Cylindrical Algebraic Decomposition I: The Basic Algorithm

Abstract

Given a set of r-variale integral polynomials, a cylindrical algebraic decomposition (cad) of euclidean r-space E T partitions ET into connected subsets compatible with the zeros of the polynomials. Collins (1975) gave an algorithm for cad construction as part of a new decision procedure for real closed fields. This algorithm has since been implemented and applied to diverse problems (optimization, curve display). New applications of it have been proposed (program verification, motion planrnng), Part I of the present paper has several purposes, FirsL, it provides an exposition of the essential aspects of the algorithm. Second, it corrects mi.p.or errors in the 1975 paper, and develops certain concepts introduced there. Third. it provides a framework fOI" the adjacency algorithm presented in Part n. ]n addition. it surveys the applications of cad's und provides a detailed example of the operation of the algorithm.Key\'iO,'ds: polynomial zeros, computer .....lgebru. computational geometry, semi-algebraic geometry. real closed fields, decision procedures, real algebraic geometry.

Encoding Relative Orientation and Mereotopology Relations with Geometric Constraints in CLP(QS)

“…Moreover, at each iteration, the solver produces increasingly better configurations as it attempts to converge on the solution; we can directly study and visualise these intermediate configurations as a spatial history of configurations [17,18] giving further insight into the nature of the problem at hand. However, a key limitation is that 2 Tarski famously proved that the theory of real-closed fields is decidable via quantifier elimination (see [2,12,13] for an overview and algorithms); i.e. in a finite amount of time we can determine the consistency (or inconsistency) of any formula consisting of quantifiers (∀, ∃) over the reals, and polynomial equations and inequalities combined using logical connectors (∧, ∨, ¬).…”

“…Thus, the task of determining whether a set of spatial constraints is consistent becomes equivalent to determining the satisfiability of a system of polynomial constraints with variables ranging over reals. 2 We have investigated a range of polynomial constraint solving methods including CLP(R) for linear constraints, and SAT Modulo Theories and quantifier elimination by Cylindrical Algebraic Decomposition (CAD) for general systems of non-linear constraints [3,31,32]. However, solving such constraints is computationally intractable in general (i.e.…”

“…However, solving such constraints is computationally intractable in general (i.e. the theory of real-closed fields; for example, the CAD algorithm has double exponential complexity, O(a b n ), in the number of variables in the polynomial constraints, n) [2], and thus far, prominent symbolic algebraic methods do not scale to real world problems [24]. In this paper we address this intractability by investigating the use of constructive geometric constraint solving for spatial reasoning within the context of CLP(QS).…”

Reachability Problems on Extended O-Minimal Hybrid Automata

“…In [14] it is proved the decidability of o-minimal(OF(R)) automata. Decidability depends on the fact that the theory (OF(R)) admits quantifier elimination [17,4] i.e. each formula in the theory is equivalent to a quantifier free one that can be algorithmically determined.…”

A New Class of Decidable Hybrid Systems

“…, y n ) is equivalent to the (quantifier-free) formula false. There are different methods to perform quantifier elimination, e.g., [3,15]. All the examples considered in this paper have been solved using the tool Redlog [4].…”