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arxiv: 2605.06268 · v1 · submitted 2026-05-07 · 💻 cs.LO

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Graded Monad Coalgebras for Continuous-Time Transition Systems

Elena Di Lavore, Jonas Forster, Mario Rom\'an

Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:39 UTC · model grok-4.3

classification 💻 cs.LO
keywords graded monadscoalgebrascontinuous-time transition systemsmodal logicsFeller-Dynkin processesbranching-time semanticstrace semanticsdistributive laws
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The pith

Graded coalgebras of graded monads model continuous-time transition systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Standard functor coalgebras describe transition systems that change in discrete steps. This paper proposes graded coalgebras of graded monads to handle systems that evolve continuously over time. The authors develop the supporting theory, including graded distributive laws and conditions under which terminal coalgebras exist. They define branching-time and trace semantics for these systems and connect the trace semantics to Feller-Dynkin processes. Coalgebraic modal logics are constructed to reason about process behaviors and to give criteria for state invariance and expressivity.

Core claim

By equipping monads and coalgebras with a grading that tracks time, continuous-time transition systems can be modeled coalgebraically. This includes the existence of terminal coalgebras under appropriate conditions, the definition of both branching-time and trace semantics linked to Feller-Dynkin processes, and the development of modal logics that characterize invariance and expressivity for these processes.

What carries the argument

Graded monads and their coalgebras, where the grading encodes the continuous passage of time, along with graded distributive laws that allow composition.

Load-bearing premise

The grading on monads and coalgebras suffices to capture continuous time evolution without requiring extra structure or discretization of time.

What would settle it

Finding a continuous-time transition system that cannot be represented as a graded coalgebra for any graded monad, or a case where terminal coalgebras fail to exist despite the proposed conditions holding.

read the original abstract

Functor coalgebras capture a wide range of transition systems that must however evolve in discrete steps. We introduce graded coalgebras of graded monads and propose them to model continuous-time transition systems. We develop the theory of graded coalgebras-including graded distributive laws between graded monads-and we give conditions for the existence of terminal coalgebras. We define both branching-time and trace semantics, linking them to recent work on Feller-Dynkin processes. Finally, we develop coalgebraic modal logics for both process semantics and state criteria for invariance and expressivity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper introduces graded coalgebras of graded monads to model continuous-time transition systems, develops the associated theory (including graded distributive laws and conditions for terminal coalgebras), defines branching-time and trace semantics with links to Feller-Dynkin processes, and constructs coalgebraic modal logics for process semantics together with invariance and expressivity criteria.

Significance. If the central constructions are sound, the work extends coalgebraic semantics from discrete to continuous time in a uniform categorical framework, providing a potential bridge between coalgebra theory and established continuous stochastic process models; the explicit links to Feller-Dynkin processes and the development of modal logics are notable strengths.

major comments (1)
  1. [Abstract and §1] Abstract and §1: The claim that graded coalgebras of graded monads model continuous-time transition systems requires that the grading monoid (presumably (ℝ≥0, +)) together with graded distributive laws and terminal-coalgebra conditions suffice to capture non-discretized continuous dynamics. Standard coalgebraic constructions live in discrete categories such as Set; the manuscript must verify that the concrete graded functor (e.g., a graded distribution monad) preserves the required limits/colimits in the category of measurable or Polish spaces and that the resulting coalgebra semantics coincides with the usual semigroup of transition kernels, otherwise the modeling claim does not hold.
minor comments (2)
  1. [Abstract] Abstract: The statement 'we give conditions for the existence of terminal coalgebras' should be expanded in the introduction to indicate whether these conditions are novel or adaptations of existing results for graded monads.
  2. The paper should include a brief comparison table or paragraph contrasting the new graded coalgebra semantics with existing coalgebraic models of timed or hybrid systems to clarify the incremental contribution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We appreciate the positive evaluation of the work's potential to bridge coalgebraic methods with continuous-time models. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: The claim that graded coalgebras of graded monads model continuous-time transition systems requires that the grading monoid (presumably (ℝ≥0, +)) together with graded distributive laws and terminal-coalgebra conditions suffice to capture non-discretized continuous dynamics. Standard coalgebraic constructions live in discrete categories such as Set; the manuscript must verify that the concrete graded functor (e.g., a graded distribution monad) preserves the required limits/colimits in the category of measurable or Polish spaces and that the resulting coalgebra semantics coincides with the usual semigroup of transition kernels, otherwise the modeling claim does not hold.

