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Cancellative Convex Semilattices

Authors Ana Sokolova , Harald Woracek

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Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Theory of computation → Logic and verification
  • Theory of computation → Probabilistic computation
Keywords
  • convex semilattice
  • cancellativity
  • Riesz space

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Abstract

Convex semilattices are algebras that are at the same time a convex algebra and a semilattice, together with a distributivity axiom. These algebras have attracted some attention in the last years as suitable algebras for probability and nondeterminism, in particular by being the Eilenberg-Moore algebras of the nonempty finitely-generated convex subsets of the distributions monad. 
A convex semilattice is cancellative if the underlying convex algebra is cancellative. Cancellative convex algebras have been characterized by M. H. Stone and by H. Kneser: A convex algebra is cancellative if and only if it is isomorphic to a convex subset of a vector space (with canonical convex algebra operations). 
We prove an analogous theorem for convex semilattices: A convex semilattice is cancellative if and only if it is isomorphic to a convex subset of a Riesz space, i.e., a lattice-ordered vector space (with canonical convex semilattice operations).

Cite As Get BibTex

Ana Sokolova and Harald Woracek. Cancellative Convex Semilattices. In 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 342, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CALCO.2025.12

Author Details

Ana Sokolova
  • University of Salzburg, Austria
Harald Woracek
  • TU Wien, Austria

