Abstract
A language L is prefix-closed if, whenever a word w is in L, then every prefix of w is also in L. We define suffix-, factor-, and subword-closed languages in an analogous way, where by factor we mean contiguous subsequence, and by subword we mean scattered subsequence. We study the state complexity (which we prefer to call quotient complexity) of operations on prefix-, suffix-, factor-, and subword-closed languages. We find tight upper bounds on the complexity of the subword-closure of arbitrary languages, and on the complexity of boolean operations, concatenation, star, and reversal in each of the four classes of closed languages. We show that repeated applications of positive closure and complement to a closed language result in at most four distinct languages, while Kleene closure and complement give at most eight.
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Notes
In contrast to some authors, we use a set of initial states, since we require the reverse of an nfa to be an nfa.
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This research was supported by the Natural Sciences and Engineering Research Council of Canada grant OGP0000871, by VEGA grant 2/0183/11, and by grant APVV-0035-10.
A much shorter preliminary version of this work appeared as an arXiv preprint at http://arxiv.org/abs/0912.1034, and in F. Ablayev, E.W. Mayr, eds., Proc. 5th International Computer Science Symposium in Russia (CSR 2010). Volume 6072 of LNCS, Springer (2010), pp. 84–95.
Most of this work was done while C. Zou was at the David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada N2L 3G1.
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Brzozowski, J., Jirásková, G. & Zou, C. Quotient Complexity of Closed Languages. Theory Comput Syst 54, 277–292 (2014). https://doi.org/10.1007/s00224-013-9515-7
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DOI: https://doi.org/10.1007/s00224-013-9515-7