〈R, +, <, 1〉 Is Decidable in 〈R, +, <, Z〉

摘要

We show that it is decidable whether or not a relation on the reals definable in the structure 〈R,+,<.Z〉 can be defined in the structure 〈R,+,<,1〉. This result is achieved by obtaining a topological characterization of 〈R, +, <, 1〉-definable relations in the family of 〈R,+, <, Z〉-defmable relations and then by following Muchnik's approach of showing that this characterization can be expressed in the logic of 〈R,+,<, 1〉.