Labelled natural deduction for a bundled branching temporal logic

We give a sound and complete labelled natural deduction system for a bundled branching temporal logic, 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. We give both a classical and an intuitionistic version of our labelled natural deduction system for the until-free version of BCTL*, and carry out a proof-theoretical analysis of the intuitionistic system: we prove that derivations reduce to a normal form, which allows us to give a purely syntactical proof of consistency (for both the intuitionistic and classical versions) of the deduction system.