Farkas Certificates and Minimal Witnesses for Probabilistic Reachability Constraints

Abstract

This paper introduces Farkas certificates for lower and upper bounds on minimal and maximal reachability probabilities in Markov decision processes (MDP), which we derive using an MDP-variant of Farkas' Lemma. The set of all such certificates is shown to form a polytope whose points correspond to witnessing subsystems of the model and the property. This allows translating the problem of finding minimal witnesses to the problem of finding vertices with a maximal number of zeros. While computing such vertices is computationally hard in general, we derive new heuristics from our formulations that exhibit competitive performance compared to state-of-the-art techniques and apply to more situations. As an argument that asymptotically better algorithms cannot be hoped for, we show that the decision version of finding minimal witnesses is NP-complete even for acyclic Markov chains.

Inductive Synthesis for Probabilistic Programs Reaches New Horizons

“…If a hole is absent in a CE, its value is irrelevant. To avoid the cost of finding optimal CEs-an NP-hard problem [19]-we consider greedy CEs that are similar to [9]. However, our greedy CEs are aware of the parameters, and try to limit the occurrence of parameters in the CE.…”

Witnessing Subsystems for Probabilistic Systems with Low Tree Width

“…To justify that the probability to reach the goal is higher than some threshold in a DTMC one can return a set of traces of the system whose probability exceeds the threshold [21]. Another notion of counterexamples for probabilistic reachability constraints are witnessing subsystems (also called critical subsystems) [16,35,36]. The idea is to justify a lower bound on the reachability probability by providing a subsystem which by itself already exceeds the threshold.…”

“…Computing minimal witnessing subsystems in terms of number of states is computationally difficult. The corresponding decision problem, henceforth called the witness problem, is NP-complete already for acyclic DTMCs [16]. Known algorithms rely on mixed-integer linear programming (MILP) [16,35,36] or vertex enumeration [16].…”

“…The corresponding decision problem, henceforth called the witness problem, is NP-complete already for acyclic DTMCs [16]. Known algorithms rely on mixed-integer linear programming (MILP) [16,35,36] or vertex enumeration [16]. On the other hand, the problem is in P for DTMCs whose underlying graph is a tree [16].…”

SWITSS: Computing Small Witnessing Subsystems

“…This paper presents SWITSS, a novel tool for the computation of small witnessing subsystems for reachability properties in Markovian models. Following [11], SWITSS proceeds by reduction to finding points of a polyhedron containing a large number of zero entries. These points also serve as certificates [22] for the fact that the computed subsystem indeed constitutes a witness.…”