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We introduce the two substructural propositional logics KL, KL+, which usedisjunction, fusion and a unary, (quasi-)exponential connective. For both weprove strong completeness with respect to the interpretation in Kleene algebrasand a variant thereof. We also prove strong completeness for language models,where each logic comes with a different interpretation. We show that for bothlogics the cut rule is admissible and both have a decidable consequencerelation.

Kleene Algebras, Regular Languages and Substructural Logics