We investigate the computational complexity of balance problems for − -circuits computing finite sets of natural numbers. These problems naturally build on problems for integer expressions and integer circuits studied by Stockmeyer and Meyer (1973), McKenzie and Wagner (2007), and Glaßer et al (2010).

Our work shows that the balance problem for − -circuits is undecidable which is the first natural problem for integer circuits or related constraint satisfaction problems that admits only one arithmetic operation and is proven to be undeciable.

Starting from this result we precisely characterize the complexity of balance problems for proper subsets of − . These problems turn out to be complete for one of the classes L, NL, and NP.