State Complexity of Multiple Concatenation

TL;DR

This paper investigates the state complexity of multiple concatenation of regular languages, providing new bounds, witness languages, and solving an open problem regarding alphabet size and automata recognition.

Contribution

It introduces simpler proofs for upper bounds, identifies minimal alphabet sizes for witnesses, and resolves an open problem on state complexity bounds for multiple concatenation.

Findings

Witness languages meeting the upper bound are described with simpler proofs.

One symbol can be saved in the alphabet for certain automata.

The ternary alphabet is proven optimal for concatenation of three 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.