# Formalizing the Dynamics of Potentiality (P_target Evolution) ## 1. Objective: Defining How Potentiality Evolves Nodes [[releases/archive/Information Ontology 1/0100_IO_Kappa_Refinement]] and [[releases/archive/Information Ontology 1/0101_IO_Transition_Rule_v2]] introduced the Target Probability Vector `P_target(i, S)` as a richer representation of Potentiality (κ) within the state `State(i, S) = { ε(i, S), (P_target(i, S), Θ_val(i, S)) }`. While the transition rule uses `P_target` to determine the outcome of a state change `ε`, we also need rules governing how `P_target` itself evolves from `S` to `S+1`. This represents the **dynamics of potentiality** – how the likelihoods of future transitions change based on the system's history and context. This is crucial for implementing the simulation plan [[releases/archive/Information Ontology 1/0102_IO_Simulation_Plan_v2]]. ## 2. Factors Influencing `P_target` Evolution The evolution `P_target(i, S) → P_target(i, S+1)` should plausibly depend on: 1. **Current Actual State `ε(i, S)`:** The state currently occupied might influence future potentials. 2. **Stability `Θ_val(i, S)`:** The stability of the current state should impact the potential to transition away from it or back to it. 3. **Whether a Change Occurred:** The update rule might differ depending on whether `ε(i, S+1) == ε(i, S)` or not. 4. **Neighboring States / Context (K, M, CA):** The local informational environment could slowly adapt the intrinsic potentials over time. 5. **Entropy (Η):** A baseline tendency towards exploration might prevent probabilities from becoming permanently fixed at 0 or 1. ## 3. Proposed Update Mechanisms for `P_target` Let `P_target(i, S) = [p_0(S), p_1(S), ..., p_{k-1}(S)]`. We need rules for `p_k(S+1)`. **Mechanism 1: Reset on Change, Reinforce on Stability (Simple)** This is a basic mechanism linking `P_target` evolution directly to `Θ_val` dynamics. * **If `ε` Changed (`ε(i, S+1) ≠ ε(i, S)`):** * The stability `Θ_val(i, S+1)` is reset to `Θ_base`. * Reset `P_target(i, S+1)` to a baseline distribution. Options for baseline: * (a) Uniform: `p_k(S+1) = 1/N_states` (Maximum potential entropy). * (b) Biased towards new state: Slightly increase `p_k` corresponding to the *new* `ε(i, S+1)` and distribute the rest uniformly/proportionally. * **If `ε` Maintained (`ε(i, S+1) == ε(i, S)`):** * The stability `Θ_val(i, S+1)` increases. * Reinforce the probability corresponding to the current state. Let `k_current = ε(i, S)`. * `p_{k_current}(S+1) = p_{k_current}(S) + Δp_inc * f(Θ_val(i, S))` (Increase probability of staying, possibly dependent on current stability). * For `k ≠ k_current`: `p_k(S+1) = p_k(S) - Δp_inc * f(Θ_val(i, S)) / (N_states - 1)` (Decrease others proportionally). * Normalize `P_target(i, S+1)` to ensure `Σ p_k = 1`. * `Δp_inc` is a small learning rate for potentiality. `f(Θ_val)` could make reinforcement stronger for already stable states. * *Pros:* Simple, directly links potential evolution to stability history (Θ). * *Cons:* Might be too simplistic; doesn't explicitly include context (K, M, CA) influence on the *intrinsic* potential over time. Resetting to uniform might erase learned potential too easily. **Mechanism 2: Gradual Adaptation to Context (Incorporating K/M/CA)** This mechanism allows the intrinsic potentials `P_target` to slowly adapt based on persistent environmental influences. * **Core Idea:** Introduce a slow "drift" term to the `P_target` update rule that nudges the probabilities towards values favored by the local context (neighbors, causal inputs), even when the state `ε` isn't changing. * **Contextual Target `P_context`:** Calculate a "context-favored" probability distribution `P_context(i, S)` based on weighted causal inputs (similar to the M bias calculation in [[releases/archive/Information Ontology 1/0101_IO_Transition_Rule_v2]]). * `p_{context, k} ∝ Σ_{j | (j → i) ∈ E, ε(j,S)=k} w(j → i, S)` (Normalize). * **Update Rule (Combined with Mechanism 1):** * Calculate the stability-based update `P_intermediate` as in Mechanism 1 (reinforce if stable, reset/bias if changed). * Apply a small contextual drift: `P_target(i, S+1) = (1 - λ_adapt) * P_intermediate + λ_adapt * P_context(i, S)` * `λ_adapt` is a small adaptation rate parameter (e.g., 0.01). This allows the intrinsic potentials to slowly track persistent contextual biases. * *Pros:* Allows potentiality itself to adapt and learn based on environment/interactions (K/M/CA influence). More biologically/cognitively plausible? * *Cons:* More complex, introduces new parameter `λ_adapt`. Needs careful tuning to avoid overly rapid adaptation erasing intrinsic potential. **Mechanism 3: Η Influence on `P_target`** Ensure Η prevents probabilities from becoming permanently fixed. * **Rule:** After applying Mechanism 1 or 2, enforce minimum probabilities. * `p_k(S+1) = max(p_k(S+1), p_min)` (where `p_min` is a very small probability, e.g., `1e-9`). * Renormalize `P_target(i, S+1)` after applying the floor. * *Pros:* Maintains exploratory potential, prevents system getting permanently stuck. * *Cons:* Adds another parameter `p_min`. ## 4. Choosing and Refining the Mechanism * **Recommendation:** Start with **Mechanism 1 (Reset/Reinforce)** due to its simplicity and direct link to Θ dynamics. Implement and test this first. * **Refinement:** If Mechanism 1 proves too rigid or fails to capture adaptive potential, introduce **Mechanism 2 (Gradual Adaptation)** with a small `λ_adapt`. * **Η Floor (Mechanism 3):** Likely necessary in most implementations to prevent numerical issues and maintain theoretical openness. The precise functional forms (e.g., how `Δp_inc` depends on `Θ_val`, the exact reset distribution, the normalization method) will require careful theoretical consideration and tuning during simulation [[releases/archive/Information Ontology 1/0102_IO_Simulation_Plan_v2]]. ## 5. Conclusion: Modeling the Potential's Flow Formalizing the dynamics of the Target Probability Vector `P_target` is essential for a complete IO transition rule using the refined state representation [[releases/archive/Information Ontology 1/0100_IO_Kappa_Refinement]]. This involves defining how the intrinsic potential probabilities evolve based on whether the actual state `ε` changes or persists, driven by stability (`Θ_val`), and potentially adapting slowly to the surrounding informational context (K, M, CA). The proposed mechanisms provide concrete starting points for implementing these dynamics, allowing simulations to explore not just the actualization of potential, but the evolution of potentiality itself within the IO framework. This addresses a key requirement for moving the formalism forward.