Verified Quadratic Virtual Substitution for Real Arithmetic
release_e2oldaaqozdl7luolm62svg7he
by
Matias Scharager, Katherine Cordwell, Stefan Mitsch, André Platzer
2021
Abstract
This paper presents a formally verified quantifier elimination (QE) algorithm
for first-order real arithmetic by linear and quadratic virtual substitution
(VS) in Isabelle/HOL. The Tarski-Seidenberg theorem established that the
first-order logic of real arithmetic is decidable by QE. However, in practice,
QE algorithms are highly complicated and often combine multiple methods for
performance. VS is a practically successful method for QE that targets formulas
with low-degree polynomials. To our knowledge, this is the first work to
formalize VS for quadratic real arithmetic including inequalities. The proofs
necessitate various contributions to the existing multivariate polynomial
libraries in Isabelle/HOL, including a method for re-indexing variables in a
polynomial. Our framework is modularized and easily expandable (to facilitate
integrating future optimizations), and could serve as a basis for developing a
general-purpose QE algorithm. Further, as our formalization is designed with
practicality in mind, we export our development to SML and test the resulting
code on 378 benchmarks from the literature, comparing to Redlog, Z3,
Mathematica, and SMT-RAT.
In text/plain
format
Archived Content
There are no accessible files associated with this release. You could check other releases for this work for an accessible version.
Know of a fulltext copy of on the public web? Submit a URL and we will archive it
2105.14183v1
access all versions, variants, and formats of this works (eg, pre-prints)