``` --- FILE: AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md --- **Conceptual Speculations for Autaxic Generative Engine Expansion (Beyond v0.1)** **Document ID:** AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1 **Project:** 6.2: Formal Mathematical & Computational Modeling (Conceptual Future) **WBS Ref:** Conceptual Exploration (Beyond Defined Scope) **Title:** Conceptual Speculations on Expanded Proto-property Space, Cosmic Algorithm Rules, Dynamic Stability, and Emergent Phenomena for AGE **Status:** Speculative Draft **Version:** 1.4 (Adding Speculation on Vacuum State and Tension Landscape) **Author:** Principal Investigator (with Autologos Assistance) **Date:** 2025-06-29 **Based On:** * `AUTX_A1_AGEv0.1_Primitives_V1.md` (v0.1 Primitives) * `AUTX_A1_AGEv0.1_CoreRules_V1.md` (v0.1 Core Rules) * `D-P6.7-1_Unified_Framework_v1.9.md` (Broader Conceptual Framework) * `AUTX_Master_Plan_v2.0_Part_V_Research_Projects.md` (Future Project Directions) * `AUTX_A1_AGEv0.1_Simulation_Analysis_Report_V1.md` (Insights from v0.1 Dynamics) * `AUTX_A1_AGEv0.1_OC_TensionMin_V1.md` (v0.1 Tension Definition) --- ### **1.0 Introduction** The AGE v0.1 "Toy Model" establishes a minimal foundation with a limited set of proto-properties and core rules to explore basic emergent structure via Ontological Closure. This document moves beyond the v0.1 scope to speculate on how the model could be expanded in future research phases (e.g., P6.3, P6.4) to incorporate richer dynamics, complexity, and potentially higher-level emergent phenomena. This involves considering a broader conceptual space for proto-properties, imagining additional rule types for the "Cosmic Algorithm", exploring dynamic rule application, differentiating sources of relational tension, conceptualizing the underlying "Vacuum State" from which genesis occurs, and viewing the system's evolution through the lens of a "Tension Landscape". These elements could give rise to complex structures and behaviors like hierarchies and proto-identities. This version adds speculation on the nature of the Vacuum State and the metaphorical framework of the Tension Landscape, building upon the previous speculation on emergent phenomena and hierarchies. ### **2.0 Expanded Proto-property Space** While v0.1 uses only `ProtoPolarity` and `ProtoValence` for Distinctions, and simple `Type`/`Strength` for Relations, the conceptual framework (`D-P6.7-1_Unified_Framework_v1.9.md`) hints at a much richer set of intrinsic attributes. Expanding this space allows for more nuanced interactions and potentially models for emergent properties like memory, persistence, or functional specialization. * **2.1 Speculative Proto-properties for Distinctions (D-nodes):** * `TemporalInertia`: (Float >= 0) A resistance to change or annihilation. Higher inertia means a Distinction is less likely to be affected by Annihilation rules or have its properties mutated. Could contribute negatively to Tension if considered "stability potential". * `InteractionRadius`: (Float >= 0) Defines a "conceptual distance" within which this Distinction can potentially form relations or influence other Distinctions, even if not directly connected. Could influence `FormationRule` applicability or introduce new `InfluenceRule` types. * `InformationContent`: (Complex data structure/hash) Represents a non-relational "payload" carried by the Distinction. Could be mutable. Could influence compatibility criteria in `FormationRule` or be subject to its own `MutationRule`. Could be a source of tension if inconsistent internally or with connected nodes. * `Flexibility`: (Float 0.0-1.0) The inverse of rigidity. A measure of how easily the Distinction's mutable state (like `current_relation_count`) or even its proto-properties can be altered by interactions or rules. Could affect the *outcome* of rule applications rather than just applicability. * `ProcessingCapacity`: (Integer >= 0) A limit on the number or complexity of relations/interactions a Distinction can actively participate in simultaneously, distinct from `ProtoValence` which might be a fixed potential. Unsatisfied capacity could contribute to tension differently than unsatisfied valence. * **2.2 Speculative Proto-properties for Relations (R-edges):** * `TemporalDecay`: (Float >= 0) A rate at which the relation's `Strength` diminishes over time or simulation steps if not reinforced. Could contribute to tension if relations decay below a certain threshold required for stability. * `DirectionalityBias`: (Enum: `Strict`, `Flexible`, `Ambidextrous`) Refines the concept of a directed edge. `Flexible` might allow for a reverse relation (`target -> source`) to form more easily, `Ambidextrous` might imply a potential for symmetry or feedback loops. Affects `FormationRule` and potentially new `SymmetrizationRule`. * `InformationChannelType`: (Enum/Tag) Defines the *kind* of information or influence that flows across the relation (e.g., 'causal', 'logical', 'inhibitory', 'reinforcing'). Could introduce complex interaction dynamics and require rules that depend on matching channel types. * `ContextSensitivity`: (Boolean/Reference to Subgraph Type) The relation's behavior (strength, decay, formation/annihilation conditions) depends on the properties or structure of the local subgraph it is part of. Requires the engine to evaluate local context. * **2.3 Dynamic Proto-property Interactions: Mechanisms** * Beyond simply being static attributes or inputs to rules, proto-properties could dynamically influence *each other* within a Distinction or Relation, or influence the properties of *connected* entities over time. This introduces internal, non-relational dynamics within entities, adding another layer of complexity. * **Proposed Mechanisms:** * **Direct Influence:** Property X of Entity A directly alters Property Y of Entity B (where A and B are related). *Example:* The `ProtoPolarity` of a connected node influences the `ProtoPolarity` of a node with low `TemporalInertia`. * **Accumulation/History:** A property changes based on the history of interactions or rule applications involving the entity. *Example:* `TemporalInertia` increases over time if the Distinction is part of stable subgraphs or resists annihilation attempts. `Flexibility` decreases if a Distinction is repeatedly involved in high-tension configurations without successful resolution. * **Threshold/Trigger:** A property changes drastically or triggers a rule application only when another property or external condition reaches a threshold. *Example:* `InformationContent` becomes inconsistent, triggering a `SplitRule` only when accumulated `InformationTension` exceeds a limit. `TemporalDecay` of a relation accelerates if its `Strength` drops below a threshold. * **Environmental/Contextual Modulation:** A property's value or its influence on rules/other properties is modified by the properties or structure of the local subgraph the entity is embedded in. *Example:* The effective `InteractionRadius` of a Distinction is reduced if it is in a densely connected, high-tension region. * **Feedback Loops:** Changes in one property trigger changes in another, which in turn affects the first, either positively (reinforcement) or negatively (damping/oscillation). *Example:* High `ProcessingCapacity` allows a node to participate in more relations, increasing `current_relation_count`. If `current_relation_count` approaches `ProcessingCapacity`, it could reduce `Flexibility`, making it harder to form *new* relations but perhaps increasing `Strength` of existing ones, which could reduce `TemporalDecay`, creating a positive feedback loop for connection strength. * **Internal State Machines:** A Distinction's mutable state could follow a simple state machine where property values transition based on inputs (rule applications, interactions). *Example:* A Distinction could be in states like 'Idle', 'Seeking Connection', 'Connected', 'Overloaded', 'Degrading', with property values (like effective `ProtoValence` or susceptibility to `AnnihilationRule`) changing based on the current state. ### **3.0 Expanded Cosmic Algorithm Rules** The v0.1 rules (Genesis, Formation, Annihilation) are sufficient for basic structure formation but lack mechanisms for transformation, adaptation, or complex pattern interaction. Future rules could introduce these capabilities, drawing inspiration from biological processes, computational systems, or logical operations. * **3.1 Speculative Rule Types (Beyond Core v0.1):** * `MergeRule(D_set)`: If a set of Distinctions (`D_set`) forms a highly coherent, tightly bound subgraph (e.g., very low local tension, high relation density, satisfying certain pattern criteria), they can merge into a *single new Distinction*. This new Distinction inherits composite properties from the merged set and represents an emergent higher-order entity. Reduces node count, creates hierarchy. * *Applicability:* Subgraph S meets specific coherence/density criteria. * *Transformation:* Remove D_set nodes and their internal relations. Create new D_new. Re-map external relations to D_new. Define D_new's proto-properties based on D_set (e.g., sum valence, average polarity, combine information