On state complexity for subword-closed languages
Jérôme Guyot
TL;DR
This paper investigates state complexity for subword-closed (downward-closed) and superword-closed (upward-closed) languages under two operators: the $k$-th root operator $\sqrt[k]{L}$ (and the union $\sqrt[*]{L}$) and the substitution operator $L^{a \leftarrow K}$. It proves exponential lower bounds for the $k$-th root on the superword-closed side and a quadratic upper bound for a specific instance of the square root in the subword-closed case, while the substitution operator yields exponential lower bounds in general and quadratic upper bounds under restrictions such as disjoint alphabets or directed input language $L$. These results advance the understanding of state complexity within subregular classes and have implications for verification tasks that rely on well-quasi-ordered domains. The work combines quotient-based techniques, divisibility arguments, and constructive analyses of substitutions to map the growth of state complexity across these operations.
Abstract
This paper investigates the state complexities of subword-closed and superword-closed languages, comparing them to regular languages. We focus on the square root operator and the substitution operator. We establish an exponential lower bound for superword-closed languages for the k-th root. For subword-closed languages we analyze in detail a specific instance of the square root problem for which a quadratic complexity is proven. For the substitution operator, we show an exponential lower bound for the general substitution. We then find some conditions for which we prove a quadratic upper bound.