Autonomous Agents and Multi-Agent Systems

, Volume 30, Issue 5, pp 793–825 | Cite as

Big Brother Logic: visual-epistemic reasoning in stationary multi-agent systems

  • Olivier Gasquet
  • Valentin Goranko
  • François Schwarzentruber


We consider multi-agent scenarios where each agent controls a surveillance camera in the plane, with fixed position and angle of vision, but rotating freely. The agents can thus observe the surroundings and each other. They can also reason about each other’s observation abilities and knowledge derived from these observations. We introduce suitable logical languages for reasoning about such scenarios which involve atomic formulae stating what agents can see, multi-agent epistemic operators for individual, distributed and common knowledge, as well as dynamic operators reflecting the ability of cameras to turn around in order to reach positions satisfying formulae in the language. We also consider effects of public announcements. We introduce several different but equivalent versions of the semantics for these languages, discuss their expressiveness and provide translations in PDL style. Using these translations we develop algorithms and obtain complexity results for model checking and satisfiability testing for the basic logic BBL that we introduce here and for some of its extensions. Notably, we show that even for the extension with common knowledge, model checking and satisfiability testing remain in PSPACE. We also discuss the sensitivity of the set of validities to the admissible angles of vision of the agents’ cameras. Finally, we discuss some further extensions: adding obstacles, positioning the cameras in 3D or enabling them to change positions. Our work has potential applications to automated reasoning, formal specification and verification of observational abilities and knowledge of multi-robot systems.


Visual-epistemic logical reasoning Multi-agent systems Surveillance cameras Observational abilities Knowledge Model checking Satisfiability testing 



This work has been partially supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02. The work of Valentin Goranko was partially supported by the CIMI (Centre International de Mathematiques et d’Informatique) Excellence Program during his visit to Toulouse as a scientific expert. We thank the anonymous reviewers for the careful reading and valuable comments and criticism which contributed to the improvement of the paper.


  1. 1.
    Balbiani, P., Gasquet, O., & Schwarzentruber, F. (2013). Agents that look at one another. Logic Journal of IGPL, 21(3), 438–467.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Balbiani, P., Goranko, V., Kellerman, R., & Vakarelov, D. (2007). Logical theories of fragments of elementary geometry. In M. Aiello, J. van Benthem, & I. Pratt-Hartmann (Eds.), Handbook of spatial logics (pp. 343–428). Heidelberg: Springer.CrossRefGoogle Scholar
  3. 3.
    Balbiani, P., Van Ditmarsch, H., Herzig, A., & De Lima, T. (2010). Tableaux for public announcement logic. Journal of Logic and Computation, 20(1), 55–76.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ben-Or, M., Kozen, D., & Reif, J. (1986). The complexity of elementary algebra and geometry. Journal of Computer and System Sciences, 32(2), 251–264.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal logic. Cambridge: Cambridge University Press.CrossRefzbMATHGoogle Scholar
  6. 6.
    Bustamante, A. L., Molina, J. M., & Patricio, M. A. (2010). Multi-camera and multi-modal sensor fusion, an architecture overview. In Proceedings of the DCAI’2010 (pp. 301–308).Google Scholar
  7. 7.
    Canny, J. (1988). Some algebraic and geometric computations in PSPACE. In Proceedings of STOC’88 (pp. 460–467). New York: ACM.Google Scholar
  8. 8.
    Fagin, R., Halpern, J., Moses, Y., & Vardi, M. (1995). Reasoning about knowledge. Cambridge: MIT Press.zbMATHGoogle Scholar
  9. 9.
    García, J., Carbó, J., & Molina, J. M. (2005). Agent-based coordination of cameras. International Journal of Computer Science & Applications, 2(1), 33–37.Google Scholar
  10. 10.
    Gasquet, O., Goranko, V., & Schwarzentruber, F. (2014). Big brother logic: Logical modeling and reasoning about agents equipped with surveillance cameras in the plane. In Proceedings of the AAMAS’2014 (pp. 325–332)Google Scholar
  11. 11.
    Goranko, V., Merker, M., & Thomassen, C. (2014). Directed graphs with restricted angles of vision. Manuscript.Google Scholar
  12. 12.
    Harel, D., Kozen, D., & Tiuryn, J. (2000). Dynamic logic. Cambridge: MIT Press.zbMATHGoogle Scholar
  13. 13.
    Plaza, J. (2007). Logics of public communications. Synthese, 158(2), 165–179.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Schwarzentruber, F. (2011). Seeing, knowledge and common knowledge. In Logic, rationality, and interaction (pp. 258–271). Berlin: Springer.Google Scholar
  15. 15.
    Sipser, M. (2006). Introduction to the theory of computation (Vol. 2). Boston: Thomson Course Technology.zbMATHGoogle Scholar
  16. 16.
    Tarski, A. (1951). A decision method for elementary algebra and geometry. Berlin: Springer.zbMATHGoogle Scholar
  17. 17.
    van Ditmarsch, H., van der Hoek, W., & Kooi, B. (2008). Dynamic epistemic logic. Dordecht: Springer.zbMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.IRITUniversité Paul SabatierToulouseFrance
  2. 2.Stockholm UniversityStockholmSweden
  3. 3.University of JohannesburgJohannesburgSouth Africa
  4. 4.IRISAENS RennesBruzFrance

Personalised recommendations