Strict Canonical Constructive Systems

We define the notions of a canonical inference rule and a canonical constructive system in the framework of strict single-conclusion Gentzen-type systems (or, equivalently, natural deduction systems), and develop a corresponding general non-deterministic Kripke-style semantics. We show that every strict constructive canonical system induces a class of non-deterministic Kripke-style frames, for which it is strongly sound and complete. This non-deterministic semantics is used for proving a strong form of the cut-elimination theorem for such systems, and for providing a decision procedure for them. These results identify a large family of basic constructive connectives, including the standard intuitionistic connectives, together with many other independent connectives.