content). * `SplitRule(D)`: If a single Distinction `D` becomes internally inconsistent (e.g., information content conflicting with external relations) or accumulates too much "tension burden", it can split into two or more new Distinctions. Increases node count, potentially resolves local tension by distributing properties. * *Applicability:* Single Distinction D meets internal inconsistency/high local tension criteria. * *Transformation:* Remove D. Create D_new_1, D_new_2 (or more). Distribute D's properties and external relations among the new nodes. Define properties of D_new_1, D_new_2. * `PropertyMutationRule(Entity)`: A rule that modifies the proto-properties (or mutable state) of a Distinction or Relation based on its history, local context, or accumulated "stress" (local tension). Introduces adaptation. * *Applicability:* Entity (D or R) meets mutation criteria (e.g., exposed to high tension for extended steps, specific neighboring configuration). * *Transformation:* Modify one or more of the Entity's proto-properties or mutable state variables within defined ranges. * `ContextualFormationRule(D1, D2, Subgraph_Context)`: A variation of `FormationRule` where compatibility is not just based on `D1`/`D2` properties, but also on the structure or properties of the immediate subgraph containing `D1` and `D2`. Allows for relations to form only in specific environments. * *Applicability:* D1, D2 meet base criteria AND Subgraph_Context (e.g., a path exists via a third node, or specific relation types are present) meets criteria. * *Transformation:* Add relation R(D1, D2) with context-dependent properties. * `ReinforcementRule(R)`: A rule that increases the `Strength` (or decreases `TemporalDecay`) of a Relation if it is part of a stable subgraph, contributes to tension reduction, or is frequently traversed (if "traversal" is introduced as a concept). Models learning or strengthening of useful connections. * *Applicability:* Relation R meets reinforcement criteria. * *Transformation:* Modify R's properties (Strength, TemporalDecay). * `InhibitionRule(R_set)`: A rule that weakens or annihilates a set of Relations if they collectively contribute to high local tension or instability, or if they are redundant. Models pruning or forgetting. * *Applicability:* Set of Relations R_set meets inhibition/redundancy criteria. * *Transformation:* Weaken or remove Relations in R_set. * **3.2 Dynamic or Context-Dependent Rule Application: Strategies** * Instead of a fixed rule application order (`Genesis -> Formation -> Annihilation`), the sequence or probability of rule application could be dynamic, potentially even differing in different regions of the graph. * **Proposed Strategies:** * **Tension-Driven Prioritization:** Rules that are applicable and would result in the greatest *reduction* of local or global tension are prioritized. This requires calculating the hypothetical tension change for all applicable rule instances. * **Probabilistic Selection:** Each applicable rule instance (or type of rule) has a probability of being applied in a given step, potentially modulated by local or global state (e.g., high global tension increases probability of Annihilation rules). * **Local Rule Sets:** Different subgraphs or types of Distinctions/Relations might "carry" or "enable" different sets of rules. A `MergeRule` might only be applicable *within* a subgraph identified as a potential candidate, and its application might originate from a specific "organizing" node type. * **History-Dependent Selection:** The rules applied in previous steps or the history of tension in a region influences which rules are prioritized now. * **Resource/Capacity Limited Application:** Rule application isn't instantaneous or guaranteed. It might consume some abstract "processing resource" or be limited by the `ProcessingCapacity` of involved nodes. High-tension states might 'free up' or 'generate' more of this resource, increasing the rate of rule application, creating a feedback loop where tension drives change. * **Emergent Rule Application Order:** Instead of a predefined order, the 'most active' rules (those with the most applicable instances, or highest potential tension reduction) are applied first, or rules are applied in parallel wherever their application instances don't conflict. Conflict resolution (e.g., two rules trying to modify the same entity) becomes part of the system's logic. ### **4.0 Conceptualizing the Vacuum State (S0)** The "Vacuum State" (S0) is the conceptual background from which the Autaxic Generative Engine operates. In v0.1, it is