Neurosymbolic Learning of Interpretable World Models

Riccardo Andreoni; Massimiliano Ronzani; Chiara Ghidini; Alessandro Daniele; Emile van Krieken

cs.AIOct 14, 2026

Neurosymbolic Learning of Interpretable World Models

Authors: Riccardo Andreoni, Massimiliano Ronzani, Chiara Ghidini, Alessandro Daniele, Emile van Krieken

Organizations: Fondazione Bruno Kessler · Free University of Bozen-Bolzano · Vrije Universiteit Amsterdam

Abstract

World models aim to learn compact representations of an environment from observations, enabling agents to predict future states and plan their behaviour. However, most existing approaches learn black-box representations, with limited ability to incorporate prior knowledge about the world. We introduce Concept-Grounded World Models (CGWMs), a neurosymbolic world model design that instead learns interpretable concept representations. We learn these concepts from observations using explicit and complete knowledge, expressed in temporal logic, that specifies the valid world states and the temporal relations governing their dynamics. Furthermore, we directly use this knowledge by formulating planning as differentiable reasoning, optimising the agent's actions by gradient ascent to satisfy a goal. We experimentally compare CGWM with a purely neural baseline and a neurosymbolic baseline on two game environments. We find that CGWMs ground concepts much more accurately, even over long horizons, resulting in significantly more robust planning both in and out of distribution.

Figures & tables

Figure 1: CGWMs are interpretable world models that ground concepts in images and use a symbolic transition system. This improves data efficiency, provides zero-shot transfer to new environments and allows for flexible and safe planning for new goals.

Figure 2: Overview of CGWM during learning.

Figure 3: Overview of CGWM during planning.

Figure 4: Initial observations of the environments: (a) Ice Slider, (b) Two Rooms (hard), (c) Two Rooms (lava dense), and (d) Two Rooms (periodic door), where the door has closed on the agent.

Figure 5: CGWM with log-product, learns accurate grounding from weak supervision and knowledge. F1 on the concepts along the imagined trace, for the largest and smallest training sets.

	Two Rooms					Ice Slider
	easy	medium	hard	lava dense	periodic door	$T{=}20$	$T{=}T^{*}$
CGWM (Gödel Trick)	100.0 (0.0)	100.0 (0.0)	96.3 (0.8)	88.4 (2.8)	94.9 (1.0)	100.0 (0.0)	100.0 (0.0)
CGWM (log-product)	100.0 (0.0)	100.0 (0.0)	90.4 (1.9)	68.6 (5.5)	84.3 (3.3)	99.8 (0.4)	90.4 (1.7)
LeWM	95.8 (1.2)	37.8 (2.8)	22.0 (9.0)	9.6 (6.9)	7.2 (3.6)	69.1 (12.6)	3.6 (2.9)

Table 1: CGWMs learn representations that allow for robust planning both in and out of distribution. Percentage of planning instances solved in the environment, mean and standard deviation (in parentheses) over 10 seeds. For CGWM, the semantics used for planning is in parentheses.

Figure 6: CGWM can plan over long horizons. Fraction of instances solved, grouped by the length of their optimal plan, with the number of instances per group in parentheses.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Operator	Abbreviation	Description
Weak next ( $\mathsf{X}_{w}$ )	$\mathsf{X}_{w}\varphi\equiv\mathsf{X}\varphi\lor\lnot\mathsf{X}\top$	If there is a next instant, $\varphi$ holds at it.
Eventually ( $\mathsf{F}$ )	$\mathsf{F}\varphi\equiv\top\mathbin{\mathsf{U}}\varphi$	$\varphi$ holds at some instant of the trace.
Globally ( $\mathsf{G}$ )	$\mathsf{G}\varphi\equiv\lnot\mathsf{F}\lnot\varphi$	$\varphi$ holds at every instant of the trace.
Release ( $\mathbin{\mathsf{R}}$ )	$\varphi_{1}\mathbin{\mathsf{R}}\varphi_{2}\equiv\lnot(\lnot\varphi_{1}\mathbin{\mathsf{U}}\lnot\varphi_{2})$	$\varphi_{2}$ holds up to and including the first instant at which $\varphi_{1}$ holds; if $\varphi_{1}$ never holds, $\varphi_{2}$ holds throughout the trace.
Strong release ( $\mathbin{\mathsf{M}}$ )	$\varphi_{1}\mathbin{\mathsf{M}}\varphi_{2}\equiv(\varphi_{1}\mathbin{\mathsf{R}}\varphi_{2})\land\mathsf{F}\varphi_{1}$	As release, but $\varphi_{1}$ is required to hold at some instant.
Weak until ( $\mathbin{\mathsf{W}}$ )	$\varphi_{1}\mathbin{\mathsf{W}}\varphi_{2}\equiv(\varphi_{1}\mathbin{\mathsf{U}}\varphi_{2})\lor\mathsf{G}\varphi_{1}$	As until, but $\varphi_{2}$ is not required to hold; if it never does, $\varphi_{1}$ holds throughout the trace.

Appendix

Table 2: Derived temporal operators of ltl f .

