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Relational Neurosymbolic Markov Models

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A new class of Markov models provably satisfies relational logical constraints in sequential tasks.

desk verdict Genuinely new model class with a clever RBPF, but the advertised 'exact' inference rests on an unproven and likely false cluster factorization; the paper needs major revision before its guarantees are credible. read the letter →

arxiv 2412.13023 v1 pith:B2HNKGDB submitted 2024-12-17 cs.AI cs.LG

Lennert De Smet , Gabriele Venturato , Luc De Raedt , Giuseppe Marra This is my paper · ORCID
classification cs.AIcs.LG
keywords neurosymbolicAIhiddenMarkovmodelsrelationallogicRao-Blackwellisedparticlefilterdifferentiableinferenceprobabilisticprogrammingsequentialgenerativeout-of-distributiongeneralisation
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

The paper introduces relational neurosymbolic Markov models (NeSy-MMs), sequential probabilistic models in which each hidden state has a subsymbolic neural part and a symbolic relational-logic part, and transitions can be logical, neural, or a mixture. The central claim is that this factorisation lets a single end-to-end differentiable model guarantee relational logical constraints, such as safety or game rules, at every time step of a sequence. To make the idea practical, the paper develops a differentiable Rao-Blackwellised particle filter that replaces resampling with exact inference over the finite symbolic variables and feeds those exact probabilities into unbiased gradient estimators. The experiments show the approach generating rule-following image sequences, classifying trajectories with partially unknown enemy behaviour, imposing new constraints at test time, and generalising out of distribution better than transformer and deep-HMM baselines.

What carries the argument

The load-bearing object is the factorised transition in Eq. (3), written as $p_\varphi(S_0\mid N_0)p(N_0)p(Z_0\mid S_0)\prod_t p_\varphi(S_{t+1}\mid S_t,N_{t+1})p(N_{t+1})p(Z_{t+1}\mid S_{t+1})$, with $N_t$ the neural/subsymbolic part, $S_t$ the relational symbolic part, and $Z_t$ observations. Inference rests on a Rao-Blackwellised particle filter: rather than resampling, the filter computes the conditional $p_\varphi(X_{t+1}\mid x_t,Z_{t+1})$ exactly when the state is finite, and splits that conditional into independent clusters by Eq. (8), a factorisation into a product over clusters that are claimed to be conditionally independent given the observation. Exact finite probabilities make unbiased discrete gradient estimators such as RLOO applicable, while infinite-domain variables are handled by differentiable particle filtering.

What would settle it

Take the discriminative task's two-enemy setting and compute the exact joint posterior over the two enemies' locations after observing a hit. If conditioning on the hit couples the enemies, then the cluster product in Eq. (8) will not hold exactly and the filter's 'exact' local inference is approximate; the paper does not report such a check.

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Extended reading notes

Core claim

The paper's central claim is that the Markovian factorisation over neurosymbolic states $X_t=(N_t,S_t)$, with transition $p_\varphi(S_{t+1}\mid S_t,N_{t+1})$ governed by relational logic and neural predicates, is the first deep sequential probabilistic model that integrates relational logical constraints and provably satisfies them. Under this factorisation, relations constrain both the inside of one time slice and the movement from one slice to the next, so a property such as "the agent is safe" can be guaranteed throughout the whole trajectory. The paper further claims that its Rao-Blackwellised particle filter, combined with cluster factorisation and state-of-the-art gradient estimation, scales inference and learning to time horizons well beyond those reachable by existing exact and approximate neurosymbolic systems, while still training neural components end to end.

Load-bearing premise

The whole inference scheme assumes that, once the observations are fixed, the symbolic state splits into conditionally independent clusters, so each cluster can be solved separately with no error.

Editorial extensions

If this is right

  • Logical constraints such as safety properties can be guaranteed at every time step of a generated or classified sequence, not just in a single static inference.
  • NeSy-MMs handle both generative tasks, such as producing image sequences that follow rules and actions, and discriminative tasks, such as classifying trajectories with partially unknown neural transitions.
  • New constraints can be imposed at test time without retraining, so a model can refuse to enter a forbidden region mid-trajectory while still following the rest of the instructions.
  • The relational symbolic state gives better out-of-distribution generalisation than transformer and deep-HMM baselines in the paper's discriminative experiments, and scales to sequence lengths where exact neurosymbolic baselines time out.