    Authors: We agree that substantiating the modeling claim for non-discretized continuous dynamics benefits from explicit discussion of the concrete setting. Our manuscript develops the abstract theory of graded coalgebras, graded distributive laws, and terminal coalgebra conditions, while linking the trace semantics to Feller-Dynkin processes (which are defined via semigroups of kernels on Polish spaces). The abstract framework is intended to apply when instantiated in categories such as Meas or Pol with suitable graded monads. However, the current version does not contain an explicit verification of (co)limit preservation for a concrete graded monad (e.g., a graded Giry monad) nor a direct comparison of the coalgebra semantics to the semigroup of transition kernels. In the revision we will add a concise discussion in §1 (or a short appendix) outlining the instantiation, referencing standard results on monads preserving relevant (co)limits in measurable spaces, and explaining how the grading and terminal coalgebra yield the continuous-time semantics. This will be a clarification rather than a new technical development. revision: yes

Circularity Check

0 steps flagged

No circularity: new graded coalgebra definitions and external links to Feller-Dynkin processes

full rationale

The paper introduces graded monads and graded coalgebras as novel constructions, develops their theory (distributive laws, terminal coalgebras, branching/trace semantics), and explicitly links the resulting semantics to independent prior work on Feller-Dynkin processes. No derivation step reduces a central claim to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The modeling proposal for continuous-time systems is presented as an independent categorical construction rather than a tautological re-expression of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work relies on standard category-theoretic assumptions for monads and coalgebras; no free parameters or invented entities beyond the new graded structures are mentioned in the abstract.

axioms (1)
  • standard math Standard definitions and properties of monads, functors, and coalgebras in category theory
    The paper builds directly on functor coalgebras and monads, invoking their usual categorical properties.
invented entities (1)
  • Graded coalgebras of graded monads no independent evidence
    purpose: Modeling continuous-time transition systems
    New structure introduced to extend discrete models to continuous time.

pith-pipeline@v0.9.0 · 5380 in / 1173 out tokens · 42461 ms · 2026-05-08T04:39:18.975452+00:00 · methodology

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Works this paper leans on

61 extracted references · 50 canonical work pages

  1. [1]

    The Quantum Monad on Relational Structures

    Samson Abramsky, Rui Soares Barbosa, Nadish de Silva, and Octavio Zapata. The Quantum Monad on Relational Structures . In Kim G. Larsen, Hans L. Bodlaender, and Jean-Francois Raskin, editors, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017) , volume 83 of Leibniz International Proceedings in Informatics (LIPIcs) , p...

    work page doi:10.4230/lipics.mfcs.2017.35 2017

  2. [2]

    Adamek and J

    J. Adamek and J. Rosicky. Locally Presentable and Accessible Categories . London Mathematical Society Lecture Note Series. Cambridge University Press, 1994

    1994

  3. [3]

    Jiří Adámek, Stefan Milius, and Lawrence S. Moss. Introduction , page 1–11. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 2025

    2025

  4. [4]

    In: Proceedings of the 12th International Conference on Computer Aided Verification (CA V’00)

    Christel Baier, Boudewijn R. Haverkort, Holger Hermanns, and Joost - Pieter Katoen. Model checking continuous-time markov chains by transient analysis. In E. Allen Emerson and A. Prasad Sistla, editors, Computer Aided Verification, 12th International Conference, CAV 2000 , volume 1855 of Lecture Notes in Computer Science , pages 358--372. Springer, 2000. ...

    work page doi:10.1007/10722167 2000

  5. [5]

