On approximation by ⊕-OBDDs

We show that for every ∨-OBDD of quasipolynomial size there is a ⊕-OBDD of quasipolynomial size computing the same function up to a small fraction of the inputs. We also prove that the final step in the approximation cannot be improved and discuss possibilities of proving non-approximability results and connections to lower bounds on matrix rigidity.