• Corpus ID: 215416119

New Opportunities for the Formal Proof of Computational Real Geometry? (Extended Abstract)

@inproceedings{Abraham2020NewOF,
  title={New Opportunities for the Formal Proof of Computational Real Geometry? (Extended Abstract)},
  author={Erika 'Abrah'am and James H. Davenport and Matthew England and Gereon Kremer and Zak Tonks},
  booktitle={PAAR+SC²@IJCAI},
  year={2020}
}
The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such questions have not been formalised. We present a thesis that a new algorithm for ascertaining satisfiability of formulae over the reals via Cylindrical Algebraic Coverings [Abraham, Davenport, England, Kremer, \emph{Deciding the Consistency of Non-Linear Real… 
View PDF on arXiv
10 Citations
Background Citations
5
Methods Citations
1
Results Citations
1

Figures from this paper

10 Citations

A mathematician's view of the unreasonable ineffectiveness of mathematics in biology

On the Implementation of Cylindrical Algebraic Coverings for Satisfiability Modulo Theories Solving

This work shows how this new implementation of cylindrical algebraic coverings simplifies proof generation for nonlinear real arithmetic problems in cvc5 and announces some very encouraging experimental results that position cVC5 at the very front of currently available SMT solvers for QF_NRA.

Varieties of Doubly-Exponential Behaviour in Cylindrical Algebraic Decomposition

It is almost fifty years since cylindrical algebraic decomposition was introduced: it was far better than previous ideas, but the algorithm was doubly exponential in the number of variables. Various…

An SMT solver for non-linear real arithmetic inside maple

It is reported on work-in-progress to create an SMT-solver inside Maple for non-linear real arithmetic (NRA) and some new work is presented on the identification of minimal conflicting cores from the coverings.

Cylindrical Algebraic Coverings for Quantifiers

This work reformulate and extend the cylindrical algebraic coverings method to allow for checking the validity of arbitrary nonlinear arithmetic formulas, adding support for both quantifiers and arbitrary Boolean structure.

SC-Square: Overview to 2021

This extended abstract was written to accompany an invited talk at the 2021 SC-Square Workshop, where the author was asked to give an overview of SC-Square progress to date. The author first reminds…

Cooperating Techniques for Solving Nonlinear Real Arithmetic in the cvc5 SMT Solver (System Description)

The cvc5 SMT solver solves quantifier-free nonlinear real arithmetic problems by combining the cylindrical algebraic coverings method with incremental linearization in an abstraction-refinement loop to create a complete algebraic decision procedure that leverages efficient heuristics for refining candidate models.

Proving UNSAT in SMT: The Case of Quantifier Free Non-Linear Real Arithmetic

We discuss the topic of unsatisfiability proofs in SMT, particularly with reference to quantifier free non-linear real arithmetic. We outline how the methods here do not admit trivial proofs and how…

Algebraic Stability Analysis of Particle Swarm Optimization Using Stochastic Lyapunov Functions and Quantifier Elimination

This paper presents a computational procedure and shows that this approach leads to a reevaluation and extension of previously known stability regions for PSO using a Lyapunov approach under stagnation assumptions.

Deciding the Consistency of Non-Linear Real Arithmetic Constraints with a Conflict Driven Search Using Cylindrical Algebraic Coverings

References

SHOWING 1-10 OF 37 REFERENCES

Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination

A formalization of discrete real closed fields in the Coq proof assistant of an algebraic proof of quantifier elimination based on pseudo-remainder sequences following the standard computer algebra literature on the topic.

Deciding the Consistency of Non-Linear Real Arithmetic Constraints with a Conflict Driven Search Using Cylindrical Algebraic Coverings

Cylindrical Algebraic Decomposition with Equational Constraints

A NEW DECISION METHOD FOR ELEMENTARY ALGEBRA

A. Tarski [4] has given a decision method for elementary algebra. In essence this comes to giving an algorithm for deciding whether a given finite set of polynomial inequalities has a solution. Below…

A Poly-algorithmic Quantifier Elimination Package in Maple

The challenges and scope such an implementation of QuantifierElimination written in Maple gives in terms of the desires of the Satisfiability Modulo Theory (SMT) community, and other frequent uses of QE are explored, noting Maple’s status as a Mathematical toolbox.

Better Answers to Real Questions

The complexity of quantifier elimination and cylindrical algebraic decomposition

A simple and constructive proof that quantifier elimination in real algebra is doubly exponential, even when there is only one free variable and all polynomials in the quantified input are linear is given.

Truth table invariant cylindrical algebraic decomposition

New results on quantifier elimination over real closed fields and applications to constraint databases

  • S. Basu
  • Computer Science, Mathematics
    JACM
  • 1999
A new algorithm for quantifier elimination in the first order theory of real closed fields that improves the complexity of the best known algorithm for this problem till now and leads to simpler and constructive proofs for these inexpressibility results than the ones known before.

Validity proof of Lazard's method for CAD construction