    Coalgebraic behavioral metrics

    Paolo Baldan, Filippo Bonchi, Henning Kerstan, and Barbara K \" o nig. Coalgebraic behavioral metrics. Log. Methods Comput. Sci. , 14(3), 2018. https://doi.org/10.23638/LMCS-14(3:20)2018 doi:10.23638/LMCS-14(3:20)2018

    work page doi:10.23638/lmcs-14(3:20)2018 2018

  6. [6]

    Distributive laws

    Jon Beck. Distributive laws. In Seminar on triples and categorical homology theory , pages 119--140. Springer, 1969

    1969

  7. [7]

    Introduction to bicategories

    Jean B \'e nabou. Introduction to bicategories. In Reports of the Midwest Category Seminar , volume 47 of Lecture Notes in Mathematics , pages 1--77, Berlin, Heidelberg, 1967. Springer Berlin Heidelberg. https://doi.org/10.1007/BFb0074299 doi:10.1007/BFb0074299

    work page doi:10.1007/bfb0074299 1967

  8. [8]

    A survey of modal logics characterising behavioural equivalences for non-deterministic and stochastic systems

    Marco Bernardo and Stefania Botta. A survey of modal logics characterising behavioural equivalences for non-deterministic and stochastic systems. Math. Struct. Comput. Sci. , 18(1):29--55, 2008. https://doi.org/10.1017/S0960129507006408 doi:10.1017/S0960129507006408

    work page doi:10.1017/s0960129507006408 2008

  9. [9]

    Behavioural equivalences for continuous-time Markov processes

    Linan Chen, Florence Clerc, and Prakash Panangaden. Behavioural equivalences for continuous-time Markov processes. Mathematical Structures in Computer Science , 33(4–5):222–258, 2023. https://doi.org/10.1017/S0960129523000099 doi:10.1017/S0960129523000099

    work page doi:10.1017/s0960129523000099 2023

  10. [10]

    A behavioural pseudometric for continuous-time markov processes

    Linan Chen, Florence Clerc, and Prakash Panangaden. A behavioural pseudometric for continuous-time markov processes. In Parosh Aziz Abdulla and Delia Kesner, editors, Foundations of Software Science and Computation Structures - FOSSACS Hamilton, ON, Canada, May 3-8, 2025, Proceedings , volume 15691 of Lecture Notes in Computer Science , pages 24--44. Spri...

    work page doi:10.1007/978-3-031-90897-2 2025

  11. [11]

    Generic infinite traces and path-based coalgebraic temporal logics

    Corina C \^ rstea. Generic infinite traces and path-based coalgebraic temporal logics. Electronic Notes in Theoretical Computer Science , 264(2):83--103, 2010

    2010

  12. [12]

    From branching to linear time, coalgebraically

    Corina C \^ rstea. From branching to linear time, coalgebraically. Fundamenta Informaticae , 150(3-4):379--406, 2017

    2017

  13. [13]

    Modal logics are coalgebraic1

    Corina Cîrstea, Alexander Kurz, Dirk Pattinson, Lutz Schröder, and Yde Venema. Modal logics are coalgebraic1. The Computer Journal , 54(1):31--41, 02 2009. https://doi.org/10.1093/comjnl/bxp004 doi:10.1093/comjnl/bxp004

    work page doi:10.1093/comjnl/bxp004 2009

  14. [14]

    Continuous stochastic logic characterizes bisimulation of continuous-time markov processes

    Jos \' e e Desharnais and Prakash Panangaden. Continuous stochastic logic characterizes bisimulation of continuous-time markov processes. J. Log. Algebraic Methods Program. , 56(1-2):99--115, 2003. https://doi.org/10.1016/S1567-8326(02)00068-1 doi:10.1016/S1567-8326(02)00068-1

    work page doi:10.1016/s1567-8326(02)00068-1 2003

  15. [15]

