Abstract
In this paper, we propose an additional application area of the subset space semantics of modal logic in terms of cooperating agents. While the original conception reflects both the knowledge acquisition process and the accompanying topological effect for a single agent, we show how a slight extension of that system can be utilized for modeling agents which, in a strict sense, cooperate for knowledge. In so doing, the agents will come in by means of so-called effort functions. These functions shall represent those of the agents’ actions which are targeted at more knowledge of the whole group. Our investigations result in a particular multi-agent version of the well-known logic of subset spaces, which allows us to reason about qualitative aspects of cooperation like the dominance of a joint commitment over any individual effort. On the technical side, a soundness and completeness theorem for one of the logics arising from that will be proved.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
This is in accordance with the general setting in the context of subset spaces.
- 2.
The part this operator plays in the novel system will become apparent later.
- 3.
If the effort functions shall depend on knowledge states alone, then topological nexttime logic, see [9], would enter the field. This would lead to a somewhat more complicated but related system.
- 4.
In other words, the schema \((p \rightarrow \mathsf {C}_i p)\wedge (\mathsf {C}_i p\rightarrow p)\) is \(\mathsf {CALSS}_n\)-derivable.
- 5.
This point of view is derived by analogy with sequencing from the theory of parallel programming.
- 6.
One or another proof of such a kind can be found in the literature; see, as regards a fully completed version for \(\mathsf {LSS}\), [4]. Note that the special circumstances of each individual case require an appropriate adjustment, which is most often non-trivial.
References
Aiello, M., Pratt-Hartmann, I.E., van Benthem, J.F.A.K.: Handbook of Spatial Logics. Springer, Dordrecht (2007)
Balbiani, P., Ditmarsch, H., Kudinov, A.: Subset space logic with arbitrary announcements. In: Lodaya, K. (ed.) ICLA 2013. LNCS, vol. 7750, pp. 233–244. Springer, Heidelberg (2013). doi:10.1007/978-3-642-36039-8_21
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic, Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press, Cambridge (2001)
Dabrowski, A., Moss, L.S., Parikh, R.: Topological reasoning and the logic of knowledge. Ann. Pure Appl. Logic 78, 73–110 (1996)
Ditmarsch, H., Knight, S., Özgün, A.: Arbitrary announcements on topological subset spaces. In: Bulling, N. (ed.) EUMAS 2014. LNCS (LNAI), vol. 8953, pp. 252–266. Springer, Heidelberg (2015). doi:10.1007/978-3-319-17130-2_17
Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning about Knowledge. MIT Press, Cambridge (1995)
Georgatos, K.: Knowledge theoretic properties of topological spaces. In: Masuch, M., Pólos, L. (eds.) Logic at Work 1992. LNCS, vol. 808, pp. 147–159. Springer, Heidelberg (1994). doi:10.1007/3-540-58095-6_11
Goldblatt, R.: Logics of Time and Computation. CSLI Lecture Notes, 2nd edn., vol. 7. Center for the Study of Language and Information, Stanford (1992)
Heinemann, B.: Topological nexttime logic. In: Kracht, M., de Rijke, M., Wansing, H., Zakharyaschev, M. (eds.) Advances in Modal Logic 1, vol. 87, pp. 99–113. CSLI Publications, Kluwer, Stanford, CA (1998)
Heinemann, B.: A PDL-like logic of knowledge acquisition. In: Diekert, V., Volkov, M.V., Voronkov, A. (eds.) CSR 2007. LNCS, vol. 4649, pp. 146–157. Springer, Heidelberg (2007). doi:10.1007/978-3-540-74510-5_17
Heinemann, B.: Topology and knowledge of multiple agents. In: Geffner, H., Prada, R., Machado Alexandre, I., David, N. (eds.) IBERAMIA 2008. LNCS (LNAI), vol. 5290, pp. 1–10. Springer, Heidelberg (2008). doi:10.1007/978-3-540-88309-8_1
Heinemann, B.: Subset spaces modeling knowledge-competitive agents. In: Zhang, S., Wirsing, M., Zhang, Z. (eds.) KSEM 2015. LNCS (LNAI), vol. 9403, pp. 3–14. Springer, Heidelberg (2015). doi:10.1007/978-3-319-25159-2_1
Heinemann, B.: Augmenting subset spaces to cope with multi-agent knowledge. In: Artemov, S., Nerode, A. (eds.) LFCS 2016. LNCS, vol. 9537, pp. 130–145. Springer, Heidelberg (2016). doi:10.1007/978-3-319-27683-0_10
Heinemann, B.: Topological facets of the logic of subset spaces (with emphasis on canonical models). J. Logic Comput. (2016). 22 pp. doi:10.1093/logcom/exv087
Krommes, G.: A new proof of decidability for the modal logic of subset spaces. In: Ten Cate, B. (ed.) Proceedings of the Eighth ESSLLI Student Session, Vienna, Austria, pp. 137–147, August 2003
Meyer, J.J.C., van der Hoek, W.: Epistemic Logic for AI and Computer Science, Cambridge Tracts in Theoretical Computer Science, vol. 41. Cambridge University Press, Cambridge (1995)
Moss, L.S., Parikh, R.: Topological reasoning and the logic of knowledge. In: Moses, Y. (ed.) Theoretical Aspects of Reasoning about Knowledge (TARK 1992), pp. 95–105. Morgan Kaufmann, Los Altos (1992)
Wáng, Y.N., Ågotnes, T.: Multi-agent subset space logic. In: Proceedings 23rd IJCAI, pp. 1155–1161. AAAI (2013)
Wáng, Y.N., Ågotnes, T.: Subset space public announcement logic. In: Lodaya, K. (ed.) ICLA 2013. LNCS, vol. 7750, pp. 245–257. Springer, Heidelberg (2013). doi:10.1007/978-3-642-36039-8_22
Weiss, M.A., Parikh, R.: Completeness of certain bimodal logics for subset spaces. Stud. Logica 71, 1–30 (2002)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this paper
Cite this paper
Heinemann, B. (2016). A Subset Space Perspective on Agents Cooperating for Knowledge. In: Lehner, F., Fteimi, N. (eds) Knowledge Science, Engineering and Management. KSEM 2016. Lecture Notes in Computer Science(), vol 9983. Springer, Cham. https://doi.org/10.1007/978-3-319-47650-6_40
Download citation
DOI: https://doi.org/10.1007/978-3-319-47650-6_40
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-47649-0
Online ISBN: 978-3-319-47650-6
eBook Packages: Computer ScienceComputer Science (R0)