Abstract
We develop a domain-theoretic Differential Calculus for locally Lipschitz functions on finite dimensional real spaces with imprecise input/output. The inputs to these functions are hyper-rectangles and the outputs are compact real intervals. This extends the domain of application of Interval Analysis and exact arithmetic to the derivative. A new notion of a tie for these functions is introduced, which in one dimension represents a modification of the notion previously used in the one-dimensional framework. A Scott continuous sub-differential for these functions is then constructed, which satisfies a weaker form of calculus compared to that of the Clarke sub-gradient. We then adopt a Program Logic viewpoint using the equivalence of the category of stably locally compact spaces with that of semi-strong proximity lattices. We show that given a localic approximable mapping representing a locally Lipschitz map with imprecise input/output, a localic approximable mapping for its sub-differential can be constructed, which provides a logical formulation of the sub-differential operator.
Highlights
A well-known hurdle in numerical computation is caused by accumulation of round-off errors in floating point arithmetic, which can create havoc and lead to catastrophic errors in compound calculations
In safety and critical systems, where reliability of numerical computation is of utmost importance, one way to avoid the pitfalls of floating point arithmetic is to use interval analysis or exact arithmetic
The induced logical framework cannot be employed for the class of functions with imprecise input/output used in exact computation since, as already pointed out, this class necessarily contains general extensions of real-valued locally Lipschitz maps of finite dimensional Euclidean spaces
Summary
A well-known hurdle in numerical computation is caused by accumulation of round-off errors in floating point arithmetic, which can create havoc and lead to catastrophic errors in compound calculations. In [13], a point free framework for sub-differentiation of real-valued locally Lipschitz functions on finite dimensional Euclidean spaces has been developed which provides a Stone duality for the Clarke gradient and enables a program logic view of differentiation. The induced logical framework cannot be employed for the class of functions with imprecise input/output used in exact computation since, as already pointed out, this class necessarily contains general extensions of real-valued locally Lipschitz maps of finite dimensional Euclidean spaces. We formulate a new notion of a tie of functions with imprecise input/output, which, in one dimension, represents a modification of the corresponding notion in [12] This allows us to develop a Scott continuous subdifferential for functions with hyper-rectangles in Rn as inputs and compact intervals in R as output, which are used in exact computation. The basic Stone duality results developed in [13] are extended to sub-differentiation of such interval maps
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