    Monoidal streams for dataflow programming

    Elena Di Lavore , Giovanni de Felice , and Mario Rom\' a n. Monoidal streams for dataflow programming. In Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science , LICS '22, New York, NY, USA, 2022. Association for Computing Machinery. https://doi.org/10.1145/3531130.3533365 doi:10.1145/3531130.3533365

    work page doi:10.1145/3531130.3533365 2022

  16. [16]

    Graded monads and graded logics for the linear time - branching time spectrum

    Ulrich Dorsch, Stefan Milius, and Lutz Schr \" o der. Graded monads and graded logics for the linear time - branching time spectrum. In Wan J. Fokkink and Rob van Glabbeek, editors, 30th International Conference on Concurrency Theory, CONCUR 2019, Amsterdam, The Netherlands, August 27-30, 2019 , volume 140 of LIPIcs , pages 36:1--36:16. Schloss Dagstuhl -...

    work page doi:10.4230/lipics.concur.2019.36 2019

  17. [17]

    o der, Harsh Beohar, and Barbara K\

    Chase Ford, Stefan Milius, Lutz Schr\" o der, Harsh Beohar, and Barbara K\" o nig. Graded monads and behavioural equivalence games. In Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science , LICS '22, New York, NY, USA, 2022. Association for Computing Machinery. https://doi.org/10.1145/3531130.3533374 doi:10.1145/3531130.3533374

    work page doi:10.1145/3531130.3533374 2022

  18. [18]

    Model enumeration of two-variable logic with quadratic delay complexity

    Jonas Forster, Lutz Schr \" o der, and Paul Wild. Conformance games for graded semantics. In 40th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2025, Singapore, June 23-26, 2025 , pages 555--567. IEEE , 2025. https://doi.org/10.1109/LICS65433.2025.00048 doi:10.1109/LICS65433.2025.00048

    work page doi:10.1109/lics65433.2025.00048 2025

  19. [19]

    o der, Paul Wild, Harsh Beohar, Sebastian Gurke, and Karla Messing. Graded semantics and graded logics for eilenberg-moore coalgebras. In Barbara K \

    Jonas Forster, Lutz Schr \" o der, Paul Wild, Harsh Beohar, Sebastian Gurke, and Karla Messing. Graded semantics and graded logics for eilenberg-moore coalgebras. In Barbara K \" o nig and Henning Urbat, editors, Coalgebraic Methods in Computer Science CMCS 2024 , volume 14617 of Lecture Notes in Computer Science , pages 114--134. Springer, 2024. https://...

    work page doi:10.1007/978-3-031-66438-0 2024

  20. [20]

    Towards a formal theory of graded monads

    Soichiro Fujii, Shin-ya Katsumata, and Paul-Andr \'e Melli \`e s. Towards a formal theory of graded monads. In Bart Jacobs and Christof L \"o ding, editors, Foundations of Software Science and Computation Structures , pages 513--530, Berlin, Heidelberg, 2016. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-49630-5_30 doi:10.1007/978-3-662-49630-5_30

    work page doi:10.1007/978-3-662-49630-5_30 2016

  21. [21]

    In: Proceedings of the 21st ACM SIGPLAN International Con- ference on Functional Programming

    Marco Gaboardi, Shin-ya Katsumata, Dominic Orchard, Flavien Breuvart, and Tarmo Uustalu. Combining effects and coeffects via grading. In Proceedings of the 21st ACM SIGPLAN International Conference on Functional Programming , ICFP 2016, page 476–489, New York, NY, USA, 2016. Association for Computing Machinery. https://doi.org/10.1145/2951913.2951939 doi:...

    work page doi:10.1145/2951913.2951939 2016

  22. [22]

    A categorical approach to probability theory

    Michèle Giry. A categorical approach to probability theory. In Categorical aspects of topology and analysis , pages 68--85. Springer, 1982

    1982

  23. [23]

    Probabilistic reliability engineering

    Boris Gnedenko and Igor A Ushakov. Probabilistic reliability engineering . John Wiley & Sons, 1995

    1995

  24. [24]

