Better Bounded Bisimulation Contractions (Preprint)

TL;DR

This paper introduces rooted $k$-contractions, a new form of bounded bisimulation contraction that preserves $k$-bisimilarity and is more succinct than traditional methods, improving the minimality and efficiency of model reductions.

Contribution

The paper proposes rooted $k$-contractions, a novel bounded bisimulation contraction that guarantees minimality and preserves $k$-bisimilarity, addressing limitations of existing bounded contraction approaches.

Findings

Rooted $k$-contractions preserve $k$-bisimilarity.

Rooted $k$-contractions are minimal among $k$-bisimulation contractions.

Rooted $k$-contractions can be exponentially more succinct than standard $k$-contractions.

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.