Free-Variable Semantic Tableaux for the Logic of Fuzzy Inequalities

We present a free-variable tableau calculus for the logic of fuzzy inequalities F∀, which is an extension of infinite-valued first-order Lukasiewicz logic L∀. The set of all L∀-sentences provable in the hypersequent calculus of Baaz and Metcalfe for L∀ is embedded into the set of all F∀-sentences provable in the given tableau calculus. We prove NPcompleteness of the problem of checking tableau closability and propose an algorithm, which is based on unification, for solving the problem.