# PP-1999-07: Hybrid Logics. Characterization, Interpolation and Complexity

PP-1999-07: Areces, Carlos and Blackburn, Patrick and Marx, Maarten (1999) Hybrid Logics. Characterization, Interpolation and Complexity. [Report]

## Abstract

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).

Item Type: Report PP-1999-07 Prepublication (PP) Series 1999 12 Oct 2016 14:36 12 Oct 2016 14:36 https://eprints.illc.uva.nl/id/eprint/7