Non-deterministic Connectives in Propositional Godel Logic

We define the notion of a canonical Godel system in the framework of single-conclusion hypersequent calculi. A corresponding general (nondeterministic) Godel valuation semantics is developed, as well as a (non-deterministic) linear intuitionistic Kripke-frames semantics. We show that every canonical Godel system induces a class of Godel valuations (and of Kripke frames) for which it is strongly sound and complete. The semantics is used to identify the canonical systems that enjoy (strong) cut-admissibility, and to provide a decision procedure for these systems. The results of this paper characterize, both proof-theoretically and semantically, a large family of (non-deterministic) connectives that can be added to propositional Godel logic.