Unrestricted State Complexity of Binary Operations on Regular Languages

TL;DR

This paper investigates how the state complexity of various binary operations on regular languages changes when the languages are over different alphabets, revealing new formulas for these complexities.

Contribution

It provides the first analysis of state complexity for binary operations on regular languages over different alphabets, extending known results for same-alphabet cases.

Findings

Union and symmetric difference have complexity mn + m + n + 1.

Intersection has complexity mn.

Difference has complexity mn + m.

Abstract

I study the state complexity of binary operations on regular languages over different alphabets. It is well known that if $L'_m$ and $L_n$ are languages restricted to be over the same alphabet, with $m$ and $n$ quotients, respectively, the state complexity of any binary boolean operation on $L'_m$ and $L_n$ is $mn$, and that of the product (concatenation) is $(m-1)2^n +2^{n-1}$. In contrast to this, I show that if $L'_m$ and $L_n$ are over their own different alphabets, the state complexity of union and symmetric difference is $mn+m+n+1$, that of intersection is $mn$, that of difference is $mn+m$, and that of the product is $m2^n+2^{n-1}$.