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A History Based Logic for Dynamic Preference Updates

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.

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Notes

  1. 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. 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. 3.

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

  4. 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. 5.

    We refer the reader to Moss (2015) for a general overview of reduction algorithms in dynamic epistemic logic.

  6. 6.

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

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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.

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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

$$\begin{aligned} t(p)&= p \\ t(\lnot \varphi )&= \lnot t(\varphi )\\ t(\varphi \wedge \psi )&= t(\varphi ) \wedge t(\psi )\\ t(\varphi \vee \psi )&= t(\varphi ) \vee t(\psi )\\ t(K_i \varphi )&= K_i t(\varphi )\\ t(\Diamond _i \varphi )&= \Diamond _i t(\varphi )\\ t(\bigcirc \varphi )&= {\bigcirc } t(\varphi )\\ t (\varphi U \psi )&= t(\varphi ) U t(\psi ) \end{aligned}$$

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

$$\begin{aligned} t( [\varphi !] \psi ) : = T ( [t(\varphi )!] t(\psi )) \end{aligned}$$

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

$$\begin{aligned} T([\varphi !] p)&= p\\ T([\varphi ! ] \lnot \psi )&= \lnot T([\varphi !] \psi ) \\ T([\varphi ! ] (\psi \wedge \chi ))&= T([\varphi !] \psi ) \wedge T([\varphi !] \chi )\\ T([\varphi ! ] (\psi \vee \chi ))&= T([\varphi !] \psi ) \vee T([\varphi !] \chi )\\ T([\varphi ! ] K_i \psi )&= K_i T([\varphi !] \psi ) \\ T([\varphi ! ] {\bigcirc } \psi )&= {\bigcirc } T([\varphi !] \psi )\\ T( [\varphi ! ] (\psi U \chi ) )&= (T([\varphi ! ] \psi )) U (T([\varphi ! ] \chi )) \\ T([\varphi ! ] \Diamond _i \psi )&= (\lnot \varphi \wedge \Diamond _i T([\varphi ! ] \psi ) ) \vee ( \Diamond _i (\varphi \wedge T([\varphi !] \psi ))). \end{aligned}$$

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.

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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

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Keywords

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