Weierstrass approximations by Lukasiewicz formulas with one quantified variable

The logic 3L of continuous piecewise linear functions with rational coefficients has enough expressive power to formalize Weierstrass approximation theorem. Thus, up to any prescribed error, every continuous (control) function can be approximated by a formula of 3L. As shown in this paper, 3L is just infinite-valued Lukasiewicz propositional logic with one quantified propositional variable. We evaluate the computational complexity of the derision problem for 3L. Enough background material is provided for all readers wishing to acquire a deeper understanding of the rapidly growing literature on Lukasiewicz propositional logic and its applications.