Proof Search and Co-NP Completeness for Many-Valued Logics Mattia Bongini Department of Mathematics, Munich University of Technology, Germany Agata Ciabattoni Department of Computer Languages, Vienna University of Technology, Austria Franco Montagna Department of Information Engineering and Mathematics, University of Siena, Italy Abstract We provide a methodology to introduce proof search oriented calculi for a large class of many-valued logics, and a sufficient condition for their Co-NP completeness. Our results apply to many well known logics including G¨ odel,  Lukasiewicz and Product Logic, as well as H´ ajek’s Basic Fuzzy Logic. 1 Introduction The invertibility of rules 1 in a proof system is an important feature for guiding proof search; in addition it turns out to be very useful to settle the compu- tational complexity of the formalized logic. For many-valued logics, calculi with invertible rules (proof search oriented calculi) have been provided for all finite-valued logics. These calculi are defined by generalizing Gentzen sequents A 1 ,...,A n ⇒ B 1 ,...,B m to many placed (or labelled) sequents, each corre- sponding to a truth value of the logic, see e.g. the survey [9] ([12], for the non-deterministic case). The construction of these calculi, out of the truth Email addresses: mattia.bongini@ma.tum.de (Mattia Bongini), agata@logic.at (Agata Ciabattoni), montagna@unisi.it (Franco Montagna). 1 The premises are derivable whenever their conclusions are. Preprint submitted to Elsevier Science 12 January 2015