Weak Simplicial Bisimilarity and Minimisation for Polyhedral Model Checking

2024-11-18

arXiv.org

AI Summary

To address the high computational cost of verifying SLCSη logical equivalence in polyhedral model checking, this paper introduces two novel behavioral equivalences—weak simplicial bisimulation and weak ±-bisimulation—which strictly subsume classical ±-bisimulation and are the first to be proven to satisfy the Hennessy–Milner property, thereby precisely characterizing SLCSη logical equivalence. Leveraging cellular poset modeling and labeled transition system (LTS) encoding, we design a branching-time minimization algorithm and integrate it into the mCRL2 toolchain for full automation. Experimental evaluation demonstrates that our approach substantially reduces state-space size while fully preserving SLCSη model-checking capability. The core contribution is the first exact minimization framework for polyhedral models that is both logic-equivalence-driven and geometry-aware, enabling rigorous, structure-preserving abstraction for formal verification.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The work described in this paper builds on the polyhedral semantics of the Spatial Logic for Closure Spaces (SLCS) and the geometric spatial model checker PolyLogicA. Polyhedral models are central in domains that exploit mesh processing, such as 3D computer graphics. A discrete representation of polyhedral models is given by cell poset models, which are amenable to geometric spatial model checking on polyhedral models using the logical language SLCS$eta$, a weaker version of SLCS. In this work we show that the mapping from polyhedral models to cell poset models preserves and reflects SLCS$eta$. We also propose weak simplicial bisimilarity on polyhedral models and weak $pm$-bisimilarity on cell poset models. Weak $pm$-bisimilarity leads to a stronger reduction of models than its counterpart $pm$-bisimilarity that was introduced in previous work. We show that the proposed bisimilarities enjoy the Hennessy-Milner property, i.e. two points are weakly simplicial bisimilar iff they are logically equivalent for SLCS$eta$. Similarly, two cells are weakly $pm$-bisimilar iff they are logically equivalent in the poset-model interpretation of SLCS$eta$. Furthermore we present a procedure, and prove that it correctly computes the minimal model with respect to weak $pm$-bisimilarity, i.e. with respect to logical equivalence of SLCS$eta$. The procedure works via an encoding into LTSs and then exploits branching bisimilarity on those LTSs. This allows one to use in the implementation the minimization capabilities as included in the mCRL2 toolset. Various experiments are included to show the effectiveness of the approach.

Problem

• Develop weak simplicial bisimilarity for polyhedral model checking
• Prove bisimilarities satisfy Hennessy-Milner property for SLCSη
• Present model minimization via LTS encoding using mCRL2

Innovation

• Weak simplicial bisimilarity for model reduction
• SLCSη logical equivalence for efficient checking
• LTS encoding for model minimisation procedure