Reading between the lines

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

  • Editorial: if the cluster-factorisation independence in Eq. (8) is not exact for a given observation model, then what the paper calls exact local inference is an approximation; an explicit check on the discriminative game task would settle this for that benchmark.
  • Editorial: the same construction could be applied to neural language generation, where a relational symbolic state tracking predicates over generated tokens would let a decoder guarantee global logical constraints, provided the cluster independence holds for the constraints used.
  • Editorial: one testable extension is to replace exact cluster inference with approximate inference for the continuous parts, trading the exactness guarantee for applicability to relational dynamics with many continuous variables.
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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

4 major / 4 minor

Summary. The paper introduces relational neurosymbolic Markov models (NeSy-MMs), a class of sequential probabilistic models that combine Markov dynamics with relational logical constraints and neural parameterizations. The model is defined by the factorization in Eq. (3), and inference is performed with a Rao-Blackwellised particle filter that conditions on observations through exact local NeSy inference, aided by a cluster factorization of the state space. The authors also propose a gradient-estimation scheme based on RLOO and report experiments on generative image-sequence modeling and discriminative trajectory classification, claiming strong guarantees, scalability beyond existing NeSy systems, and improved out-of-distribution generalization.

Significance. If the technical claims are correct, this is a valuable step toward sequential neurosymbolic models: it addresses a real scalability bottleneck, supports both discriminative and generative tasks, and demonstrates that relational constraints can be injected into deep Markov models. The paper also contributes two new benchmarks, reports hyperparameters and seeds, and states that code is provided, which are strengths for reproducibility. However, the central guarantees of exactness and unbiased learning currently rest on an unstated conditional-independence assumption and on a recursive factorization that is not generally valid, so the significance is contingent on fixing or carefully qualifying these points.

