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Expressivity of bisimulation pseudometrics over analytic state spaces

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Pith's one-line read For continuous Markov decision processes, behavioral distance is exactly logical distance: the paper proves a quantitative Hennessy–Milner theorem for MDPs over analytic state spaces.

desk verdict A substantial new expressivity result for bisimulation pseudometrics on continuous MDPs, slowed by one unproved coupling-lifting step in Theorem 28. read the letter →

arxiv 2505.23635 v1 pith:GISTROAV submitted 2025-05-29 cs.LO

Daniel Luckhardt , Harsh Beohar , Clemens Kupke This is my paper
classification cs.LO
keywords MarkovdecisionprocessanalyticspaceperfectmeasureWassersteindistanceKantorovich-RubinsteindualityquantitativeHennessy-Milnertheorembisimulationpseudometricmodallogic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to prove a quantitative Hennessy–Milner theorem for Markov decision processes with continuous, analytic state spaces: the bisimulation pseudometric between states, defined as the least fixed point of a Wasserstein-style distance lifting, coincides with the logical distance given by a quantitative modal logic whose formulas test expected transitions and rewards. If the theorem is right, then for any two states in such an MDP, behavioral similarity is exactly indistinguishability by all formulas in that logic, at any fixed discount factor $c\in[0,1]$. The significance is that this equivalence holds without assuming a Polish topology: the paper works purely measure-theoretically with universally measurable distances and perfect measures, and it derives the bisimulation metric via Kleene's fixed-point theorem rather than contraction arguments. The central result is Corollary 31, for the language $L$ of Lemma 30, with the theory map made topologisable so that Stone–Weierstrass approximation converts logical agreement into metric equality.

What carries the argument

The machinery has three load-bearing pieces. First, a fibration of predicates over analytic spaces: $\mathrm{Pred}(X)$ is the set of universally measurable functions $X\to[0,1]$, and $\mathrm{lsPred}(X)$ is the subfibration of lower semi-measurable functions; the direct-image left adjoint exists for Suslin-level predicates, which is what makes the coupling-based lifting possible. Second, the distance lifting $\sigma_{\mathrm{MDP}}$ built from the Wasserstein lifting of the Giry functor, $\hat\sigma(d)(m,n)=\inf_{c\in K(m,n)}\int d\,dc$, composed with reward differences by a convex combination with parameter $c$. Third, the theory map $qTh:X\to [0,1]^L$ sending each state to its vector of formula values; topologisability of $qTh$ lets the proof restrict attention to a standard subspace $A\subseteq [0,1]^L$ and apply Stone–Weierstrass to approximate short predicates by formulas. The chain is: push the logical distance along the theory map, apply Kantorovich–Rubinstein duality to rewrite the Wasserstein cost as a supremum over short predicates, then approximate those predicates by modal formulas.

What would settle it

To refute the expressivity half, one would exhibit an analytic-space MDP, two states $x$ and $y$, an action $a$, and a discount $c$ for which the pushforward measures $qTh_*(m_{x,a})$ and $qTh_*(m_{y,a})$ admit a coupling that is not the image of any coupling of $m_{x,a}$ and $m_{y,a}$; then the step equating equations (14) and (15) collapses, and one can check directly whether $\mathrm{bd}_c(x,y)>dL_c(x,y)$. Equivalently, a pair of states with identical values on every formula of $L$ but with positive bisimulation distance would refute Corollary 31.

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Extended reading notes

Core claim

The paper's central claim is that the behavioral distance on a continuous MDP is fully captured by a quantitative modal logic. Concretely, for an MDP coalgebra $\gamma:(X,\mathcal{A})\to B_{\mathrm{MDP}}(X,\mathcal{A})$ in the category of analytic spaces, with language $L$ generated by truth, negation, conjunction, scalar addition and subtraction $\_+r$, $\_-r$ with $r\in[0,1]$, and a diamond modality $\Diamond_a$ for each action $a$, the bisimulation pseudometric $\mathrm{bd}_c$, defined as the least fixed point of $\gamma\circ\sigma_{\mathrm{MDP}}$ with discount $c$, equals the logical distance $dL_c(x,y)=\sup_{\varphi\in L}|\varphi(x)-\varphi(y)|$. Adequacy ($\mathrm{bd}\ge dL$) holds whenever the function symbols are interpreted as nonexpansive functions; expressivity ($\mathrm{bd}\le dL$) requires perfect transition measures, a topologisable theory map, and scalar addition in the signature. The proof combines Kantorovich–Rubinstein duality for perfect measures with a Stone–Weierstrass lemma over the compact-open topology on the space of formulas.

