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-Decidability over the Reals Sicun Gao, Jeremy Avigad, and Edmund M. Clarke

Summary: -Decidability over the Reals
Sicun Gao, Jeremy Avigad, and Edmund M. Clarke
Carnegie Mellon University, Pittsburgh, PA 15213
February 10, 2012
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 n-sentences, when
F is in a Type 2 complexity class C closed under function composi-
tion and polynomial-time reduction. This stands in sharp contrast to
the well-known undecidability of 1-sentences 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 first-order theories over reals, and have
immediate practical relevance in application domains such as formal


Source: Avigad, Jeremy - Departments of Mathematical Sciences & Philosophy, Carnegie Mellon University


Collections: Multidisciplinary Databases and Resources; Mathematics