Fractals from Regular Behaviours

Authors Todd Schmid , Victoria Noquez , Lawrence S. Moss

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Author Details

Todd Schmid
  • University College, London, UK
Victoria Noquez
  • St. Mary’s College of California, Moraga, CA, USA
Lawrence S. Moss
  • Indiana University, Bloomington, IN, USA


We would like to thank Alexandra Silva and Dylan Thurston for helpful discussions. The images in Figures 1 and 2 were made using and

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Todd Schmid, Victoria Noquez, and Lawrence S. Moss. Fractals from Regular Behaviours. In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We are interested in connections between the theory of fractal sets obtained as attractors of iterated function systems and process calculi. To this end, we reinterpret Milner’s expressions for processes as contraction operators on a complete metric space. When the space is, for example, the plane, the denotations of fixed point terms correspond to familiar fractal sets. We give a sound and complete axiomatization of fractal equivalence, the congruence on terms consisting of pairs that construct identical self-similar sets in all interpretations. We further make connections to labelled Markov chains and to invariant measures. In all of this work, we use important results from process calculi. For example, we use Rabinovich’s completeness theorem for trace equivalence in our own completeness theorem. In addition to our results, we also raise many questions related to both fractals and process calculi.

Subject Classification

ACM Subject Classification
  • Theory of computation → Process calculi
  • fixed-point terms
  • labelled transition system
  • fractal
  • final coalgebra
  • equational logic
  • completeness


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