**
An Algebraic Approach for the Unsatisfiability of Nonlinear Constraints
**

###
Ashish Tiwari

####
Presented at
CSL 2005, Oxford, UK, Aug 22-25, 2005.
© Springer-Verlag. The final version will appear at the
the Springer LNCS site.

### Abstract

We describe a simple algebraic semi-decision procedure for
detecting unsatisfiability of
a (quantifier-free) conjunction of nonlinear equalities and inequalities.
The procedure consists of Gr\"obner basis computation plus extension rules
that introduce new definitions, and hence it can be described as a
critical-pair completion-based logical procedure. This procedure is
shown to be sound and refutationally complete.
When projected onto the linear case, our procedure
reduces to the Simplex method for solving linear constraints.
If only finitely many new definitions are introduced, then the procedure
is also terminating. Such terminating, but potentially incomplete,
procedures are used in ``incompleteness-tolerant'' applications.
pdf.
Springer
link .

#### Slides

Slides of the presentation at CSL'05 :
postscript
pdf.

#### Errata

Paul Jackson correctly pointed out that Corollary 1, as stated
in the paper, is wrong. The correct statement should be:
P=0 AND V > 0 AND W >= 0
is unsatisfiable iff
there is a polynomial p'+c\mu \in Ideal(P) \cap Cone[V,W]
such that p'\in Cone[V,W] and c > 0 and \mu \in [V].

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#### BibTeX Entry

@inproceedings{Tiwari05:CSL,
author = "A. Tiwari",
title = "An Algebraic Approach for the Unsatisfiability of Nonlinear Constraints",
booktitle = "Computer Science Logic, 14th Annual Conf., CSL 2005",
EDITOR = "Ong, L.",
pages = "248--262",
series = {LNCS},
volume = 3634,
PUBLISHER = {Springer},
MONTH = aug,
YEAR = 2005
}

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