Proof theory for locally finite many-valued logics: Semi-projective logics

Abstract

We extend the methodology in Baaz and Fermüller (1999)  [5] to systematically construct analytic calculi for semi-projective logics—a large family of (propositional) locally finite many-valued logics. Our calculi, defined in the framework of sequents of relations, are proof search oriented and can be used to settle the computational complexity of the formalized logics. As a case study we derive sequent calculi of relations for Nilpotent Minimum logic and for Hajek’s Basic Logic extended with the $n$ -contraction axiom ( $n\ge 1$ ). The introduced calculi are used to prove that the decidability problem in these logics is Co-NP complete.