## Abstract

History based models suggest a process-based approach to epistemic and temporal reasoning. In this work, we introduce preferences to history based models. Motivated by game theoretical observations, we discuss how preferences can dynamically be updated in history based models. Following, we consider arrow update logic and event calculus, and give history based models for these logics. This allows us to relate dynamic logics of history based models to a broader framework.

This is a preview of subscription content, access via your institution.

## Notes

- 1.
This is important in order to prevent some technical problems such as evaluating the truth of a formula at time point

*n*, for example, when the finite history*h*is shorter than*n*. We are thankful to the anonymous referee for pointing this out. - 2.
However, the axiom \({K_i \bigcirc } \varphi \rightarrow \bigcirc K_i \varphi \) is not sufficient to establish the completeness of frames with respect to perfect recall (van der Meyden and Wong 2003; van der Meyden 1994). The additional axiom required for this task is a complicated one:

\(K_i \varphi _1 \wedge \bigcirc (K_i \varphi _2 \wedge \lnot K_i \varphi _3) \rightarrow \lnot K_i \lnot ( (K_i \varphi _1) U ( (K_i \varphi _2) U \lnot \varphi _3 ) )\).

- 3.
For a detailed exposition of such reductions in the context of dynamic epistemic logic, we refer the reader to Moss (2015).

- 4.
It is important to notice that the time element introduces a second dimension for the epistemic issues we discuss. Different combinations of quantification over histories and time stamps may suggest new approaches to

*deletion-based*dynamic logics, including sabotage logics (van Benthem 2005). As such, the current system can be used to develop extensions of those systems. - 5.
We refer the reader to Moss (2015) for a general overview of reduction algorithms in dynamic epistemic logic.

- 6.
See a relevant video at http://youtu.be/QX_oy9614HQ, and the Wikipedia entry at (http://en.wikipedia.org/wiki/Stanford_marshmallow_experiment).

## References

Anderson, G., McCusker, G., & Pym, D. (2016). A logic for the compliance budget. In Q. Zhu, T. Alpcan, E. Panaousis, M. Tambe, & W. Casey (Eds.),

*Proceedings, GameSec 2016—Decision and game theory for security*(pp. 370–381).Baltag, A., Moss, L., & Solecki, S. (1998). The logic of public announcements and common knowledge and private suspicions. In I. Gilboa (Ed.),

*Proceedings of the 7th conference on theoretical aspects of rationality and knowledge, TARK 98*(pp. 43–56).Başkent, C. (2011). A logic for strategy updates. In H. van Ditmarsch & J. Lang (Eds.),

*Proceedings of the third international workshop on logic, rationality and interaction (LORI-3), volume LNCS 6953*(pp. 382–383).Blackburn, P., de Rijke, M., & de Venema, Y. (2001).

*Modal logic. Cambridge Tracts in Theoretical Computer Science*. Cambridge: Cambridge University Press.Brandenburger, A. (2014).

*The language of game theory*. Singapore: World Scientific Publishing.Fagin, R., Geanakoplos, J., Halpern, J. Y., & Vardi, M. Y. (1999). The hierarchical approach to modeling knowledge and common knowledge.

*International Journal of Game Theory*,*28*, 331–365.Fagin, R., Halpern, J. Y., Moses, Y., & Vardi, M. Y. (1995).

*Reasoning about knowledge*. Cambridge: MIT Press.Fagin, R., Halpern, J. Y., & Vardi, M. Y. (1991). A model-theoretic analysis of knowledge.

*Journal of the Association for Computing Machinery*,*38*(2), 382–428.Gabbay, D. M., Kurucz, A., Wolter, F., & Zakharyaschev, M. (2003).

*Many dimensional modal logics: Theory and applications*. Amsterdam: Elsevier.Gerbrandy, J. (1999).

*Bisimulations on planet Kripke*. PhD thesis, Institute of Logic, Language and Computation; Universiteit van Amsterdam.Halpern, J. Y. (2008). Computer science and game theory: A brief survey. In S. N. Durlauf & L. E. Blume (Eds.),

*Palgrave dictionary of economics*. London: Palgrave MacMillan.Halpern, J. Y. & Pass, R. (2017). A knowledge-based analysis of the blockchain protocol. In J. Lang (Ed.),

*Proceedings of the sixteenth conference on theoretical aspects of rationality and knowledge. TARK 2018, EPTCS 251*(pp. 324–335).Halpern, J. Y., & Vardi, M. Y. (1989). The complexity of reasoning about knowledge and time. I. Lower bounds.

*Journal of Computer and System Sciences*,*38*(1), 195–237.Halpern, J. Y., Vardi, M. Y., & van der Meyden, R. (2004). Complete axiomatization for reasoning about knowledge and time.

*SIAM Journal of Computing*,*33*(2), 674–703.Hanson, S. O. (2001). Preference logic. In D. Gabbay & F. Guenthner (Eds.),

*Handbook of philosophical logic*(Vol. 4, pp. 319–393). Dordrecht: Kluwer.Harsanyi, J. C. (1967). Games with incomplete information played by ‘Bayesian’ players: I. The basic model.