Load-bearing premise

The equality between the bisimulation distance and the logical distance relies on a lifting step: it assumes that whenever the formula-profiles of two states' transition probabilities are paired, that paired profile really comes from pairing the two original random transitions; the paper cites a regularity theorem for this step but gives no proof.

Editorial extensions

If this is right

  • If Corollary 31 is correct, the bisimulation pseudometric on continuous MDPs is determined by formula evaluations: two states are $\varepsilon$-close behaviorally exactly when no formula of $L$ separates them by more than $\varepsilon$.
  • Because the equality holds for every $c\in[0,1]$, the discount factor may be set to $0$ or $1$, covering purely reward-based and purely transition-based comparisons, which the contraction-based definition of the earlier bisimulation metric could not handle.
  • Adequacy holds for any language whose function symbols are nonexpansive, so adding operators such as scalar multiplication or convex combination preserves $\mathrm{bd}\ge dL$; expressivity then follows whenever the theory map is topologisable.
  • The alternative language with a separate reward modality and expected-transition modality is shown to be expressive but not adequate, because truncated addition is not nonexpansive; this explains why the paper's single diamond modality, which mixes the transition expectation and the reward, is needed.
  • Since the least fixed point is obtained by Kleene iteration, the definition of $\mathrm{bd}_c$ does not rely on the transition kernel being a contraction, which broadens the class of MDPs for which the metric is defined.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper's claims: if $\mathrm{bd}_c=dL_c$ is stable under Lipschitz embeddings, then representation-learning pipelines that map states to formula-value vectors preserve behavioral distances up to the Lipschitz constant; this would give a formal justification for using logical features as state encodings in continuous MDPs.
  • The unproved disintegration step suggests a targeted stress test: exhibit two perfect measures on an analytic space whose theory-pushforwards admit a coupling that is not the image of any coupling of the original measures; such an example would isolate where the Stone–Weierstrass approximation argument needs a regularity condition beyond analyticity.
  • A natural extension, which the authors list as future work, would be to replace the compact-open topology on the formula space with a measurable Stone–Weierstrass theorem; if such a theorem existed, the topologisability assumption could likely be dropped from Theorem 28.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper develops bisimulation pseudometrics for Markov reward processes and Markov decision processes whose state spaces are analytic measurable spaces, viewed as coalgebras in the category Ana. The metric is defined as the least fixed point of a functional built from a coupling-based Wasserstein lifting, formalized through a fibration of universally measurable predicates. The paper then introduces a quantitative modal logic with truth, conjunction, negation, scalar addition and subtraction, and one diamond per action; the diamond semantics is a convex combination of the expected value of the formula and the action reward. The main claim, Corollary 31, is that for every MDP in Ana and every discount factor c in [0,1], the bisimulation pseudometric equals the logical distance induced by this language, i.e. a quantitative Hennessy-Milner theorem. Adequacy is proved in Theorem 22, and expressivity is attacked through a general theorem (Theorem 28) assuming topologisability of the theory map and perfectness of the transition measures.

Significance. If the main theorem is correct, this is a valuable contribution to the quantitative coalgebraic semantics of continuous-state MDPs: it replaces Polish-space assumptions with analytic measurable spaces, works with universally measurable predicates rather than lower semi-continuous ones, and provides an expressive quantitative modal logic. The categorical framework, based on two fibrations and a coupling-based lifting, is original, and the adequacy proof is clean and well structured. The paper also gives a self-contained treatment of Kantorovich-Rubinstein duality for perfect measures, which is of independent interest. However, the expressivity proof contains a load-bearing unproved equality of coupling infima and a separate algebraic inconsistency with the discount factor, and the proof of Theorem 13 uses a semicontinuity claim that is not correct as written. These issues are substantial but appear repairable within the manuscript's scope.

