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

Jeremy Avigad·Yimu Yin
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Abstract

In 1985, 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 $2^0_{O(n)}$, where $n$ is the length of the input formula and $2_k^x$ denotes $k$-fold iterated exponentiation.

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Jeremy Avigad, Yimu Yin. 2006-10-19. Quantifier elimination for the reals with a predicate for the powers of two. https://arxiv.org/abs/cs/0610117

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