References

  1. Alejandro Aguirre and Lars Birkedal. Step-indexed logical relations for countable nondeterminism and probabilistic choice. Proc. ACM Program. Lang., 7(POPL), 2023. URL: https://doi.org/10.1145/3571195.
  2. Christel Baier and Joost-Pieter Katoen. Principles of model checking. MIT Press, 2008. Google Scholar
  3. Filippo Bonchi, Ana Sokolova, and Valeria Vignudelli. The theory of traces for systems with nondeterminism and probability. In Proc. LICS'19, pages 1-14, 2019. URL: https://doi.org/10.1109/LICS.2019.8785673.
  4. Filippo Bonchi, Ana Sokolova, and Valeria Vignudelli. Presenting Convex Sets of Probability Distributions by Convex Semilattices and Unique Bases. In Proc. CALCO'21, volume 211 of LIPIcs, pages 11:1-11:18, 2021. URL: https://doi.org/10.4230/LIPIcs.CALCO.2021.11.
  5. Filippo Bonchi, Ana Sokolova, and Valeria Vignudelli. The theory of traces for systems with nondeterminism, probability, and termination. Logical Methods in Computer Science, Volume 18, Issue 2, June 2022. URL: https://doi.org/10.46298/lmcs-18(2:21)2022.
  6. Pablo S. Castro, Prakash Panangaden, and Doina Precup. Equivalence relations in fully and partially observable markov decision processes. In Proc. IJCAI'09, pages 1653-1658, 2009. Google Scholar
  7. Ep de Jonge and Arnoud C.M. van Rooij. Introduction to Riesz spaces, volume 78 of Math. Cent. Tracts. Centrum voor Wiskunde en Informatica (CWI), Amsterdam, 1977. Google Scholar
  8. Christian Dehnert, Sebastian Junges, Joost-Pieter Katoen, and Matthias Volk. A storm is coming: A modern probabilistic model checker. In Proc. CAV 2017, volume 10427 of LNCS, pages 592-600, 2017. URL: https://doi.org/10.1007/978-3-319-63390-9_31.
  9. Ernst-Erich Doberkat. Eilenberg-Moore algebras for stochastic relations. Inform. and Comput., 204(12):1756-1781, 2006. URL: https://doi.org/10.1016/j.ic.2006.09.001.
  10. Ernst-Erich Doberkat. Erratum and addendum: Eilenberg-Moore algebras for stochastic relations [mr2277336]. Inform. and Comput., 206(12):1476-1484, 2008. URL: https://doi.org/10.1016/j.ic.2008.08.002.
  11. Jean Goubault-Larrecq. Prevision domains and convex powercones. In Proc. FOSSACS'08, volume 4962 of LNCS, pages 318-333, 2008. URL: https://doi.org/10.1007/978-3-540-78499-9_23.
  12. Alexandre Goy and Daniela Petrisan. Combining probabilistic and non-deterministic choice via weak distributive laws. In Proc. LICS'20, pages 454-464. ACM, 2020. URL: https://doi.org/10.1145/3373718.3394795.
  13. Hans A. Hansson. Time and probability in formal design of distributed systems. PhD thesis, Uppsala University, 1991. Google Scholar
  14. Jerry I. den Hartog and Erik P. de Vink. Mixing up nondeterminism and probability: A preliminary report. In Proc. PROBMIV'98. ENTCS 22, 1998. Google Scholar
  15. Holger Hermanns, Jan Krcál, and Jan Kretínský. Probabilistic bisimulation: Naturally on distributions. In Proc. CONCUR'14, volume 8704 of LNCS, pages 249-265, 2014. URL: https://doi.org/10.1007/978-3-662-44584-6_18.
  16. Holger Hermanns, Augusto Parma, Roberto Segala, Björn Wachter, and Lijun Zhang. Probabilistic logical characterization. Information and Computation, 209(2):154-172, 2011. URL: https://doi.org/10.1016/J.IC.2010.11.024.
  17. Chris Heunen, Ohad Kammar, Sam Staton, and Hongseok Yang. A convenient category for higher-order probability theory. CoRR, abs/1701.02547, 2017. URL: https://arxiv.org/abs/1701.02547.
  18. Jennifer Hyndman, James B. Nation, and Joy Nishida. Congruence lattices of semilattices with operators. Studia Logica, 104(2):305-316, 2016. URL: https://doi.org/10.1007/s11225-015-9641-0.
  19. Bart Jacobs. Coalgebraic trace semantics for combined possibilitistic and probabilistic systems. Electr. Notes Theor. Comput. Sci., 203(5):131-152, 2008. URL: https://doi.org/10.1016/J.ENTCS.2008.05.023.
  20. Bart Jacobs. Convexity, duality and effects. In Theoretical computer science, volume 323 of IFIP Adv. Inf. Commun. Technol., pages 1-19. Springer, Berlin, 2010. URL: https://doi.org/10.1007/978-3-642-15240-5_1.
  21. Leslie P. Kaelbling, Michael L Littman, and Anthony R. Cassandra. Planning and Acting in Partially Observable Stochastic Domains. Artif. Intell., 1998. Google Scholar
  22. Klaus Keimel and Gordon D. Plotkin. Mixed powerdomains for probability and nondeterminism. Log. Methods Comput. Sci., 13(1):84, 2017. Id/No 2. URL: https://doi.org/10.23638/LMCS-13(1:2)2017.
