Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Synthesizing switching logic using constraint solving

  • 169 Accesses

  • 12 Citations


For a system that can operate in multiple different modes, we define the switching logic synthesis problem as follows: given a description of the dynamics in each mode of the system, find the conditions for switching between the modes so that the resulting system satisfies some desired properties. In this paper, we present an approach for solving the switching logic synthesis problem in the case when (1) the dynamics in each mode of the system are given using differential equations and, hence, the synthesized system is a hybrid system, and (2) the desired property is a safety property. Our approach for solving the switching logic synthesis problem, called the constraint-based approach, consists of two steps. In the first constraint generation step, the synthesis problem is reduced to satisfiability of a quantified formula over the theory of reals. In the second constraint solving step, the quantified formula is solved. This paper focuses on constraint generation. The constraint generation step is based on the concept of a controlled inductive invariant. The search for controlled inductive invariant is cast as a constraint solving problem. The controlled inductive invariant is then used to arrive at the maximally liberal switching logic. We prove that the synthesized switching logic always gives us a well-formed and safe hybrid system. When the system, the safety property, and the controlled inductive invariant are all expressed only using polynomials, the generated constraint is an \({\exists\forall}\) formula in the theory of reals, whose satisfiability is decidable.

This is a preview of subscription content, log in to check access.


  1. 1

    Alur R., Courcoubetis C., Halbwachs N., Henzinger T.A., Ho P.-H., Nicollin X., Olivero A., Sifakis J., Yovine S.: The algorithmic analysis of hybrid systems. Theor. Comput. Sci. 138(3), 3–34 (1995)

  2. 2

    Asarin E., Bournez O., Dang T., Maler O., Pnueli A.: Effective synthesis of switching controllers for linear systems. Proc. IEEE 88(7), 1011–1025 (2000)

  3. 3

    Blanchini F.: Set invariance in control. Automatica 35, 1747–1767 (1999)

  4. 4

    Burns K., Gidea M.: Differential Geometry and Topology: With a view to dynamical systems. Chapman & Hall, London (2005)

  5. 5

    Chaudhuri, S., Solar-Lezama, A.: Smooth interpretation. In: ACM Conference on Programming Language Design and Implementation PLDI (2010)

  6. 6

    Colón, M.: Schema-guided synthesis of imperative programs by constraint solving. In: LOPSTR, pp. 166–181 (2004)

  7. 7

    Cury J., Krogh B., Niinomi T.: Supervisory controllers for hybrid systems based on approximating automata. IEEE Trans. Aut. Control 43, 564–568 (1998)

  8. 8

    Gulwani, S., Srivastava, S., Venkatesan, R.: Program analysis as constraint solving. In: Proceedings of ACM Conference on Programming Language Design and Implementation PLDI, pp. 281–292 (2008)

  9. 9

    Gulwani, S., Tiwari, A.: Constraint-based approach for analysis of hybrid systems. In: CAV, volume 5123 of LNCS, pp. 190–203. Springer (2008)

  10. 10

    Hong, H.: Quantifier elimination procedure by cylindrical algebraic decomposition (1995).,

  11. 11

    Jha, S., Gulwani, S., Seshia, S., Tiwari, A.: Synthesizing switching logic for safety and dwell-time requirements. In: ACM/IEEE International Conference on Cyber-Physical Systems, ICCPS (2010)

  12. 12

    Koo, T., Sastry, S.: Mode switching synthesis for reachability specification. In: Proceedings of HSCC 2001, LNCS 2034, pp. 333–346 (2001)

  13. 13

    Liberzon D., Morse A.S.: Benchmark problems in stability and design of switched systems. IEEE Control Syst. Mag. 19, 59–70 (1999)

  14. 14

    Lustig, Y., Vardi, M.: Synthesis from component libraries. In: Proc. FoSSaCS, pp. 395–409 (2009)

  15. 15

    Manna Z., Waldinger R.: A deductive approach to program synthesis. ACM TOPLAS 2(1), 90–121 (1980)

  16. 16

    Manon, P., Valentin-Roubinet, C.: Controller synthesis for hybrid systems with linear vector fields. In: Proceedings of IEEE Symposium on Intell. Control, pp. 17–22 (1999)

  17. 17

    Moor, T., Raisch, J.: Discrete control of switched linear systems. In: Proceedings of European Control Conference on ECC’99 (1999)

  18. 18

    Platzer A.: Differential-algebraic dynamic logic for differential-algebraic programs. J. Log. Comput. 20(1), 309–352 (2010) Advance Access published on (November 18 2008)

  19. 19

    Prajna, S., Jadbabaie, A.: Safety verification of hybrid systems using barrier certificates. In: Proceedings of HSCC, volume 2993 of LNCS, pp. 477–492 (2004)

  20. 20

    Prajna, S., Jadbabaie, A., Pappas, G.: A framework for worst-case and stochastic safety verification using barrier certificates. IEEE Trans. Automat. Contr. 52(8) (2007)

  21. 21

    Sankaranarayanan, S., Sipma, H., Manna, Z.: Constructing invariants for hybrid systems. In: Proceedings of HSCC, volume 2993 of LNCS, pp. 539–554 (2004)

  22. 22

    Shapiro E.Y.: Algorithmic Program DeBugging. MIT Press, Cambridge (1983)

  23. 23

    Solar-Lezama, A., Tancau, L., Bodík, R., Seshia, S., Saraswat, V.: Combinatorial sketching for finite programs. In: ASPLOS (2006)

  24. 24

    Taly, A., Gulwani, S., Tiwari, A.: Synthesizing switching logic using constraint solving. In: Proceedings of 10th International Conference on Verification, Model Checking and Abstract Interpretation, VMCAI, volume 5403 of LNCS, pp. 305–319. Springer (2009)

  25. 25

    Taly, A., Tiwari, A.: Deductive verification of continuous dynamical systems. In: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2009), volume 4 of Leibniz International Proceedings in Informatics (LIPIcs), pp. 383–394. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2009)

  26. 26

    Tarski A.: A Decision Method for Elementary Algebra and Geometry. 2nd edn. University of California Press, California (1948)

  27. 27

    Tomlin C., Lygeros L., Sastry S.: A game-theoretic approach to controller design for hybrid systems. Proc. IEEE 88(7), 949–970 (2000)

Download references

Author information

Correspondence to Ashish Tiwari.

Additional information

Research supported in part by the National Science Foundation under grants CNS-0720721, CSR-EHCS-0834810, CSR-0917398 and CCF-1017483 and by NASA under Grant NNX08AB95A. Work done when the first author was visiting SRI International

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Taly, A., Gulwani, S. & Tiwari, A. Synthesizing switching logic using constraint solving. Int J Softw Tools Technol Transfer 13, 519–535 (2011).

Download citation


  • Formal methods
  • Controller synthesis
  • Hybrid systems