major comments (3)
  1. [Theorem 28, Eq. (14) to Eq. (15)] The equality of the infimum over couplings of m_{x,a}, m_{y,a} of the integral of dL with the infimum over couplings of their pushforwards along qTh of the integral of d~L is asserted without proof. One inequality is immediate: every coupling of the original measures pushes forward to a coupling of the pushforward measures with the same cost. The reverse direction requires a coupling-lifting lemma: every coupling of qTh_*m_{x,a} and qTh_*m_{y,a} must be obtainable as the pushforward of some coupling of m_{x,a} and m_{y,a}. This is equivalent to the existence of regular conditional distributions of the transition measures along qTh. The parenthetical citation of [18, Thm. 6] does not by itself establish this lifting, and no such lemma is stated or proved in the paper. Since the subsequent argument bounds the pushforward infimum, the direction bd <= dL depends on this step. Please add a complete proof of the equality (or of the needed inequality direction) and verify the required measurability and regularity conditions.
  2. [Theorem 13, proof using Sion's minimax theorem] In the proof of Theorem 13, the function f(x,c) = c(d~_x), with d~_x = d_i for x in [i,i+1), is claimed to be upper semicontinuous in x. With the usual topology on [0,Infinity), the preimage f(.,c)^{-1}([0,r)) is a union of half-open intervals of the form [i,i+1), which is not open in general. For example, if c(d_0) >= r and c(d_1) < r, then the point 1 belongs to the preimage but no neighbourhood of 1 is contained in it. Thus the hypotheses of Sion's minimax theorem (Lemma 44) are not satisfied as written. Since Theorem 13 is used both for the existence of the least fixed point (Corollary 16) and in the reduction at the start of the proof of Theorem 28, this proof needs to be repaired or replaced by a valid argument.
  3. [Theorem 28, Eqs. (14) and (18), discount-factor algebra] The discount factor is handled inconsistently in the proof of Theorem 28. Lemma 11 gives sigmaMRP_X(d)((m,r),(n,s)) = c * inf_{kappa in K(m,n)} integral d dkappa + (1-c)|r-s|, but Eq. (14) writes the expression as inf over couplings of integral dL dc + c r^a_xy, dropping the leading factor c on the coupling term and replacing the reward coefficient 1-c by c. Moreover, the manipulation in Eq. (18), from sup_phi integral JphiK d(m_{x,a}-m_{y,a}) + c r^a_xy to sup_phi (integral JphiK dm_{x,a} + c r^x_a) - (integral JphiK dm_{y,a} + c r^y_a), is not algebraically valid when r^a_xy = |r^x_a - r^y_a|; the right-hand side can be smaller by 2c|r^x_a - r^y_a|. This makes the chain of inequalities leading to (13) impossible to follow as written and must be corrected.
minor comments (6)
  1. [Theorem 28, Eq. (15)] The second pushforward measure is written as qTh(m_{y,a}); it should be qTh_*(m_{y,a}).
  2. [Theorem 28, notation] The same symbol c is used both for the discount factor and for an arbitrary coupling in Eqs. (14) and (15); please use different letters, e.g. kappa for couplings, to avoid ambiguity.
  3. [Reference [4]] The companion paper [4] is cited for the fact that the predicate lifting improves universally measurable predicates to Borel predicates, but it is listed as an unpublished manuscript with a placeholder arXiv number. Since this fact is used in the main development, the proof should be included in the appendix or the companion should be made publicly available.
  4. [Lemma 30] The sentence 'Recalling that [0,1] is compact, thus the assumptions of Theorem 29 are fulfilled' is terse; it would be clearer to state explicitly that the relevant index sets are [0,1] with the usual topology and the countable set Sigma with the discrete topology, both second countable Hausdorff.
  5. [Appendix G.2, proof of Lemma 27] The measurability argument for the function h defined as an infimum over y is compressed into the sentence 'g is nothing but exists_{pr1}(lambda xy. d(x,y)-g(y))'; a few more details or a reference would help the reader verify this step.
  6. [Reference [4] title] The title of reference [4] contains a typo: 'Henneysey-milner' should be 'Hennessy-Milner'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: bd_c and dL are independently defined; the equality is a proved least-fixpoint/Stone-Weierstrass theorem, and the only self-citation [4] is redundant with Appendix Corollary 43.

