pith. sign in

arxiv: 2604.25355 · v2 · pith:LZHW2GCUnew · submitted 2026-04-28 · 💻 cs.LO

From Coalgebraic Determinization to Belief Construction for Partial Observability

Mayuko Kori , Kazuki Watanabe This is my paper

Pith reviewed 2026-05-21 00:23 UTC · model grok-4.3

classification 💻 cs.LO
keywords coalgebraic determinizationbelief constructionpartial observabilityPOMDPmonad liftingslice categoriesweighted transition systemssemimodule monad
0
0 comments X

The pith

The semantics of a partially observable system coincides with that of the corresponding belief coalgebra obtained via monad lifting and belief decomposition.

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

This paper builds a coalgebraic framework for turning partially observable systems into fully observable ones while keeping the same semantics. It lifts a monad to slice categories and adds a belief decomposition that groups states by their observations. These steps combine with existing coalgebraic determinization to produce a belief coalgebra whose behavior matches the original partial system. The authors then isolate conditions under which this belief coalgebra further matches a fully observable system, recovering the known POMDP-to-belief-MDP equivalence and adding a new result for weighted transition systems.

Core claim

We show that the semantics of a partially observable system coincides with that of the corresponding belief coalgebra. We then study when the latter further agrees with the semantics of its fully observable counterpart, and use this to identify conditions under which the semantics of a partially observable system coincides with that of the corresponding fully observable belief system. As consequences, we recover the standard equivalence between POMDPs and belief MDPs, and obtain a new equivalence result for weighted transition systems with the semimodule monad.

What carries the argument

Belief decomposition that reorganizes states according to observations, combined with the lifting of a monad to slice categories and coalgebraic determinization.

If this is right

  • Semantics of partially observable systems reduces to semantics of the corresponding belief coalgebra.
  • Standard equivalence between POMDPs and belief MDPs is recovered inside the coalgebraic setting.
  • A new equivalence result holds for weighted transition systems equipped with the semimodule monad.
  • Conditions are given under which partial-observability semantics matches that of a fully observable belief system.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same monad-lifting and decomposition steps may apply to other coalgebraic models beyond MDPs and weighted systems.
  • The framework could support compositional verification of systems with partial observations by reusing existing coalgebraic tools.
  • Concrete examples of weighted transition systems could be checked to confirm the new equivalence result holds in practice.

Load-bearing premise

The lifting of a monad to slice categories together with the introduced belief decomposition preserves the relevant semantics when combined with coalgebraic determinization.

What would settle it

A concrete partially observable system for which the constructed belief coalgebra produces different observable behaviors than the original system, even after the monad lift and decomposition steps are applied.

read the original abstract

The belief construction is a fundamental technique for transforming partially observable systems to fully observable ones while preserving the relevant semantics. It plays a central role in the analysis of partially observable systems, in particular partially observable Markov decision processes (POMDPs), which is a central model in artificial intelligence and formal verification. In this paper, we develop a coalgebraic framework for the belief construction. To handle observations categorically, we lift a monad to slice categories and introduce a belief decomposition that reorganizes states according to their observations. This allows us to introduce a coalgebraic generalization of the belief construction, obtained by combining the belief decomposition with the coalgebraic determinization of Silva, Bonchi, Bonsangue, and Rutten. In this framework, we show that the semantics of a partially observable system coincides with that of the corresponding belief coalgebra. We then study when the latter further agrees with the semantics of its fully observable counterpart, and use this to identify conditions under which the semantics of a partially observable system coincides with that of the corresponding fully observable belief system. As consequences, we recover the standard equivalence between POMDPs and belief MDPs, and obtain a new equivalence result for weighted transition systems with the semimodule monad.

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

0 major / 3 minor

Summary. The paper develops a coalgebraic framework for the belief construction in partially observable systems. It lifts a monad to slice categories, introduces a belief decomposition that reorganizes states by observations, and combines this with coalgebraic determinization. The central results are that the semantics of a partially observable system coincides with that of the corresponding belief coalgebra, together with conditions under which the belief coalgebra further agrees with the semantics of its fully observable counterpart. As consequences the framework recovers the standard equivalence between POMDPs and belief MDPs and yields a new equivalence result for weighted transition systems under the semimodule monad.

