Computational complexities of axiomatic extensions of monoidal t-norm based logic

We study the computational complexity of some axiomatic extensions of the monoidal t-Norm based logic (MTL), namely NM corresponding to the logic of the so-called nilpotent minimum t-norm (due to Fodor [8]); and SMTL corresponding to left-continuous strict t-norms, introduced by Esteva (and others) in [4] and [5]. In particular, we show that the sets of 1-satisfiable and positively satisfiable formulae of both NM and SMTL are NP-complete, while the set of 1tautologies of NM and the set of positive tautologies of both NM and SMTL are co-NP-complete. The set of 1-tautologies of SMTL is only shown to be co-NP-hard, and it remains open if this set is in co-NP. Also, some results on the relations between these sets are obtained.