represented primarily by an empty graph and the `p_genesis` parameter. Expanding this concept provides a richer foundation for the system's initial conditions and ongoing generation. * **4.1 S0 as a Potential Field:** S0 isn't just "nothingness," but a state of pure, undifferentiated potential. It can be conceptualized as a field carrying inherent, perhaps mutable, properties that bias the creation of distinctions and relations. * **Properties of S0 Field:** * `GenesisPotentialDensity`: (Float >= 0) Determines the overall rate of spontaneous creation of new distinctions (related to `p_genesis`). This density might not be uniform across the conceptual "space" of S0. * `ProtoPropertyDistributionBias`: (Functions/Distributions) Instead of random assignment, S0 might have inherent biases for the distribution of initial proto-properties (`ProtoPolarity`, `ProtoValence`, etc.) of newly generated distinctions. This could model inherent "tendencies" or "laws" of the vacuum itself. * `RelationalPrecursors`: S0 might contain not just the potential for distinctions, but also for specific *types* of relations, influencing which kinds of connections are more likely to form initially via Genesis (if Genesis creates D/R pairs) or bias `FormationRule`. * `InherentTension`: (Float >= 0) S0 itself might possess an intrinsic level of tension or instability that drives the process of genesis. Genesis events could be seen as a local "release" or transformation of this primordial tension into structured relational tension. * **4.2 Dynamic Vacuum:** The properties of S0 might not be static. They could change over time, or even be influenced by the emergent structure (the graph G) within it. * *Example:* High tension or specific emergent patterns in the graph G might locally or globally deplete/replenish/alter the `GenesisPotentialDensity` or `ProtoPropertyDistributionBias` of the surrounding S0. This creates a feedback loop between the manifested reality (G) and the potential space (S0). * **4.3 Genesis as Phase Transition:** Genesis (`GenesisRule`) can be seen as a local phase transition from the undifferentiated potential of S0 into discrete, related entities (D and R). The `p_genesis` parameter (or a more complex function of S0's properties and local G state) governs the probability of this transition occurring at any given conceptual 'point' in S0 or near existing G. ### **5.0 The Tension Landscape as a Conceptual Framework** Viewing the system's evolution as navigating a "Tension Landscape" provides a powerful metaphor for understanding the role of tension and OC. * **5.1 State Space Dimensions:** The landscape exists in a high-dimensional state space. Each dimension could represent: * The presence/absence of a specific potential distinction or relation. * The value of a proto-property or mutable state variable for each entity. * Topology of the graph (connectivity patterns). * **5.2 Landscape Elevation (Tension):** The "height" or "elevation" of the landscape at any point in this state space is given by the total `RelationalTension(G)` (or a vector of differentiated tension types). The system seeks lower elevations. * **5.3 Movement (Rule Application):** The "movement" or "trajectory" of the system across this landscape is driven by the application of the Cosmic Algorithm rules. Each applicable rule instance corresponds to a specific "step" or "vector" in this state space, moving the system from graph state $G_t$ to $G_{t+1}$. * Rules that reduce tension move the system "downhill". * Rules that increase tension (like Genesis adding a new, potentially high-valence, polarized node) move the system "uphill". * Rules that maintain tension move the system along "contours". * **5.4 Features of the Landscape:** * **Local Minima:** Points (graph states) where no *single* applicable rule can move the system to a lower tension state. This corresponds to Tension Minimization OC (`is_tension_minimum_oc`). These might be shallow depressions or deep valleys. * **Fixed Points:** A special case of local minima where *no* structural rules (Formation, Annihilation, potentially others) are applicable at all. The system is completely static. This corresponds to Fixed Point OC (`is_fixed_point_oc`). These are like the absolute "bottom" of a valley where movement stops. * **Limit Cycles and Attractors:** With dynamic properties, mutable state, and dynamic rule application, the system might not settle into a static point but cycle through a sequence of states, or be drawn into a region of state space (an attractor) without settling to a single point. These represent