Abstract
In the literature, different axiomatizations of Public Announcement Logic (PAL) have been proposed. Most of these axiomatizations share a “core set” of the so-called “reduction axioms”. In this paper, by designing non-standard Kripke semantics for the language of PAL, we show that the proof system based on this core set of axioms does not completely axiomatize PAL without additional axioms and rules. In fact, many of the intuitive axioms and rules we took for granted could not be derived from the core set. Moreover, we also propose and advocate an alternative yet meaningful axiomatization of PAL without the reduction axioms. The completeness is proved directly by a detour method using the canonical model where announcements are treated as merely labels for modalities as in normal modal logics. This new axiomatization and its completeness proof may sharpen our understanding of PAL and can be adapted to other dynamic epistemic logics.
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Notes
For the simplicity of the exposition, we only consider the single agent case in this paper, all of our results and techniques apply to the multi-agent case as well.
Note that the usual rule of replacement of logical equivalents as in Plaza (1989) and many other works is stronger than our RE in the sense that it is not restricted to the non-modality occurrences and can be viewed as a combination of our RE and !RE rules. The separation of RE and !RE helps us to pinpoint exactly the rules that are needed to make PA complete.
Due to the connection with the recursive definition of the syntactic relativization, van Benthem (2011) advocates the name “recursion axioms” than the reduction axioms, since the reduction may not be the main goal. In this paper we stick to the usual name.
PALwith common knowledge operator is such an example (van Ditmarsch et al. 2007).
Gerbrandy (1999) mentioned that he has to abandon NEC! in order to cope with private updates which do not preserve S5 frame properties. Therefore the usual canonical model method does not work any more in the absence of \(\mathtt{NEC}\mathrm{!}\).
Here we say an inference rule \(\dfrac{\phi }{\psi }\) is derivable from a system \(\mathbf S \) if \(\psi \) can be derived by using \(\phi \), the axiom schemata and inference rules of \(\mathbf S \). An inference rule is admissible in S if the set of theorems stays the same when this rule is added to S. Given a system, a derivable rule is clearly admissible but an admissible rule may not be derivable.
We need to adapt the proof just a little bit to fit !K in the proof instead of \(\mathtt{!K}^\prime \) used in van Ditmarsch et al. (2007).
van Benthem (2007) views the reduction axioms as postulates of abstract updates thus opens a new kind of correspondence study in modal logic: between axioms and updates.
In some dynamic epistemic logics, such as van Benthem et al. (2006), the valuation of basic propositions can also be updated in a systematic way based on the previous valuation. However, this does not change the picture dramatically. Such logics with factual changes also validate axioms similar to INV thus still suffering from the loss of uniform substitution.
We conjecture that weakening PFUNC to \(\langle {\psi }\rangle \top \leftrightarrow \psi \) is enough to make the system complete. The functionality property can be guaranteed under the presence of INV, NM, and PR.
Recall the famous muddy children example, cf. e.g., van Ditmarsch et al. (2007).
See Goldblatt and Jackson (2012) for a more detailed discussion on the reason for the undecidability.
The semantics for PAL as in Gerbrandy and Groeneveld (1997) is based on “possibilities” which are essentially bisimulation classes of pointed Kripke models. The public/private announcement operator is defined as a function mapping one possibility to another by essentially deleting epistemic relations, thus every announcement is executable. Essentially, the formulas are interpreted in a “universal” model where each point stands for a class of Kripke models.
In the temporal epistemic setting based on linear temporal logic, the axiom is usually presented as \(\Diamond \bigcirc \phi \rightarrow \bigcirc \Diamond \phi \) where \(\bigcirc \) is the next moment operator. In such a setting, there is no difference between the “box” and “diamond” forms of the \(\bigcirc \) operator since it is assumed that there is always a unique next moment.
Halpern et al. (2004) discussed the general properties of perfect recall and no learning, which can be simplified significantly in the setting of synchronous systems.
The best way to understand this is by looking at the “box versions” of these two axioms: \([a]\square \phi \rightarrow \square [a]\phi \) and \(\square [a]\phi \rightarrow [a]\square \phi \).
The first author conjectures that this difference may require new techniques in axiomatizing ETL (with fixed point operators) on structures with properties like no miracles.
Based on this result, Proposition 3 in van Benthem et al. (2009) also gives a characterization of PAL-generated ETL models where the uniform no miracles can be reduced to no miracles.
Note that this semantics is very similar to the first non-standard semantics given in Sect. 3 except the clause for \([\psi ]\phi \). Here we use the context accumulation inspired by the composition axiom.
For example, Liu (2008) discussed memory-less agents.
We have shown two general ways to reduce PAL to EL: “inside-out” (by using RE) and “outside-in” (by using !COMP). The composition axiom plays an important role when RE is not available as we have shown in Theorem 24. In some other cases, the composition may not be possible but RE is available cf. e.g., van Benthem and Minică (2009).
References
Aucher, G., & Herzig, A. (2011). Dynamic formal epistemology, Chap. Exploring the power of converse events. Berlin: Springer.
Baltag, A., & Moss, L. (2004). Logics for epistemic programs. Synthese, 139(2), 165–224.
Baltag, A., Moss, L., & Solecki, S. (1998). The logic of public announcements, common knowledge, and private suspicions. In Proceedings of TARK ’98 (pp. 43–56). Los Altos, CA: Morgan Kaufmann Publishers.
Blackburn, P., de Rijke, M., & Venema, Y. (2002). Modal logic. Cambridge: Cambridge University Press.
