Labelled natural deduction for a bundled branching temporal logic ANDREA MASINI, LUCA VIGANÒ and MARCO VOLPE, Department of Computer Science, University ofVerona, Italy. E-mail: andrea.masini@univr.it; luca.vigano@univr.it; marco.volpe@univr.it Abstract 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. Keywords: Temporal logic, proof theory, natural deduction, labelled deduction. 1 Introduction The importance of temporal logic in computer science has become clear since the seminal work of Pnueli in 1977 [32]. Interesting applications include its use as a tool for the specification and verification of programs and protocols [5], in the study and development of temporal databases [9], as a framework within which to define the semantics of temporal expressions in natural language [25], and as a language for encoding temporal knowledge in artificial intelligence [20]. Many branching temporal logics have been proposed, varying both in the set of the operators used and in the semantics adopted (see [12, 23] for a survey). In particular, the branching-time logic CTL ∗ (full computation tree logic [14]) has been shown to be especially useful in developing and checking the correctness of reactive systems (see, e.g. [19, 26]). In spite of its great relevance, the problem of presenting a satisfactory deduction system or even a Hilbert-style axiomatization for such a logic has been, partially, solved only recently in [36]. The main difficulty encountered in finding a finitary axiomatization of CTL ∗ (and, in fact, such an axiomatization is still unknown, as discussed in, e.g. [36]) resides in the extreme difficulty to master the so-called limit-closure property of the standard CTL ∗ validity semantics. For this reason, a number of interesting sublogics of CTL ∗ have been proposed in the literature. Amongst these logics, a special role is played by BCTL ∗ [38]. The logic BCTL ∗ , which coincides with the logic ∀LTFC described in [41], 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 limit-closure property.