dynamic forms of stability, where the system is trapped in a low-tension *dynamic pattern* rather than a static structure. These would appear as recurring paths or bounded regions in the tension landscape. * **Ridges and Barriers:** Configurations that are difficult to transition through due to high tension or lack of applicable rules. * **Plateaus:** Regions where tension is relatively constant, even as the graph structure changes. * **Funnels:** Regions of state space where dynamics consistently drive the system towards a particular attractor (a stable state or cycle). * **5.5 Dynamic Landscape:** With dynamic proto-properties (Sec 2.3) and dynamic rule application strategies (Sec 3.2), the very shape of the tension landscape (the tension value at each state, and which transitions are possible/prioritized) can change over time based on the system's history and current state. This means the system is navigating a landscape that is itself constantly shifting and evolving, making the dynamics complex and potentially leading to novel emergent behaviors. Different differentiated tension types (Sec 6.1) can be seen as different "layers" or "dimensions" of the landscape's elevation, which might influence rule application priorities differently. ### **6.0 Implications for Ontological Closure and Tension** An expanded proto-property space and rule set, combined with dynamic rule application, dynamic vacuum, and the perspective of the tension landscape, would significantly complicate the definitions and checks for Ontological Closure. * **6.1 Differentiated Tension Sources and Localization:** * The simple scalar `Tension(G)` could be broken down into distinct components or localized measures. * *Components:* `ValenceTension`, `PolarityTension`, `InformationTension` (from inconsistent `InformationContent`), `CapacityTension` (from overloaded `ProcessingCapacity`), `DecayTension` (from weak relations), `InertiaTension` (from rigid nodes in dynamic regions), `VacuumTension` (tension inherent in the surrounding S0, potentially influencing local genesis). * *Localization:* Instead of global tension, define `LocalTension(Subgraph S)` or `NodeTension(D)`. Tension could be calculated for individual nodes, relations, or specific subgraphs. * *Interaction:* These different tension types could interact. High `InformationTension` in a node might increase its `LocalTension`, making it a target for `SplitRule`, which in turn might decrease its `CapacityTension` but increase `ValenceTension` on the new nodes. * This allows for the system to have multiple, potentially conflicting, sources of "dissatisfaction" or inconsistency, leading to more complex dynamics where resolving one type of tension might increase another, or where different regions of the graph experience different kinds of tension. OC could then involve minimizing specific tension types or achieving a balance between them, rather than just minimizing a single global score. Local tension minima in specific components or regions could define emergent stable subgraphs or "proto-entities". * **6.2 Dynamic Stability Concepts:** * **Fixed Point OC:** Becomes much harder to achieve or even define. With `MutationRule`, `ReinforcementRule`, `TemporalDecay`, dynamic properties, dynamic rule application order, and a dynamic vacuum, the graph state might *never* become structurally static or have zero applicable rules in a predictable, repeatable way. A "Fixed Point" might need to be redefined as a statistical equilibrium, a fixed distribution of properties/structures, or a state where the *set of actively applying rules* becomes static, even if the graph itself changes slightly. In the Tension Landscape, this is the absolute bottom of a static valley. * **Tension Minimization OC:** Remains a valuable concept but the tension landscape becomes vastly more complex, with potentially many local minima and dynamic changes in the landscape itself (due to dynamic properties, rules, and vacuum). The rules navigate this higher-dimensional landscape. Finding a "global" minimum might be impossible or irrelevant; the focus shifts to navigating towards *sufficiently low* tension states that represent viable configurations, potentially defined by the *balance* of different tension types. These are the local valleys in the landscape. * **Limit Cycles & Attractors:** These become more plausible and potentially more meaningful outcomes with dynamic properties and rules. Stable patterns might manifest as recurring sequences of states (Limit Cycles) or as regions in state space that the system tends to return to (Attractors), rather than single fixed points. Detecting and classifying these becomes a key challenge, moving the definition of OC towards identifying stable *dynamics* rather than static states. This aligns with the idea that "reality" is a process, not a fixed object. In the Tension Landscape, these are dynamic patterns of movement within low-tension regions, like orbiting within a valley or traversing a stable cycle of peaks and troughs. ### **7.0 Speculative Emergent Phenomena and Hierarchies** Building upon the expanded conceptual space, we can speculate on how these elements might interact to produce higher-level emergent phenomena not explicitly programmed into the base rules. * **7.1 Formation of Proto-Identities / Stable Subgraphs:** * Combinations of `FormationRule` (especially `ContextualFormationRule`), `ReinforcementRule`, and `AnnihilationRule` can lead to the formation of tightly bound subgraphs with low internal tension (low local tension, specific balance of differentiated tension types). * Nodes within these subgraphs might develop high `TemporalInertia` and low `Flexibility` via `PropertyMutationRule` or `Accumulation/History` mechanisms as they participate in stable configurations. * These stable subgraphs could be candidates for application of the `MergeRule`, transitioning from a collection of nodes to a single higher-order Distinction. This corresponds to finding and collapsing local minima in the tension landscape, creating a new node that represents that minimum. * **7.2 Hierarchical Organization:** * The `MergeRule` provides a direct mechanism for creating hierarchical layers. A merged Distinction (`D_new`) exists at a higher level of abstraction, representing the pattern of its constituent nodes. * This `D_new` can then participate in relations with other Distinctions (either base level or also merged), forming higher-order patterns. Rules like `FormationRule` or `ContextualFormationRule` could apply between these higher-order Distinctions, building nested structures. * `SplitRule` provides a mechanism for decomposition, allowing a higher-order Distinction to break back down into its constituents if it becomes unstable or internally inconsistent (e.g., high `InformationTension` or `CapacityTension`). This represents instability or conflicting tension within a merged entity, potentially pushing it back "up" a local peak in the landscape until it splits. * **7.3 Adaptive Behavior and Learning:** * `PropertyMutationRule` allows entities to change their intrinsic biases (`ProtoPolarity`, `ProtoValence`, `InteractionRadius`, etc.) based on their history or environment (local subgraph context). * `ReinforcementRule` strengthens relations that are part of stable or tension-reducing configurations. `InhibitionRule` prunes relations associated with instability or high tension. * Dynamic Rule Application strategies (like Tension-Driven Prioritization or History-Dependent Selection) allow the system's *behavior* (which rules are applied) to adapt based on its state and history, effectively learning to navigate the tension landscape more efficiently or in ways that favor certain types of structure. This is like the system learning preferred "paths" or "algorithms" for moving through the landscape based on past success (tension reduction). * Combined, these mechanisms could lead to systems that adapt their structure and dynamics over time based on their interactions, exhibiting primitive forms of learning. * **7.4 Proto-Memory and Persistence:** * `TemporalInertia` provides a direct mechanism for persistence, making certain nodes or relations more resistant to change. * `Accumulation/History` mechanisms allow properties to encode information about past interactions or states. * Stable subgraphs (`Proto-Identities`) can act as persistent structures that maintain information or relational patterns over time, functioning as basic memory units. These persistent structures represent deep, stable features within the evolving graph and its tension landscape trajectory. * **7.5 Proto-Communication and Influence:** * `InformationContent` and `InformationChannelType` on relations allow for the transmission of non-structural data or influence across the graph. * `InteractionRadius` defines zones of influence beyond direct connection. * Rules could be designed to react to or transform `InformationContent`, or to prioritize interactions based on `InformationChannelType`, leading to complex propagation and processing of information across the network. This could be a basis for modeling signaling or influence dynamics, adding new vectors