Quantifier elimination for the reals with a predicate for the powers of two

In [5], van den Dries showed that the theory of the reals with a predicate for the integer powers of two admits quantifier elimination in an expanded language, and is hence decidable. He gave a model-theoretic argument, which provides no apparent bounds on the complexity of a decision procedure. We provide a syntactic argument that yields a procedure that is primitive recursive, although not elementary. In particular, we show that it is possible to eliminate a single block of existential quantifiers in time 20 O(n) , where n is the length of the input formula and 2 k denotes k-fold iterated exponentiation.