    Trace semantics via generic observations

    Sergey Goncharov. Trace semantics via generic observations. In Reiko Heckel and Stefan Milius, editors, Algebra and Coalgebra in Computer Science , pages 158--174, Berlin, Heidelberg, 2013. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-40206-7_13 doi:10.1007/978-3-642-40206-7_13

    work page doi:10.1007/978-3-642-40206-7_13 2013

  25. [25]

    An adequate while-language for hybrid computation

    Sergey Goncharov and Renato Neves. An adequate while-language for hybrid computation. In Proceedings of the 21st International Symposium on Principles and Practice of Declarative Programming , PPDP '19, New York, NY, USA, 2019. Association for Computing Machinery. https://doi.org/10.1145/3354166.3354176 doi:10.1145/3354166.3354176

    work page doi:10.1145/3354166.3354176 2019

  26. [26]

    Simulations and bisimulations for coalgebraic modal logics

    Daniel Gor \'i n and Lutz Schr \"o der. Simulations and bisimulations for coalgebraic modal logics. In Reiko Heckel and Stefan Milius, editors, Algebra and Coalgebra in Computer Science , pages 253--266, Berlin, Heidelberg, 2013. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-40206-7_19 doi:10.1007/978-3-642-40206-7_19

    work page doi:10.1007/978-3-642-40206-7_19 2013

  27. [27]

    Generic trace semantics via coinduction

    Ichiro Hasuo, Bart Jacobs, and Ana Sokolova. Generic trace semantics via coinduction. Logical Methods in Computer Science , Volume 3, Issue 4, Nov 2007. URL: https://lmcs.episciences.org/864, https://doi.org/10.2168/LMCS-3(4:11)2007 doi:10.2168/LMCS-3(4:11)2007

    work page doi:10.2168/lmcs-3(4:11)2007 2007

  28. [28]

    Memoryful geometry of interaction: from coalgebraic components to algebraic effects

    Naohiko Hoshino, Koko Muroya, and Ichiro Hasuo. Memoryful geometry of interaction: from coalgebraic components to algebraic effects. In Thomas A. Henzinger and Dale Miller, editors, Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), C...

    work page doi:10.1145/2603088.2603124 2014

  29. [29]

    Trace Semantics for Coalgebras , volume =

    Bart Jacobs. Trace semantics for coalgebras. Electronic Notes in Theoretical Computer Science , 106:167--184, 2004. Proceedings of the Workshop on Coalgebraic Methods in Computer Science (CMCS). https://doi.org/10.1016/j.entcs.2004.02.031 doi:10.1016/j.entcs.2004.02.031

    work page doi:10.1016/j.entcs.2004.02.031 2004

  30. [30]

    Hyper normalisation and conditioning for discrete probability distributions

    Bart Jacobs. Hyper normalisation and conditioning for discrete probability distributions. Logical Methods in Computer Science , Volume 13, Issue 3, Aug 2017. URL: https://lmcs.episciences.org/2009, https://doi.org/10.23638/LMCS-13(3:17)2017 doi:10.23638/LMCS-13(3:17)2017

    work page doi:10.23638/lmcs-13(3:17)2017 2017

  31. [31]

    Trace semantics via determinization

    Bart Jacobs, Alexandra Silva, and Ana Sokolova. Trace semantics via determinization. Journal of Computer and System Sciences , 81(5):859--879, 2015. 11th International Workshop on Coalgebraic Methods in Computer Science, CMCS 2012 (Selected Papers). https://doi.org/10.1016/j.jcss.2014.12.005 doi:10.1016/j.jcss.2014.12.005

    work page doi:10.1016/j.jcss.2014.12.005 2015

  32. [32]

    Coalgebraic trace semantics for probabilistic transition systems based on measure theory

    Henning Kerstan and Barbara K \"o nig. Coalgebraic trace semantics for probabilistic transition systems based on measure theory. In Maciej Koutny and Irek Ulidowski, editors, CONCUR 2012 -- Concurrency Theory , pages 410--424, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-32940-1_29 doi:10.1007/978-3-642-32940-1_29

    work page doi:10.1007/978-3-642-32940-1_29 2012

  33. [33]