major comments (4)
  1. [§4.2, Eqs. (8)-(9)] The factorization p(X | Z) = ∏_i p(X_i | Z) is presented as an equality without stating the required conditional-independence assumption. In general, conditioning on a shared observation couples the variables. In the discriminative experiment (§5.2, Appendix D.2), the observation hit(T) is a logical disjunction over all enemies, so conditioning on hit=true induces dependence among enemy locations via explaining away. Concretely, with two enemies each independently at adjacent cell A or non-adjacent cell B, prior uniform, and hit = (E1=A or E2=A), the exact posterior satisfies P(E1=A, E2=A | hit) = 1/3, while the product of marginals is 4/9. Thus Eq. (9) is not a theorem in this setting. Either the factorization is an approximation, which would contradict the 'exact' wording in §4.1 and the gradient derivation in Appendix C, or the method applies only when observations decompose per cluster, which would exclude the main discriminative experiment. The authors need to state the assumption, prove it for their settings, or explicitly label the factorization as approximate and adjust the guarantee claims accordingly.
  2. [§4.2, Eq. (10)] The notion of 'maximal number of clusters' B is undefined, and no procedure is given for computing the clusters or for verifying that the factors in Eq. (10) are well-defined conditional distributions. Even if one takes the clusters to be connected components of some factor graph of p(X | Z), variables within a cluster need not become independent after conditioning on Z, so the refined factorization p(F_i, I_i | x_t, Z) = p(F_i | I_i, x_t, Z) p(I_i | x_t, Z) is always true, but the cross-cluster product in Eq. (9) is the nontrivial step. The paper should specify the cluster-identification algorithm and prove the factorization under explicit conditions, or state that the cluster product is an approximation.
  3. [Appendix C, Eqs. (16)-(17)] The recursive factorization of p(x_{0:T} | Z_{0:T}) in Eq. (16) is not generally correct. The equation writes p(x_{0:T} | Z_{0:T}) = p(x_T | x_{T-1}, Z_{t+1}) p(x_{0:T-1} | Z_{0:T-1}), but the second factor is the filtering distribution at time T-1, not the conditional distribution of the past given the full observation sequence including Z_T. The past states are generally dependent on future observations. For example, with T=1, deterministic transition x_1 = x_0, and observation z_1 = x_1, p(x_0 | z_0, z_1) differs from p(x_0 | z_0), so the product on the right-hand side is not the joint posterior. Consequently, Eq. (17), which computes log p(x_{0:T} | Z_{0:T}) as a sum of per-step proposal conditionals p(x_t | x_{t-1}, Z_t), substitutes the proposal log-density for the target log-density. The RLOO estimator in Eq. (15) then is not unbiased for the objective in Eq. (11). The authors need to derive the gradient estimator with proper importance weighting or explicitly present the method as an approximate, biased estimator.
  4. [Appendix D.2 and §5.1] The logic program for the generative task does not encode wall constraints. Section 5.1 states that the agent moves in a grid 'surrounded by walls', and the claims in §5.3 say that NeSy-MMs 'perfectly adhere to the mechanics of the game' and 'provably satisfy' constraints. However, the rules in Appendix D.2 move the agent by one cell for each action with no check that the target cell is inside the grid or is not a wall. As printed, the program permits moves that walk through or outside the walls. The authors should either provide the actual program including wall checks, or qualify the guarantee claim to the movement rules as written rather than the full environment mechanics.
minor comments (4)
  1. [§4.2, Eq. (10)] Equation (10) is missing a comma in the conditioning set: it should read p(F_i | I_i, x_t, Z_{t+1}), not p(F_i | I_i x_t, Z_{t+1}).
  2. [Appendix C, Eq. (16)] The first factor in Eq. (16) uses the time index Z_{t+1} although the current time is T; this should be Z_T, which would at least make the notation internally consistent.
  3. [§4.1] The text says the RBPF assumes p(X_{t+1} | X_t, Z_{t+1}) 'can be computed exactly' and later describes the results as 'exact local inference'. Given the cluster-factorization issue in §4.2, these statements should be qualified to avoid overclaiming.
  4. [§5.2 and Table 1] The class-balance percentages in Table 1 are useful, but the paper does not state how the agent death label is determined in the generated data; adding one sentence on the death condition would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the constraints are built into the model by design, and the inference novelty rests on component algorithms with independent published status; the main weakness is an unproved factorisation, not a circular reduction.

full rationale

The derivation chain is not circular. Eq. (3) defines NeSy-MMs as a Markov factorisation over neurosymbolic states, and the 'provably satisfy' claim is a direct property of encoding logical constraints such as `observe(safe0:T, true)` into the model: constraints are inputs, not outputs fitted to the experiments, so satisfying them is a design guarantee rather than a prediction extracted from data. The inference method combines a standard Rao-Blackwellised particle filter (Murphy and Russell 2001), exact NeSy inference from published systems (Kisa et al. 2014; De Smet et al. 2023), and the RLOO/CatLog gradient estimators (Kool et al. 2019; De Smet, Sansone, and Zuidberg Dos Martires 2023); although some of these references are self-citations, they are component methods with independent published derivations and are not invoked to forbid alternatives or to force the paper's central conclusion. The learned hit probability in the discriminative experiment is a fitted parameter, but it does not masquerade as a prediction of the main claims. The most serious flaw is elsewhere: Section 4.2, Eq. (8)-(9), asserts that p(X | Z) factorises over clusters without proving conditional independence, and conditioning on a shared observation such as the binary `hit` variable couples all enemy locations, so the 'exact' local inference is generally approximate. That is a correctness/approximation risk, not circularity, and it does not raise the circularity score.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central derivation depends on the conditional-independence assumption for clusters (Eq. 8), the completeness of the provided logic programs, and standard exact-inference machinery for finite symbolic states. The only explicit fitted scalar in the experiments is the hit-success probability; the enemy policy is a learned neural network. No new physical entities are introduced.

