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Distinguishability Operations and Closures on Regular Languages

Given a regular language $L$, we study the language of words $\mathsf{D}(L)$, that distinguish between pairs of different left-quotients of $L$. We characterize this distinguishability operation, show that its iteration has always a fixed point, and we generalize this result to operations derived from closure operators and Boolean operators. We give an upper bound for the state complexity of the distinguishability operation, and prove its tightness. We show that the set of minimal words that can be used to distinguish between different left-quotients of a language $L$ has at most $n-1$ elements, where $n$ is the state complexity of $L$, and we also study the properties of its iteration. We generalize the results for the languages of words that distinguish between pairs of different right-quotients and two-sided quotients of a language $L$.