Predict the relative tightness of memory and partition-refined IMC abstractions

Determine, a priori, which of the memory-dependent or partition-refined interval Markov chain abstractions yields tighter expected-cost bounds for a given stochastic system and abstraction-complexity budget.

Background

The paper develops memory-dependent and no-memory interval Markov chain abstractions as alternatives to classic partition refinement for obtaining sound expected-cost bounds of continuous-state stochastic systems. Numerical experiments show that memory can outperform partition refinement in some examples, but the relative effectiveness depends on the concrete dynamics, noise, partition, memory length, and computational budget.

The authors derive separate upper bounds for the tightening effects of memory and partition refinement, but these bounds do not establish an ordering of the exact robust expected-cost interval widths. Consequently, the unresolved problem is to predict which abstraction strategy will be tighter before carrying out the computation, particularly when the two strategies are compared under a fixed complexity budget.

References

We note that predicting a-priori which approach will yield tighter bounds remains unresolved and is a compelling direction for future research.

— Memory-Dependent Interval Markov Chain Abstractions of Stochastic Dynamics  (2610.02950 - Zutphen et al., 2 Oct 2026) in Section 4.2, subsection “Simple example system”