2019 # 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 mag…

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“…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.…”

confidence: 99%

“…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.…”

confidence: 99%

“…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.…”

confidence: 99%