Unrestricted State Complexity of Binary Operations on Regular Languages

  • Janusz BrzozowskiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9777)


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 \(mn+1\), that of difference is \(mn+m+1\), and that of the product is \(m2^n+2^{n-1}\).


Boolean operation Concatenation Different alphabets Most complex languages Product Quotient complexity Regular language State complexity Stream Unrestricted complexity 



I am very grateful to Sylvie Davies, Bo Yang Victor Liu and Corwin Sinnamon for careful proofreading and constructive comments. I thank Marek Szykuła for contributing the important Remark 1 and Proposition 1.


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© IFIP International Federation for Information Processing 2016

Authors and Affiliations

  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

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