*Management Science*,*14*(3), 159–182.Hodges, W. (2013) Logic and games. In Zalta, E. N. (Ed.),

*The Stanford encyclopedia of philosophy*. Retrieved 2009, from http://plato.stanford.edu/archives/spr2009/entries/logic-games.Kooi, B., & Renne, B. (2011). Arrow update logic.

*The Review of Symbolic Logic*,*4*(4), 536–559.Kurtonina, N., & de Rijke, M. (1997). Bisimulations for temporal logic.

*Journal of Logic, Language and Information*,*6*, 403–425.Leyton-Brown, K., & Shoham, Y. (2008).

*Essentials of game theory*. San Rafael: Morgan & Claypool.Lorini, E., & Moisan, F. (2011). An epistemic logic of extensive games.

*Electronic Notes in Theoretical Computer Science*,*278*, 245–260.Moss, L. S. (2015). Dynamic epistemic logic. In H. van Ditmarsch, J. Y. Halpern, W. van der Hoek, & B. Kooi (Eds.),

*Handbook of epistemic logic*(pp. 261–312). London: College Publications.Osborne, M. J., & Rubinstein, A. (1994).

*A course in game theory*. Cambridge: MIT Press.Osherson, D., & Weinstein, S. (2012). Preference based on reasons.

*The Review of Symbolic Logic*,*5*(1), 122–147.Pacuit, E. (2007). Some comments on history based structures.

*Journal of Applied Logic*,*5*(4), 613–624.Pacuit, E., Parikh, R., & Cogan, E. (2006). The logic of knowledge based obligation.

*Synthese*,*149*(2), 311–341.Parikh, R., & Ramanujam, R. (2003). A knowledge based semantics of messages.

*Journal of Logic, Language and Information*,*12*(4), 453–467.Ramanujam, R., & Simon, S. (2008). Dynamic logic on games with structured strategies. In G. Brewka, & J. Lang (Eds.),

*Proceedings of the 11th international conference on principles of knowledge representation and reasoning. KR-08*(pp. 49–58).Renne, B., Sack, J., & Yap, A. (2016). Logics of temporal-epistemic actions.

*Synthese*,*193*(3), 813–849.Sack, J. (2008). Temporal languages for epistemic programs.

*Journal of Logic, Language and Information*,*17*(2), 183–216.van Benthem, J. (2005). An essay on sabotage and obstruction. In D. Hutter (Ed.),

*Mechanizing mathematical reasoning*(pp. 268–276). Berlin: Springer.van Benthem, J. (2014).

*Logic in games*. Cambridge: MIT Press.van Benthem, J., Gerbrandy, J., & Pacuit, E. (2007). Merging frameworks for interaction: Del and etl. In D. Samet (Ed.),

*Proceedings of tark 2007*.van Benthem, J., Girard, P., & Roy, O. (2009). Everything else being equal: A modal logic for ceteris paribus preferences.

*Journal of Philosophical Logic*,*38*(1), 83–125.van Benthem, J., & Liu, F. (2007). Dynamic logic of preference upgrade.

*Journal of Applied Non-Classical Logics*,*17*(2), 157–182.van der Meyden, R. (1994). Axioms for knowledge and time in distributed systems with perfect recall. In

*Proceedings of IEEE symposium on logic in computer science*(pp. 448–457).van der Meyden, R., & Wong, K. (2003). Complete axiomatization for reasoning about knowledge and branching time.

*Studia Logica*,*75*(1), 93–123.van Ditmarsch, H., van der Hoek, W., & Kooi, B. (2007).

*Dynamic Epistemic Logic*. Berlin: Springer.

## Acknowledgements

We acknowledge the input and feedback of David Pym and Gabriella Anderson for the early versions of this work. The work was carried out under the EPSRC project ALPUIS.

## Author information

### Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Appendix: An Alternative Method to Define the Reduction Function t

### Appendix: An Alternative Method to Define the Reduction Function *t*

In what follows, we present an alternative method the define the reduction function *t* discussed in the proof of Lemma 4.3.

We define a translation function *t* as follows

For update free formulas \(\varphi , \psi \), we define a function *T* as

where *T* is defined on \([\varphi !] \psi \) for update-free formulas in the following

It is a straight-forward procedural argument to see that *t* indeed reduces formulas in the language of HBPL* to update-free formulas.

Let us prove it for the formula \([\varphi !][\psi !]\chi \). By definition we have \(t([\varphi !][\psi !]\chi ) = T ( [t(\varphi )!] t([\psi !]\chi ) )\). By induction hypothesis, \(t(\varphi )\) and \(t([\psi !]\chi )\) are update-free. Then, *T* becomes applicable. By induction for *T* with the update-free formula \(t([\psi !]\chi )\), it follows that \(T ( [t(\varphi )!] t([\psi !]\chi ) )\) is update-free.

## Rights and permissions

## About this article

### Cite this article

Başkent, C., McCusker, G. A History Based Logic for Dynamic Preference Updates.
*J of Log Lang and Inf* **29, **275–305 (2020). https://doi.org/10.1007/s10849-019-09307-1

Published:

Issue Date:

### Keywords

- History based models
- Preference logic
- Dynamic logic
- Arrow update logic
- Product update models