The Decidable Properties of Subrecursive Functions

Author Mathieu Hoyrup

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Mathieu Hoyrup

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Mathieu Hoyrup. The Decidable Properties of Subrecursive Functions. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 108:1-108:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


What can be decided or semidecided about a primitive recursive function, given a definition of that function by primitive recursion? What about subrecursive classes other than primitive recursive functions? We provide a complete and explicit characterization of the decidable and semidecidable properties. This characterization uses a variant of Kolmogorov complexity where only programs in a subrecursive programming language are allowed. More precisely, we prove that all the decidable and semidecidable properties can be obtained as combinations of two classes of basic decidable properties: (i) the function takes some particular values on a finite set of inputs, and (ii) every finite part of the function can be compressed to some extent.
  • Rice theorem
  • subrecursive class
  • decidable property
  • Kolmogorov complexity
  • compressibility


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  1. Andrea Asperti. The intensional content of rice’s theorem. In George C. Necula and Philip Wadler, editors, Proceedings of the 35th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2008, San Francisco, California, USA, January 7-12, 2008, pages 113-119. ACM, 2008. URL:
  2. G. S. Ceitin. Algorithmic operators in constructive metric spaces. Trudy Matematiki Instituta Steklov, 67:295-361, 1962. English translation: American Mathematical Society Translations, series 2, 64:1-80, 1967. Google Scholar
  3. Johanna N. Y. Franklin, Noam Greenberg, Frank Stephan, and Guohua Wu. Anti-complex sets and reducibilities with tiny use. J. Symb. Log., 78(4):1307-1327, 2013. URL:
  4. Richard M. Friedberg. Un contre-exemple relatif aux fonctionnelles récursives. Comptes Rendus de l'Académie des Sciences, 247:852-854, 1958. Google Scholar
  5. David Gajser. Verifying time complexity of turing machines. Theoretical Computer Science, 600:86-97, 2015. URL:
  6. Mathieu Hoyrup and Cristóbal Rojas. On the information carried by programs about the objects they compute. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 447-459. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL:
  7. Dexter Kozen. Indexings of subrecursive classes. Theoretical Computer Science, 11(3):277-301, 1980. URL:
  8. Georg Kreisel, Daniel Lacombe, and Joseph R. Schœ nfield. Fonctionnelles récursivement définissables et fonctionnelles récursives. Comptes Rendus de l'Académie des Sciences, 245:399-402, 1957. Google Scholar
  9. Ming Li and Paul M. B. Vitanyi. An Introduction to Kolmogorov Complexity and Its Applications. Springer-Verlag, Berlin, 1993. Google Scholar
  10. Albert R. Meyer and Dennis M. Ritchie. The complexity of loop programs. In Proceedings of the 1967 22Nd National Conference, ACM'67, pages 465-469, New York, NY, USA, 1967. ACM. URL:
  11. Henry G. Rice. Classes of recursively enumerable sets and their decision problems. Transactions of the American Mathematical Society, 74(2):pp. 358-366, 1953. URL:
  12. Norman Shapiro. Degrees of computability. Transactions of the American Mathematical Society, 82:281-299, 1956. Google Scholar
  13. Klaus Weihrauch. Computable Analysis. Springer, Berlin, 2000. Google Scholar