Counting Subwords and Regular Languages

@article{Colbourn2018CountingSA,
  title={Counting Subwords and Regular Languages},
  author={Charles J. Colbourn and Ryan E. Dougherty and Thomas Finn Lidbetter and Jeffrey Shallit},
  journal={ArXiv},
  year={2018},
  volume={abs/1804.11175}
}
Let x and y be words. We consider the languages whose words z are those for which the numbers of occurrences of x and y, as subwords of z, are the same (resp., the number of x’s is less than the number of y’s, resp., is less than or equal). We give a necessary and sufficient condition on x and y for these languages to be regular, and we show how to check this condition efficiently. 
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