Practical Approximate Quantifier Elimination for Non-linear Real Arithmetic (Long Version)
Résumé
Quantifier Elimination (QE) concerns finding a quantifier-
free formula that is semantically equivalent to a quantified formula in
a given logic. For the theory of non-linear arithmetic over reals (NRA),
QE is known to be computationally challenging. In this paper, we show
how QE over NRA can be solved approximately and efficiently in prac-
tice using a Boolean combination of constraints in the linear arithmetic
over reals (LRA). Our approach works by approximating the solution
space of a set of NRA constraints when all real variables are bounded.
It combines adaptive dynamic gridding with application of Handelman’s
Theorem to obtain the approximation efficiently via a sequence of lin-
ear programs (LP). We provide rigorous approximation guarantees, and
also proofs of soundness and completeness (under mild assumptions) of
our algorithm. Interestingly, our work allows us to bootstrap on earlier
work (viz. [41]) and solve quantified SMT problems over a combination
of NRA and other theories, that are beyond the reach of state-of-the-
art solvers. We have implemented our approach in a preprocessor for
Z3 called POQER. Our experiments show that POQER+Z3EG outper-
forms state-of-the-art SMT solvers on non-trivial problems, adapted from
a suite of benchmarks.
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