The logic of KM belief update is contained in the logic of AGM belief revision
This work clarifies the theoretical relationship between two foundational frameworks for belief change, which is important for researchers in logic and AI.
This paper demonstrates that the logic of KM belief update is contained within the logic of AGM belief revision. Specifically, it shows that every axiom of the modal logic derived from KM update ($\mathcal L_{KM}$) is a theorem of the modal logic derived from AGM revision ($\mathcal L_{AGM}$), implying AGM revision is a special case of KM update. For the strong version of KM update, the difference between the two logics is reduced to a single axiom concerning unsurprising information.
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.