Composition and orbits of language operations: finiteness and upper bounds

Abstract

We consider a set of eight natural operations on formal languages (Kleene closure, positive closure, complement, prefix, suffix, factor, subword, and reversal), and compositions of them. If x and y are compositions, we say x is equivalent to y if they have the same effect on all languages L. We prove that the number of equivalence classes of these eight operations is finite. This implies that the orbit of any language L under the elements of the monoid is finite and bounded, independent of L. This generalizes previous results about complement, Kleene closure, and positive closure. We also estimate the number of distinct languages generated by various subsets of these operations.

Keywords:

• formal language
• Kleene closure
• complement
• automaton
• orbit

2010 AMS Subject Classifications:

• 68Q45
• 20M35
• 20M05
• 54A05

Acknowledgements

We thank John Brzozowski, Ross Willard, and the referees for their comments.