Interpolants in Nonlinear Theories Over the Reals

We develop algorithms for computing Craig interpolants for first-order formulas over real numbers with a wide range of nonlinear functions, including transcendental functions and differential equations. We transform proof traces from $$\delta $$-complete decision procedures into interpolants that consist of Boolean combinations of linear constraints. The algorithms are guaranteed to find the interpolants between two formulas A and B whenever $$A \wedge B$$ is not $$\delta $$-satisfiable. At the same time, by exploiting $$\delta $$-perturbations one can parameterize the algorithm to find interpolants with different positions between A and B. We show applications of the methods in control and robotic design, and hybrid system verification.

[1]  Kenneth L. McMillan.  Widening and Interpolation , 2011, SAS.

[2]  Marsha Chechik,et al.  Ufo: A Framework for Abstraction- and Interpolation-Based Software Verification , 2012, CAV.

[3]  Edmund M. Clarke,et al.  δ-Complete Decision Procedures for Satisfiability over the Reals , 2012, IJCAR.

[4]  Alberto Griggio,et al.  Efficient Interpolant Generation in Satisfiability Modulo Linear Integer Arithmetic , 2011, TACAS.

[5]  Pavel Pudlák,et al.  Lower bounds for resolution and cutting plane proofs and monotone computations , 1997, Journal of Symbolic Logic.

[6]  Liyun Dai,et al.  Generating Non-linear Interpolants by Semidefinite Programming , 2013, CAV.

[7]  Kenneth L. McMillan,et al.  Lazy Abstraction with Interpolants , 2006, CAV.

[8]  Thomas Wies,et al.  Flow-Sensitive Fault Localization , 2013, VMCAI.

[9]  Edmund M. Clarke,et al.  Proof Generation from Delta-Decisions , 2014, 2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing.

[10]  Dejan Jovanović,et al.  Solving Non-linear Arithmetic , 2012, IJCAR.

[11]  Kenneth L. McMillan,et al.  Interpolants from Z3 proofs , 2011, 2011 Formal Methods in Computer-Aided Design (FMCAD).

[12]  Edmund M. Clarke,et al.  Parameter Synthesis for Cardiac Cell Hybrid Models Using δ-Decisions , 2014, CMSB.

[13]  Edmund M. Clarke,et al.  dReal: An SMT Solver for Nonlinear Theories over the Reals , 2013, CADE.

[14]  Madan Musuvathi,et al.  A Combination Method for Generating Interpolants , 2005, CADE.

[15]  Frédéric Benhamou,et al.  Continuous and Interval Constraints , 2006, Handbook of Constraint Programming.

[16]  Klaus Weihrauch,et al.  Computable Analysis: An Introduction , 2014, Texts in Theoretical Computer Science. An EATCS Series.

[17]  Bernd Becker,et al.  Craig Interpolation in the Presence of Non-linear Constraints , 2011, FORMATS.

[18]  Jochen Hoenicke,et al.  SMTInterpol: An Interpolating SMT Solver , 2012, SPIN.

[19]  Martin Fränzle,et al.  Efficient Solving of Large Non-linear Arithmetic Constraint Systems with Complex Boolean Structure , 2007, J. Satisf. Boolean Model. Comput..

[20]  Edmund M. Clarke,et al.  Parameter Synthesis for Cardiac Cell Hybrid Models Using Delta-Decisions , 2014, ArXiv.

[21]  Viorica Sofronie-Stokkermans,et al.  Constraint solving for interpolation , 2007, J. Symb. Comput..

[22]  Daniel Kroening,et al.  An Interpolating Sequent Calculus for Quantifier-Free Presburger Arithmetic , 2010, IJCAR.

[23]  Edmund M. Clarke,et al.  Satisfiability modulo ODEs , 2013, 2013 Formal Methods in Computer-Aided Design.

[24]  Aws Albarghouthi,et al.  Spatial Interpolants , 2015, ESOP.

[25]  Thomas Wies,et al.  Explaining inconsistent code , 2013, ESEC/FSE 2013.

[26]  Daniel Kroening,et al.  Interpolant Strength , 2010, VMCAI.

[27]  Edmund M. Clarke,et al.  Delta-Decidability over the Reals , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

[28]  Frank S. de Boer,et al.  Automated Verification of Recursive Programs with Pointers , 2012, IJCAR.

[29]  Kenneth L. McMillan.  Interpolants and Symbolic Model Checking , 2007, VMCAI.

[30]  Kenneth L. McMillan,et al.  An interpolating theorem prover , 2005, Theor. Comput. Sci..

[31]  William Craig,et al.  Linear reasoning. A new form of the Herbrand-Gentzen theorem , 1957, Journal of Symbolic Logic.

[32]  Klaus Weihrauch,et al.  A Tutorial on Computable Analysis , 2008 .

[33]  George E. Collins,et al.  Partial Cylindrical Algebraic Decomposition for Quantifier Elimination , 1991, J. Symb. Comput..

[34]  Thomas Wies,et al.  Error Invariants , 2012, FM.

[35]  Marsha Chechik,et al.  Whale: An Interpolation-Based Algorithm for Inter-procedural Verification , 2012, VMCAI.

[36]  Wei Chen,et al.  dReach: δ-Reachability Analysis for Hybrid Systems , 2015, TACAS.

[37]  Sriram Sankaranarayanan,et al.  Simulation-guided lyapunov analysis for hybrid dynamical systems , 2014, HSCC.

[38]  Kenneth L. McMillan,et al.  Interpolation and SAT-Based Model Checking , 2003, CAV.

[39]  Matthias Althoff,et al.  Online Verification of Automated Road Vehicles Using Reachability Analysis , 2014, IEEE Transactions on Robotics.

[40]  Peter Csaba Ölveczky,et al.  Designing and verifying distributed cyber-physical systems using Multirate PALS: An airplane turning control system case study , 2015, Sci. Comput. Program..

[41]  Alberto Griggio,et al.  The MathSAT5 SMT Solver , 2013, TACAS.

[42]  Kenneth L. McMillan,et al.  Generalizing DPLL to Richer Logics , 2009, CAV.

[43]  Xin Chen,et al.  Flow*: An Analyzer for Non-linear Hybrid Systems , 2013, CAV.