Case Study: Reachability and Scalability in a Unified Combat-Command-and-Control Model

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12448)


Reachability analysis computes an envelope encompassing the reachable states of a hybrid automaton within a given time horizon. It is known to be a computationally intensive task. In this case study paper, we consider the application of reachability analysis on a mathematical model unifying two key warfighting functions: Combat, and Command-and-Control (C2). Reachability here has a meaning of whether, given a range of initial combat forces and a C2 network and various uncertainties, one side can survive combat with intact forces while the adversary is diminished to zero. These are questions which arise in military Operations Research (OR). This paper is the first to utilize the notions of a hybrid automaton and reachability analysis in the area of OR. We explore the applicability and scalability of Taylor-model based reachability techniques in this domain. Our experiments demonstrate the potential of reachability analysis in the context of OR.


Hybrid automata Reachability analysis Operations research Combat Command and control 



The authors would like to thank Alexander C. Kalloniatis from Joint and Operations Analysis Division, Defence Science and Technology Group for many productive discussions.

This research was collaboration between the Commonwealth of Australia represented by the Defence Science and Technology Group and Australian National University, where this work was initiated, through a Defence Science Partnerships agreement. The research was conducted under the auspices of the Modelling Complex Warfighting initiative and was supported in part by the Air Force Office of Scientific Research under award number FA2386-17-1-4065. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the United States Air Force.