    Coalgebraic trace semantics for continuous probabilistic transition systems

    Henning Kerstan and Barbara König. Coalgebraic trace semantics for continuous probabilistic transition systems. Logical Methods in Computer Science , Volume 9, Issue 4, Dec 2013. URL: https://lmcs.episciences.org/859, https://doi.org/10.2168/LMCS-9(4:16)2013 doi:10.2168/LMCS-9(4:16)2013

    work page doi:10.2168/lmcs-9(4:16)2013 2013

  34. [34]

    Coalgebraic modelling of timed processes

    Marco Kick. Coalgebraic modelling of timed processes . PhD thesis, University of Edinburgh, School of Informatics, 2003. URL: http://hdl.handle.net/1842/24771

    2003

  35. [35]

    Coalgebraic semantics for timed processes

    Marco Kick, John Power, and Alex Simpson. Coalgebraic semantics for timed processes. Information and Computation , 204(4):588--609, 2006. Seventh Workshop on Coalgebraic Methods in Computer Science 2004. https://doi.org/10.1016/j.ic.2005.11.003 doi:10.1016/j.ic.2005.11.003

    work page doi:10.1016/j.ic.2005.11.003 2006

  36. [36]

    Probability theory: a comprehensive course

    Achim Klenke. Probability theory: a comprehensive course . Springer, 2008. https://doi.org/10.1007/978-1-84800-048-3 doi:10.1007/978-1-84800-048-3

    work page doi:10.1007/978-1-84800-048-3 2008

  37. [37]

    Coalgebraic modal logic beyond sets

    Bartek Klin. Coalgebraic modal logic beyond sets. Electronic Notes in Theoretical Computer Science , 173:177--201, 2007. Proceedings of the 23rd Conference on the Mathematical Foundations of Programming Semantics (MFPS XXIII). https://doi.org/10.1016/j.entcs.2007.02.034 doi:10.1016/j.entcs.2007.02.034

    work page doi:10.1016/j.entcs.2007.02.034 2007

  38. [38]

    Coalgebraic trace semantics via forgetful logics

    Bartek Klin and Jurriaan Rot. Coalgebraic trace semantics via forgetful logics. Logical Methods in Computer Science , Volume 12, Issue 4, Apr 2017. URL: https://lmcs.episciences.org/2622, https://doi.org/10.2168/LMCS-12(4:10)2016 doi:10.2168/LMCS-12(4:10)2016

    work page doi:10.2168/lmcs-12(4:10)2016 2017

  39. [39]

    Coalgebraic semantics of modal logics: An overview

    Clemens Kupke and Dirk Pattinson. Coalgebraic semantics of modal logics: An overview. Theoretical Computer Science , 412(38):5070--5094, 2011. CMCS Tenth Anniversary Meeting. https://doi.org/10.1016/j.tcs.2011.04.023 doi:10.1016/j.tcs.2011.04.023

    work page doi:10.1016/j.tcs.2011.04.023 2011

  40. [40]

    Functorial remarks on the general concept of chaos

    William Lawvere. Functorial remarks on the general concept of chaos. Preprint series \# 87, Institute for Mathematics and its Applications, University of Minnesota, July 1984

    1984

  41. [41]

    Lew, Marco F

    Alexander K. Lew, Marco F. Cusumano-Towner, Benjamin Sherman, Michael Carbin, and Vikash K. Mansinghka. Trace types and denotational semantics for sound programmable inference in probabilistic languages. Proc. ACM Program. Lang. , 4(POPL), December 2019. https://doi.org/10.1145/3371087 doi:10.1145/3371087

    work page doi:10.1145/3371087 2019

  42. [42]

    2025 , issue_date =

    Jack Liell-Cock and Sam Staton. Compositional imprecise probability: A solution from graded monads and markov categories. Proc. ACM Program. Lang. , 9(POPL), January 2025. https://doi.org/10.1145/3704890 doi:10.1145/3704890

    work page doi:10.1145/3704890 2025

  43. [43]

