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Normality and Differentiability

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Semantics and Syntax: A Legacy of Alan Turing

By transferring to the world of functions computable by finite automata the classical theorem of numerical analysis establishing that every non-decreasing real valued function is almost everywhere differentiable, we obtain a characterization of the property of Borel normality. We consider functions mapping infinite sequences to infinite sequences and a notion of differentiability that, on the class of non-decreasing real valued functions, coincides with standard differentiability. We prove that the following are equivalent, for a real x in [0,1]:

(1) x is normal to base b.

(2) Every non-decreasing function computable by a finite automaton mapping infinite sequences to infinite sequences is differentiable at the expansion of x in base b.

(3) Every non-decreasing function computable by a finite automaton in base b mapping real numbers to real numbers is differentiable at x.

Joint work with Vernica Becher, Universidad de Buenos Aires.

This talk is part of the Isaac Newton Institute Seminar Series series.

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