Generators and Bases for Monadic Closures

Authors Stefan Zetzsche, Alexandra Silva, Matteo Sammartino



PDF
Thumbnail PDF

File

LIPIcs.CALCO.2023.11.pdf
  • Filesize: 0.79 MB
  • 19 pages

Document Identifiers

Author Details

Stefan Zetzsche
  • Amazon Web Services, London, UK
  • University College London, UK
Alexandra Silva
  • Cornell University, Ithaca, NY, USA
  • University College London, UK
Matteo Sammartino
  • Royal Holloway, University of London, UK
  • University College London, UK

Cite AsGet BibTex

Stefan Zetzsche, Alexandra Silva, and Matteo Sammartino. Generators and Bases for Monadic Closures. In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CALCO.2023.11

Abstract

It is well-known that every regular language admits a unique minimal deterministic acceptor. Establishing an analogous result for non-deterministic acceptors is significantly more difficult, but nonetheless of great practical importance. To tackle this issue, a number of sub-classes of non-deterministic automata have been identified, all admitting canonical minimal representatives. In previous work, we have shown that such representatives can be recovered categorically in two steps. First, one constructs the minimal bialgebra accepting a given regular language, by closing the minimal coalgebra with additional algebraic structure over a monad. Second, one identifies canonical generators for the algebraic part of the bialgebra, to derive an equivalent coalgebra with side effects in a monad. In this paper, we further develop the general theory underlying these two steps. On the one hand, we show that deriving a minimal bialgebra from a minimal coalgebra can be realized by applying a monad on an appropriate category of subobjects. On the other hand, we explore the abstract theory of generators and bases for algebras over a monad.

Subject Classification

ACM Subject Classification
  • Theory of computation → Abstract machines
Keywords
  • Monads
  • Category Theory
  • Generators
  • Automata
  • Coalgebras
  • Bialgebras

