# Symbolic Time and Space Tradeoffs for Probabilistic Verification

We present a faster symbolic algorithm for the following central problem in probabilistic verification: Compute the maximal end-component (MEC) decomposition of Markov decision processes (MDPs). This problem generalizes the SCC decomposition problem of graphs and closed recurrent sets of Markov chains. The model of symbolic algorithms is widely used in formal verification and model-checking, where access to the input model is restricted to only symbolic operations (e.g., basic set operations and computation of one-step neighborhood). For an input MDP with n vertices and m edges, the classical symbolic algorithm from the 1990s for the MEC decomposition requires O(n^2) symbolic operations and O(1) symbolic space. The only other symbolic algorithm for the MEC decomposition requires O(n √(m)) symbolic operations and O(√(m)) symbolic space. A main open question is whether the worst-case O(n^2) bound for symbolic operations can be beaten. We present a symbolic algorithm that requires O(n^1.5) symbolic operations and O(√(n)) symbolic space. Moreover, the parametrization of our algorithm provides a trade-off between symbolic operations and symbolic space: for all 0<ϵ≤ 1/2 the symbolic algorithm requires O(n^2-ϵ) symbolic operations and O(n^ϵ) symbolic space (O hides poly-logarithmic factors). Using our techniques we present faster algorithms for computing the almost-sure winning regions of ω-regular objectives for MDPs. We consider the canonical parity objectives for ω-regular objectives, and for parity objectives with d-priorities we present an algorithm that computes the almost-sure winning region with O(n^2-ϵ) symbolic operations and O(n^ϵ) symbolic space, for all 0 < ϵ≤ 1/2.

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