On the State Complexity of the Shuffle of Regular Languages

TL;DR

This paper establishes an upper bound on the state complexity of the shuffle operation on regular languages, provides partial results on tightness, and discusses the conditions under which this bound can be achieved.

Contribution

It introduces a new upper bound for the state complexity of the shuffle of regular languages and analyzes conditions for its tightness and reachability.

Findings

Upper bound on state complexity: f(m,n)

Existence of witness languages meeting the bound for certain m, n

All states in the subset automaton can be distinguishable with a small alphabet

Abstract

We investigate the shuffle operation on regular languages represented by complete deterministic finite automata. We prove that $f(m,n)=2^{mn-1} + 2^{(m-1)(n-1)}(2^{m-1}-1)(2^{n-1}-1)$ is an upper bound on the state complexity of the shuffle of two regular languages having state complexities $m$ and $n$, respectively. We also state partial results about the tightness of this bound. We show that there exist witness languages meeting the bound if $2\le m\le 5$ and $n\ge2$, and also if $m=n=6$. Moreover, we prove that in the subset automaton of the NFA accepting the shuffle, all $2^{mn}$ states can be distinguishable, and an alphabet of size three suffices for that. It follows that the bound can be met if all $f(m,n)$ states are reachable. We know that an alphabet of size at least $mn$ is required provided that $m,n \ge 2$. The question of reachability, and hence also of the tightness of the bound $f(m,n)$ in general, remains open.