Theory of linear and integer programming
Theory of linear and integer programming
Real quantifier elimination is doubly exponential
Journal of Symbolic Computation
Partial Cylindrical Algebraic Decomposition for quantifier elimination
Journal of Symbolic Computation
REDLOG: computer algebra meets computer logic
ACM SIGSAM Bulletin
Concrete Mathematics: A Foundation for Computer Science
Concrete Mathematics: A Foundation for Computer Science
A New Approach for Automatic Theorem Proving in Real Geometry
Journal of Automated Reasoning
QEPCAD B: a program for computing with semi-algebraic sets using CADs
ACM SIGSAM Bulletin
Convex Optimization
Efficient solving of quantified inequality constraints over the real numbers
ACM Transactions on Computational Logic (TOCL)
Differential Dynamic Logic for Hybrid Systems
Journal of Automated Reasoning
Logical Verification and Systematic Parametric Analysis in Train Control
HSCC '08 Proceedings of the 11th international workshop on Hybrid Systems: Computation and Control
Computing Differential Invariants of Hybrid Systems as Fixedpoints
CAV '08 Proceedings of the 20th international conference on Computer Aided Verification
IJCAR '08 Proceedings of the 4th international joint conference on Automated Reasoning
KeYmaera: A Hybrid Theorem Prover for Hybrid Systems (System Description)
IJCAR '08 Proceedings of the 4th international joint conference on Automated Reasoning
Aligator: A Mathematica Package for Invariant Generation (System Description)
IJCAR '08 Proceedings of the 4th international joint conference on Automated Reasoning
Extending a resolution prover for inequalities on elementary functions
LPAR'07 Proceedings of the 14th international conference on Logic for programming, artificial intelligence and reasoning
Verifying nonlinear real formulas via sums of squares
TPHOLs'07 Proceedings of the 20th international conference on Theorem proving in higher order logics
TACAS'08/ETAPS'08 Proceedings of the Theory and practice of software, 14th international conference on Tools and algorithms for the construction and analysis of systems
Verification of object-oriented software: The KeY approach
Verification of object-oriented software: The KeY approach
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
Integrating ICP and LRA solvers for deciding nonlinear real arithmetic problems
Proceedings of the 2010 Conference on Formal Methods in Computer-Aided Design
Speeding up cylindrical algebraic decomposition by gröbner bases
CICM'12 Proceedings of the 11th international conference on Intelligent Computer Mathematics
IJCAR'12 Proceedings of the 6th international joint conference on Automated Reasoning
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
Logical analysis of hybrid systems: a complete answer to a complexity challenge
DCFS'12 Proceedings of the 14th international conference on Descriptional Complexity of Formal Systems
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
dReal: an SMT solver for nonlinear theories over the reals
CADE'13 Proceedings of the 24th international conference on Automated Deduction
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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.