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**1 - 2**of**2**### Complete axiomatization of the stutter-invariant fragment of the linear time µ-calculus

, 2009

"... The logic µ(U) is the fixpoint extension of the “Until”-only fragment of linear-time temporal logic. It also happens to be the stutterinvariant fragment of linear-time µ-calculus µ(♦). We provide complete axiomatizations of µ(U) on the class of finite words and on the class of ω-words. We introduce ..."

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The logic µ(U) is the fixpoint extension of the “Until”-only fragment of linear-time temporal logic. It also happens to be the stutterinvariant fragment of linear-time µ-calculus µ(♦). We provide complete axiomatizations of µ(U) on the class of finite words and on the class of ω-words. We introduce for this end another logic, which we call µ(♦Γ), and which is a variation of µ(♦) where the Next time operator is replaced by the family of its stutter-invariant counterparts. This logic has exactly the same expressive power as µ(U). Using already known results for µ(♦), we first prove completeness for µ(♦Γ), which finally allows us to obtain completeness for µ(U).

### Complete Axiomatization of the Stutter-invariant Fragment of the Linear Time µ-calculus

"... The logic µTL(U) is the fixpoint extension of the “Until”-only fragment of linear-time temporal logic. It also happens to be the stutter-invariant fragment of linear-time µ-calculus µTL. We provide complete axiomatizations of µTL(U) on the class of finite words and on the class of ω-words. We introd ..."

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The logic µTL(U) is the fixpoint extension of the “Until”-only fragment of linear-time temporal logic. It also happens to be the stutter-invariant fragment of linear-time µ-calculus µTL. We provide complete axiomatizations of µTL(U) on the class of finite words and on the class of ω-words. We introduce for this end another logic, which we call µTL(♦Γ), and which is a variation of µTL where the Next time operator is replaced by the family of its stutter-invariant counterparts. This logic has exactly the same expressive power as µTL(U). Using already known results for µTL, we first prove completeness for µTL(♦Γ), which finally allows us to obtain completeness for µTL(U).