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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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  1. Jos C. M. Baeten. A brief history of process algebra. Theor. Comput. Sci., 335(2-3):131-146, 2005. URL:
  2. Stefan Banach. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae, 3:133-181, 1922. Google Scholar
  3. Jan A. Bergstra, Alban Ponse, and Scott A. Smolka, editors. Handbook of Process Algebra. North-Holland / Elsevier, 2001. URL:
  4. Prasit Bhattacharya, Lawrence S. Moss, Jayampathy Ratnayake, and Robert Rose. Fractal sets as final coalgebras obtained by completing an initial algebra. In Franck van Breugel, Elham Kashefi, Catuscia Palamidessi, and Jan Rutten, editors, Horizons of the Mind. A Tribute to Prakash Panangaden - Essays Dedicated to Prakash Panangaden on the Occasion of His 60th Birthday, volume 8464 of Lecture Notes in Computer Science, pages 146-167. Springer, 2014. Google Scholar
  5. Stephen L. Bloom and Zoltán Ésik. Iteration Theories - The Equational Logic of Iterative Processes. EATCS Monographs on Theoretical Computer Science. Springer, 1993. URL:
  6. Stephen L. Bloom and Zoltán Ésik. Solving polynomial fixed point equations. In Mathematical foundations of computer science 1994 (Košice, 1994), volume 841 of Lecture Notes in Comput. Sci., pages 52-67. Springer, Berlin, 1994. Google Scholar
  7. Graeme C. Boore. Directed graph iterated function systems. PhD thesis, University of St. Andrews, 2011. URL:
  8. G. A. Edgar and R. Daniel Mauldin. Multifractal decompositions of digraph recursive fractals. Proceedings of the London Mathematical Society, s3-65(3):604-628, 1992. URL:
  9. Gerald A. Edgar. Measure, Topology, and Fractal Geometry. Springer New York, NY, 1 edition, 1990. URL:
  10. Kenneth J. Falconer. The Geometry of Fractal Sets. Cambridge Tracts in Mathematics. Cambridge University Press, 1986. URL:
  11. C. A. R. Hoare. Communicating sequential processes. Commun. ACM, 21(8):666-677, 1978. URL:
  12. A. J. C. Hurkens, Monica McArthur, Yiannis N. Moschovakis, Lawrence S. Moss, and Glen T. Whitney. Erratum: "The logic of recursive equations". J. Symbolic Logic, 64, 1999. Google Scholar
  13. John E. Hutchinson. Fractals and self similarity. Indiana University Mathematics Journal, 30(5):713-747, 1981. URL:
  14. Henning Kerstan and Barbara König. Coalgebraic Trace Semantics for Continuous Probabilistic Transition Systems. Logical Methods in Computer Science, Volume 9, Issue 4, December 2013. URL:
  15. S. C. Kleene. Representation of events in nerve nets and finite automata. In Claude Shannon and John McCarthy, editors, Automata Studies, pages 3-41. Princeton University Press, Princeton, NJ, 1956. Google Scholar
  16. Tom Leinster. A general theory of self-similarity. Adv. Math., 226(4):2935-3017, 2011. Google Scholar
  17. Benoit B. Mandelbrot. Fractals: Form, Chance, and Dimension. Mathematics Series. W. H. Freeman, 1977. URL:
  18. R Daniel Mauldin and S. C. Williams. Hausdorff dimension in graph directed constructions. Transactions of the American Mathematical Society, 309(2):811-829, 1988. URL:
  19. Alexandru Mihail and Radu Miculescu. Generalized ifss on noncompact spaces. Fixed Point Theory and Applications, 1(584215), 2010. URL:
  20. Stefan Milius and Lawrence S. Moss. The category-theoretic solution of recursive program schemes. Theor. Comput. Sci., 366(1-2):3-59, 2006. URL:
  21. Stefan Milius and Lawrence S. Moss. Equational properties of recursive program scheme solutions. Cah. Topol. Géom. Différ. Catég., 50(1):23-66, 2009. Google Scholar
  22. Robin Milner. A complete inference system for a class of regular behaviours. J. Comput. Syst. Sci., 28(3):439-466, 1984. URL:
  23. Dusko Pavlovic and Martín Hötzel Escardó. Calculus in coinductive form. In Thirteenth Annual IEEE Symposium on Logic in Computer Science, Indianapolis, Indiana, USA, June 21-24, 1998, pages 408-417. IEEE Computer Society, 1998. Google Scholar
  24. Dusko Pavlovic and Vaughan R. Pratt. The continuum as a final coalgebra. Theor. Comput. Sci., 280(1-2):105-122, 2002. Google Scholar
  25. Alexander Moshe Rabinovich. A complete axiomatisation for trace congruence of finite state behaviors. In Stephen D. Brookes, Michael G. Main, Austin Melton, Michael W. Mislove, and David A. Schmidt, editors, Mathematical Foundations of Programming Semantics, 9th International Conference, New Orleans, LA, USA, April 7-10, 1993, Proceedings, volume 802 of Lecture Notes in Computer Science, pages 530-543. Springer, 1993. URL:
  26. Alexandra Silva and Ana Sokolova. Sound and complete axiomatization of trace semantics for probabilistic systems. In Michael W. Mislove and Joël Ouaknine, editors, Twenty-seventh Conference on the Mathematical Foundations of Programming Semantics, MFPS 2011, Pittsburgh, PA, USA, May 25-28, 2011, volume 276 of Electronic Notes in Theoretical Computer Science, pages 291-311. Elsevier, 2011. URL:
  27. Eugene W. Stark and Scott A. Smolka. A complete axiom system for finite-state probabilistic processes. In Gordon D. Plotkin, Colin Stirling, and Mads Tofte, editors, Proof, Language, and Interaction, Essays in Honour of Robin Milner, pages 571-596. The MIT Press, 2000. Google Scholar
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