Unrestricted state complexity of binary operations on regular languages

Abstract

I study the state complexity of binary operations on regular languages over different alphabets. It is well known that if L’m and Ln 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 Ln is mn, and that of the product (concatenation) is (m-1)2n +2n-1. In contrast to this, I show that if L’m and Ln are over their own different alphabets, the state complexity of union and symmetric difference is mn +m+ n + 1, that of intersection is mn+1, that of difference is mn + m + 1, and that of the product is m2n + 2n-1.