The state complexity of L^2 and L^k

TL;DR

This paper investigates the worst-case state complexity of language powers, revealing specific formulas for general and unary alphabets, thereby advancing understanding of automata state requirements for language concatenations.

Contribution

The paper provides exact worst-case state complexity formulas for L^2 and L^k, including the unary case, improving theoretical understanding of automata state growth.

Findings

State complexity of L^2 is n*2^n - 2^{n-1} for general alphabets.

State complexity of L^k is kn - k + 1 for unary alphabets, for all k >= 2.

Results clarify how automata size scales with language power and alphabet type.

Abstract

We show that if M is a DFA with n states over an arbitrary alphabet and L = L(M), then the worst-case state complexity of L^2 is n*2^n - 2^{n-1}. If, however, M is a DFA over a unary alphabet, then the worst-case state complexity of L^k is kn-k+1 for all k >= 2.