Accelerating Parametric Probabilistic Verification

Abstract

We present a novel method for computing reachability probabilities of parametric discrete-time Markov chains whose transition probabilities are fractions of polynomials over a set of parameters. Our algorithm is based on two key ingredients: a graph decomposition into strongly connected subgraphs combined with a novel factorization strategy for polynomials. Experimental evaluations show that these approaches can lead to a speed-up of up to several orders of magnitude in comparison to existing approaches. PreliminariesDefinition 1 (Discrete-time Markov chain). A discrete-time Markov chain (DTMC) is a tuple D = (S, I, P ) with a non-empty finite set S of states, an initial

Efficient Parametric Model Checking Using Domain Knowledge

“…A parametric Markov chain is an MC comprising transition probabilities P(s, s ) and/or rewards ρ(s) defined as rational functions over a set of continuous variables [20], [34], [36].…”

“…Parametric model checking (PMC) [20], [34], [36] is a formal technique for the analysis of Markov chains with transitions probabilities specified as rational functions over a set of continuous variables. When the analysed Markov chains model software systems, these variables represent configurable parameters of the software or environment parameters unknown until runtime.…”

“…PMC is supported by the model checkers PARAM [33], PRISM [40] and Storm [23]. However, despite significant advances in recent years [20], [34], [36], the current PMC techniques (which these model checkers implement) are computationally very expensive, generate expressions that are often extremely large and inefficient to evaluate, and do not support the analysis of parametric Markov chains modelling important classes of software systems.…”

Are Parametric Markov Chains Monotonic?

“…If assumptions are needed (l. [21][22][23][24][25][26], then we push the three assumptions to the Queue. Therefore, l. 27, should not be executed, so we continue to the next iteration of the while loop after adding assumptions (l. 26) At the end, we check for global monotonicity (l. [29][30][31][32][33][34][35][36][37].…”

“…This method contracts each SCC into a set of states, one for each entry state of the SCC. Applied to pMCs [31], it preserves the reachability probabilities of target set T . For each SCC, all nonentry states (i.e., states without incoming transitions from outside the SCC) are eliminated by state elimination [18,27] and transitions between entry states are deleted [19].…”