正则语言的Parikh映象：复杂性及其应用
Parikh Images of Regular Languages: Complexity and Applications
摘要：
我们证明了在一个k大小的字母表上具有n个状态的NFA语言的Parikh图像可以被描述为至多具有k个生成元和总大小为2^{O（k^2 log n）}的线性集的有限并集，即，多项式对于所有固定的k >= 1。在此之前，我们不知道生成器的数量是否可以与n无关，并且总大小的最佳上限是n的指数。此外，我们给出了一个算法执行这样的翻译时间2^{O（k^2 log（kn））}。我们的证明利用了一个以前未知的连接到凸集理论，并建立了一个范式定理的半线性集，这是独立的利益。为了补充这些结果，我们表明，我们的上界是紧的，结果不能扩展到上下文无关的语言。我们给出四个应用：（1）整数规划的一个新的多项式片段，（2）NFA的Parikh象的精确隶属复杂度，（3）半线性集的多项式PAC-可学习性的一个公开问题的答案，（4）离散时间反向有界计数器系统上LTL模型检验的一个最优算法.
英文摘要：
We show that the Parikh image of the language of an NFA with n states over an alphabet of size k can be described as a finite union of linear sets with at most k generators and total size 2^{O(k^2 log n)}, i.e., polynomial for all fixed k >= 1. Previously, it was not known whether the number of generators could be made independent of n, and best upper bounds on the total size were exponential in n. Furthermore, we give an algorithm for performing such a translation in time 2^{O(k^2 log(kn))}. Our proof exploits a previously unknown connection to the theory of convex sets, and establishes a normal form theorem for semilinear sets, which is of independent interests. To complement these results, we show that our upper bounds are tight and that the results cannot be extended to context-free languages. We give four applications: (1) a new polynomial fragment of integer programming, (2) precise complexity of membership for Parikh images of NFAs, (3) an answer to an open question about polynomial PAC-learnability of semilinear sets, and (4) an optimal algorithm for LTL model checking over discrete-timed reversal-bounded counter systems.
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