The Range of State Complexities of Languages Resulting from the Cut Operation
We investigate the state complexity of languages resulting from the cut operation of two regular languages represented by minimal deterministic finite automata with m and n states. We show that the entire range of complexities, up to the known upper bound, can be produced in the case when the input alphabet has at least two symbols. Moreover, we prove that in the unary case, only complexities up to \(2m-1\) and between n and \(m+n-2\) can be produced, while if \(2m\le n-1\), then the complexities from 2m up to \(n-1\) cannot be produced.
We thank Juraj Šebej and Jozef Jirásek Jr. for their help on border values in our theorems. Moreover, also thanks to Galina Jirásková for her support and to all who helped us to improve the presentation of the paper.
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