A connection between the Cantor–Bendixson derivative and the well-founded semantics of finite logic programs

Results of Schlipf (J Comput Syst Sci 51:64–86, 1995) and Fitting (Theor Comput Sci 278:25–51, 2001) show that the well-founded semantics of a finite predicate logic program can be quite complex. In this paper, we show that there is a close connection between the construction of the perfect kernel of a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Pi^0_1$\end{document} class via the iteration of the Cantor–Bendixson derivative through the ordinals and the construction of the well-founded semantics for finite predicate logic programs via Van Gelder’s alternating fixpoint construction. This connection allows us to transfer known complexity results for the perfect kernel of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Pi^0_1$\end{document} classes to give new complexity results for various questions about the well-founded semantics \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathit{wfs}}(P)$\end{document} of a finite predicate logic program P.