Abstract
Bisimulations are standard in modal logic and, more generally, in the theory of state-transition systems. The quotient structure of a Kripke model with respect to the bisimulation relation is called a bisimulation contraction. The bisimulation contraction is a minimal model bisimilar to the original model, and hence, for (image-)finite models, a minimal model modally equivalent to the original. Similar definitions exist for bounded bisimulations (k-bisimulations) and bounded bisimulation contractions. Two finite models are k-bisimilar if and only if they are modally equivalent up to modal depth k. However, the quotient structure with respect to the k-bisimulation relation does not guarantee a minimal model preserving modal equivalence to depth k. In this paper, we remedy this asymmetry to standard bisimulations and provide a novel definition of bounded contractions called rooted k-contractions. We prove that rooted k-contractions preserve k-bisimilarity and are minimal with this property. Finally, we show that rooted k-contractions can be exponentially more succinct than standard k-contractions.