Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition

Collins, George E.

doi:10.1007/978-3-7091-9459-1_4

Cited by 383 publications

(541 citation statements)

References 16 publications

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“…In the "worst case", the augmented projection (Collins, 1975) must be used to produce the formulas F and G and no simplification takes place, so that S P ⊂ S F and S Q ⊂ S G . Thus, barring coincidental common factors, B ⊂ A.…”

Section: The Obvious Approach and Its Flawsmentioning

confidence: 99%

“…In Collins' original algorithm for quantifier elimination by CAD (Collins, 1975), a CAD of free variable space is produced that is sign-invariant with respect to the projection factor set. Definition 2.…”

Section: Sign-invariant Cadsmentioning

confidence: 99%

“…This decomposition is truth invariant with respect to the input formula, meaning that the formula is either identically true or identically false in each cell of the decomposition. The method determines the truth of the input formula for each cell of the CAD, and then uses the CAD to construct a solution formula-a quantifier-free formula that is equivalent to the input formula (see Collins, 1975;Collins and Hong, 1991;Hong, 1992). Simple equivalent formulas can be constructed from a truth-invariant CAD (see Hong, 1992;Brown and Collins, 1996), which motivates the consideration of both quantified and unquantified input formulas.…”

Section: Introductionmentioning

confidence: 99%

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Simple CAD Construction and its Applications

Brown

2001

Journal of Symbolic Computation

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This paper presents a method for the simplification of truth-invariant cylindrical algebraic decompositions (CADs). Examples are given that demonstrate the usefulness of the method in speeding up the solution formula construction phase of the CADbased quantifier elimination algorithm. Applications of the method to the construction of truth-invariant CADs for very large quantifier-free formulas and quantifier elimination of non-prenex formulas are also discussed.

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Section: The Obvious Approach and Its Flawsmentioning

confidence: 99%

Section: Sign-invariant Cadsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Simple CAD Construction and its Applications

Brown

2001

Journal of Symbolic Computation

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“…This latter question will be resolved by an algorithm T that decides the truth in R; +, · , 0, 1 of all first-order sentences in +, · , 0, and 1. Such an algorithm was devised by Alfred Tarski in 1931 (see [40,41,42]) with improved versions devised later by G. E. Collins in 1975 (see [11]), and by several others in later years (see e.g. [9]).…”

Section: Corollary 92mentioning

confidence: 99%

Equations on real intervals

Taylor

2006

Algebra univers.

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A (finite or infinite) set Σ of equations, in operation symbols Ft (t ∈ T ) and variables x i , is said to be compatible with R iff there exist continuous operations F A t on R such that the algebra A = (R; F A t ) t∈T satisfies the equations Σ (with the variables x i understood as universally quantified). It is proved that there is no algorithm to decide R-compatibility for all finite Σ.If the definition is restricted to C 1 idempotent operations F A t , then there does exist an algorithm for compatibility.

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“…We are interested in constructing a semi-algebraic deformation of σ to in R 3 \ + , if we know that a continuous deformation exists. Consider the cylindrical algebraic decomposition[6] of η and + and refine it to a triangulation. Contractibility of η can be decided by considering only the incidence structure of the triangulation.…”

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confidence: 99%

Motion Planning for Multiple Robots

Aronov¹,

Berg²,

Stappen³

et al. 1999

Discrete Comput Geom

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We study the motion-planning problem for pairs and triples of robots operating in a shared workspace containing n obstacles. A standard way to solve such problems is to view the collection of robots as one composite robot, whose number of degrees of freedom is d, the sum of the numbers of degrees of freedom of the individual robots. We show that it is sufficient to consider a constant number of robot systems whose number of degrees of freedom is at most d − 1 for pairs of robots, and d − 2 for triples. (The result for a pair assumes that the sum of the number of degrees of freedom of the robots constituting the pair reduces by at least one if the robots are required to stay in contact; for triples a similar assumption is made. Moreover, for triples we need to assume that a solution with positive clearance exists.)We use this to obtain an O(n d ) time algorithm to solve the motion-planning problem for a pair of robots; this is one order of magnitude faster than what the standard method would give. For a triple of robots the running time becomes O(n d−1 ), which is two orders of magnitude faster than the standard method. We also apply our method to the case of a collection of bounded-reach robots moving in a low-density environment. Here the running time of our algorithm becomes O(n log n) both for pairs and triples.

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Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition

Cited by 383 publications

References 16 publications

Simple CAD Construction and its Applications

Simple CAD Construction and its Applications

Equations on real intervals

Motion Planning for Multiple Robots

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