Efficient CSL Model Checking Using Stratification

For continuous-time Markov chains, the model-checking problem with respect to continuous-time stochastic logic (CSL) has been introduced and shown to be decidable by Aziz, Sanwal, Singhal and Brayton in 1996. Their proof can be turned into an approximation algorithm with worse than exponential complexity. In 2000, Baier, Haverkort, Hermanns and Katoen presented an efficient polynomial-time approximation algorithm for the sublogic in which only binary until is allowed. In this paper, we propose such an efficient polynomial-time approximation algorithm for full CSL. The key to our method is the notion of stratified CTMCs with respect to the CSL property to be checked. On a stratified CTMC, the probability to satisfy a CSL path formula can be approximated by a transient analysis in polynomial time (using uniformization). We present a measure-preserving, linear-time and -space transformation of any CTMC into an equivalent, stratified one. This makes the present work the centerpiece of a broadly applicable full CSL model checker. Recently, the decision algorithm by Aziz et al. was shown to work only for stratified CTMCs. As an additional contribution, our measure-preserving transformation can be used to ensure the decidability for general CTMCs.

💡 Research Summary

The paper addresses the long‑standing challenge of efficiently model‑checking Continuous Stochastic Logic (CSL) on continuous‑time Markov chains (CTMCs) when the path formulas contain multiple nested “until” operators. While the original decidability result by Aziz, Sanwal, Singhal and Brayton (1996) yields an algorithm whose complexity is worse than exponential, and while Baier, Haverkort, Hermanns and Katoen (2000) provided a polynomial‑time approximation for the binary‑until fragment, no practical polynomial‑time method existed for the full CSL fragment (k ≥ 2).

The authors introduce the notion of a stratified CTMC with respect to a given CSL path formula ϕ = f₁ U I₁ f₂ U I₂ … U Iₖ₋₁ fₖ. A CTMC is stratified if every transition respects the order imposed by the sequence of atomic propositions f₁,…,fₖ: a state whose minimal label (according to the order) is ⊥ (no relevant label) or the final label fₖ must be absorbing, and any transition from a state with minimal label fᵢ (i < k) can only go to a state whose minimal label is fⱼ with j ≥ i. This structural restriction guarantees that any path violating the prescribed order can never satisfy the formula, while paths that respect the order can be evaluated by a simple recursive decomposition.

To apply this idea to arbitrary CTMCs, the paper presents a linear‑time, linear‑space transformation that turns any CTMC into an equivalent stratified one while preserving the probability of satisfying ϕ. The transformation proceeds in two steps:

1. Formula automaton construction – For the given CSL formula a deterministic finite automaton (DFA) B_ϕ is built. Its alphabet is the powerset of the set of atomic propositions {f₁,…,fₖ}. The DFA has k + 1 “progress” states q₁,…,qₖ and a sink state ⊥. Transitions encode the requirement that the next observed label must be at least as large (in the order) as the current one; otherwise the automaton jumps to ⊥.

2. Product CTMC – The original CTMC C = (S,R,L) is combined with B_ϕ via a synchronous product. A new state is a pair (s,q) with s ∈ S and q ∈ Q. The rate from (s,q) to (s′,q′) is R(s,s′) if the DFA transition δ(q, L(s′)) = q′, otherwise zero. The sink and final DFA states become absorbing in the product. By construction, the product CTMC C′ = C ⊗ B_ϕ satisfies the stratification constraints with respect to ϕ, and a rigorous proof shows that Prₛ(ϕ) = Pr′_{(s,q₁)}(ϕ).

Having obtained a stratified CTMC, the probability of a multiple‑until formula can be computed by uniformization, a classic technique for transient analysis of CTMCs. Uniformization replaces the original CTMC by a discrete‑time Markov chain with transition matrix P = I + R/λ, where λ ≥ maxₛ E(s) (E(s) is the exit rate). The number of jumps occurring in a time interval I =