Significance. If the derivations hold, the work supplies a uniform categorical treatment of belief construction that generalizes across coalgebraic models. The recovery of the classical POMDP equivalence serves as a sanity check, while the new result for weighted transition systems demonstrates that the framework produces non-trivial consequences beyond the motivating example. The use of standard monad and coalgebra operations plus a single new decomposition keeps the construction parsimonious and potentially reusable for other partial-observability settings in verification and AI.

minor comments (3)
  1. §3.2: the definition of the belief decomposition would benefit from an explicit commutative diagram showing how the observation map interacts with the slice-category lift; the current prose description leaves the universal property implicit.
  2. Theorem 5.3: the statement of the agreement condition between the belief coalgebra and the fully observable counterpart should list the required assumptions on the monad (e.g., preservation of weak pullbacks) in the theorem itself rather than only in the surrounding text.
  3. The running example of weighted transition systems in §6.1 uses the semimodule monad without recalling its definition; a one-sentence reminder would aid readers who are not experts in that monad.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript. The referee's summary accurately captures the central contributions: the coalgebraic lifting of monads to slice categories, the belief decomposition, the combination with determinization, the equivalence of semantics between partially observable systems and belief coalgebras, and the recovery of the POMDP equivalence together with the new result for semimodule monads. We are pleased that the referee recognizes the parsimonious nature of the construction and its potential reusability.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via independent categorical constructions

full rationale

The paper introduces a monad lifting to slice categories and a belief decomposition as new operations, then combines them with the coalgebraic determinization construction from Silva et al. (independent prior work). The central claim that the semantics of the partially observable system coincides with the belief coalgebra is established directly from these definitions and the standard coalgebraic semantics, without reducing to fitted parameters, self-definitional loops, or load-bearing self-citations. The further agreement conditions with fully observable systems and the recovered POMDP equivalence are derived as consequences of the same constructions. No step equates a prediction or result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on standard properties of monads, coalgebras, and slice categories from prior literature; the belief decomposition is introduced as an organizational device without new numerical parameters or external data fits.

axioms (2)
  • standard math Monads can be lifted to slice categories while preserving the relevant algebraic structure
    Invoked to handle observations categorically before introducing the belief decomposition
  • standard math Coalgebras and their semantics are preserved under the determinization construction of Silva et al.
    Used as the base for combining with the new belief decomposition
invented entities (1)
  • belief decomposition no independent evidence
    purpose: reorganizes states according to their observations to enable the coalgebraic belief construction
    Newly introduced device that groups states by observations before applying determinization

pith-pipeline@v0.9.0 · 5749 in / 1422 out tokens · 48922 ms · 2026-05-21T00:23:45.027545+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

  1. [1]

    Moss, and Lurdes Sousa

    Jir \' Ad \' a mek, Stefan Milius, Lawrence S. Moss, and Lurdes Sousa. Well-pointed coalgebras. Log. Methods Comput. Sci. , 9(3), 2013

    work page 2013

  2. [2]

    Initial algebras and terminal coalgebras: the theory of fixed points of functors

    Ji r \' Ad \'a mek, Stefan Milius, and Lawrence Stuart Moss. Initial algebras and terminal coalgebras: the theory of fixed points of functors. Cambridge University Press , 2025

    work page 2025

  3. [3]

    Weakest preconditions in fibrations

    Alejandro Aguirre, Shin - ya Katsumata, and Satoshi Kura. Weakest preconditions in fibrations. Math. Struct. Comput. Sci. , 32(4):472--510, 2022

    work page 2022

  4. [4]

    Search and explore: symbiotic policy synthesis in POMDP s

    Roman Andriushchenko, Alexander Bork, Milan Ceska, Sebastian Junges, Joost - Pieter Katoen, and Filip Mac \' a k. Search and explore: symbiotic policy synthesis in POMDP s. Formal Methods Syst. Des. , 68(1):4, 2026

    work page 2026

  5. [5]

    A coalgebraic perspective on predictive processing

    Manuel Baltieri, Filippo Torresan, and Tomoya Nakai. A coalgebraic perspective on predictive processing. arXiv preprint arXiv:2508.16877 , 2025

    work page arXiv 2025

  6. [6]

    Minimization via duality

    Nick Bezhanishvili, Clemens Kupke, and Prakash Panangaden. Minimization via duality. In WoLLIC , Lecture Notes in Computer Science, pages 191--205. Springer, 2012

    work page 2012

  7. [7]

    Bonsangue, Georgiana Caltais, Jan Rutten, and Alexandra Silva

    Filippo Bonchi, Marcello M. Bonsangue, Georgiana Caltais, Jan Rutten, and Alexandra Silva. A coalgebraic view on decorated traces. Math. Struct. Comput. Sci. , 26(7):1234--1268, 2016

    work page 2016

  8. [8]

    Distribution bisimilarity via the power of convex algebras

    Filippo Bonchi, Alexandra Silva, and Ana Sokolova. Distribution bisimilarity via the power of convex algebras. Log. Methods Comput. Sci. , 17(3), 2021

    work page 2021

  9. [9]

    The theory of traces for systems with nondeterminism, probability, and termination

    Filippo Bonchi, Ana Sokolova, and Valeria Vignudelli. The theory of traces for systems with nondeterminism, probability, and termination. Log. Methods Comput. Sci. , 18(2), 2022

    work page 2022

  10. [10]

    Under-approximating expected total rewards in POMDP s

    Alexander Bork, Joost - Pieter Katoen, and Tim Quatmann. Under-approximating expected total rewards in POMDP s. In TACAS (2) , Lecture Notes in Computer Science, pages 22--40. Springer, 2022

    work page 2022

  11. [11]

