On Countermodels in Basic Logic

In [3] the tautology problem for Hajek’s Basic Logic BL is proved to be coNP-complete by showing that if a formula φ is not a tautology of BL then there exists an integer m > 0, polynomially bounded by the length of φ, such that φ fails to be a tautology in the infinite-valued logic mA L corresponding to the ordinal sum of m copies of the A Lukasiewicz t-norm. In this paper we state that if φ is not a tautology of BL then it already fails to be a tautology of a finite set of finitevalued logics, defined by taking the ordinal sum of m copies of k-valued A Lukasiewicz logics, for effectively determined integers m, k > 0 only depending on polynomialtime computable features of φ. This result allows the definition of a calculus for mA L along the lines of [1, 2], while the analysis of features of functions associated with formulas of mA L constitutes a step toward the characterization of finitely generated free BL-algebras as algebras of [0, 1]-valued functions.