It is well-known that the winning region of a parity game with $n$ nodes and $k$ priorities can be computed as a $k$-nested fixpoint of a suitable function; straightforward computation of this nested fixpoint requires $\mathcal{O}(n^{\frac{k}{2}})$ iterations of the function. Calude et al.'s recent quasipolynomial-time parity game solving algorithm essentially shows how to compute the same fixpoint in only quasipolynomially many iterations by reducing parity games to quasipolynomially sized safety games. Universal graphs have been used to modularize this transformation of parity games to equivalent safety games that are obtained by combining the original game with a universal graph. We show that this approach naturally generalizes to the computation of solutions of systems of \emph{any} fixpoint equations over finite lattices; hence, the solution of fixpoint equation systems can be computed by quasipolynomially many iterations of the equations. We present applications to modal fixpoint logics and games beyond relational semantics. For instance, the model checking problems for the energy $\mu$-calculus, finite latticed $\mu$-calculi, and the graded and the (two-valued) probabilistic $\mu$-calculus -- with numbers coded in binary -- can be solved via nested fixpoints of functions that differ substantially from the function for parity games but still can be computed in quasipolynomial time; our result hence implies that model checking for these $\mu$-calculi is in QP. Moreover, we improve the exponent in known exponential bounds on satisfiability checking.

翻译：众所周知, 以 $ 节点和 $ k$ 优先级的对等游戏的赢区域可以被计算成 一个合适的函数的 $k$ 受命的固定点; 这个嵌入的固定点的简单计算需要 $mathcal{O} (n\frac{k ⁇ 2\\\\\\\\\美元) 函数的迭代。 Calude et al. 的最近准Polynomioal- 时间对等游戏解算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算。 通用图形将平价游戏转换为等值的安全游戏。 通用图形已被使用, 将平价游戏的变换算算法化为等值的游戏, 将原始游戏和通用图形组合的平价调算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算法, 。 固定点平位数可以由 精度变数计算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算法, 。 。 。 。 。 美元 。 美元 。 。, 平级算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算算法, 。