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Jiri Adamek, Filippo Bonchi, Mathias Hülsbusch, Barbara König, Stefan Milius, and Alexandra Silva. A coalgebraic perspective on minimization and determinization. In International Conference on Foundations of Software Science and Computational Structures, pages 58-73. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-28729-9_4.
  2. Jiri Adamek, Horst Herrlich, and George E Strecker. Abstract and concrete categories: The joy of cats. Reprints in Theory and Applications of Categories, 2009. Google Scholar
  3. Jiri Adamek, Stefan Milius, Lurdes Sousa, and Thorsten Wißmann. Finitely presentable algebras for finitary monads. Theory and Applications of Categories, 34(37):1179-1195, 2019. Google Scholar
  4. Jiri Adamek and Jiri Rosicky. Locally Presentable and Accessible Categories, volume 189. Cambridge University Press, 1994. URL: https://doi.org/10.1017/CBO9780511600579.
  5. Dana Angluin. Learning regular sets from queries and counterexamples. Information and Computation, 75(2):87-106, 1987. URL: https://doi.org/10.1016/0890-5401(87)90052-6.
  6. Michael A Arbib and Ernest G Manes. Fuzzy machines in a category. Bulletin of the Australian Mathematical Society, 13(2):169-210, 1975. URL: https://doi.org/10.1017/S0004972700024412.
  7. André Arnold, Anne Dicky, and Maurice Nivat. A note about minimal non-deterministic automata. Bulletin of the EATCS, 47:166-169, 1992. Google Scholar
  8. Steve Awodey. Category Theory. Oxford University Press, Inc., 2010. Google Scholar
  9. Jon Beck. Distributive laws. In Seminar on Triples and Categorical Homology Theory, pages 119-140. Springer, 1969. URL: https://doi.org/10.1007/BFb0083084.
  10. Sebastian Berndt, Maciej Liśkiewicz, Matthias Lutter, and Rüdiger Reischuk. Learning residual alternating automata. In Thirty-First AAAI Conference on Artificial Intelligence, 2017. URL: https://doi.org/10.1609/aaai.v31i1.10891.
  11. Filippo Bonchi, Marcello M Bonsangue, Helle H Hansen, Prakash Panangaden, Jan Rutten, and Alexandra Silva. Algebra-coalgebra duality in brzozowski’s minimization algorithm. ACM Transactions on Computational Logic (TOCL), 15(1):1-29, 2014. URL: https://doi.org/10.1145/2490818.
  12. Filippo Bonchi, Marcello M Bonsangue, Jan Rutten, and Alexandra Silva. Brzozowski’s algorithm (co)algebraically. In Logic and Program Semantics, pages 12-23. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-29485-3_2.
  13. Aldridge K Bousfield. Constructions of factorization systems in categories. Journal of Pure and Applied Algebra, 9(2-3):207-220, 1977. URL: https://doi.org/10.1016/0022-4049(77)90067-6.
  14. Janusz A Brzozowski. Canonical regular expressions and minimal state graphs for definite events. In Proc. Symposium of Mathematical Theory of Automata, volume 12, pages 529-561, 1962. Google Scholar
  15. Janusz A. Brzozowski and Hellis Tamm. Theory of átomata. Theor. Comput. Sci., 539:13-27, 2014. URL: https://doi.org/10.1016/j.tcs.2014.04.016.
  16. François Denis, Aurélien Lemay, and Alain Terlutte. Residual finite state automata. In Annual Symposium on Theoretical Aspects of Computer Science, pages 144-157. Springer, 2001. URL: https://doi.org/10.1007/3-540-44693-1_13.
  17. Yann Esposito, Aurélien Lemay, François Denis, and Pierre Dupont. Learning probabilistic residual finite state automata. In International Colloquium on Grammatical Inference, pages 77-91. Springer, 2002. URL: https://doi.org/10.1007/3-540-45790-9_7.
  18. Bart Jacobs. A bialgebraic review of deterministic automata, regular expressions and languages. In Algebra, Meaning, and Computation, pages 375-404. Springer, 2006. URL: https://doi.org/10.1007/11780274_20.
  19. Bart Jacobs. Bases as coalgebras. In Algebra and Coalgebra in Computer Science, pages 237-252. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22944-2_17.
  20. Bart Jacobs. Introduction to Coalgebra: Towards Mathematics of States and Observation. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 2016. URL: https://doi.org/10.1017/CBO9781316823187.
  21. Bart Jacobs, Alexandra Silva, and Ana Sokolova. Trace semantics via determinization. In International Workshop on Coalgebraic Methods in Computer Science, pages 109-129. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-32784-1_7.
  22. Alexander Kurz. Logics for Coalgebras and Applications to Computer Science. PhD thesis, Ludwig-Maximilians-Universität München, 2000. Google Scholar
  23. Serge Lang. Algebra. Graduate Texts in Mathematics, 2002. URL: https://doi.org/10.1007/978-1-4613-0041-0.
  24. Saunders Mac Lane. Categories for the Working Mathematician, volume 5. Springer, 2013. URL: https://doi.org/10.1007/978-1-4757-4721-8.
  25. Saunders MacLane. Duality for groups. Bulletin of the American Mathematical Society, 56(6):485-516, 1950. Google Scholar
  26. Eugenio Moggi. Computational Lambda-Calculus and Monads. University of Edinburgh, Department of Computer Science, Laboratory for Foundations of Computer Science, 1988. Google Scholar
  27. Eugenio Moggi. An Abstract View of Programming Languages. University of Edinburgh, Department of Computer Science, Laboratory for Foundations of Computer Science, 1990. Google Scholar
  28. Eugenio Moggi. Notions of computation and monads. Information and Computation, 93(1):55-92, 1991. URL: https://doi.org/10.1016/0890-5401(91)90052-4.
  29. Robert S. R. Myers, Jiri Adamek, Stefan Milius, and Henning Urbat. Coalgebraic constructions of canonical nondeterministic automata. Theoretical Computer Science, 604:81-101, 2015. URL: https://doi.org/10.1016/j.tcs.2015.03.035.
  30. Anil Nerode. Linear automaton transformations. Proceedings of the American Mathematical Society, 9(4):541-544, 1958. URL: https://doi.org/10.2307/2033204.
  31. Louis Parlant, Jurriaan Rot, Alexandra Silva, and Bas Westerbaan. Preservation of Equations by Monoidal Monads. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020), volume 170 of Leibniz International Proceedings in Informatics (LIPIcs), pages 77:1-77:14. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.MFCS.2020.77.
  32. Emily Riehl. Factorization systems, 2008. URL: https://math.jhu.edu/~eriehl/factorization.pdf.
  33. Jan Rutten. Universal coalgebra: A theory of systems. Theoretical Computer Science, 249(1):3-80, 2000. URL: https://doi.org/10.1016/S0304-3975(00)00056-6.
  34. Jan Rutten. The method of coalgebra: Exercises in coinduction, 2019. Google Scholar
  35. Gavin J Seal. Tensors, monads and actions. Theory and Applications of Categories, 28(15):403-433, 2013. Google Scholar
  36. Alexandra Silva, Filippo Bonchi, Marcello M Bonsangue, and Jan Rutten. Generalizing the powerset construction, coalgebraically. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010), volume 8, pages 272-283. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2010. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2010.272.
  37. Ross Street. The formal theory of monads. Journal of Pure and Applied Algebra, 2(2):149-168, 1972. URL: https://doi.org/10.1016/0022-4049(72)90019-9.
  38. Ross Street. Weak distributive laws. Theory and Applications of Categories, 22:313-320, 2009. Google Scholar
  39. Daniele Turi. Functorial Operational Semantics. PhD thesis, Vrije Universiteit Amsterdam, 1996. Google Scholar
  40. Daniele Turi and Gordon Plotkin. Towards a mathematical operational semantics. In Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science, pages 280-291. IEEE, 1997. URL: https://doi.org/10.1109/LICS.1997.614955.
  41. Gerco van Heerdt. An abstract automata learning framework. Master’s thesis, Radboud University Nijmegen, 2016. Google Scholar
  42. Gerco van Heerdt. CALF: Categorical Automata Learning Framework. PhD thesis, University College London, 2020. Google Scholar
  43. Gerco van Heerdt, Matteo Sammartino, and Alexandra Silva. Learning automata with side-effects. In Coalgebraic Methods in Computer Science, pages 68-89. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-57201-3_5.
  44. Jean Vuillemin and Nicolas Gama. Efficient equivalence and minimization for non deterministic xor automata. Technical report, Ecole Normale Supérieure, 2010. Google Scholar
  45. Thorsten Wißmann. Minimality notions via factorization systems and examples. Logical Methods in Computer Science, 18(3), 2022. URL: https://doi.org/10.46298/lmcs-18(3:31)2022.
  46. Stefan Zetzsche, Gerco van Heerdt, Matteo Sammartino, and Alexandra Silva. Canonical automata via distributive law homomorphisms. Electronic Proceedings in Theoretical Computer Science, 351:296-313, 2021. URL: https://doi.org/10.4204/eptcs.351.18.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail