Compositionality and Systematic Generalization › Worked Example: A Factorial Dynamical World

Let's make all of this concrete with the smallest possible factorial world. The state is a single scalar $x$. The world is described by two independent mechanisms:

• Mechanism A (intrinsic dynamics), with three variants indexed by $m \in \{0, 1, 2\}$, each with a decay $\alpha_m$ and a curvature $\kappa_m$.
• Mechanism B (action response), with three variants indexed by $n \in \{0, 1, 2\}$, each with an action gain $\beta_n$ and a state-action coupling $\gamma_n$.

The transition is:

$$x_{t+1} = \underbrace{\alpha_m x_t + \kappa_m \tanh(x_t)}_{\text{mechanism A}} \;+\; \underbrace{\beta_n a_t + \gamma_n\, a_t\, x_t}_{\text{mechanism B}} \;+\; \varepsilon_t,$$

where:

• $x_t$: the scalar state at time $t$
• $a_t$: the action taken at time $t$
• $\alpha_m, \kappa_m$: the decay and curvature parameters of mechanism A variant $m$
• $\tanh(x_t)$: the hyperbolic tangent of the state, which bounds the curvature contribution to magnitude at most $|\kappa_m|$; the full intrinsic term can still grow linearly through $\alpha_m x_t$
• $\beta_n, \gamma_n$: the action gain and the state-action coupling of mechanism B variant $n$
• $\varepsilon_t$: small process noise added to the state at each transition, with standard deviation $0.02$ in the implementation below

The structural form of this equation is the entire point of the exercise. The right-hand side is a literal sum of a mechanism-A term and a mechanism-B term, with no interaction between them, so the true dynamics satisfy the additive factorization exactly. This is the ideal case for a factorized model: its inductive bias is not an approximation of the truth but a restatement of it. The two mechanisms are also semantically distinct, one governing how the state evolves on its own and one governing how actions perturb it, so the factorization matches a natural physical reading of the world rather than a mere algebraic convenience.

There are $3 \times 3 = 9$ worlds. We will train on six of them and hold out the anti-diagonal: $(m{=}2, n{=}0)$, $(m{=}1, n{=}1)$, and $(m{=}0, n{=}2)$.

To see the composition rule in action, fix $x_t = 2$ and $a_t = -1$, and ignore noise. Mechanism A with $m=1$ contributes $0.85 \times 2 - 0.3 \times \tanh(2) = 1.70 - 0.29 \approx 1.41$, using $\tanh(2) \approx 0.96$. Mechanism B with $n=0$ contributes $0.8 \times (-1) = -0.8$, giving $x_{t+1} \approx 1.41 - 0.80 = 0.61$. Mechanism B with $n=1$ instead contributes $1.6 \times (-1) + 0.2 \times (-1) \times 2 = -1.6 - 0.4 = -2.0$, giving $x_{t+1} \approx 1.41 - 2.0 = -0.59$. The two worlds share the intrinsic term exactly and differ only in the action response. If a model has adequately identified both contributions on the state-action support used by the held-out rollout, the valid additive rule determines their combination; learning a separate categorical world embedding supplies no corresponding constraint on $(1,1)$.

The arithmetic makes the data-sharing argument tangible. Every training world with $m=1$ in our split, namely $(1,0)$ and $(1,2)$, provides observations of the same intrinsic term $0.85 x - 0.3\tanh(x)$, and every world with $n=1$, namely $(0,1)$ and $(2,1)$, provides observations of the same action-response term. The held-out world $(1,1)$ asks the model to add two familiar contributions. In the true simulator the pairing introduces no interaction term, but finite data, limited excitation, optimization, and off-support component error can still leave the learned contributions inaccurate.

The composition rule can be drawn directly: each mechanism contributes a curve over states, and the held-out prediction is simply their sum.