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 $O(n \sqrt{m})$ symbolic operations and $O(\sqrt{m})$ 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 $\widetilde{O}(n^{1.5})$ symbolic operations and $\widetilde{O}(\sqrt{n})$ symbolic space. Moreover, the parametrization of our algorithm provides a trade-off 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 poly-logarithmic factors). Using our techniques we present faster algorithms for computing the almost-sure 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 almost-sure winning region with $\widetilde{O}(n^{2-\epsilon})$ symbolic operations and $\widetilde{O}(n^{\epsilon})$ symbolic space, for all $0 < \epsilon \leq 1/2$.
翻译:在概率校验中,我们为以下中心问题提出了一个更快捷的象征算法 : 计算最高端成份( 最大端成份( 最高端成份( 最高端成份( 最高端成) ) 象征价值( 最高端成份( 最高端成) 象征价值( MDPs ) 。 这个问题一般化了 SCC 分解图形和关闭的Markov 链。 在正式校验中, 象征性算法模式被广泛使用, 进入输入模型仅限于象征性操作( 例如, 基本集成操作和计算一阶区) 。 对于符号操作而言, 我们最坏的ODO( 最低端成) 象征性算值( 最低端成) 最低端值( 最低端成) 最低端值( 最低端成( 最低端成) 最低端值( 最低端值) 和最高空域域( 最低端成份) 最低度算( 最低端成品( 最高端成品) 最低度操作( 最低度) ( 最低端值) 最低端值( 最低端值) 最低点算( 最高值) 最低值) 和最高值) 最低值算( 最低值) 需要( 最高值) 最低操作( 最低点) ( O) 和最高值)