Abstract
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 undecidable.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.
Highlights
In 1973, Stockmeyer and Meyer [18] defined and studied membership and equivalence problems for integer expressions
The membership problem for integer expressions asks whether some given number is contained in the set described by a given integer expression, whereas the equivalence problem for integer expressions asks whether two given integer expression describe the same set
Each input gate of such a circuit is labeled with a natural number, the inner gates compute set operations and arithmetic operations (∪, ∩, +, ·)
Summary
In 1973, Stockmeyer and Meyer [18] defined and studied membership and equivalence problems for integer expressions. The question is whether there is an assignment of the variables such that all equations are satisfied These constraint satisfaction problems have the peculiarity that expressions describe sets of integers whereas variables can only store singleton sets of natural numbers. As the circuits only work over the domain of finite subsets of N, it suggests itself to allow the input gates of a circuit to compute arbitrary finite subsets of N and singleton sets (cf Dose [5] where the analogous step was made for constraint satisfaction problems). We show that BC(·) is NL-complete, BC(−) is NP-complete, and BC(∅) ∈ L
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