 
Summary: Decidability over the Reals
Sicun Gao, Jeremy Avigad, and Edmund M. Clarke
Carnegie Mellon University, Pittsburgh, PA 15213
February 10, 2012
Abstract
Given any collection F of Type 2 computable functions over reals,
we show that there exists an algorithm that, given any LF sentence
containing only bounded quantifiers and any positive rational number
, decides either " is true", or "a strengthening of is false". This
"decision problem" resides in (P
n )C
for bounded nsentences, when
F is in a Type 2 complexity class C closed under function composi
tion and polynomialtime reduction. This stands in sharp contrast to
the wellknown undecidability of 1sentences in LF when F extends
arithmetic by sine. In fact, both boundedness and "robustness" are
necessary for decidability in this general setting. We believe our results
provide a theoretical basis for the use of numerical methods in deci
sion procedures for nonlinear firstorder theories over reals, and have
immediate practical relevance in application domains such as formal
