On the Positive Calculus of Relations with Transitive Closure

Author Damien Pous

Thumbnail PDF


  • Filesize: 0.49 MB
  • 16 pages

Document Identifiers

Author Details

Damien Pous

Cite AsGet BibTex

Damien Pous. On the Positive Calculus of Relations with Transitive Closure. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Binary relations are such a basic object that they appear in many places in mathematics and computer science. For instance, when dealing with graphs, program semantics, or termination guarantees, binary relations are always used at some point. In this survey, we focus on the relations themselves, and we consider algebraic and algorithmic questions. On the algebraic side, we want to understand and characterise the laws governing the behaviour of the following standard operations on relations: union, intersection, composition, converse, and reflexive-transitive closure. On the algorithmic side, we look for decision procedures for equality or inequality of relations. After having formally defined the calculus of relations, we recall the existing results about two well-studied fragments of particular importance: Kleene algebras and allegories. Unifying those fragments yields a decidable theory whose axiomatisability remains an open problem.
  • Relation Algebra
  • Kleene Algebra
  • Allegories
  • Automata
  • Graphs


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Hajnal Andréka, Szabolcs Mikulás, and István Németi. The equational theory of kleene lattices. Theor. Comput. Sci., 412(52):7099-7108, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2011.09.024.
  2. H. Andréka and D.A. Bredikhin. The equational theory of union-free algebras of relations. Algebra Universalis, 33(4):516-532, 1995. URL: http://dx.doi.org/10.1007/BF01225472.
  3. Valentin M. Antimirov. Partial derivatives of regular expressions and finite automaton constructions. Theor. Comput. Sci., 155(2):291-319, 1996. URL: http://dx.doi.org/10.1016/0304-3975(95)00182-4.
  4. Stephen L. Bloom, Zoltán Ésik, and Gheorghe Stefanescu. Notes on equational theories of relations. algebra universalis, 33(1):98-126, Mar 1995. URL: http://dx.doi.org/10.1007/BF01190768.
  5. Paul Brunet and Damien Pous. Petri automata for kleene allegories. In 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, Japan, July 6-10, 2015, pages 68-79. IEEE Computer Society, 2015. URL: http://dx.doi.org/10.1109/LICS.2015.17.
  6. Paul Brunet and Damien Pous. Algorithms for kleene algebra with converse. J. Log. Algebr. Meth. Program., 85(4):574-594, 2016. URL: http://dx.doi.org/10.1016/j.jlamp.2015.07.005.
  7. J. H. Conway. Regular algebra and finite machines. Chapman and Hall, 1971. Google Scholar
  8. Enric Cosme-Llópez and Damien Pous. K4-free graphs as a free algebra. In Kim G. Larsen, Hans L. Bodlaender, and Jean-François Raskin, editors, 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017, August 21-25, 2017 - Aalborg, Denmark, volume 83 of LIPIcs, pages 76:1-76:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.MFCS.2017.76.
  9. R. Diestel. Graph Theory. Graduate Texts in Mathematics. Springer Verlag, 2005. Google Scholar
  10. Zoltán Ésik and L. Bernátsky. Equational properties of kleene algebras of relations with conversion. Theor. Comput. Sci., 137(2):237-251, 1995. URL: http://dx.doi.org/10.1016/0304-3975(94)00041-G.
  11. P. Freyd and A. Scedrov. Categories, Allegories. North Holland, 1990. Google Scholar
  12. Ian Hodkinson and Szabolcs Mikulás. Axiomatizability of reducts of algebras of relations. Algebra Universalis, 43(2):127-156, 2000. URL: http://dx.doi.org/10.1007/s000120050150.
  13. Dexter Kozen. A completeness theorem for kleene algebras and the algebra of regular events. In Proceedings of the Sixth Annual Symposium on Logic in Computer Science (LICS '91), Amsterdam, The Netherlands, July 15-18, 1991, pages 214-225. IEEE Computer Society, 1991. URL: http://dx.doi.org/10.1109/LICS.1991.151646.
  14. Dexter Kozen. A completeness theorem for kleene algebras and the algebra of regular events. Inf. Comput., 110(2):366-390, 1994. URL: http://dx.doi.org/10.1006/inco.1994.1037.
  15. Daniel Krob. Complete systems of b-rational identities. Theor. Comput. Sci., 89(2):207-343, 1991. URL: http://dx.doi.org/10.1016/0304-3975(91)90395-I.
  16. Donald Monk. On representable relation algebras. Michigan Math. J., 11(3):207-210, 09 1964. URL: http://dx.doi.org/10.1307/mmj/1028999131.
  17. T. Murata. Petri nets: Properties, analysis and applications. Proc. of the IEEE, 77(4):541-580, Apr 1989. URL: http://dx.doi.org/10.1109/5.24143.
  18. Yoshiki Nakamura. Partial derivatives on graphs for kleene allegories. In 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, Reykjavik, Iceland, June 20-23, 2017, pages 1-12. IEEE Computer Society, 2017. URL: http://dx.doi.org/10.1109/LICS.2017.8005132.
  19. C. A. Petri. Fundamentals of a theory of asynchronous information flow. In IFIP Congress, pages 386-390, 1962. Google Scholar
  20. Damien Pous. Informatique Mathématique: une photographie en 2016, chapter Algèbres de relations. E. Baudrier and L. Mazo (Eds), 2016. Course notes for the EJCIM research school. Google Scholar
  21. V. Redko. On defining relations for the algebra of regular events. Ukr. Mat. Z., 16:120-, 1964. Google Scholar
  22. Arto Salomaa. Two complete axiom systems for the algebra of regular events. J. ACM, 13(1):158-169, 1966. URL: http://dx.doi.org/10.1145/321312.321326.
  23. Larry J. Stockmeyer and Albert R. Meyer. Word problems requiring exponential time: Preliminary report. In Alfred V. Aho, Allan Borodin, Robert L. Constable, Robert W. Floyd, Michael A. Harrison, Richard M. Karp, and H. Raymond Strong, editors, Proceedings of the 5th Annual ACM Symposium on Theory of Computing, April 30 - May 2, 1973, Austin, Texas, USA, pages 1-9. ACM, 1973. URL: http://dx.doi.org/10.1145/800125.804029.
  24. A. Tarski and S. Givant. A Formalization of Set Theory without Variables, volume 41 of Colloquium Publications. American Mathematical Society, Providence, Rhode Island, 1987. Google Scholar
  25. Alfred Tarski. On the calculus of relations. J. Symbolic logic, 6:73-89, 1941. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail