Real functions incrementally computable by finite automata
Michal Konečný2004, Theoretical Computer Science
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Abstract
This is an investigation into exact real number computation using the incremental approach of Potts (Ph.D. where numbers are represented as inÿnite streams of digits, each of which is a M obius transformation. The objective is to determine for each particular system of digits which functions R → R can be computed by a ÿnite transducer and ultimately to search for the most ÿnitely expressible M obius representations of real numbers. The main result is that locally such functions are either not continuously di erentiable or equal to some M obius transformation. This is proved using elementary properties of ÿnite transition graphs and M obius transformations. Applying the results to the standard signed-digit representations, we can classify functions that are ÿnitely computable in such a representation and are continuously di erentiable everywhere except for ÿnitely many points. They are exactly those functions whose graph is a fractured line connecting ÿnitely many points with rational coordinates.
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