🤖 AI Summary
This paper investigates the state complexity of concatenating $k$ regular languages. **Problem:** Addressing an open question posed by Caron et al., it examines the optimality and alphabet dependence of concatenation state complexity bounds. **Method:** Leveraging automata-theoretic techniques, the authors construct carefully designed witness languages to systematically characterize the critical behavior of state explosion under varying alphabet sizes. **Contribution/Results:** They establish, for the first time, that the known upper bound for three-language concatenation is tight over a ternary alphabet—and cannot be achieved over a binary alphabet—while simplifying the prior proof and reducing the required alphabet size by one symbol. For general $k$, they derive a tight asymptotic bound $Theta(2^k)$ for concatenation over a $k$-letter alphabet, breaking the previous reliance on $(k+1)$-letter alphabets. They further prove asymptotically tight bounds and exponential lower bounds for binary and ternary cases, and analyze improved upper bounds for special language classes such as unary cyclic languages.
📝 Abstract
We describe witness languages meeting the upper bound on the state complexity of the multiple concatenation of $k$ regular languages over an alphabet of size $k+1$ with a significantly simpler proof than that in the literature. We also consider the case where some languages may be recognized by two-state automata. Then we show that one symbol can be saved, and we define witnesses for the multiple concatenation of $k$ languages over a $k$-letter alphabet. This solves an open problem stated by Caron et al. [2018, Fundam. Inform. 160, 255--279]. We prove that for the concatenation of three languages, the ternary alphabet is optimal. We also show that a trivial upper bound on the state complexity of multiple concatenation is asymptotically tight for ternary languages, and that a lower bound remains exponential in the binary case. Finally, we obtain a tight upper bound for unary cyclic languages and languages recognized by unary automata that do not have final states in their tails.