Equations over sets of integers with addition only

Abstract

Systems of equations of the form X=Y+Z and X=C are considered, in which the unknowns are sets of integers, the plus operator denotes element-wise sum of sets, and C is an ultimately periodic constant. For natural numbers, such equations are equivalent to language equations over a one-symbol alphabet using concatenation and regular constants. It is shown that such systems are computationally universal: for every recursive (r.e., co-r.e.) set S⊆N there exists a system with a unique (least, greatest) solution containing a component T with S={n|16n+13∈T}. Testing solution existence for these equations is Π10-complete, solution uniqueness is Π20-complete, and finiteness of the set of solutions is Σ30-complete. A similar construction for integers represents any hyper-arithmetical set S⊆Z by a set T⊆Z satisfying S={n|16n∈T}, whereas testing solution existence is Σ11-complete.