The logic of KM belief update is contained in the logic of AGM belief revision

Giacomo Bonanno

cs.AIFeb 26, 2026

The logic of KM belief update is contained in the logic of AGM belief revision

Authors: Giacomo Bonanno

Organizations: University of California, Davis, USA

Abstract

For each axiom of KM belief update we provide a corresponding axiom in a modal logic containing three modal operators: a unimodal belief operator $B$ , a bimodal conditional operator $>$ and the unimodal necessity operator $\square$ . We then compare the resulting logic to the similar logic obtained from converting the AGM axioms of belief revision into modal axioms and show that the latter contains the former. Denoting the latter by $\mathcal L_{AGM}$ and the former by $\mathcal L_{KM}$ we show that every axiom of $\mathcal L_{KM}$ is a theorem of $\mathcal L_{AGM}$ . Thus AGM belief revision can be seen as a special case of KM belief update. For the strong version of KM belief update we show that the difference between $\mathcal L_{KM}$ and $\mathcal L_{AGM}$ can be narrowed down to a single axiom, which deals exclusively with unsurprising information, that is, with formulas that were not initially disbelieved.

Figures & tables

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

	$\neg\square\neg(\phi\wedge\psi)\,\rightarrow\,\neg\square\neg\phi$	$(C_{\neg\square\neg})$
		(see Remark 3 )
	$\big(\neg\square\neg(\phi\wedge\psi)\wedge B(\phi>\psi)\big)\,\rightarrow\,\left(\neg\square\neg\phi\wedge B(\phi>\psi)\right)$	1, PL
	$\left(\neg\square\neg\phi\wedge B(\phi>\psi)\right)\rightarrow\neg B(\phi>\neg\psi)$	$\text{axiom }(A^{*5b}_{\diamond 3b})$
	$\big(\neg\square\neg(\phi\wedge\psi)\wedge B(\phi>\psi)\big)\,\rightarrow\,\neg B(\phi>\neg\psi)$	2, 3, PL
	$\chi\rightarrow(\psi\rightarrow\chi)$	tautology
	$B(\phi>\chi)\rightarrow B\left(\phi>(\psi\rightarrow\chi)\right)$	$\text{5, rule }(RM_{B>})$

Appendix

Table 1

	$\neg\square\neg(\phi\wedge\psi)\wedge B\left((\phi\wedge\psi)>\chi\right)$
	$\rightarrow\,B\big(\phi>(\psi\rightarrow\chi)\big)$	$\text{axiom }(A*7)$
	$\neg\square\neg(\phi\wedge\psi)\wedge B\left((\phi\wedge\psi)>\chi\right)\wedge B(\phi>\psi)$
	$\rightarrow\,\big(B(\phi>\psi)\wedge B\big(\phi>(\psi\rightarrow\chi)\big)\big)$	15, PL
	$\big(B(\phi>\psi)\wedge B(\phi>(\psi\rightarrow\chi)\big)\,\rightarrow\,B(\phi>\chi)$	$\text{axiom }(A^{*1}_{\diamond 0})$
	$\big(\neg\square\neg(\phi\wedge\psi)\wedge B(\phi>\psi)\wedge B\left((\phi\wedge\psi)>\chi\right)\big)\,\rightarrow\,B(\phi>\chi)$	16, 17, PL
	$\big(\neg\square\neg(\phi\wedge\psi)\wedge B(\phi>\psi)\big)\,\rightarrow\,\big(B\left((\phi\wedge\psi)>\chi\right)\rightarrow B(\phi>\chi)\big)$	18, PL

Appendix

Table 2

	$\phi\leftrightarrow\big((\phi\vee\psi)\wedge\phi\big)$	tautology
	$B(\phi>\chi)\leftrightarrow B\big(\left((\phi\vee\psi)\wedge\phi\right)>\chi\big)$	$\text{1, rule }(R^{*6}_{\diamond 4})$
	$B(\phi>\chi)\rightarrow B\big(\left((\phi\vee\psi)\wedge\phi\right)>\chi\big)$	2, PL
	$\phi\rightarrow\big((\phi\vee\psi)\wedge\phi\big)$	tautology
	$\neg\square\neg\phi\rightarrow\neg\square\neg\left((\phi\vee\psi)\wedge\phi\right)$	$\text{1, rule }(RM_{\neg\square\neg})$
		$(\text{see Remark \ref{REM:derivedAxRul}})$
	$\neg\square\neg\phi\wedge B(\phi>\chi)$

Appendix

Table 3

	$\neg\square\neg\phi\wedge B(\phi>\chi)\wedge\neg\square\neg\psi\wedge B(\psi>\chi)$
	$\rightarrow\,B\big((\phi\vee\psi)>(\phi\rightarrow\chi)\big)\wedge B\big((\phi\vee\psi)>(\psi\rightarrow\chi)\big)$	8, 9, PL
	$B\big((\phi\vee\psi)>(\phi\rightarrow\chi)\big)\wedge B\big((\phi\vee\psi)>(\psi\rightarrow\chi)\big)$
	$\rightarrow B\big((\phi\vee\psi)>(\phi\rightarrow\chi\wedge\psi\rightarrow\chi)\big)$	$\text{axiom }(C_{B})$
	$(\phi\rightarrow\chi)\wedge(\psi\rightarrow\chi)\,\rightarrow\left((\phi\vee\psi)\rightarrow\chi\right)$	tautology
	$B\big((\phi\vee\psi)>(\phi\rightarrow\chi\wedge\psi\rightarrow\chi)\big)$
	$\rightarrow B\big((\phi\vee\psi)>\left((\phi\vee\psi)\rightarrow\chi\right)\big)$	$\text{12, rule }(RM_{B>})$

