Symbolic Time and Space Tradeoffs for Probabilistic Verification
Abstract
We present a faster symbolic algorithm for the following central problem in probabilistic verification: Compute the maximal endcomponent (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 modelchecking, where access to the input model is restricted to only symbolic operations (e.g., basic set operations and computation of onestep 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 $O(n \sqrt{m})$ symbolic operations and $O(\sqrt{m})$ symbolic space. A main open question is whether the worstcase $O(n^2)$ bound for symbolic operations can be beaten. We present a symbolic algorithm that requires $\widetilde{O}(n^{1.5})$ symbolic operations and $\widetilde{O}(\sqrt{n})$ symbolic space. Moreover, the parametrization of our algorithm provides a tradeoff between symbolic operations and symbolic space: for all $0<\epsilon \leq 1/2$ the symbolic algorithm requires $\widetilde{O}(n^{2\epsilon})$ symbolic operations and $\widetilde{O}(n^{\epsilon})$ symbolic space ($\widetilde{O}$ hides polylogarithmic factors). Using our techniques we present faster algorithms for computing the almostsure winning regions of $\omega$regular objectives for MDPs. We consider the canonical parity objectives for $\omega$regular objectives, and for parity objectives with $d$priorities we present an algorithm that computes the almostsure winning region with $\widetilde{O}(n^{2\epsilon})$ symbolic operations and $\widetilde{O}(n^{\epsilon})$ symbolic space, for all $0 < \epsilon \leq 1/2$.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 arXiv:
 arXiv:2104.07466
 Bibcode:
 2021arXiv210407466C
 Keywords:

 Computer Science  Logic in Computer Science;
 Computer Science  Computer Science and Game Theory
 EPrint:
 Accepted at LICS'21