We give a sound and complete labeled natural deduction system for an interesting fragment of CTL∗, namely the until-free version of BCTL∗. The logic BCTL∗ is obtained by referring to a more general semantics than that of CTL∗, where we only require that the set of paths in a model is closed under taking suffixes (i.e. is suffix-closed) and is closed under putting together a finite prefix of one path with the suffix of any other path beginning at the same state where the prefix ends (i.e. is fusion-closed). In other words, this logic does not enjoy the so-called limit-closure property of the standard CTL∗ validity semantics.
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