Complexity theory of real functions
Complexity theory of real functions
Solving Polynomial Systems Using a Branch and Prune Approach
SIAM Journal on Numerical Analysis
Computable analysis: an introduction
Computable analysis: an introduction
Quantified constraints under perturbation
Journal of Symbolic Computation
Bounded Model Checking Using Satisfiability Solving
Formal Methods in System Design
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
The complexity of quantifier elimination and cylindrical algebraic decomposition
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Decision Procedures: An Algorithmic Point of View
Decision Procedures: An Algorithmic Point of View
SAT Modulo ODE: A Direct SAT Approach to Hybrid Systems
ATVA '08 Proceedings of the 6th International Symposium on Automated Technology for Verification and Analysis
Lipschitz Continuous Ordinary Differential Equations are Polynomial-Space Complete
CCC '09 Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity
Solving Non-linear Polynomial Arithmetic via SAT Modulo Linear Arithmetic
CADE-22 Proceedings of the 22nd International Conference on Automated Deduction
MetiTarski: An Automatic Theorem Prover for Real-Valued Special Functions
Journal of Automated Reasoning
The image computation problem in hybrid systems model checking
HSCC'07 Proceedings of the 10th international conference on Hybrid systems: computation and control
Including ordinary differential equations based constraints in the standard CP framework
CP'10 Proceedings of the 16th international conference on Principles and practice of constraint programming
CalCS: SMT solving for non-linear convex constraints
Proceedings of the 2010 Conference on Formal Methods in Computer-Aided Design
Integrating ICP and LRA solvers for deciding nonlinear real arithmetic problems
Proceedings of the 2010 Conference on Formal Methods in Computer-Aided Design
Improving SAT modulo ODE for hybrid systems analysis by combining different enclosure methods
SEFM'11 Proceedings of the 9th international conference on Software engineering and formal methods
An interval-based SAT modulo ODE solver for model checking nonlinear hybrid systems
International Journal on Software Tools for Technology Transfer (STTT) - VSTTE 2009
A fast linear-arithmetic solver for DPLL(T)
CAV'06 Proceedings of the 18th international conference on Computer Aided Verification
Delta-Decidability over the Reals
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
Computation in real closed infinitesimal and transcendental extensions of the rationals
CADE'13 Proceedings of the 24th international conference on Automated Deduction
dReal: an SMT solver for nonlinear theories over the reals
CADE'13 Proceedings of the 24th international conference on Automated Deduction
Verification of annotated models from executions
Proceedings of the Eleventh ACM International Conference on Embedded Software
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We introduce the notion of "δ-complete decision procedures" for solving SMT problems over the real numbers, with the aim of handling a wide range of nonlinear functions including transcendental functions and solutions of Lipschitz-continuous ODEs. Given an SMT problem ϕ and a positive rational number δ, a δ-complete decision procedure determines either that ϕ is unsatisfiable, or that the "δ-weakening" of ϕ is satisfiable. Here, the δ-weakening of ϕ is a variant of ϕ that allows δ-bounded numerical perturbations on ϕ. We establish the existence and complexity of δ-complete decision procedures for bounded SMT over reals with functions mentioned above. We propose to use δ-completeness as an ideal requirement for numerically-driven decision procedures. As a concrete example, we formally analyze the DPLL〈ICP〉 framework, which integrates Interval Constraint Propagation in DPLL(T), and establish necessary and sufficient conditions for its δ-completeness. We discuss practical applications of δ-complete decision procedures for correctness-critical applications including formal verification and theorem proving.