Algebraic Invariants for Linear Hybrid Automata

We exhibit an algorithm to compute the strongest algebraic (or polynomial) invariants that hold at each location of a given unguarded linear hybrid automaton (i.e., a hybrid automaton having only unguarded transitions, all of whose assignments are given by affine expressions, and all of whose continuous dynamics are given by linear differential equations). Our main tool is a control-theoretic result of independent interest: given such a linear hybrid automaton, we show how to discretise the continuous dynamics in such a way that the resulting automaton has precisely the same algebraic invariants.

[1]  Deepak Kapur,et al.  When Is a Formula a Loop Invariant? , 2015, Logic, Rewriting, and Concurrency.

[2]  Sriram Sankaranarayanan,et al.  Automatic invariant generation for hybrid systems using ideal fixed points , 2010, HSCC '10.

[3]  Enric Rodríguez-Carbonell,et al.  Generating all polynomial invariants in simple loops , 2007, J. Symb. Comput..

[4]  Paul Pettersson,et al.  Tools and Algorithms for the Construction and Analysis of Systems: 28th International Conference, TACAS 2022, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022, Munich, Germany, April 2–7, 2022, Proceedings, Part II , 1998, TACAS.

[5]  Zachary Kincaid,et al.  Non-linear reasoning for invariant synthesis , 2017, Proc. ACM Program. Lang..

[6]  André Platzer,et al.  Characterizing Algebraic Invariants by Differential Radical Invariants , 2014, TACAS.

[7]  Gerardo Lafferriere,et al.  Symbolic Reachability Computation for Families of Linear Vector Fields , 2001, J. Symb. Comput..

[8]  Ashish Tiwari,et al.  Generating Polynomial Invariants for Hybrid Systems , 2005, HSCC.

[9]  Henny B. Sipma,et al.  Constructing invariants for hybrid systems , 2008, Formal Methods Syst. Des..

[10]  Thomas A. Henzinger,et al.  The theory of hybrid automata , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

[11]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

[12]  Pascal Koiran,et al.  Quantum automata and algebraic groups , 2005, J. Symb. Comput..

[13]  Enric Rodríguez-Carbonell,et al.  Automatic generation of polynomial invariants of bounded degree using abstract interpretation , 2007, Sci. Comput. Program..

[14]  Hirokazu Anai,et al.  Reach Set Computations Using Real Quantifier Elimination , 2001, HSCC.

[15]  Joël Ouaknine,et al.  Polynomial Invariants for Affine Programs , 2018, LICS.

[16]  Michele Boreale.  Complete Algorithms for Algebraic Strongest Postconditions and Weakest Preconditions in Polynomial ODE'S , 2018, SOFSEM.