Deduction Modulo Real Algebra and Computer Algebra

We show how deductive, real algebraic, and computer algebraic methods can be combined for verifying hybrid systems in an automated theorem proving approach. In particular, we highlight the interaction of deductive and algebraic reasoning that is used for handling the joint discrete and continuous behaviour of hybrid systems. Systematically, we derive a canonical tableau procedure modulo from the calculus of differential dynamic logic. We delineate the nondeterminisms in the tableau procedure carefully and analyse their practical impact in the presence of computationally expensive handling of real algebraic constraints. Based on experience with larger case studies, we analyse proof strategies for dealing with the practical challenges for integrated algebraic and deductive verification of hybrid systems. To overcome the complexity pitfalls of integrating real arithmetic, we propose the iterative background closure and iterative inflation order strategies, with which we achieve substantial computational improvements.

[1]  George E. Collins,et al.  Interval Arithmetic in Cylindrical Algebraic Decomposition , 2002, J. Symb. Comput..

[2]  Martin Giese,et al.  Incremental Closure of Free Variable Tableaux , 2001, IJCAR.

[3]  Andrew Adams,et al.  Computer Algebra Meets Automated Theorem Proving: Integrating Maple and PVS , 2001, TPHOLs.

[4]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I: Introduction. Preliminaries. The Geometry of Semi-Algebraic Sets. The Decision Problem for the Existential Theory of the Reals , 1992, J. Symb. Comput..

[5]  Henny B. Sipma,et al.  Deductive Verification of Hybrid Systems Using STeP , 1998, HSCC.

[6]  Andrej Bauer,et al.  Analytica – An Experiment in Combining Theorem Proving and Symbolic Computation , 1996, Journal of Automated Reasoning.

[7]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[8]  Dima Grigoriev,et al.  Solving Systems of Polynomial Inequalities in Subexponential Time , 1988, J. Symb. Comput..

[9]  Philipp Rümmer,et al.  A Sequent Calculus for Integer Arithmetic with Counterexample Generation , 2007, VERIFY.

[10]  James H. Davenport,et al.  The complexity of quantifier elimination and cylindrical algebraic decomposition , 2007, ISSAC '07.

[11]  D. Grigor'ev.  Complexity of deciding Tarski algebra , 1988 .

[12]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part III: Quantifier Elimination , 1992, J. Symb. Comput..

[13]  Bruno Buchberger,et al.  A survey of the Theorema project , 1997, ISSAC.

[14]  D. Gabbay,et al.  Handbook of tableau methods , 1999 .

[15]  Melvin Fitting,et al.  First-Order Logic and Automated Theorem Proving , 1990, Graduate Texts in Computer Science.

[16]  Erika Ábrahám,et al.  Verification of hybrid systems: formalization and proof rules in PVS , 2001, Proceedings Seventh IEEE International Conference on Engineering of Complex Computer Systems.

[17]  George E. Collins,et al.  Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975, Automata Theory and Formal Languages.

[18]  Jean-Jacques Risler.  Some Aspects of Complexity in Real Algebraic Geometry , 1988, J. Symb. Comput..

[19]  Bernhard Beckert.  Equality and Other Theories , 1999 .

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

[21]  Rüdiger Loos,et al.  Applying Linear Quantifier Elimination , 1993, Comput. J..

[22]  Cesare Tinelli,et al.  Cooperation of Background Reasoners in Theory Reasoning by Residue Sharing , 2003, Journal of Automated Reasoning.

[23]  James H. Davenport,et al.  Real Quantifier Elimination is Doubly Exponential , 1988, J. Symb. Comput..

[24]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part II: The General Decision Problem. Preliminaries for Quantifier Elimination , 1992, J. Symb. Comput..

[25]  André Platzer,et al.  Combining Deduction and Algebraic Constraints for Hybrid System Analysis , 2007, VERIFY.

[26]  Claude Kirchner,et al.  Theorem Proving Modulo , 2003, Journal of Automated Reasoning.

[27]  Volker Weispfenning,et al.  The Complexity of Linear Problems in Fields , 1988, Journal of symbolic computation.