We show a new simple algorithm that solves the model-checking problem for recursion schemes: check whether the tree generated by a given higher-order recursion scheme is accepted by a given alternating parity automaton. The algorithm amounts to a procedure that transforms a recursion scheme of order $n$ to a recursion scheme of order $n-1$, preserving acceptance, and increasing the size only exponentially. After repeating the procedure $n$ times, we obtain a recursion scheme of order $0$, for which the problem boils down to solving a finite parity game. Since the size grows exponentially at each step, the overall complexity is $n$-EXPTIME, which is known to be optimal. More precisely, the transformation is linear in the size of the recursion scheme, assuming that the arity of employed nonterminals and the size of the automaton are bounded by a constant; this results in an FPT algorithm for the model-checking problem. Our transformation is a generalization of a previous transformation of the author (2020), working for reachability automata in place of parity automata. The step-by-step approach can be opposed to previous algorithms solving the considered problem "in one step", being compulsorily more complicated.

中文翻译：

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逐步检查高阶模型

我们展示了一种新的简单算法，该算法解决了递归方案的模型检查问题：检查由给定的高阶递归方案生成的树是否被给定的交替奇偶校验自动机接受。该算法相当于将$ n $阶递归方案转换为$ n-1 $阶递归方案，保留接受并仅按指数增加大小的过程。重复过程$ n $次后，我们获得了阶次为$ 0 $的递归方案，为此问题可归结为求解有限奇偶博弈。由于大小在每一步都呈指数增长，因此总体复杂度为$ n $ -EXPTIME，这是最佳的。更准确地说，转换在递归方案的大小上是线性的，假设使用的非终结符的稀疏性和自动机的大小受一个常数限制；这将导致针对模型检查问题的FPT算法。我们的转换是作者（2020）先前转换的概括，致力于可及性自动机代替奇偶校验自动机。逐步方法可能与以前的算法“一步一步”解决所考虑的问题相反，因为后者强制性地更加复杂。