Cylindrical algebraic decomposition I: the basic algorithm
SIAM Journal on Computing
Solving systems of strict polynomial inequalities
Journal of Symbolic Computation
Solution formula construction for truth invariant cad's
Solution formula construction for truth invariant cad's
IJCAR'12 Proceedings of the 6th international joint conference on Automated Reasoning
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This paper presents an algorithm that, roughly speaking, constructs a single open cell from a cylindrical algebraic decomposition (CAD). The algorithm takes as input a point and a set of polynomials, and computes a description of an open cylindrical cell containing the point in which the input polynomials have constant non-zero sign, provided the point is sufficiently generic. The paper reports on a few example computations carried out by a test implementation of the algorithm, which demonstrate the functioning of the algorithm and illustrate the sense in which it is more efficient than following the usual "open CAD" approach. Interest in the problem of computing a single cell from a CAD is motivated by a 2012 paper of Jovanovic and de Moura that require solving this problem repeatedly as a key step in NLSAT system. However, the example computations raise the possibility that repeated application of the new method may in fact be more efficient than the usual open CAD approach, both in time and space, for a broad range of problems.