Appendix

Table 4

	$B(\phi\vee\neg\phi)\,\rightarrow\,\big(B(\phi\rightarrow\psi)\leftrightarrow B\left((\phi\vee\neg\phi)>(\phi\rightarrow\psi)\right)\big)$	$\text{axiom }(A\diamond 2)$
	$\phi\vee\neg\phi$	tautology
	$B(\phi\vee\neg\phi)$	$\text{2, rule }(N_{B})$
	$B(\phi\rightarrow\psi)\leftrightarrow B\left((\phi\vee\neg\phi)>(\phi\rightarrow\psi)\right)$	$\text{1, 3, rule }(MP)$
	$\neg\square\neg((\phi\vee\neg\phi)\wedge\phi)\wedge B\left((\phi\vee\neg\phi)\wedge\phi)>\psi\right)$
	$\rightarrow B((\phi\wedge\neg\phi)>(\phi\rightarrow\psi))$	$\text{axiom }(A\diamond 5)$
	$\phi\rightarrow(\phi\vee\neg\phi)\wedge\phi$	tautology

Appendix

Table 5

Belief Revision

Belief revision studies how agents change, retain, or retract beliefs when new evidence conflicts with what they already accept.

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Study on Quantitative Dynamic Epistemic Logic for Belief Revision

Belief revision is a process in which an agent begins to believe in something she previously did not. I begin the paper by presenting, based on (Gärdenfors, 1998; Hansson, 1999), postulates for belief revision that constitute the basis of the AGM theory. I will then briefly show the semantics of a modal logic introduced in (van Ditmarsch, 2005), which I call

$P$'. This logic formalizes static epistemic states and has greater expressive power than AGM in doing so because it captures the quantitative notion of "degrees of conviction". The third step is to introduce revision operators on $P$ and, mostly following (van Ditmarsch, 2005), obtain the Dynamic Epistemic Logic (DEL) I call

P*

'. It models processes of belief revision in several ways. Original results are presented in the following two sections. The first one of these sections revolves around a formalization of AGM postulates within

P*

by proving some theorems related to the satisfaction of those postulates by revisions defined in

P*

. The last section features an analysis of

P*

's revisions that go beyond the mere satisfaction of postulates. I compare their formal behavior with respect to some philosophical criteria. At last, I conclude that the functions presented in (van Ditmarsch, 2005) are not good formalizations of the philosophical intuition behind AGM. Instead, it is captured by the function

*^0

originally defined in this paper (but highly inspired by (van Benthem, 2007)). An implementation of this function is also provided.

Felipe Nunes de Souza Camargo

State University of Campinas (UNICAMP), Campinas-SP, Brasil

Belief Revision Dynamic Epistemic Logic Epistemic Logic

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May 4, 2026

A Study of Belief Revision Postulates in Multi-Agent Systems (Extended Version)

We investigate the belief revision problem in epistemic planning, i.e., what will be the beliefs of all agents in a multi-agent system after an agent gains the belief in some state property. Based on the standard representation in epistemic planning of agents' beliefs via a single multi-agent Kripke model, we generalize the classical AGM belief revision postulates to the multi-agent setting, with the aim to provide a formal framework for evaluating dynamic epistemic reasoning frameworks in which the beliefs of all agents as the result of actions are computed. As an example of a simple operator that satisfies all of the generalized AGM postulates, we present generalized full-meet multi-agent belief revision. We moreover define a generalization of the standard postulates for iterated revision, present a more sophisticated, event model based revision operator, and discuss the potential issues in defining an epistemic operator on Kripke models that can satisfy all of the generalized postulates for iterated multi-agent belief revision.

Michael Thielscher, Tran Cao Son

University of New South Wales, Sydney, Australia · New Mexico State University, Las Cruces, New Mexico, USA

Multi-Agent Reasoning

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Jun 1, 2026

An Abstract Worlds Semantic Framework for Belief Change Operators

This article proposes a set-theoretic framework for belief change, called Abstract Worlds Semantics, in which no logical syntax is assumed. Inspired by Grove's (1988) results, our approach treats worlds as primitive elements, over which world contraction and world revision operators are defined. This semantic framework enables a unified analysis of belief change models. Within this framework, we unify classical and non-prioritized belief change constructions by defining versatile operators. When classical propositional logic is considered, our framework provides a homogeneous account of AGM, KM, and Multiple Change models. In summary, AWS systematizes belief change frameworks and operators, simplifying and generalizing belief change theory over belief sets.

Daniel Grimaldi, M. Vanina Martinez, Ricardo O. Rodriguez

Departamento de Computación Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires, Argentina · Instituto de Investigación en Ciencias de la Computación UBA-CONICET, Argentina · Artificial Intelligence Research Institute (IIIA-CSIC), Spain

Belief Revision

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Abstract

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