full rationale

The derivation is not circular. bd_c is the least fixed point of gamma o sigma_MDP (Corollary 16), while dL is the supremum over formulae (Definition 17, Eq. 9); the diamond clause (Eq. 7) reuses the same discount c and the transition/reward data, but that shared syntax is exactly what adequacy requires and does not define bd in terms of dL. The main equality bd = dL (Corollary 31) is obtained by showing dL is a fixed point (Theorem 28) and invoking the least-fixed-point property, a substantive argument using Kantorovich-Rubinstein duality and Stone-Weierstrass. There are no fitted parameters and no data-driven predictions, and c appears symmetrically on both sides. The only self-citation is [4] for the fact that predicate lifting sends universally measurable predicates to Borel predicates; the same fact is proved in the appendix (Corollary 43), so the citation is redundant and not load-bearing. The reviewer-flagged gap at Eqs. (14)-(15) — replacing inf over couplings of m_{x,a}, m_{y,a} by inf over couplings of their pushforwards along qTh — is a missing disintegration lemma (the parenthetical '[18, Thm. 6]' is not a proof of lifting arbitrary pushforward couplings), and is a correctness concern, not a circularity: it does not presuppose bd <= dL or the equality to be proved. Hence no circular step can be exhibited; score 2 only for the minor redundant self-citation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numbers are fitted to data in this paper. The discount factor c is a universally quantified parameter of the pseudometric and the logic; the main theorem is proved for every c in [0,1]. The axioms listed are the measure-theoretic and analyticity assumptions the proofs rely on, including the critical disintegration assumption in the expressivity proof.

assumptions (5)
  • domain assumption The working state spaces are analytic spaces, i.e., the category Ana is used instead of all measurable spaces.
    Section 3.1 restricts to analytic spaces so that Suslin sets are preserved under direct images and every probability measure is perfect. This is more general than Polish spaces but less general than arbitrary measurable spaces.
  • domain assumption Probability measures on analytic spaces are perfect.
    Used in Theorem 13 and Lemma 27; stated in the proof of Theorem 45 in Appendix D.1. The perfectness property is automatic for analytic spaces, but it is a load-bearing premise of the fixed-point and duality arguments.
  • domain assumption Regular conditional distributions exist for perfect measures on analytic spaces along countably generated maps, and every coupling of pushforward measures can be lifted to a coupling of the original measures.
    Invoked in the proof of Theorem 28 when passing from equation (14) to equation (15), with a citation to [18, Thm. 6]. This is the most fragile step of the expressivity proof.
  • standard math Sion's minimax theorem, Banach-Alaoglu theorem, and the Riesz-Markov-Kakutani representation theorem for finitely additive measures.
    Used in Appendix D.1 to prove the omega-cpo continuity of the fixed-point functional (Theorem 13).
  • standard math Stone-Weierstraß theorem for compact R2 spaces.
    Used in the proof of Theorem 28 to approximate short predicates by formulas on a compact set; stated as Lemma 46 in Appendix E.

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Pith. "Pith review of Expressivity of bisimulation pseudometrics over analytic state spaces." pith.science (2026). https://pith.science/paper/GISTROAV

@misc{pith2026250523635,
  author       = {Pith},
  title        = {Pith review of: Expressivity of bisimulation pseudometrics over analytic state spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GISTROAV}},
  note         = {Machine review of arXiv:2505.23635}
}
read the original abstract

A Markov decision process (MDP) is a state-based dynamical system capable of describing probabilistic behaviour with rewards. In this paper, we view MDPs as coalgebras living in the category of analytic spaces, a very general class of measurable spaces. Note that analytic spaces were already studied in the literature on labelled Markov processes and bisimulation relations. Our results are twofold. First, we define bisimulation pseudometrics over such coalgebras using the framework of fibrations. Second, we develop a quantitative modal logic for such coalgebras and prove a quantitative form of Hennessy-Milner theorem in this new setting stating that the bisimulation pseudometric corresponds to the logical distance induced by modal formulae.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.