Fractal Completeness Techniques in Topological Modal Logic: Koch Curve, Limit Tree, and the Real Line

This paper explores the connection between fractal geometry and topological modal logic. In the early 1940’s, Tarski showed that the modal logic S4 can be interpreted in topological spaces. Since then, many interesting completeness results in the topological semantics have come to light, and renewed interest in this semantics can be seen in such recent papers as [1], [2], [3], [4], and [7]. In this paper we introduce the use of fractal techniques for proving completeness of S4 and non-trivial extensions of S4 for a variety of topological spaces. The main results of the paper are completeness of S4 for the binary tree with limits (or Wilson tree), and completeness of S4 for the Koch Curve, a well known fractal curve. An important corollary is a new and very much simplified proof of completeness of S4 for the real line, R (originally proved by Tarski and McKinsey in [9]). These results are best seen as examples of the power of the fractal techniques introduced. The main technique is developed to relate formally the somewhat peculiar non-Hausdorff tree topologies with more familiar Euclidean and other metric topologies. As we argue in the paper, the techniques developed here can be usefully applied to a wide range of completeness problems in the topological semantics. This paper explores the connection between fractal geometry and topological modal logic. The main result of the paper is a proof of completeness of the modal logic S4 with respect to the Koch Curve, a well known fractal curve. An important corollary is a new proof of completeness of S4 for the real line, R. The latter result was originally obtained by Tarski and McKinsey in [9], and much simplified and refined by Mints et al. and van Benthem et al.. Our new proof uses fractal techniques, that, as we will argue in the last section, are the paper’s main contribution to the topological semantics for modal logic. Completeness for both the Koch Curve and R are best seen as examples of the power of the fractal techniques introduced. The results of Section 4 and the techniques above are not tailor-made for solving completeness of S4 for the real line or for the slightly wider problem of completeness of S4 with respect to interesting classes of metric topological models. Rather, they should be seen as tools for obtaining completeness results for a larger variety of languages and with respect to the full range of Euclidean and other metric topologies. The main technique is developed to relate formally the somewhat peculiar non-Hausdorff tree topologies with more familiar Euclidean ∗University of California, Berkeley. Email: [email protected] †Colorado State University. Email: [email protected] 1Examples, variations, and refinements of this proof can be found in [1], [4], and [7].