 But there is also another side to this coin. By adding modalities to a system, one increases its expressivity, but this may also have the undesirable consequence of raising its  computational complexity.  Indeed, a modal language allows us to describe a certain class of models. If the language is fairly simple, then deciding whether a given formula is true in a given world of a given model (the model-checking problem) and deciding whether a given formula is true in all worlds of all models in a given class (the validity problem) are simple tasks. Now suppose a more expressive language is used. It is then possible to distinguish models that were, from the first language’s perspective, the same (see, e.g., the appendix on the non-definability of distributed and common knowledge within \(\cL_{\left\{ \oK{1}, \ldots, \oK{n} \right\}}\)). However, intuitively, we might then need to make a stronger effort to see those differences: we might need more time to make the calculations, and we might need more space to save intermediate results. In a single sentence, expressivity and complexity go hand in hand, and an increase in the first typically produces an increase in the second.