  23. Hellmuth Kneser. Konvexe Räume. Arch. Math., 3:198-206, 1952. URL: https://doi.org/10.1007/BF01899364.
  24. Marta Z. Kwiatkowska, Gethin Norman, and David Parker. Prism: Probabilistic symbolic model checker. In Computer Performance Evaluation / TOOLS, volume 2324 of LNCS, pages 200-204, 2002. URL: https://doi.org/10.1007/3-540-46029-2_13.
  25. Wilhelmus A.J. Luxemburg and Adriaan C. Zaanen. Riesz spaces. Vol. I, volume 1 of North-Holland Math. Libr. Elsevier (North-Holland), Amsterdam, 1971. Google Scholar
  26. Matteo Mio. Upper-expectation bisimilarity and łukasiewicz μ-calculus. In Proc. FOSSACS'14, volume 8412 of LNCS, pages 335-350, 2014. URL: https://doi.org/10.1007/978-3-642-54830-7_22.
  27. Matteo Mio, Ralph Sarkis, and Valeria Vignudelli. Combining nondeterminism, probability, and termination: Equational and metric reasoning. In Proc. LICS'21, pages 1-14, 2021. URL: https://doi.org/10.1109/LICS52264.2021.9470717.
  28. Matteo Mio and Valeria Vignudelli. Monads and quantitative equational theories for nondeterminism and probability. In Proc. CONCUR'20, volume 171 of LIPIcs, pages 28:1-28:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.CONCUR.2020.28.
  29. Michael W. Mislove. Nondeterminism and probabilistic choice: Obeying the laws. In Proc CONCUR'00, volume 1877 of LNCS, pages 350-364, 2000. URL: https://doi.org/10.1007/3-540-44618-4_26.
  30. Michael W. Mislove, Joel Ouaknine, and James Worrell. Axioms for probability and nondeterminism. In Proc. EXPRESS'03, volume 96 of ENTCS, pages 7-28. Elsevier, 2003. URL: https://doi.org/10.1016/J.ENTCS.2004.04.019.
  31. Aloïs Rosset, Helle H. Hansen, and Jörg Endrullis. Algebraic presentation of semifree monads. In Proc. CMCS'22, volume 13225 of LNCS, pages 110-132, 2022. URL: https://doi.org/10.1007/978-3-031-10736-8_6.
  32. Stuart Russell and Peter Norvig. Artificial Intelligence: A Modern Approach. Prentice Hall, 2009. Google Scholar
  33. Roberto Segala. Modeling and verification of randomized distributed real-time systems. PhD thesis, MIT, 1995. Google Scholar
  34. Roberto Segala and Nancy Lynch. Probabilistic simulations for probabilistic processes. Nordic Journal of Computing, 2(2):250-273, 1995. Google Scholar
  35. Jonathan D.H. Smith. Modes, modals, and barycentric algebras: a brief survey and an additivity theorem. Demonstr. Math., 44(3):571-587, 2011. URL: https://doi.org/10.1515/dema-2013-0332.
  36. Ana Sokolova and Harald Woracek. Congruences of convex algebras. J. Pure Appl. Algebra, 219(8):3110-3148, 2015. URL: https://doi.org/10.1016/j.jpaa.2014.10.005.
  37. Sam Staton, Hongseok Yang, Frank Wood, Chris Heunen, and Ohad Kammar. Semantics for probabilistic programming: higher-order functions, continuous distributions, and soft constraints. In Proc. LICS'16, pages 525-534, 2016. URL: https://doi.org/10.1145/2933575.2935313.
  38. Marshall H. Stone. Postulates for the barycentric calculus. Ann. Mat. Pura Appl. (4), 29:25-30, 1949. URL: https://doi.org/10.1007/BF02413910.
  39. Tadeusz Świrszcz. Monadic functors and convexity. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 22:39-42, 1974. Google Scholar
  40. Regina Tix, Klaus Keimel, and Gordon D. Plotkin. Semantic domains for combining probability and non-determinism. ENTCS, 222:3-99, 2009. URL: https://doi.org/10.1016/j.entcs.2009.01.002.
  41. Daniele Varacca. Probability, Nondeterminism and Concurrency: Two Denotational Models for Probabilistic Computation. PhD thesis, Univ. Aarhus, 2003. BRICS Dissertation Series, DS-03-14. Google Scholar
  42. Daniele Varacca and Glynn Winskel. Distributing probabililty over nondeterminism. MSCS, 16(1):87-113, 2006. URL: https://doi.org/10.1017/S0960129505005074.
  43. Moshe Y. Vardi. Automatic verification of probabilistic concurrent finite state programs. In Foundations of Computer Science, 1985., 26th Annual Symposium on, pages 327-338. IEEE, 1985. URL: https://doi.org/10.1109/SFCS.1985.12.
  44. Noam Zilberstein, Dexter Kozen, Alexandra Silva, and Joseph Tassarotti. A demonic outcome logic for randomized nondeterminism. Proc. ACM Program. Lang., 9(POPL), 2025. URL: https://doi.org/10.1145/3704855.
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