Cylindrical Algebraic Decomposition using validated numerics

Strzeboński, Adam

doi:10.1016/j.jsc.2006.06.004

Cited by 102 publications

(96 citation statements)

References 13 publications

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“…For each, it displays the number of times the factorisation subprocedure is invoked in Z3, the number of times the polynomial argument is irreducible, the percentage of irreducible polynomials, and the percentage of runtime spent in the factorisation subprocedure. 9 For univariate benchmarks, we observed that the overhead of polynomial factorisation is quite significant. Moreover, our RCF procedure in Z3 does not seem to benefit from factorisation as a preprocessing step even when polynomials can be factored.…”

Section: Univariate Factorisationsmentioning

confidence: 90%

“…This command is one of the best and most sophisticated general-purpose tools for deciding RCF sentences, containing highly-tuned implementations of a vast array of approaches to making RCF decisions [8,9]. Using Mathematica's Reduce to decide all of these 2,776 RCF sentences, we see that 253.33 seconds is spent in total.…”

Section: Metitarski Proof Search In More Detailmentioning

confidence: 99%

“…∀ t ∈ (0, ∞), v ∈ (0, ∞) ((1.565 + 0.313 v) cos(1.16 t) + (0.01340 + 0.00268 v) sin(1.16 t)) e −1.34 t − (6.55 + 1.31 v) e −0.318 t + v + 10 ≥ 0 Internally, MetiTarski is a resolution theorem prover integrated with various decision procedures for the theory of real-closed fields (RCF) [2,7,9]. MetiTarski reduces its input problem to a series of logical combinations of polynomial inequalities, which are further reduced to true or false by an RCF decision procedure.…”

Section: Introductionmentioning

confidence: 99%

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Real Algebraic Strategies for MetiTarski Proofs

Passmore

Paulson

Moura

2012

Lecture Notes in Computer Science

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Abstract. MetiTarski [1] is an automatic theorem prover that can prove inequalities involving sin, cos, exp, ln, etc. During its proof search, it generates a series of subproblems in nonlinear polynomial real arithmetic which are reduced to true or false using a decision procedure for the theory of real closed fields (RCF). These calls are often a bottleneck: RCF is fundamentally infeasible. However, by studying these subproblems, we can design specialised variants of RCF decision procedures that run faster and improve MetiTarski's performance.

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Section: Univariate Factorisationsmentioning

confidence: 90%

Section: Metitarski Proof Search In More Detailmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Real Algebraic Strategies for MetiTarski Proofs

Passmore

Paulson

Moura

2012

Lecture Notes in Computer Science

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“…The root structure function r : R n × Z + maps any point x ∈ R n to the set of roots of the polynomials in the (n + 1)-dimensional lifted cylinder above it (for convenience, consider ±∞ to be roots). This is a standard part of the CAD algorithm (see [75], [101]), although it is a computationally expensive operation. For some applications, it is possible to improve efficiency by identifying only root intervals rather than exact roots, using root separation/gap theorems.…”

Section: B Algorithm Assumptionsmentioning

confidence: 99%

Simple and Efficient Algorithms for Computing Smooth, Collision-free Feedback Laws Over Given Cell Decompositions

Lindemann

LaValle

2009

The International Journal of Robotics Research

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Abstract-This paper presents a novel approach to computing feedback laws in the presence of obstacles. Instead of computing a trajectory between a pair of initial and goal states, our algorithms compute a vector field over the entire state space; all trajectories obtained from following this vector field are guaranteed to asymptotically reach the goal state. As a result, the vector field globally solves the navigation problem and provides robustness to disturbances in sensing and control. The vector field's integral curves (system trajectories) are guaranteed to avoid obstacles and are C ∞ smooth. We construct a vector field with these properties by partitioning the space into simple cells, defining local vector fields for each cell, and smoothly interpolating between them to obtain a global vector field. We present an algorithm that computes these feedback controls for a kinematic point robot in an arbitrary dimensional space with piecewise linear boundary; the algorithm requires minimal preprocessing of the environment and is extremely fast during execution. For many practical applications in two-dimensional environments, full computation can be done in milliseconds. We also present an algorithm for computing feedback laws over cylindrical algebraic decompositions, thereby solving a smooth feedback version of the generalized piano movers' problem.

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“…QQ Partial cylindrical algebraic decomposition (PCAD) [3] in QEPCAD B [9]; QM QE based on partial CAD [3] and validated numerics [21] in Mathematica; QR c Partial CAD [3] in Redlog [10]; QR s Virtual substitution [4] in Redlog [10], falling back to QR c ; QC Harrison's implementation of Cohen-Hörmander quantifier elimination; QH Proof-producing quantifier elimination [11] in HOL Light.…”

Section: Implementations We Compare Six Implementations Of Qe In Expmentioning

confidence: 99%

Real World Verification

Platzer

Quesel

Rümmer

2009

Lecture Notes in Computer Science

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Abstract. Scalable handling of real arithmetic is a crucial part of the verification of hybrid systems, mathematical algorithms, and mixed analog/digital circuits. Despite substantial advances in verification technology, complexity issues with classical decision procedures are still a major obstacle for formal verification of real-world applications, e.g., in automotive and avionic industries. To identify strengths and weaknesses, we examine state of the art symbolic techniques and implementations for the universal fragment of real-closed fields: approaches based on quantifier elimination, Gröbner Bases, and semidefinite programming for the Positivstellensatz. Within a uniform context of the verification tool KeYmaera, we compare these approaches qualitatively and quantitatively on verification benchmarks from hybrid systems, textbook algorithms, and on geometric problems. Finally, we introduce a new decision procedure combining Gröbner Bases and semidefinite programming for the real Nullstellensatz that outperforms the individual approaches on an interesting set of problems.

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Cylindrical Algebraic Decomposition using validated numerics

Cited by 102 publications

References 13 publications

Real Algebraic Strategies for MetiTarski Proofs

Real Algebraic Strategies for MetiTarski Proofs

Simple and Efficient Algorithms for Computing Smooth, Collision-free Feedback Laws Over Given Cell Decompositions

Real World Verification

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