Real quantifier elimination is doubly exponential
Journal of Symbolic Computation
Solving systems of polynomial inequalities in subexponential time
Journal of Symbolic Computation
REDLOG: computer algebra meets computer logic
ACM SIGSAM Bulletin
A Computing Procedure for Quantification Theory
Journal of the ACM (JACM)
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
QEPCAD B: a program for computing with semi-algebraic sets using CADs
ACM SIGSAM Bulletin
Efficient projection orders for CAD
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
An algorithm for the computation of the radical of an ideal
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Verifying nonlinear real formulas via sums of squares
TPHOLs'07 Proceedings of the 20th international conference on Theorem proving in higher order logics
A proof-producing decision procedure for real arithmetic
CADE' 20 Proceedings of the 20th international conference on Automated Deduction
An algebraic approach for the unsatisfiability of nonlinear constraints
CSL'05 Proceedings of the 19th international conference on Computer Science Logic
Synthesis for PCTL in parametric Markov decision processes
NFM'11 Proceedings of the Third international conference on NASA Formal methods
Virtual substitution for SMT-solving
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
SMT-RAT: an SMT-compliant nonlinear real arithmetic toolbox
SAT'12 Proceedings of the 15th international conference on Theory and Applications of Satisfiability Testing
Abstract partial cylindrical algebraic decomposition i: the lifting phase
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
SMT for Polynomial Constraints on Real Numbers
Electronic Notes in Theoretical Computer Science (ENTCS)
Formalization of Bernstein Polynomials and Applications to Global Optimization
Journal of Automated Reasoning
dReal: an SMT solver for nonlinear theories over the reals
CADE'13 Proceedings of the 24th international conference on Automated Deduction
The strategy challenge in SMT solving
Automated Reasoning and Mathematics
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Methods for deciding quantifier-free non-linear arithmetical conjectures over *** are crucial in the formal verification of many real-world systems and in formalised mathematics. While non-linear (rational function) arithmetic over *** is decidable, it is fundamentally infeasible: any general decision method for this problem is worst-case exponential in the dimension (number of variables) of the formula being analysed. This is unfortunate, as many practical applications of real algebraic decision methods require reasoning about high-dimensional conjectures. Despite their inherent infeasibility, a number of different decision methods have been developed, most of which have "sweet spots" --- e.g., types of problems for which they perform much better than they do in general. Such "sweet spots" can in many cases be heuristically combined to solve problems that are out of reach of the individual decision methods when used in isolation. RAHD ("Real Algebra in High Dimensions") is a theorem prover that works to combine a collection of real algebraic decision methods in ways that exploit their respective "sweet-spots." We discuss high-level mathematical and design aspects of RAHD and illustrate its use on a number of examples.