The logic of KM belief update is contained in the logic of AGM belief revision
2026-02-26 • Artificial Intelligence
Artificial IntelligenceLogic in Computer Science
AI summaryⓘ
The authors translate the KM belief update rules into modal logic using three types of operators and compare this to a similar logic built from AGM belief revision rules. They find that the logic from AGM revision actually includes all the rules from KM update, meaning AGM revision can be viewed as a special case of KM update. For a stronger form of KM update, the authors identify that only one specific axiom differentiates the two logics, which concerns information that was not unexpected. This work helps clarify the relationship between two important ways of modeling belief change.
KM belief updateAGM belief revisionmodal logicbelief operatorconditional operatornecessity operatoraxiomsbelief changeunsurprising informationtheorem
Authors
Giacomo Bonanno
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.