Abstract
Subsequently, a particular extension of the bi-modal logic of subset spaces, LSS, to the case of several agents will be provided. The basic system, which originally was designed for revealing the intrinsic relationship between knowledge and topology, has been developed in several directions in recent years, not least towards a comprehensive knowledge-theoretic formalism. However, while subset spaces have been shown to be smoothly combinable with various epistemic concepts in the single-agent case, adjusting them to general multi-agent scenarios has brought about rather unsatisfactory results up to now. This is due to reasons inherent in the system so that one is led to consider more special cases. In the present paper, such a widening of LSS to the multi-agent setting is proposed. The peculiarity is here given by the case that the agents are supplied with certain knowledge-enabling functions allowing, in particular, for comparing their respective knowledge. It turns out that such circumstances can be modeled in corresponding logical terms to a considerable extent.
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
Notes
- 1.
If those knowledge-enabling functions shall depend on knowledge states alone, which is clearly worthy of discussion, then topological nexttime logic, see [8], would enter the field. This would lead to a somewhat more complicated but related system.
- 2.
See footnote 1 above for an alternative way of modeling.
- 3.
That is to say, the necessitation rule for each of the \(\mathsf {C}_i\)’s is added to the proof rules for \(\mathsf {LSS}\).
- 4.
One or another proof of that kind can be found in the literature; see, e.g., [4] for a fully completed proof regarding \(\mathsf {LSS}\) and [9] for a particular multi-agent variation. Thus, it is really sufficient to confine ourselves to the case-specific issues here (which are quite difficult enough in themselves).
- 5.
Note that such a proof is necessary here, since a Sahlqvist argument is insufficient because \(\mathsf {ALSS}\) is a non-normal logic.
References
Aiello, M., Pratt-Hartmann, I.E., van Benthem, J.F.A.K.: Handbook of Spatial Logics. Springer, Dordrecht (2007)
Balbiani, P., van Ditmarsch, H., Kudinov, A.: Subset space logic with arbitrary announcements. In: Lodaya, K. (ed.) Logic and Its Applications. LNCS, vol. 7750, pp. 233–244. Springer, Heidelberg (2013)
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. Log. 78, 73–110 (1996)
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.) Knowledge Representation and Uncertainty, Logic at Work. Lecture Notes in Artificial Intelligence, vol. 808, pp. 147–159. Springer, Heidelberg (1994)
Goldblatt, R.: Logics of Time and Computation. CSLI Lecture Notes, vol. 7, 2nd edn. 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. CSLI Publications, vol. 87, pp. 99–113. Kluwer, Stanford (1998)
Heinemann, B.: Logics for multi-subset spaces. J. Appl. Non-Class. Log. 20(3), 219–240 (2010)
Heinemann, B.: Coming upon the classic notion of implicit knowledge again. In: Buchmann, R., Kifor, C.V., Yu, J. (eds.) KSEM 2014. LNCS, vol. 8793, pp. 1–12. Springer, Heidelberg (2014)
Heinemann, B.: Subset spaces modeling knowledge-competitive agents. In: Zhang, S., Wirsing, M., Zhang, Z. (eds.) KSEM 2015. LNCS, vol. 9403, pp. 1–12. Springer, Heidelberg (2015). doi:10.1007/978-3-319-25159-2_1
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, pp. 137–147. Austria, Vienna, August 2003
Lomuscio, A., Ryan, M.: Ideal agents sharing (some!) knowledge. In: Prade, H. (ed.) ECAI 1998, 13th European Conference on Artificial Intelligence, pp. 557–561. John Wiley & Sons Ltd, Chichester (1998)
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, CA (1992)
Wáng, Y.N., Ågotnes, T.: Subset space public announcement logic. In: Lodaya, K. (ed.) Logic and Its Applications. LNCS, vol. 7750, pp. 245–257. Springer, Heidelberg (2013)
Acknowledgement
I would like to take this opportunity to thank the anonymous referees very much for their detailed reviews which, among other things, contain valuable comments on the system presented here as well as suggestions for alternative approaches to multi-agent subset spaces being worth considering.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Heinemann, B. (2016). Augmenting Subset Spaces to Cope with Multi-agent Knowledge. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2016. Lecture Notes in Computer Science(), vol 9537. Springer, Cham. https://doi.org/10.1007/978-3-319-27683-0_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-27683-0_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-27682-3
Online ISBN: 978-3-319-27683-0
eBook Packages: Computer ScienceComputer Science (R0)