Reasoning about Rare-Event Reachability in Stochastic Vector Addition Systems via Affine Vector Spaces

TL;DR

This paper introduces two novel heuristics, ISR and SDP, for efficiently estimating rare-event probabilities in stochastic vector addition systems, especially in chemical reaction networks, by constructing solution spaces for probabilistic model checking.

Contribution

The paper presents two new heuristics, ISR and SDP, that improve rare-event probability estimation in VAS by constructing solution spaces, enabling faster and more efficient model checking.

Findings

Both methods are deterministic and fast.

They demonstrate significant performance improvements on challenging CRN models.

They provide lower-bound probability estimates for rare events.

Abstract

Rare events in Stochastic Vector Addition System (VAS) are of significant interest because, while extremely unlikely, they may represent undesirable behavior that can have adverse effects. Their low probabilities and potentially extremely large state spaces challenge existing probabilistic model checking and stochastic rare-event simulation techniques. In particular, in Chemical Reaction Networks (CRNs), a chemical kinetic language often represented as VAS, rare event effects may be pathological. We present two novel heuristics for priority-first partial state space expansion and trace generation tuned to the transient analysis of rare-event probability in VAS: Iterative Subspace Reduction (ISR) and Single Distance Priority (SDP). Both methods construct a closed vector space containing all solution states. SDP then simply prioritizes shorter distances to this ``solution space'', while ISR constructs a set of nested subspaces, where short and highly-probable satisfying traces are likely to pass through in sequence. The resulting partial state graph from each method contains likely traces to rare-event states, allowing efficient probabilistic model checking to compute a lower-bound probability of a rare event of interest. These methods are deterministic, fast, and demonstrate marked performance on challenging CRN models.