Are Parametric Markov Chains Monotonic?

Abstract

This paper presents a simple algorithm to check whether reachability probabilities in parametric Markov chains are monotonic in (some of) the parameters. The idea is to construct-only using the graph structure of the Markov chain and local transition probabilities-a pre-order on the states. Our algorithm cheaply checks a sufficient condition for monotonicity. Experiments show that monotonicity in several benchmarks is automatically detected, and monotonicity can speed up parameter synthesis up to orders of magnitude faster than a symbolic baseline.

Finding Provably Optimal Markov Chains

“…Thus, while global monotonicity considers the derivative of the entire solution function, local monotonicity (in s) only considers the derivative of the first transition (emanating from s). Local monotonicity of parameter x in every state implies global monotonicity of x, as shown in [27]. As checking global monotonicity is co-ETR hard [27], a practical approach is to check sufficient conditions for monotonicity.…”

“…Local monotonicity of parameter x in every state implies global monotonicity of x, as shown in [27]. As checking global monotonicity is co-ETR hard [27], a practical approach is to check sufficient conditions for monotonicity. These conditions are based on constructing a pre-order on the states of the pMC; this is explained in detail in Section 4.…”

“…As discussed in Section 3.1, we aim to analyse whether for a given region R, parameter x is locally monotonic at state s. The key ingredient is a pre-order on the states of the pMC at hand that is used for checking sufficient conditions for being local monotonic. We define the pre-order and recap the "cheap" rules for efficiently determining the pre-order as adopted from [27]. We add a new, simple rule to this repertoire that lets us avoid the computationally "expensive" rules using assumptions from [27].…”

“…We define the pre-order and recap the "cheap" rules for efficiently determining the pre-order as adopted from [27]. We add a new, simple rule to this repertoire that lets us avoid the computationally "expensive" rules using assumptions from [27]. The information needed to apply this new rule readily comes from parameter lifting as we will see.…”

“…Approach. We propose to tackle the optimal synthesis problem by a deep integration of two seemingly unrelated techniques: monotonicity checking [27] and parameter lifting [25]. The former builds a partial order on the state space to check whether a pMC is (local or global) monotonic in a certain parameter, while the latter is an abstraction technique that "lifts" the parameter dependencies, obtaining interval MCs [17,21], and solves them in an iterative manner.…”