, NEXP, and beyond. McKenzie and Wagner (2003) studied related membership problems for circuits over sets of natural numbers. Our results also have consequences for these membership problems: We provide an improved upper bound for the case of {∪,∩,⁻,+,×}-circuits.