Meaning as a Dynamic Landscape

 Continuing from yesterday・・・

Dynamic Semantic Field: Meaning as a Deforming Attractor Landscape

The Lion Example (Robot-Centered Version)

At the beginning, the robots do not share any meaning.

When a lion appears, each robot generates its own internal response.
One robot may say:

“Pabu!”

while the other says:

“BuBupapa!”

Although the external event is the same, the internal states are different.

Each encounter produces an internal experience represented by a state vector

$$\mathbf{x} = (x_1, x_2, \dots, x_n)$$

whose components reflect the robot’s internal activation, such as perceived danger, motion intensity, size, and behavioral tendency.

Initially, these internal experiences are scattered:

$$\mathbf{x}^{(1)} \neq \mathbf{x}^{(2)}$$

because the robots’ internal structures are not yet aligned.

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Repeated Internal Experience

As the robots repeatedly encounter the lion, similar internal activation patterns occur again and again.

Learning strengthens the correlations within each internal experience:

$$W(t+1) = W(t) + \eta \, \mathbf{x}\mathbf{x}^T$$

The internal dynamics evolves as

$$\mathbf{x}_{t+1} = F(\mathbf{x}_t; W(t))$$

After sufficient repetition, different initial activations produced by the same external event begin to converge:

$$\mathbf{x}_t \rightarrow \mathbf{x}_{lion}$$

This means that the robots have formed a stable internal experience.

As a result:

• both robots reach the same internal state

• both produce the same behavior (escape)

• both use the same word

In geometric terms, repeated internal experiences create a stable attractor in the internal space.

This visualization represents the internal state space of the system as a circular semantic field.

Each point corresponds to an internal state

$$\mathbf{x} = (x_1, x_2, \dots, x_n)$$

In this experiment, the high-dimensional state is projected onto a 2-D plane, and the surface height represents stability.

Let the landscape be defined by a potential function:

$$z = -E(\mathbf{x}, t)$$

where

• $E$ : stability (energy)

• lower $E$ → deeper valley → more stable meaning

• $t$ : experience time

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Local Attractors

A valley corresponds to a stable state:

$$\mathbf{x}_{t+1} = F(\mathbf{x}_t)$$

An attractor $\mathbf{x}^*$ satisfies

$$F(\mathbf{x}^*) = \mathbf{x}^*$$

and nearby states converge:

$$\|\mathbf{x}_{t+1} - \mathbf{x}^*\| < \|\mathbf{x}_t - \mathbf{x}^*\|$$

Thus,

Meaning = a stable dynamical state

The deeper the valley, the stronger the attractor.

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Time-Dependent Landscape (Learning)

Unlike classical Hopfield models, the landscape is not fixed.

Learning changes the internal structure:

$$W(t+1) = W(t) + \eta \, \mathbf{x}\mathbf{x}^T$$

and the energy becomes

$$E(\mathbf{x}, t) = -\frac{1}{2} \mathbf{x}^T W(t)\mathbf{x}$$

Therefore,

$$E = E(\mathbf{x}, t)$$

The surface itself deforms over time.

This means:

• experience reshapes stability

• the semantic field evolves

• the system’s “world” changes

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Valley Model Used in the Visualization

Each valley is modeled as a Gaussian attractor:

$$z(x,y,t) = -\sum_{k} D_k(t) \exp\!\left( -\frac{(x-c_{k,x}(t))^2 + (y-c_{k,y}(t))^2} {2\sigma_k^2} \right)$$

where

• $c_k(t)$ : attractor center

• $D_k(t)$ : depth (strength of meaning)

• $\sigma_k$ : spread (generalization)

Time-dependent centers represent learning-induced drift.

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Valley Splitting (Concept Formation)

Concept differentiation appears as valley splitting.

Initially:

$$z(x,y) = -D \exp\!\left(-\frac{r^2}{2\sigma^2}\right)$$

After learning:

$$z(x,y) = -D_1 \exp\!\left(-\frac{r_1^2}{2\sigma_1^2}\right) - -D_2 \exp\!\left(-\frac{r_2^2}{2\sigma_2^2}\right)$$

A ridge emerges between the two basins.

Interpretation:

• single valley → undifferentiated meaning

• two valleys → separate concepts

• ridge → ambiguity / decision boundary

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Circular Boundary

The circular domain represents the finite internal space:

$$r = \sqrt{x^2 + y^2} \le R$$

To avoid artificial edge effects, the surface is smoothly attenuated:

$$z(x,y) \leftarrow z(x,y) \cdot w(r)$$
$$w(r) = 1 - S(r)$$

where $S(r)$ is a smooth step function near $r=R$.

This produces a closed semantic field.

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Interpretation

In this model:

• A valley is not a stored symbol

• A valley is a stable dynamical regime

• Learning = deformation of $E(\mathbf{x},t)$

• Concept formation = bifurcation of attractors

Meaning is therefore not assigned.

$$\text{Meaning} = \text{stability in a changing field}$$