free parameters (2)
  • hit success probability (learnable predicate t(_))
    In the discriminative experiment (Appendix D.2), the probability that an adjacent enemy's attack succeeds is a learnable Bernoulli parameter, not given by the NetHack rules. Its fitted value is not reported; it is part of the 'unknown enemy behavior' that the experiment is designed to learn.
  • enemy action policy (neural network weights)
    The enemy transition is a neural network with two hidden layers (Appendix D.1) predicting the next move distribution; it is learned from data. This is standard learned model capacity rather than an ad hoc scalar, listed for completeness.
assumptions (3)
  • domain assumption Clusters of state variables are conditionally independent given observations (used in Eq. 8).
    The factorization p(X|Z)=product_i p(X_i|Z) in Section 4.2 is exact only if the clusters are conditionally independent given Z; the paper does not prove or state this condition. In the discriminative task, the 'hit' observation couples enemy locations, so this assumption likely fails.
  • ad hoc to paper The printed logic programs completely and correctly encode the environment dynamics.
    The generative task's logic program (Appendix D.2) contains movement rules but no wall or boundary constraints, yet Section 5.3 claims generated trajectories 'perfectly adhere' to MiniHack mechanics. The claim relies on either unprinted rules or the absence of wall collisions in the tested action sequences.
  • domain assumption Exact inference over the finite symbolic subspace is computationally feasible for the considered experiments.
    Section 4.1 assumes exact computation of p_phi(X_{t+1}|X_t,Z_{t+1}) for the finite part of the state, using knowledge compilation or weighted model integration as cited. This is a standard background capability from the NeSy literature, and the paper does not supply new machinery for it.

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Cite this review

Pith. "Pith review of Relational Neurosymbolic Markov Models." pith.science (2026). https://pith.science/paper/B2HNKGDB

@misc{pith2026241213023,
  author       = {Pith},
  title        = {Pith review of: Relational Neurosymbolic Markov Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B2HNKGDB}},
  note         = {Machine review of arXiv:2412.13023}
}
read the original abstract

Sequential problems are ubiquitous in AI, such as in reinforcement learning or natural language processing. State-of-the-art deep sequential models, like transformers, excel in these settings but fail to guarantee the satisfaction of constraints necessary for trustworthy deployment. In contrast, neurosymbolic AI (NeSy) provides a sound formalism to enforce constraints in deep probabilistic models but scales exponentially on sequential problems. To overcome these limitations, we introduce relational neurosymbolic Markov models (NeSy-MMs), a new class of end-to-end differentiable sequential models that integrate and provably satisfy relational logical constraints. We propose a strategy for inference and learning that scales on sequential settings, and that combines approximate Bayesian inference, automated reasoning, and gradient estimation. Our experiments show that NeSy-MMs can solve problems beyond the current state-of-the-art in neurosymbolic AI and still provide strong guarantees with respect to desired properties. Moreover, we show that our models are more interpretable and that constraints can be adapted at test time to out-of-distribution scenarios.

Figures

Figures reproduced from arXiv: 2412.13023 by the authors.

Figure 1
Figure 1. Probabilistic graphical model representations of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. DeepSeaProbLog (De Smet et al. 2023) encoding [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. On the left, a logic programming description of the game Example 3.1 in the discrete-continuous probabilistic NeSy [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Example trajectories of length 4 in a 5×5 grid for the generative (a) and discriminative (b) datasets, with the corre￾sponding labels above the images. Note that for the discriminative task, the models do not take images as input but rather the symbolic state. The imag…
Figure 5
Figure 5. Figure 5: Generated trajectory for actions: right, down, left, up, left, up, right, down. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Generated trajectory for actions: right, down, left, up, right, right, up, right; but with the test-time constraint that the area to the right of the start position should not be entered. When the agent is asked to move in the unsafe area (i.e. actions in italics) it, …
Figure 7
Figure 7. Figure 7: Probabilistic graphical model view of the [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Generated trajectory for actions: right, down, left, up, right, up, left, down using the Deep-HMM. [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Generated trajectory for actions: right, down, left, up, right, up, left, down using the variational transformer. [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: Rolled-out graphical model of Figure 3, from Example 3.1. We leave out the query node [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.