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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 $$L_n$$ 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 $$m2^n+2^{n-1}$$.
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Janusz Brzozowski. Unrestricted State Complexity of Binary Operations on Regular Languages. 18th International Workshop on Descriptional Complexity of Formal Systems (DCFS), Jul 2016, Bucharest, Romania. pp.60-72, ⟨10.1007/978-3-319-41114-9_5⟩. ⟨hal-01633951⟩

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