Note: Regular languages with no conflicts (forbidden pairs) are regular but have exponential size DFA

Abstract : Many graph problems (Dominating sets, Steiner tree, etc.) are hard to optimize but finding a solution, regardless of its size, is in general trivial and polynomial. In recent papers, several authors introduced \emph{conflicts}, that are pairs of edges or vertices that cannot be both in a solution. These new constraints drastically improve the hardness: They proved that in most cases, deciding if there exists a solution is now NP-complete. In this short note we transport this problematic of conflicts in langage theory. Despite the negative results obtained in the field of discrete optimization, we show that a language LangP composed by the words of any regular language Lang that do not contain pairs of conflicting symbols is still regular. However, we show that the DFA accepting LangP that we construct has a non polynomial number of states. Nevertheless, we prove that this drawback cannot be avoided in general, even if a symbol is in conflict with at most another one, and seems to represent the price to pay for dealing with conflicts in regular languages.
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Alexis Cornet, Christian Laforest. Note: Regular languages with no conflicts (forbidden pairs) are regular but have exponential size DFA. [Research Report] LIMOS (UMR CNRS 6158), université Clermont Auvergne, France 2016. ⟨hal-01402160⟩

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