[Paper Review] A Two Step Perspective for Kripke Structure Reduction

Arpit Sharma|arXiv (Cornell University)|Oct 1, 2012

Cellular Automata and Applications21 references3 citations

TL;DR

This paper introduces Kripke Minimization Equivalence (KME) and Weak Kripke Minimization Equivalence (WKME), two novel equivalence relations for reducing Kripke structure state spaces by analyzing two-step predecessor-successor relationships. The method enables coarser state reduction than strong or divergence-sensitive stutter bisimulation while preserving linear-time and stutter-insensitive linear-time properties, with KME proven compositional under synchronous parallel composition.

ABSTRACT

This paper presents a novel theoretical framework for the state space reduction of Kripke structures. We define two equivalence relations, Kripke minimization equivalence (KME) and weak Kripke minimization equivalence (WKME). We define the quotient system under these relations and show that these relations are strictly coarser than strong (bi)simulation and divergence-sensitive stutter (bi)simulation, respectively. We prove that the quotient system obtained under KME and WKME preserves linear-time and stutter-insensitive linear-time properties. Finally, we show that KME is compositional w.r.t. synchronous parallel composition.

Motivation & Objective

• . The paper aims to address the state-space explosion problem in model checking Kripke structures by proposing more aggressive reduction techniques.
• It seeks to develop equivalence relations that are coarser than strong (bi)simulation and divergence-sensitive stutter (bi)simulation to enable larger state space reductions.
• The objective includes proving that KME and WKME preserve linear-time and stutter-insensitive linear-time properties, respectively.
• It aims to establish compositional properties of KME under synchronous parallel composition for scalable verification of distributed systems.
• The research further aims to provide a theoretical foundation for efficient algorithms that achieve reductions beyond traditional bisimulation.

Proposed method

• . KME is defined by comparing states based on their two-step reachability: two states are equivalent if all their direct predecessors can reach the same set of equivalence classes in two transitions.
• WKME extends KME by abstracting from stutter steps, requiring predecessors to reach the same equivalence classes in two or more steps, with extra steps confined within the target class.
• The quotient system is constructed by collapsing equivalent states under KME or WKME into single representative states.
• The paper proves that KME is strictly coarser than strong (bi)simulation and WKME is strictly coarser than divergence-sensitive stutter (bi)simulation.
• Preservation of linear-time properties under KME is established via a trace-based argument showing that equivalent states generate the same infinite traces.
• Compositional behavior of KME under synchronous parallel composition is proven using structural induction on the composition rule, showing that KME respects the composition operation.

Experimental results

Research questions

• RQ1 . Can a new equivalence relation be defined that enables larger state space reduction than strong (bi)simulation while preserving linear-time properties?
• RQ2 . Is it possible to define a weak variant of this equivalence that abstracts from stutter steps and still preserves stutter-insensitive linear-time properties?
• RQ3 . Does the proposed equivalence relation KME preserve the ability to verify temporal logic properties expressed in Linear Temporal Logic (LTL)?
• RQ4 . Can the KME equivalence be applied compositionally in synchronous parallel composition of Kripke structures?
• RQ5 . Is the quotient system under KME equivalent to the original system for all linear-time properties, including those not expressible in LTL/⃝?

Key findings

• . KME is strictly coarser than strong (bi)simulation, enabling larger state space reductions while preserving linear-time properties.
• . WKME is strictly coarser than divergence-sensitive stutter (bi)simulation, allowing even greater reduction potential in the weak setting.
• . The quotient system under KME preserves all linear-time properties defined over infinite words.
• . The quotient system under WKME preserves all stutter-insensitive linear-time properties, including ω-regular and siPSL-formulatable properties.
• . KME is compositional with respect to synchronous parallel composition, meaning that KME reduction can be applied before or during composition without loss of correctness.
• . The theoretical framework provides a basis for developing efficient algorithms that achieve reductions beyond traditional bisimulation, though not necessarily the minimal possible quotient.