Real functions incrementally computable by finite automata

Michal Konečný*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


This is an investigation into exact real number computation using the incremental approach of Potts (Ph.D. Thesis, Department of Computing, Imperial College, 1998), Edalat and Potts (Electronic Notes in Computer Science, Vol. 6, Elsevier Science Publishers, Amsterdam, 2000), Nielsen and Kornerup (J. Universal Comput. Sci. 1(7) (1995) 527), and Vuillemin (IEEE Trans. on Comput. 39(8) (1990) 1087) where numbers are represented as infinite streams of digits, each of which is a Möbius transformation. The objective is to determine for each particular system of digits which functions R→R can be computed by a finite transducer and ultimately to search for the most finitely expressible Möbius representations of real numbers. The main result is that locally such functions are either not continuously differentiable or equal to some Möbius transformation. This is proved using elementary properties of finite transition graphs and Möbius transformations. Applying the results to the standard signed-digit representations, we can classify functions that are finitely computable in such a representation and are continuously differentiable everywhere except for finitely many points. They are exactly those functions whose graph is a fractured line connecting finitely many points with rational coordinates.

Original languageEnglish
Pages (from-to)109-133
Number of pages25
JournalTheoretical Computer Science
Issue number1
Publication statusPublished - 5 May 2004


  • Finite automaton
  • Möbius transformation
  • Real number computation
  • Sub-self-similarity


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