  Proof-theoretic methods are developed for subsystems of Johansson's logic
obtained by extending the positive fragment of intuitionistic logic with weak
negations. These methods are exploited to establish properties of the logical
systems. In particular, cut-free complete sequent calculi are introduced and
used to provide a proof of the fact that the systems satisfy the Craig
interpolation property. Alternative versions of the calculi are later obtained
by means of an appropriate loop-checking history mechanism. Termination of the
new calculi is proved, and used to conclude that the considered logical systems
are PSPACE-complete.
