Universality in two dimensions

Nachum Dershowitz, Gilles Dowek

Research output: Contribution to journalArticlepeer-review


Turing, in his immortal 1936 paper, observed that '[human] computing is normally done by writing... symbols on [two-dimensional] paper', but noted that use of a second dimension 'is always avoidable' and that 'the two-dimensional character of paper is no essential of computation'. We propose to promote two-dimensional models of computation and exploit the naturalness of two-dimensional representations of data. In particular, programs for a two-dimensional Turing machine can be recorded most naturally on its own two-dimensional input-output grid in such a transparent fashion that schoolchildren would have no difficulty comprehending their behaviour. This two-dimensional rendering allows, furthermore, for a most perspicacious rendering of Turing's universal machine.

Original languageEnglish
Pages (from-to)143-167
Number of pages25
JournalJournal of Logic and Computation
Issue number1
StatePublished - 14 Oct 2012


  • Turing machines
  • computer-science education
  • two-dimensional models
  • universality


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