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In first-order logic, Robinson diagrams are useful for defining different kinds of maps: homomorphisms, embeddings, elementary embeddings, and isomorphisms:

For example, consider two structures $A$ and $B$ with a map $f : A \to B$ between them.

If $D$ is an $A$-diagram, let $f(D)$ be a $B$-diagram obtained by sending every parameter $p$ in every formula $\varphi(\cdots p \cdots)$ in $D$ to $f(p)$.

  • $f$ is a homomorphism if and only if $f(\text{Diag}^+(A)) \subset \text{Diag}^+(B)$.
  • $f$ is an embedding if and only if $f(\text{Diag}(A)) \subset \text{Diag}(B)$.
  • $f$ is an elementary embedding if and only if $f(\text{ElDiag}(A)) \subset \text{ElDiag}(B)$.
  • $f$ is an isomorphism if and only if $f(\text{ElDiag}(A)) = \text{ElDiag}(B)$.

I'm wondering if we can do something similar to define notions of homomorphisms, embeddings, elementary embeddings, and isomorphisms for Kripke frames and models in modal logic in such a way that they line up with the usual conventions of modal logic.

For example, there's a choice to make on how to define a diagram for unpointed Kripke frames, shown below. It's not clear to me which of the two choices (if either) is the right one.

For unpointed Kripke models as described in this answer, I think it makes most sense to do the following, which is equivalent to leveraging the standard translation of modal logic to first-order logic:

  • Use a first-order signature with $R$ as our binary relation and one unary relation $V_i$ for each variable $v_i$. Call this the positive encoding.

In this setting, the parameters are specific worlds. One counterintuitive consequence of this translation is that for a given Kripke model $A$, there's a homomorphism going to the model with the same worlds and accessibility relation as $A$, but where every primitive proposition is always true.

There's another choice we could make:

  • Use a first-order signature with $R$ as our accessibility relation and two unary vairables $V_i$ and $N_i$ for each variable $v_i$. We additionally insist that $\forall w \, (V_i(w) \iff \lnot N_i(w))$. Call this the neutral encoding.

In this setting, homomorphisms can collapse worlds or add new accessibility edges to the frame, but can't change the value at any world from false to true.

It's not clear to me that either the positive or the neutral encoding is right. What's the right way to define what a diagram is for a modal model so we get a reasonable notion of homomorphism &c?

Greg Nisbet
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The problem of defining Robinson diagrams for Kripke models in the basic modal language (one diamond) is that this language is not even able to specify how atomic properties are distributed among points (states, worlds etc.). This is because the basic modal language lacks resources to designate specific points from a model.

But an extended modal language like hybrid logic has the expressive power to do exactly this: There we have so-called nominals $i, j, \ldots$ which are formulas true at exactly one point. These can be taken to be designators of the points they are true at. Furthermore there are satisfaction operators $@_i, @_j, \ldots$. These are one-place operators from formulas to formulas such that $@_i \varphi$ is satisfied at a point exactly if $\varphi$ is satisfied at the point designated by the nominal $i$ that indexes the operator.

Hybrid logic allows us to completely describe modal models and so to define modal Robinson diagrams: Hybrid formulas $@_i p$, $\neg@_i p$ tell us how atomic information is distributed among points. Formulas of the form $@_i j$, $\neg @_i j$ tell us which points are identical to which. Finally formulas of the form $@_i \Diamond j$, $\neg @_i \Diamond j$ specify the successor relations of models.

sequitur
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