State complexity of inversion operations

The reversal operation is well-studied in the literature and the deterministic (respectively, nondeterministic) state complexity of reversal is known to be 2 n (respectively, n). We consider the inversion operation where some substring of the given string is reversed. Formally, the inversion (respectively, prefix-inversion) of a language L consists of all strings u x R v such that u x v ∈ L (respectively, all strings u R x where u x ∈ L ). We show that the nondeterministic state complexity of prefix-inversion is Θ ( n 2 ) and that of inversion is Θ ( n 3 ). We show that the deterministic state complexity of prefix-inversion is at most 2 n ⋅ log ⁡ n + n and has lower bound 2 Ω ( n log ⁡ n ). The same lower bound holds for the state complexity of inversion, but for inversion we do not have a matching upper bound. We also study the state complexity of other variants of the inversion operation.

Authors

• Da-Jung Cho
• Yo-Sub Han
• Sang-Ki Ko
• Kai Salomaa

Keywords

• State complexity
• Inversion operations
• Finite automata
• Regular languages