  1. 1.
  2. 2.
    Acebrón, J.A., Bonilla, L.L., Vicente, C.J.P., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Modern Phys. 77(1), 137 (2005)CrossRefGoogle Scholar
  3. 3.
    Ahern, R., Zuparic, M., Kalloniatis, A., Hoek, K.: Unifying warfighting functions in mathematical modelling: combat, Manoeuvre and C2. Submitted to Journal of the Operational research Society (JORS)Google Scholar
  4. 4.
    Althoff, M.: Reachability analysis and its application to the safety assessment of autonomous cars. Ph.D. thesis, Technische Universität München (2010)Google Scholar
  5. 5.
    Althoff, M.: Reachability analysis of nonlinear systems using conservative polynomialization and non-convex sets. In: Proceedings of the 16th International Conference on Hybrid Systems: Computation and Control, pp. 173–182. ACM (2013)Google Scholar
  6. 6.
    Alur, R., Courcoubetis, C., Henzinger, T.A., Ho, P.-H.: Hybrid automata: an algorithmic approach to the specification and verification of hybrid systems. In: Grossman, R.L., Nerode, A., Ravn, A.P., Rischel, H. (eds.) HS 1991-1992. LNCS, vol. 736, pp. 209–229. Springer, Heidelberg (1993). Scholar
  7. 7.
    Bak, S., Bogomolov, S., Henzinger, T.A., Johnson, T.T., Prakash, P.: Scalable static hybridization methods for analysis of nonlinear systems. In: 19th International Conference on Hybrid Systems: Computation and Control (HSCC 2016), pp. 155–164. ACMGoogle Scholar
  8. 8.
    Bansal, S., Chen, M., Herbert, S., Tomlin, C.J.: Hamilton-Jacobi reachability: a brief overview and recent advances. In: IEEE 56th Annual Conference on Decision and Control (CDC), pp. 2242–2253. IEEE (2017)Google Scholar
  9. 9.
    Benet, L., Sanders, D.: TaylorSeries.jl: Taylor expansions in one and several variables in Julia. J. Open Source Softw. 4, 1043 (2019)CrossRefGoogle Scholar
  10. 10.
    Benet, L., Sanders, D.P.: JuliaDiff/TaylorSeries.jl, March 2019.
  11. 11.
    Benet, L., Sanders, D.P.: JuliaIntervals/TaylorModels.jl, March 2019.
  12. 12.
    Berz, M., Makino, K.: Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models. Reliable Comput. 4(4), 361–369 (1998). Scholar
  13. 13.
    Bogomolov, S., et al.: Guided search for hybrid systems based on coarse-grained space abstractions. Int. J. Softw. Tools Tech. Trans. 18(4), 449–467 (2015). Scholar
  14. 14.
    Bogomolov, S., Forets, M., Frehse, G., Potomkin, K., Schilling, C.: JuliaReach: a toolbox for set-based reachability. In: 22nd ACM International Conference on Hybrid Systems: Computation and Control (HSCC 2019), pp. 39–44. ACM (2019)Google Scholar
  15. 15.
    Bogomolov, S., Mitrohin, C., Podelski, A.: Composing reachability analyses of hybrid systems for safety and stability. In: Bouajjani, A., Chin, W.-N. (eds.) ATVA 2010. LNCS, vol. 6252, pp. 67–81. Springer, Heidelberg (2010). Scholar
  16. 16.
    Bronski, J., deVille, L., Park, M.J.: Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model. Chaos 22(3), 033133 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Bünger, F.: Shrink wrapping for Taylor models revisited. Numer. Algorithms 78(4), 1001–1017 (2017). Scholar
  18. 18.
    Chen, X., Abraham, E., Sankaranarayanan, S.: Taylor model flowpipe construction for non-linear hybrid systems. In: IEEE 33rd Real-Time Systems Symposium, pp. 183–192. IEEE (2012)Google Scholar
  19. 19.
    Chen, X., Ábrahám, E., Sankaranarayanan, S.: Flow*: an analyzer for non-linear hybrid systems. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 258–263. Springer, Heidelberg (2013). Scholar
  20. 20.
    Chen, X., Sankaranarayanan, S.: Decomposed reachability analysis for nonlinear systems. In: IEEE Real-Time Systems Symposium (RTSS), pp. 13–24. IEEE (2016)Google Scholar
  21. 21.
    Dekker, A., Taylor, R.: Synchronization properties of trees in the Kuramoto model. SIAM J. Appl. Dyn. Sys. 12(2), 596–617 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    D’silva, V., Kroening, D., Weissenbacher, G.: A survey of automated techniques for formal software verification. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 27(7), 1165–1178 (2008)CrossRefGoogle Scholar
  23. 23.
    da Fonseca, J., Abud, C.: The Kuramoto model revisited. J. Stat. Mech: Theory Exp. 2018(10), 103204 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Frehse, G., et al.: SpaceEx: scalable verification of hybrid systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 379–395. Springer, Heidelberg (2011). Scholar
  25. 25.
    Girard, A., Guernic, C.L.: Efficient reachability analysis for linear systems using support functions. IFAC Proc. Vol. 41, 8966–8971 (2008)CrossRefGoogle Scholar
  26. 26.