    Flexibly graded monads and graded algebras

    Dylan McDermott and Tarmo Uustalu. Flexibly graded monads and graded algebras. In Ekaterina Komendantskaya, editor, Mathematics of Program Construction , pages 102--128, Cham, 2022. Springer International Publishing. https://doi.org/10.1007/978-3-031-16912-0_4 doi:10.1007/978-3-031-16912-0_4

    work page doi:10.1007/978-3-031-16912-0_4 2022

  44. [44]

    Generic Trace Semantics and Graded Monads

    Stefan Milius, Dirk Pattinson, and Lutz Schr\" o der. Generic Trace Semantics and Graded Monads . In Lawrence S. Moss and Pawel Sobocinski, editors, 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015) , volume 35 of Leibniz International Proceedings in Informatics (LIPIcs) , pages 253--269, Dagstuhl, Germany, 2015. Schloss Dagstuhl --...

    work page doi:10.4230/lipics.calco.2015.253 2015

  45. [45]

    Memoryful geometry of interaction II: recursion and adequacy

    Koko Muroya, Naohiko Hoshino, and Ichiro Hasuo. Memoryful geometry of interaction II: recursion and adequacy. In Rastislav Bod \' k and Rupak Majumdar, editors, Proceedings of the 43rd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2016, St. Petersburg, FL, USA, January 20 - 22, 2016 , pages 748--760. ACM , 2016. https://...

    work page doi:10.1145/2837614.2837672 2016

  46. [46]

    Barbosa, Dirk Hofmann, and Manuel A

    Renato Neves, Luis S. Barbosa, Dirk Hofmann, and Manuel A. Martins. Continuity as a computational effect. Journal of Logical and Algebraic Methods in Programming , 85(5, Part 2):1057--1085, 2016. Articles dedicated to Prof. J. N. Oliveira on the occasion of his 60th birthday. https://doi.org/10.1016/j.jlamp.2016.05.005 doi:10.1016/j.jlamp.2016.05.005

    work page doi:10.1016/j.jlamp.2016.05.005 2016

  47. [47]

    Renato Neves and Luís S. Barbosa. Languages and models for hybrid automata: A coalgebraic perspective. Theoretical Computer Science , 744:113--142, 2018. Theoretical aspects of computing. https://doi.org/10.1016/j.tcs.2017.09.038 doi:10.1016/j.tcs.2017.09.038

    work page doi:10.1016/j.tcs.2017.09.038 2018

  48. [48]

    An adequate while-language for stochastic hybrid computation

    Renato Neves, Jos\' e Proen c a, and Juliana Souza. An adequate while-language for stochastic hybrid computation. In Proceedings of the 27th International Symposium on Principles and Practice of Declarative Programming , PPDP '25, New York, NY, USA, 2025. Association for Computing Machinery. https://doi.org/10.1145/3756907.3756927 doi:10.1145/3756907.3756927

    work page doi:10.1145/3756907.3756927 2025

  49. [49]

    The Category of Markov Kernels

    Prakash Panangaden. The Category of Markov Kernels . Electronic Notes in Theoretical Computer Science , 22:171--187, January 1999. https://doi.org/10.1016/S1571-0661(05)80602-4 doi:10.1016/S1571-0661(05)80602-4

    work page doi:10.1016/s1571-0661(05)80602-4 1999

  50. [50]

    Differential dynamic logics - automated theorem proving for hybrid systems

    Andr \' e Platzer. Differential dynamic logics - automated theorem proving for hybrid systems . PhD thesis, Carl von Ossietzky University of Oldenburg, 2008. URL: http://oops.uni-oldenburg.de/1403/

    2008

  51. [51]

    Stochastic differential dynamic logic for stochastic hybrid programs

    Andr \' e Platzer. Stochastic differential dynamic logic for stochastic hybrid programs. In Nikolaj S. Bj rner and Viorica Sofronie - Stokkermans, editors, Automated Deduction - CADE-23 - 23rd International Conference on Automated Deduction, Wroclaw, Poland, July 31 - August 5, 2011. Proceedings , Lecture Notes in Computer Science, pages 446--460. Springe...

