Hybrid Logics. Characterization, Interpolation and Complexity
Carlos Areces, Patrick Blackburn, Maarten Marx
 
Hybrid languages are extended modal languages which can refer to (or even 
quantify over) worlds.  The use of strong hybrid languages dates back to at 
least \cite{prio:past67}, but recent work (for example \cite{blac:hybr98},
\cite{blac:hybr98a}) has focused  on a more constrained system called
$\mathcal{H}(\downarrow,@)$.  The purpose of the present paper is to show 
in detail that $\mathcal{H}(\downarrow,@)$ is a modally natural system.  
We begin by studying its  expressivity, and provide both model theoretic
characterizations (via a restricted notion of Ehrenfeucht-Fra\"{\i}ss\'e 
game, and an enriched notion of bisimulation) and a syntactic characteri-
zation (in terms of bounded formulas).  The key result to emerge is that 
$\mathcal{H}(\downarrow,@)$ corresponds precisely to the first-order 
fragment which is invariant for generated submodels.  We further establish 
that $\mathcal{H}(\downarrow,@)$ has (strong) interpolation, and provide 
failure results in the finite variable fragments.  We also show that weak 
interpolation holds for the sublanguage $\mathcal{H}(@)$, and provide
complexity results for $\mathcal{H}(@)$ and other fragments and variants 
(the full logic being undecidable).