Mehrdad Maleki: Differentiation in logical form

Differentiation in Logical Form 

Mehrdad Maleki

Institute for Research in Fundamental Sciences (IPM)

Tehran, Iran 

Abstract: We introduce a logical theory of differentiation for a real-valued func- tion on a finite dimensional real Euclidean space. A real-valued continuous function is represented by a localic approximable mapping between two semi-strong proximity lattices, representing the two coherent Euclidean spaces for the domain and the range of the function. Similarly, the Clarke subgradient, equivalently the L-derivative, of a locally Lipschitz map, which is non-empty, compact and convex valued, is represented by an approximable mapping. Corresponding to the notion of a single-tie of a locally Lipschitz function, used to derive the domain-theoretic L-derivative of the function, we introduce the dual notion of a single-knot of approximable mappings which gives rise to Lipschitzian approximable mappings. We then develop the notion of a strong single-tie and that of a strong knot leading to a Stone duality result for locally Lipschitz maps and Lipschitzian approximable mappings: a locally Lipschitz map belongs to a strong single-tie associated with a compact and convex set with non-empty interior if and only if the corresponding approximable mapping belongs to the strong single-knot associated with the interior of the compact convex set. The strong single-knots, in which a Lipschitzian approximable mapping belongs, are employed to define the Lipschitzian derivative of the approximable mapping. The latter is dual to the Clarke subgradient of the corresponding locally Lipschitz map defined domain- theoretically using strong single-ties. A stricter notion of strong single- knots is subsequently developed which captures approximable mappings of continuously differentiable maps providing a gradient Stone duality for these maps. Finally, we derive a calculus for Lipschitzian derivative of approximable mapping for some basic constructors and show that it is dual to the calculus satisfied by the Clarke subgradient. This is a joint work with Abbas Edalat.