    Graded semantics and graded logics for Eilenberg-Moore coalgebras

    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 CMCS , Lecture Notes in Computer Science, pages 114--134. Springer, 2024

    work page 2024

  12. [12]

    Coalgebraic semantics for nominal automata

    Florian Frank, Stefan Milius, and Henning Urbat. Coalgebraic semantics for nominal automata. In CMCS , Lecture Notes in Computer Science, pages 45--66. Springer, 2022

    work page 2022

  13. [13]

    On the compositionality of monads via weak distributive laws

    Alexandre Goy. On the compositionality of monads via weak distributive laws. (Compositionnalit \' e des monades par lois de distributivit \' e faibles) . PhD thesis, University of Paris-Saclay, France, 2021

    work page 2021

  14. [14]

    (in)finite trace equivalence of probabilistic transition systems

    Alexandre Goy and Jurriaan Rot. (in)finite trace equivalence of probabilistic transition systems. In CMCS , Lecture Notes in Computer Science, pages 100--121. Springer, 2018

    work page 2018

  15. [15]

    Introduction to Coalgebra: Towards Mathematics of States and Observation , volume 59 of Cambridge Tracts in Theoretical Computer Science

    Bart Jacobs. Introduction to Coalgebra: Towards Mathematics of States and Observation , volume 59 of Cambridge Tracts in Theoretical Computer Science . Cambridge University Press, 2016

    work page 2016

  16. [16]

    Trace semantics via determinization

    Bart Jacobs, Alexandra Silva, and Ana Sokolova. Trace semantics via determinization. J. Comput. Syst. Sci. , 81(5):859--879, 2015

    work page 2015

  17. [17]

    Traces, executions and schedulers, coalgebraically

    Bart Jacobs and Ana Sokolova. Traces, executions and schedulers, coalgebraically. In CALCO , Lecture Notes in Computer Science, pages 206--220. Springer, 2009

    work page 2009

  18. [18]

    Doctrinal adjunction

    Gregory Maxwell Kelly. Doctrinal adjunction. In Gregory M. Kelly, editor, Category Seminar , pages 257--280, Berlin, Heidelberg, 1974. Springer Berlin Heidelberg

    work page 1974

  19. [19]

    Verification and control of partially observable probabilistic systems

    Gethin Norman, David Parker, and Xueyi Zou. Verification and control of partially observable probabilistic systems. Real Time Syst. , 53(3):354--402, 2017

    work page 2017

  20. [20]

    Steps and traces

    Jurriaan Rot, Bart Jacobs, and Paul Blain Levy. Steps and traces. J. Log. Comput. , 31(6):1482--1525, 2021

    work page 2021

  21. [21]

    Gordon, and Sebastian Thrun

    Nicholas Roy, Geoffrey J. Gordon, and Sebastian Thrun. Finding approximate POMDP solutions through belief compression. J. Artif. Intell. Res. , 23:1--40, 2005

    work page 2005

  22. [22]

    Artificial Intelligence: A Modern Approach (4th Edition)

    Stuart Russell and Peter Norvig. Artificial Intelligence: A Modern Approach (4th Edition) . Pearson, 2020

    work page 2020

  23. [23]

    A survey of point-based POMDP solvers

    Guy Shani, Joelle Pineau, and Robert Kaplow. A survey of point-based POMDP solvers. Auton. Agents Multi Agent Syst. , 27(1):1--51, 2013

    work page 2013

  24. [24]

    Bonsangue, and Jan J

    Alexandra Silva, Filippo Bonchi, Marcello M. Bonsangue, and Jan J. M. M. Rutten. Generalizing determinization from automata to coalgebras. Log. Methods Comput. Sci. , 9(1), 2013

    work page 2013

  25. [25]

    Sim \ a o, Marnix Suilen, and Nils Jansen

    Thiago D. Sim \ a o, Marnix Suilen, and Nils Jansen. Safe policy improvement for POMDP s via finite-state controllers. In AAAI , pages 15109--15117. AAAI Press, 2023

    work page 2023

  26. [26]

    Preservation and reflection of bisimilarity via invertible steps

    Ruben Turkenburg, Clemens Kupke, Jurriaan Rot, and Ezra Schoen. Preservation and reflection of bisimilarity via invertible steps. In FoSSaCS , Lecture Notes in Computer Science, pages 328--348. Springer, 2023

    work page 2023

  27. [27]

    V \'a zquez-M \'a rquez

    A. V \'a zquez-M \'a rquez. Monad and comonad objects through 2-adjunctions of the type adj-mnd. arXiv: Category Theory , 2015. URL: https://api.semanticscholar.org/CorpusID:119586944

    work page 2015

  28. [28]

    A coalgebraic view on reachability

    Thorsten Wi^^c3^^9fmann, Stefan Milius, Shinya Katsumata, and J^^c3^^a9r^^c3^^a9my Dubut. A coalgebraic view on reachability. Commentationes Mathematicae Universitatis Carolinae , 60(4):605--638, 2019

    work page 2019