Marchioni, Enrico; Montagna, Franco Complexity and definability issues in Ł\(\Pi \frac 1 2\). (English) Zbl 1142.03015 J. Log. Comput. 17, No. 2, 311-331 (2007). This paper is devoted to the logic \textŁ\(\Pi\frac 1 2\), which unites Łukasiewicz logic and product logic and, in addition, has a constant interpreted by \(\frac 1 2\). It is first shown that there is a way to translate any quantifier-free formula \(\Phi\) of the language of the real closed field \(\mathcal R\) of real numbers to a formula \(t(\Phi)\) of \textŁ\(\Pi\frac 1 2\) such that \(\Phi\) is valid in \(\mathcal R\) if and only if \(t(\Phi)\) is valid in \textŁ\(\Pi\frac 1 2\). One consequence is that \textŁ\(\Pi\frac 1 2\){} and the universal theory of \(\mathcal R\) have the same complexity. Furthermore, several definability issues are discussed. For instance, if the graph of a left-continuous t-norm \(\star\) is definable by means of a (two-valued) \textŁ\(\Pi\frac 1 2\){} formula, then the logic based on \(\star\) and the corresponding residuum, is in PSPACE. Finally, it is shown for several well-known fuzzy logics which are, like e.g. BL, based on specific t-norms and their residua, that completeness theorems still hold when restricting the respective class of t-norms to those which are term-definable in \textŁ\(\Pi\frac 1 2\).We note that in the definition of an \textŁ\(\Pi\frac 1 2\){} algebra, the constant \(\frac 1 2\) was accidentally omitted. Reviewer: Thomas Vetterlein (Wien) Cited in 5 Documents MSC: 03B52 Fuzzy logic; logic of vagueness 03D15 Complexity of computation (including implicit computational complexity) Keywords:real closed fields; fuzzy logic; left-continuous t-norms; decidability PDFBibTeX XMLCite \textit{E. Marchioni} and \textit{F. Montagna}, J. Log. Comput. 17, No. 2, 311--331 (2007; Zbl 1142.03015) Full Text: DOI