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Abstract: We show that the Parikh image of the language of an NFA with n states over analphabet of size k can be described as a finite union of linear sets with atmost k generators and total size 2^{Ok^2 log n}, i.e., polynomial for allfixed k >= 1. Previously, it was not known whether the number of generatorscould be made independent of n, and best upper bounds on the total size wereexponential in n. Furthermore, we give an algorithm for performing such atranslation in time 2^{Ok^2 logkn}. Our proof exploits a previously unknownconnection to the theory of convex sets, and establishes a normal form theoremfor semilinear sets, which is of independent interests. To complement theseresults, we show that our upper bounds are tight and that the results cannot beextended to context-free languages. We give four applications: 1 a newpolynomial fragment of integer programming, 2 precise complexity ofmembership for Parikh images of NFAs, 3 an answer to an open question aboutpolynomial PAC-learnability of semilinear sets, and 4 an optimal algorithmfor LTL model checking over discrete-timed reversal-bounded counter systems.



Author: Anthony Widjaja To

Source: https://arxiv.org/







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