State complexity of combined operations with two basic operations

This paper studies the state complexity of (L1L2) , L1 L2, L ∗ 1L2, (L1 ∪L2)L3, (L1∩L2)L3, L1L2∩L3, and L1L2∪L3 for regular languages L1, L2, and L3. We first show that the upper bound proposed by [Liu, Martin-Vide, Salomaa, Yu, 2008] for the state complexity of (L1L2) R coincides with the lower bound and is thus the state complexity of this combined operation by providing some witness DFAs. Also, we show that, unlike most other cases, due to the structural properties of the result of the first operation of the combinations L1 L2, L ∗ 1L2, and (L1 ∪ L2)L3, the state complexity of each of these combined operations is close to the mathematical composition of the state complexities of the component operations. Moreover, we show that the state complexities of (L1 ∩L2)L3, L1L2 ∩ L3, and L1L2 ∪ L3 are exactly equal to the mathematical compositions of the state complexities of their component operations in the general cases. We also include a brief survey that summarizes all state complexity results for combined operations with two basic operations.