Summary: -Complete Decision Procedures for
Satisfiability over the Reals
Sicun Gao, Jeremy Avigad, and Edmund M. Clarke
Carnegie Mellon University, Pittsburgh, PA 15213
Abstract. We introduce the notion of "-complete decision procedures"
for solving SMT problems over real numbers, with the aim of handling a
wide range of nonlinear functions including transcendental functions and
solutions of Lipschitz-continuous ODEs. Given an SMT problem and a
positive rational number , a -complete decision procedure determines
either that is unsatisfiable, or that the "-weakening" of is satisfiable.
Here, the -weakening of is a variant of that allows -bounded numer-
ical perturbations on . We prove the existence of -complete decision
procedures for bounded SMT over reals with functions mentioned above.
For functions in Type 2 complexity class C, under mild assumptions, the
corresponding decision problem (bounded -SMT) is in NPC
. This stands
in sharp contrast to the well-known undecidability results. -Complete
decision procedures can exploit scalable numerical methods for handling
nonlinearity, and we propose to use this notion as an ideal requirement
for numerically-driven decision procedures. As a concrete example, we