# Formalizing Mimicry (Μ) in a 1D Binary State Model ## 1. Objective: Adding Mimicry to the Toy Model Building upon the formalization of Theta (Θ) in a 1D binary CA model [[releases/archive/Information Ontology 1/0086_Formalizing_Theta]], this node aims to incorporate a mechanism for **Mimicry (Μ)** [[releases/archive/Information Ontology 1/0007_Define_Mimicry_M]]. Μ represents the tendency for interacting states/patterns to influence others towards similar configurations (resonance, alignment, replication). Adding Μ allows us to explore the interplay between Η (random exploration), Θ (stability), and Μ (alignment) in pattern formation within this simplified context, following the refinement strategy [[releases/archive/Information Ontology 1/0085_IO_Refinement_Strategy]] and exploring mechanisms discussed in [[releases/archive/Information Ontology 1/0070_IO_Mimicry_Mechanisms]]. ## 2. Model Setup: Extending the 1D Binary CA We use the same basic setup as [[releases/archive/Information Ontology 1/0086_Formalizing_Theta]]: * 1D array of `N` nodes. * Binary states `ε(i, S) ∈ {0, 1}`. * Implicit potentiality κ. * Immediate neighbor interactions (`i-1`, `i+1`). * Discrete sequence steps `S`. * Theta strength `Θ_val(i, S)` updated as before (increases on state maintenance, resets/decays on flip). ## 3. Formalizing Mimicry (Μ): Biased State Transition Mimicry suggests that when a node considers changing state (e.g., triggered by Η), the state it transitions *to* should be influenced by its neighbors. We can implement Mechanism 5 from [[releases/archive/Information Ontology 1/0070_IO_Mimicry_Mechanisms]] (Biased Exploration). * **Neighborhood Influence:** When node `i` is triggered to potentially flip (by Η, overcoming Θ resistance), instead of simply flipping `0 ↔ 1`, it chooses its *new* state based on the majority state in its neighborhood (including itself, perhaps, or just neighbors). * **Mimicry Strength Parameter:** Introduce `p_M` (probability or strength of Mimicry influence). `0 ≤ p_M ≤ 1`. ## 4. Modified State Transition (κ → ε) Rule (Incorporating Η, Θ, Μ) Let's refine the update rule from [[releases/archive/Information Ontology 1/0086_Formalizing_Theta]]: 1. For each node `i`: 2. **Η Trigger:** Generate `r1 ~ U(0,1)`. If `r1 < p_H`, node `i` considers changing state. 3. **Θ Resistance Check:** Calculate `Θ_val(i, S)`. Calculate `P_change_allowed = ResistanceFunction(Θ_val(i, S))`. (Using `1 / (1 + α * Θ_val)` or similar from [[0086]]). Generate `r2 ~ U(0,1)`. If `r2 ≥ P_change_allowed`, the change is resisted by Θ; go to step 6 (Maintain State). 4. **Μ Influence (Determine Target State):** If change is triggered by Η and allowed by Θ: a. Examine neighbor states: `ε(i-1, S)` and `ε(i+1, S)`. b. Determine the "preferred" state `ε_preferred` based on neighbors. E.g., `ε_preferred = majority state of neighbors`. If neighbors are different (0 and 1), `ε_preferred` could be chosen randomly, or perhaps default to the node's *current* state (favoring stability slightly). Let's use: `if ε(i-1, S) == ε(i+1, S): ε_preferred = ε(i-1, S) else: ε_preferred = ε(i, S)` (defaults to current state if neighbors disagree). c. Determine the target state `ε_target`. With probability `p_M`, the target state is the preferred state: `ε_target = ε_preferred`. With probability `(1 - p_M)`, the target state is the opposite of the current state (simple flip): `ε_target = 1 - ε(i, S)`. (This `1-p_M` term allows for some non-mimetic change even when M is active). 5. **Execute State Change:** Set `ε(i, S+1) = ε_target`. Reset/decay `Θ_val(i, S+1) = Θ_base` (or decay). 6. **Maintain State (No Η trigger or Θ resistance):** Set `ε(i, S+1) = ε(i, S)`. Increment `Θ_val(i, S+1) = min(Θ_val(i, S) + ΔΘ_inc, Θ_max)`. *(Self-Correction during rule design: Initially considered making the flip *probability* depend on neighbors (closer to toy model [[0037]]), but making the *target state* depend on neighbors seems a clearer implementation of Mimicry as alignment/copying here. Added `p_M` to control the strength of this effect.)* ## 5. Expected Emergent Behaviors (Η, Θ, Μ Interplay) With Η driving random exploration, Θ providing stability, and Μ promoting local alignment, we expect richer dynamics: * **Domain Formation and Smoothing:** Μ should actively promote the growth of uniform domains (0s or 1s) by aligning nodes at the boundaries with the larger domain. Boundaries should become smoother than with Η/Θ alone. * **Pattern Competition:** Different stable patterns might emerge and compete based on their resilience (Θ) and ability to propagate (Μ). * **Parameter Dependence:** * High `p_M`: Strong alignment tendency, potentially leading to rapid homogenization or freezing into large domains, possibly suppressing Η-driven novelty if Θ is also significant. * Low `p_M`: Weak alignment, behavior closer to the Η/Θ model of [[0086]]. * Interaction with `p_H` and Θ parameters (`α`, `ΔΘ_inc`, `Θ_max`): The balance determines the outcome. E.g., high Η might overcome weak Μ/Θ; strong Θ might resist strong Μ alignment attempts if the node is stable. ## 6. Limitations and Next Steps * **Simplicity:** Still a very basic model. * **Mechanism Choice:** This "majority rule" / "biased target" is one simple way to model Μ. Resonance or template mechanisms [[releases/archive/Information Ontology 1/0070_IO_Mimicry_Mechanisms]] would require different formalisms. * **Parameter Sensitivity:** Finding interesting "edge of chaos" behavior likely requires careful tuning of `p_H`, `p_M`, and Θ parameters. * **Next Steps:** * Implement and simulate this Η/Θ/Μ model. * Analyze the phase space as parameters vary. * Explore different neighborhood definitions or Μ rules. * Consider adding a simple CA mechanism (e.g., directional bias based on past state changes). * Attempt to connect emergent patterns to simple physical analogies (e.g., domain walls, excitations). ## 7. Conclusion: Modeling Alignment Dynamics This node demonstrates how Mimicry (Μ) can be formalized alongside Entropy (Η) and Theta (Θ) within a simple computational model. By introducing a mechanism where state transitions are biased towards local neighborhood patterns, we explicitly model the principle of alignment or resonance. Simulating this model would allow exploration of how the interplay between random exploration (Η), pattern stabilization (Θ), and pattern alignment (Μ) drives self-organization and pattern formation, providing concrete insights into the potential dynamics of the conceptual IO framework. This represents further progress on Pillar 1 (Formalization) of the refinement strategy [[releases/archive/Information Ontology 1/0085_IO_Refinement_Strategy]].