    work page doi:10.1007/978-3-642-22438-6 2011

  52. [52]

    Steps and traces

    Jurriaan Rot, Bart Jacobs, and Paul Blain Levy. Steps and traces. Journal of Logic and Computation , 31(6):1482--1525, 09 2021. https://doi.org/10.1093/logcom/exab050 doi:10.1093/logcom/exab050

    work page doi:10.1093/logcom/exab050 2021

  53. [53]

    Jan J. M. M. Rutten. Universal coalgebra: a theory of systems. Theoretical Computer Science , 249(1):3--80, 2000. https://doi.org/10.1016/S0304-3975(00)00056-6 doi:10.1016/S0304-3975(00)00056-6

    work page doi:10.1016/s0304-3975(00)00056-6 2000

  54. [54]

    String Diagrams for Graded Monoidal Theories, with an Application to Imprecise Probability

    Ralph Sarkis and Fabio Zanasi. String Diagrams for Graded Monoidal Theories, with an Application to Imprecise Probability . In Corina C\^ i rstea and Alexander Knapp, editors, 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025) , volume 342 of Leibniz International Proceedings in Informatics (LIPIcs) , pages 5:1--5:23, Dagstuhl, Germ...

    work page doi:10.4230/lipics.calco.2025.5 2025

  55. [55]

    Strong Completeness of Coalgebraic Modal Logics

    Lutz Schr\" o der and Dirk Pattinson. Strong Completeness of Coalgebraic Modal Logics . In Susanne Albers and Jean-Yves Marion, editors, 26th International Symposium on Theoretical Aspects of Computer Science , volume 3 of Leibniz International Proceedings in Informatics (LIPIcs) , pages 673--684, Dagstuhl, Germany, 2009. Schloss Dagstuhl -- Leibniz-Zentr...

    work page doi:10.4230/lipics.stacs.2009.1855 2009

  56. [56]

    Expressivity of coalgebraic modal logic: The limits and beyond

    Lutz Schröder. Expressivity of coalgebraic modal logic: The limits and beyond. Theoretical Computer Science , 390(2):230--247, 2008. Foundations of Software Science and Computational Structures. https://doi.org/10.1016/j.tcs.2007.09.023 doi:10.1016/j.tcs.2007.09.023

    work page doi:10.1016/j.tcs.2007.09.023 2008

  57. [57]

    Martin L. Shooman. Reliability of computer systems and networks: fault tolerance, analysis, and design . John Wiley & Sons, 2003. https://doi.org/10.1002/047122460X doi:10.1002/047122460X

    work page doi:10.1002/047122460x 2003

  58. [58]

    A.L. Smirnov. Graded monads and rings of polynomials. Journal of Mathematical Sciences , 151:3032--3051, 2008. https://doi.org/10.1007/s10958-008-9013-7 doi:10.1007/s10958-008-9013-7

    work page doi:10.1007/s10958-008-9013-7 2008

  59. [59]

    Coalgebraic analysis of probabilistic systems

    Ana Sokolova. Coalgebraic analysis of probabilistic systems . PhD thesis, Eindhoven University of Technology, 2005. URL: https://pure.tue.nl/ws/files/2025267/200513143.pdf

    work page arXiv 2005

  60. [60]

    Probabilistic systems coalgebraically: A survey

    Ana Sokolova. Probabilistic systems coalgebraically: A survey. Theoretical Computer Science , 412(38):5095--5110, 2011. CMCS Tenth Anniversary Meeting. https://doi.org/10.1016/j.tcs.2011.05.008 doi:10.1016/j.tcs.2011.05.008

    work page doi:10.1016/j.tcs.2011.05.008 2011

  61. [61]

    Coalgebraic infinite traces and kleisli simulations

    Natsuki Urabe and Ichiro Hasuo. Coalgebraic infinite traces and kleisli simulations. Logical Methods in Computer Science , 14, 2018

    2018