New complexity results for Lukasiewicz logic

One aspect that has been poorly studied in multiple-valued logics, and in particular in Łukasiewicz logic, is the generation of instances of varying difficulty for evaluating, comparing and improving satisfiability solvers. With the ultimate goal of finding challenging benchmarks for Łukasiewicz satisfiability solvers, we start by defining a natural and intuitive class of clausal forms (simple Ł-clausal forms) and studying their complexity. Since we prove that the satisfiability problem of simple Ł-clausal forms can be solved in linear time, we then define two new classes of clausal forms (Ł-clausal forms and restricted Ł-clausal forms) that truly exploit the non-lattice operations of Łukasiewicz logic and whose satisfiability problems are NP-complete when clauses have at least three literals, and admit linear-time algorithms when clauses have at most two literals. We also define an efficient satisfiability preserving translation of Łukasiewicz logic formulas into Ł-clausal forms. Finally, we describe a random generator of Ł-clausal forms and report on an empirical investigation in which we identify an easy-hard-easy pattern and a phase transition phenomenon for Ł-clausal forms.This work was supported by Project LOGISTAR from the EU H2020 Research and Innovation Programme under Grant Agreement No. 769142, MINECO-FEDER Projects RASO (TIN2015-71799-C2-1-P) and LoCos (TIN2015-66293-R), and the UdG Project MPCUdG2016/055. Vidal acknowledges the support from the Project GA17-04630S of the Czech Science Foundation (GAČR).Peer reviewe