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 $L_m^{\prime }$ 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 $L_m^{\prime }$ 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 $L_m^{\prime }$ 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 $m{2}^n+2^{n-1}$.

13 pages, 6 figures. An earlier version is to appear in the proceedings of DCFS 2016. Two errors are corrected in the present arXiv version

Cited by in corpus (5)

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• Primitivity, Uniform Minimality and State Complexity of Boolean Operations
• Complexity of Prefix-Convex Regular Languages
• Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages
• A General Approach to State Complexity of Operations: Formalization and Limitations