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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(More)
We prove completeness of the propositional modal logic S4 for the measure algebra based on the Lebesgue-measurable subsets of the unit interval, [0, 1]. In recent talks, Dana Scott introduced a new measure-based semantics for the standard propositional modal language with Boolean connec-tives and necessity and possibility operators, and ♦. Propositional(More)
xxx Working notes. Three main things remain to do. (1) Either include Tamar's argument [now in separate document] for Theorem 5, or replace by some better construction for worst-case upper bound. (2) Put in the average-case lower bound calculus calculations needed in section 6. (3) General version of the worst-case lower bound method in section 7.3. A(More)
Consider a network linking the points of a rate-1 Poisson point process on the plane. Write ave (s) for the minimum possible mean length per unit area of such a network, subject to the constraint that the route-length between every pair of points is at most s times the Euclidean distance. We give upper and lower bounds on the function ave (s), and on the(More)
This paper brings together Dana Scott's measure-based semantics for the propositional modal logic S4, and recent work in Dynamic Topological Logic. In a series of recent talks, Scott showed that the language of S4 can be interpreted in the Lebesgue measure algebra, M, or algebra of Borel subsets of the real interval, [0, 1], modulo sets of measure zero.(More)
Probabilistic Semantics for Modal Logic We develop a probabilistic semantics for modal logic, which was introduced in recent years by Dana Scott. This semantics is intimately related to an older, topological semantics for modal logic developed by Tarski in the 1940's. Instead of interpreting modal languages in topological spaces, as Tarski did, we interpret(More)