Bonnay, D., & Égré, P. (2009). Inexact knowledge with introspection. Journal of Philosophical Logic, 38(2), 179–227.
Dégremont, C., Löwe, B., & Witzel, A. (2011). The synchronicity of dynamic epistemic logic. In TARK2011, Groningen.
Fagin, R., Halpern, J., Moses, Y., & Vardi, M. (1995). Reasoning about knowledge. Cambridge, MA: MIT Press.
Gabbay, D. M. (2002). A theory of hypermodal logics: mode shifting in modal logic. Journal of Philosophical Logic, 31(3), 211–243.
Gerbrandy, J. (1999). Bisimulations on Planet Kripke. Ph.D. thesis, University of Amsterdam.
Gerbrandy, J., & Groeneveld, W. (1997). Reasoning about information change. Journal of Logic, Language and Information, 6(2), 147–169.
Goldblatt, R., & Jackson, M. (2012). Well-structured program equivalence is highly undecidable. ACM Transactions on Computational Logic, 13(3), 26:1–26:8.
Halpern, J., van der Meyden, R., & Vardi, M. (2004). Complete axiomatizations for reasoning about knowledge and time. SIAM Journal on Computing, 33(3), 674–703.
Harel, D. (1985). Recurring dominoes: Making the highly undecidable highly understandable. In Selected papers of the international conference on “foundations of computation theory” on topics in the theory of computation (pp. 51–71). New York: Elsevier North-Holland.
Herzig, A., & Lima, T. D. (2006). Epistemic actions and ontic actions: A unified logical framework. In: Proceedings of IBERAMIA-SBIA (pp. 409–418). Berlin: Springer.
Holliday, W., Hoshi, T., & Icard, T. (2012). A uniform logic of information dynamics. In Proceedings of advances in modal logic 2012 (Vol. 9. pp. 348–367). London: College Publications.
Hoshi, T. (2009). Epistemic dynamics and protocol information. Ph.D. thesis, Stanford.
Liu, F. (2008). Changing for the better. Ph.D. thesis, University of Amsterdam.
Miller, J., & Moss, L. (2005). The undecidability of iterated modal relativization. Studia Logica, 79(3), 373–407.
Parikh, R., & Ramanujam, R. (1985). Distributed processes and the logic of knowledge. In Proceedings of conference on logic of programs (pp. 256–268). London: Springer.
Plaza, J. A. (1989). Logics of public communications. In M. L. Emrich, M. S. Pfeifer, M. Hadzikadic, & Z. W. Ras (Eds.), Proceedings of the 4th international symposium on methodologies for intelligent systems (pp. 201–216). Knoxville, TX: Oak Ridge National Laboratory.
Stalnaker, R. (1978). Assertion. In P. Cole (Ed.), Syntax and semantics (Vol. 9). New York: Academic Press.
van Benthem, J. (1999). Update as relativization. Unpublished notes.
van Benthem, J. (2007). Dynamic logic for belief revision. Journal of Applied Non-Classical Logics, 17(2), 129–155.
van Benthem, J. (2011). Logical dynamics of information and interaction. Cambridge: Cambridge University Press.
van Benthem, J. (2012). Two logical faces of belief revision. Unpublished manuscript.
van Benthem, J., Gerbrandy, J., Hoshi, T., & Pacuit, E. (2009). Merging frameworks for interaction. Journal of Philosophical Logic, 38(5), 491–526.
van Benthem, J., & Ikegami, D. (2008). Modal fixed-point logic and changing models. In A. Avron, N. Dershowitz, & A. Rabinovich (Eds.), Pillars of computer science (pp. 146–165). Heidelberg: Springer.
van Benthem, J., & Minică, S. (2009). Toward a dynamic logic of questions. In X. He, J. F. Horty, & E. Pacuit (Eds.), Proceedings of the 2nd international workshop on logic, rationality and interaction (LORI-II), Vol. 5834 of FoLLI-LNAI (pp. 27–41). Berlin: Springer.
van Benthem, J., van Eijck, J., & Kooi, B. (2006). Logics of communication and change. Information and Computation, 204(11), 1620–1662.
van der Hoek, W., & Pauly, M. (2006). Handbook of modal logic, Chap. Modal logic for games and information. Amsterdam: Elsevier.
van Ditmarsch, H. (2003). The Russian Cards problem. Studia Logica, 75(1), 31–62.
van Ditmarsch, H., van der Hoek, W., & Kooi, B. (2007). Dynamic epistemic logic (Synthese Library, 1st ed.). Berlin: Springer.
Wang, Y. (2006). Indexed semantics and its application in modelling interactive unawareness. Master’s thesis, University of Amsterdam.
Wang, Y. (2011). On axiomatizations of PAL. In Proceedings of LORI-III (pp. 314–327).
Wang, Y., & Li, Y. (2012). Not all those who wander are lost: dynamic epistemic reasoning in navigation. In Proceedings of advances in modal logic (Vol. 9. pp. 559–580). London: College Publications.
Acknowledgments
The first author is supported by SSFC grant 11CZX054 and the Major Program of National Social Science Foundation of China (No. 12&ZD119). The authors would like to thank Johan van Benthem, Hans van Ditmarsch, Meiyun Guo, Wesley Holliday, Fenrong Liu, Ram Ramanujam, Tomoyuki Yamada and anonymous reviewers of this journal for their insightful comments on earlier versions of this paper.
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Wang, Y., Cao, Q. On axiomatizations of public announcement logic. Synthese 190 (Suppl 1), 103–134 (2013). https://doi.org/10.1007/s11229-012-0233-5
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DOI: https://doi.org/10.1007/s11229-012-0233-5