Formula	Meaning
Specification formulas $\Omega_{IS}$
$\mathsf{F}\;\mathtt{at\_goal}$	The agent eventually reaches the goal.
Constraints $\mathcal{K}_{IS}$
$\mathsf{G}(\mathtt{at\_goal}\rightarrow\mathsf{Last})$	The agent is on the goal only at the last time step.
$\mathsf{G}\,\exists!g.\,\mathtt{agent}(g)$	There is exactly one agent.
$\mathsf{G}\,\exists!g.\,\mathtt{goal}(g)$	There is exactly one goal.

Appendix

Table 3: Knowledge given to CGWM in Ice Slider.

Formula	Meaning
Specification formulas $\Omega_{TR}$
$\mathsf{F}\,\mathtt{at\_goal}$	The agent eventually reaches the goal.
$\mathsf{F}\,\mathtt{on\_lava}$	The agent eventually steps on lava.
$\mathsf{F}\,\mathtt{door\_open}$	The door is eventually open.
Constraints $\mathcal{K}_{TR}$
$\mathsf{G}(\mathtt{at\_goal}\rightarrow\mathsf{Last})$	The agent is on the goal only at the last time step.

Appendix

Table 4: Knowledge given to CGWM in Two Rooms (hard).

Formula	Meaning
$\mathtt{door}^{\prime}(g)=\big(\mathtt{door}(g)\wedge\mathtt{agent}(g)\big)\vee\big(\mathsf{Y}\,\mathtt{door}(g)\wedge\neg(\mathtt{door}(g)\wedge\mathtt{pressed})\big)$	The door stays closed until the button is pressed, and alternates between open and closed afterwards, unless the agent is on it.
$\mathtt{agent}^{\prime}(g)=\bigvee_{d}\Big(\big(\mathtt{agent}(g-\delta_{d})\wedge\neg\,\mathtt{door}(g-\delta_{d})\wedge d\wedge\neg\,\mathtt{obstacle}^{\prime}(g)\big)\vee\big(\mathtt{agent}(g)\wedge\neg\,\mathtt{door}(g)\wedge d\wedge\mathtt{obstacle}^{\prime}(g+\delta_{d})\big)\Big)\vee\big(\mathtt{agent}(g)\wedge\mathtt{door}(g)\big)$	The agent moves as in Two Rooms (hard), with the door of the next time step blocking it, and it stays in $g$ forever once the door closes on it.
$c^{\prime}(g)=c(g)$ , for $c\in\{\mathtt{wall},\mathtt{lava},\mathtt{button},\mathtt{goal}\}$	Walls, lava, button and goal do not move.

Appendix

Table 5: Transition rules of Two Rooms (periodic door).

CGWM (log-product)
	Static concepts		Dynamic concepts					All concepts
$N$	$h{=}0$	$40$	$h{=}0$	$10$	$20$	$30$	$40$	$h{=}0$	$40$
3200	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)
800	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)
200	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)
50	0.97 (0.07)	0.97 (0.07)	1.00 (0.00)	0.98 (0.05)	0.97 (0.08)	0.98 (0.07)	0.98 (0.06)	0.98 (0.05)	0.98 (0.07)

Appendix

Table 6: F1 of the concepts in Two Rooms (hard), for each training set size

N

and rollout step

h

, as mean and standard deviation (in parentheses) over the seeds.

CGWM (log-product)
	Static concepts		Dynamic concepts					All concepts
$N$	$h{=}0$	$20$	$h{=}0$	$5$	$10$	$15$	$20$	$h{=}0$	$20$
3200	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)
800	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)
200	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)
50	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)	1.00 (0.00)

Appendix

Table 7: F1 of the concepts in Ice Slider, for each training set size

N

and rollout step

h

, as mean and standard deviation (in parentheses) over the seeds.

Gödel
	Two Rooms					Ice Slider
	easy	medium	hard	lava dense	periodic door	$T{=}20$	$T{=}T^{*}$
Gödel Trick	100.0 (0.0)	100.0 (0.0)	96.3 (0.8)	88.4 (2.8)	94.9 (1.0)	100.0 (0.0)	100.0 (0.0)
softmax	100.0 (0.0)	84.4 (3.4)	88.5 (2.6)	86.2 (3.5)	81.7 (4.1)	100.0 (0.0)	100.0 (0.0)
log-product
Gumbel-Softmax	100.0 (0.0)	100.0 (0.0)	90.4 (1.9)	68.6 (5.5)	84.3 (3.3)	99.8 (0.4)	90.4 (1.7)

Appendix

Table 8: Percentage of planning instances solved in the environment for each semantics and mode. Mean and standard deviation (in parentheses) over the seeds.

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State-of-the-art model-based reinforcement learning methods learn neural world models that allow policy improvement by planning in a latent space, without assumptions on the structure of the underlying environment. While expressive, these models are generally task-dependent: they learn uninterpretable latent representations that are tied to the training task and thus hard to generalize to new tasks. In this work, we present a novel world model formulation where the reward prediction only depends on a subset of structured, symbolic components of the whole latent state. Decoupling observation reconstruction and reward prediction allows us to learn world models that can adapt zero-shot, i.e. without further environment interactions, to new reward functions defined over the same symbolic state space. We discuss the main advantages and challenges of learning these neurosymbolic world models and demonstrate the strong generalisation properties of our approach over purely neural methods.

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