    Gomez-Gardenes, J., Moreno, Y., Arenas, A.: Synchronizability determined by coupling strengths and topology on complex networks. Phys. Rev. E 75, 066106 (2007)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Gupta, A.: Formal hardware verification methods: a survey. Form Method Syst. Des. 1, 151–238 (1992). In: Computer-Aided Verification. pp. 5–92. Springer CrossRefGoogle Scholar
  28. 28.
    Hasík, J.: Beyond the briefing: theoretical and practical problems in the works and legacy of John Boyd. Contemp. Secur. Policy 34(3), 583–599 (2013)CrossRefGoogle Scholar
  29. 29.
    Hong, H., Choi, M.Y., Kim, B.J.: Synchronization on small-world networks. Phys. Rev. E 65(2), 026139 (2002)CrossRefGoogle Scholar
  30. 30.
    Ichinomiya, T.: Frequency synchronization in a random oscillator network. Phys. Rev. E 70(2), 026116 (2004)CrossRefGoogle Scholar
  31. 31.
    Immler, F., et al.: ARCH-COMP19 category report: Continuous and hybrid systems with nonlinear dynamics. In: ARCH19. 6th International Workshop on Applied Verification of Continuous and Hybrid Systemsi, part of CPS-IoT Week 2019, Montreal, QC, Canada, pp. 41–61 (2019)Google Scholar
  32. 32.
    Immler, F., et al.: ARCH-COMP19 category report: continuous and hybrid systems with nonlinear dynamics. EPiC Ser. Comput. 61, 41–61 (2019)CrossRefGoogle Scholar
  33. 33.
    Joldes, M.M.: Rigorous polynomial approximations and applications. Ph.D. thesis (2011)Google Scholar
  34. 34.
    Kalloniatis, A., Hoek, K., Zuparic, M.: Network synchronisation and next generation combat models - a dynamical systems approach. In: 86th Military Operations Research Society Symposium (2018)Google Scholar
  35. 35.
    Kalloniatis, A., McLennan-Smith, T., Roberts, D.: Modelling distributed decision-making in command and control using stochastic network synchronisation. Eur. J. Oper. Res. (2020). Scholar
  36. 36.
    Kuramoto, Y.: International Symposium on Mathematical Problems in Theoretical Physics. Lecture Notes in Physics, p. 420. Springer, Heidelberg (1975). Scholar
  37. 37.
    Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Courier Corporation (2003)Google Scholar
  38. 38.
    Lanchester, F.W.: Aircraft in Warfare: The Dawn of the Fourth Arm. Constable limited (1916)Google Scholar
  39. 39.
    Leavitt, H.J.: Some effects of certain communication patterns on group performance. J. Abnorm. Soc. Psychol. 46(1), 38–50 (1951)CrossRefGoogle Scholar
  40. 40.
    Makino, K., Berz, M.: Taylor models and other validated functional inclusion methods. Int. J. Pure Appl. Math. 6, 239–316 (2003)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Meyer, P.J., Devonport, A., Arcak, M.: Tira: toolbox for interval reachability analysis. In: Proceedings of the 22nd ACM International Conference on Hybrid Systems: Computation and Control, pp. 224–229. ACM (2019)Google Scholar
  42. 42.
    Mitchell, I.M.: Comparing forward and backward reachability as tools for safety analysis. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 428–443. Springer, Heidelberg (2007). Scholar
  43. 43.
    Morse, P., Kimball, G.: Methods of Operations Research. Massachusetts Institute of Technology (1951)Google Scholar
  44. 44.
    Nedialkov, N.S.: Interval tools for ODEs and DAEs. In: 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006), p. 4. IEEE (2006)Google Scholar
  45. 45.
    Osinga, F.: “Getting” a discourse on winning and losing: a primer on Boyd’s “theory of intellectual evolution”. Contemp. Secur. Policy 34(3), 603–624 (2013)CrossRefGoogle Scholar
  46. 46.
    Pérez-Hernández, J.A., Benet, L.: Perezhz/taylorintegration.jl, February 2019.
  47. 47.
    Ray, R., Gurung, A., Das, B., Bartocci, E., Bogomolov, S., Grosu, R.: XSpeed: accelerating reachability analysis on multi-core processors. In: Piterman, N. (ed.) HVC 2015. LNCS, vol. 9434, pp. 3–18. Springer, Cham (2015). Scholar
  48. 48.
    Rogge, J.A., Aeyals, D.: Stability of phase locking in a ring of unidirectionally coupled oscillators. SIAM J. Appl. Dyn. Syst. 37, 11135–11148 (2004)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Rwth, X.C., Sankaranarayanan, S., Ábrahám, E.: Under-approximate flowpipes for non-linear continuous systems. In: Formal Methods in Computer-Aided Design (FMCAD), pp. 59–66. IEEE (2014)Google Scholar
  50. 50.
    Tam, J.H.: Application of Lanchester combat model in the Ardennes campaign. Nat. Resour. Model. 11(2), 95–116 (1998)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Newcastle UniversityNewcastle upon TyneUK
  2. 2.Australian National UniversityCanberraAustralia
  3. 3.Universidad de la RepúblicaMontevideoUruguay

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