Abstract
The notion of reactive graph generalises the one of graph by allowing the base accessibility relation to change when its edges are traversed. Can we represent these more general structures using points and arrows? We prove this can be done by introducing higher order arrows: the switches. The possibility of expressing the dependency of the future states of the accessibility relation on individual transitions by the use of higher-order relations, that is, coding meta-relational concepts by means of relations, strongly suggests the use of modal languages to reason directly about these structures. We introduce a hybrid modal logic for this purpose and prove its completeness.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Areces, C.: Logic engineering. The case of description and hybrid logics. PhD thesis, Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherlands (2000)
Areces, C., Blackburn, P., Marx, M.: A road-map on complexity for hybrid logics. In: Proceedings of the 13th International Workshop and 8th Annual Conference of the EACSL on Computer Science Logic, CSL ’99, pp. 307–321. Springer, London (1999)
Areces, C., Blackburn, P., Delany, S.R.: Bringing them all together. J. Log. Comput. 11, 657–669 (2001)
Areces, C., Carreiro, F., Figueira, S., Mera, S.: Basic model theory for memory logics. In: WoLLIC, pp. 20–34 (2011)
Barringer, H., Gabbay, D.M.: Modal and temporal argumentation networks. In: Essays in Memory of Amir Pnueli, pp. 1–25 (2010)
Blackburn, P.: Representation, reasoning, and relational structures: a hybrid logic manifesto. Log. J. IGPL 8(3), 339–365 (2000)
Blackburn, P., ten Cate, B.: Pure extensions, proof rules, and hybrid axiomatics. Stud. Log. 84(2), 277–322 (2006)
Blackburn, P., Tzakova, M.: Hybrid languages and temporal logic. Log. J. IGPL 7(1), 27–54 (1999)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)
Blackburn, P., van Benthem, J.F.A.K.,Wolter, F.: Handbook of modal logic. In: Studies in Logic and Practical Reasoning, vol. 3. Elsevier Science, New York (2006)
Bull, R.A.: An approach to tense logic. Theoria 36(3), 282–300 (1970)
Crochemore, M., Gabbay, D.M.: Reactive automata. Inf. Comput. 209(4), 692–704 (2011)
Gabbay, D.M.: A theory of hypermodal logics: mode shifting in modal logic. J. Philos. Logic 31(3), 211–243 (2002)
Gabbay, D.M.: Reactive Kripke semantics and arc accessibility. In: Carnielli, W., Dionisio, F.M., Mateus, P. (eds) Proceedings of CombLog 2004, pp. 7–20. Centre for Logic and Computation, University of Lisbon (2004)
Gabbay, D.M.: Introducing reactive Kripke semantics and arc accessibility. In: Pillars of Computer Science, pp. 292–341 (2008)
Gabbay, D.M.: Reactive Kripke models and contrary to duty obligations. In: van der Meyden, R., van der Torre, L. (eds.) DEON, Lecture Notes in Computer Science, vol. 5076, pp. 155–173. Springer, Berlin (2008)
Gabbay, D.M.: Reactive intuitionistic tableaux. Synthese 179(2), 253–269 (2011). doi:10.1007/s11229-010-9781-8
Gabbay, D.M., Kurucz, A., Wolter, F., Zakharyaschev, M.: Many-dimensional modal logics: theory and applications. In: Studies in Logic and the Foundations of Mathematics, vol. 148. Elsevier, Amsterdam (2003)
Gabbay, D.M., Marcelino, S.: Modal logics of reactive frames. Stud. Log. 93(2–3), 405–446 (2009)
Gabbay, D.M., Schlechta, K.: Defeasible inheritance systems and reactive diagrams. Log. J. IGPL 17(1), 1–54 (2009)
Gabbay, D.M., Schlechta, K.: Reactive preferential structures and nonmonotonic consequence. Rev. Symb. Log. 2(2), 414–450 (2009)
Gabbay, D.M., Schlechta, K.: An analysis of defeasible inheritance systems. In: Logical Tools for Handling Change in Agent-Based Systems, Cognitive Technologies, pp. 251–293. Springer, Berlin (2010)
Gabbay, D.M., Schlechta, K.: A theory of hierarchical consequence and conditionals. J. Logic, Lang. Inf. 19, 3–32 (2010)
Gargov, G., Goranko, V.: Modal logic with names. J. Philos. Logic 22, 607–636 (1993). doi:10.1007/BF01054038
Harel, D., Pnueli, A.: On the Development of Reactive Systems, pp. 477–498. Springer, New York (1985)
Huth, M., Ryan, M.: Logic in Computer Science: Modelling and Reasoning About Systems (1999)
Marcelino, S.: Modal logic for changing systems. Ph.D. thesis, King’s College London (2011)
Modgil, S.: Reasoning about preferences in argumentation frameworks. Artif. Intell. 173, 901–934 (2009)
Prior, A.N.: Modality and quantification in S5. J. Symb. Log. 21(1), 60–62 (1956)
Prior, A.: Past, Present and Future. Oxford University Press, Oxford (1967)
Sattler, U., Vardi, M.: The hybrid μ-calculus. In: Proceedings of IJCAR’01, Siena (2001)
van Benthem, J.: An essay on sabotage and obstruction. In: Hutter, D., Stephan, W. (eds.) Mechanizing Mathematical Reasoning, Lecture Notes in Computer Science, vol. 2605, pp. 268–276. Springer, Berlin (2005)
van Benthem, J.: Dynamic logic for belief revision. J. Appl. Non-Class. Log. 17, 129–155 (2007)
Zanardo, A.: Branching-time logic with quantification over branches: the point of view of modal logic. J. Symb. Logic 61(1), 1–39 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author Sérgio Marcelino thanks the support of FCT and EU FEDER, via the postdoc scholarship SFRH/BPD/76513/2011, the project FCT PEst-OE/EEI/LA0008/2011 of IT, the FP7-PEOPLE-2012-IRSES GetFun Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme, as well as the PQDR initiative of SGIQ.
Rights and permissions
About this article
Cite this article
Gabbay, D., Marcelino, S. Global view on reactivity: switch graphs and their logics. Ann Math Artif Intell 66, 131–162 (2012). https://doi.org/10.1007/s10472-012-9316-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10472-012-9316-8