or forces that influence movement across the tension landscape. * **7.6 Localized Function and Specialization:** * Through differential rule application (e.g., `Local Rule Sets` strategy) or context-dependent rules (`ContextualFormationRule`, `PropertyMutationRule` based on context), different regions or types of subgraphs could develop specialized structures or dynamics. * Nodes that become part of stable, high-inertia subgraphs might become "core" or "processing" units, while nodes with high `Flexibility` or `InteractionRadius` might act as "boundary" or "sensing" units. * This differential adaptation based on local conditions and rule applications could lead to the emergence of regions with distinct "functions" within the overall graph, corresponding to different types of local minima or dynamic attractors in the tension landscape that emerge in specific areas of the state space. ### **8.0 Conclusion** Exploring an expanded conceptual space for proto-properties and imagining richer rule types, combined with dynamic rule application, a dynamic vacuum, and the conceptual framework of the tension landscape, reveals the potential complexity and power of the Autaxic Generative Engine framework beyond its v0.1 instantiation. These speculative additions suggest pathways for modeling phenomena like adaptation, learning, hierarchical organization, diverse forms of dynamic stability (including cycles and attractors), and localized emergent phenomena. Detailing potential mechanisms for dynamic proto-property interactions and rule selection provides concrete (though still conceptual) avenues for formalizing how the system's internal state and history could influence its own evolution. Furthermore, speculating on how combinations of these features could give rise to proto-identities, hierarchies, and adaptive behaviors highlights the potential of the framework to model complex self-organizing systems. Conceptualizing the Vacuum State as a dynamic field of potential and viewing the system's evolution as navigating a complex, dynamic tension landscape adds significant depth to the model's explanatory power, providing intuitive metaphors for genesis, stability, and change. Formalizing and simulating these concepts in future work will be essential for testing their generative capacity and assessing their ability to produce complex, self-organizing systems driven by intrinsic consistency principles and the dynamic resolution of multiple forms of relational and intrinsic tension. --- --- FILE: AUTX-SOW-P6.2.2-TM2-OC.md --- **Statement of Work (SOW)** **SOW ID:** AUTX-SOW-P6.2.2-TM2-OC **WBS Reference:** Relevant to WBS 2.2 (P6.2), targeting _25161103619.md Phase 1, Step 3. **Project Title:** "Toy Model" (AGE v0.1) - Formalize Ontological Closure (OC) via Tension Minimization **Project Manager/Lead:** Rowan Brad Quni (PI) **Start Date:** (Assumed Immediate) **End Date:** (Start Date + 3 weeks) **Duration:** 3 weeks **1. Objective:** To formally define and specify a "Relational Tension" function for subgraphs within the AGE v0.1 framework, and to define Ontological Closure (OC) in terms of a subgraph representing a local minimum of this tension function. This will complement the "Fixed Point" OC definition already explored. **2. Scope of Work / Key Activities:** * **2.1 Define "Relational Tension" Function:** * Based on the proto-properties defined for AGE v0.1 (e.g., ProtoPolarity, ProtoValence for Distinctions (D-nodes) as per AUTX_A1_FormalNotation_AGEv0.1_V1.md and AUTX-A0-CCD-TM-001), formally define a function Tension(S) that takes a subgraph S and returns a scalar value representing its relational tension. * Consider the suggestions from AUTX-A0-CCD-TM-001 (e.g., tension from unsatisfied valences and unbalanced polarities on nodes with unsatisfied valences). * Specify the mathematical form of this function using the established notation. * **2.2 Define OC via Tension Minimization:** * Formally define what it means for a subgraph S to be at a "local minimum" of the Tension(S) function. This should consider the application of the core "Cosmic Algorithm" rules (GenesisRule, FormationRule, AnnihilationRule as defined in AUTX_A1_AGEv0.1_CoreRules_V1.md). * A possible definition: A subgraph S achieves OC via tension minimization if no single application of an available core rule to S (or its components) results in a new state S' such that Tension(S') < Tension(S). * **2.3 Documentation:** Document the formal definition of the Tension(S) function and the OC criterion based on its minimization in the AFKB (e.g., as AUTX_A1_AGEv0.1_OC_TensionMin_V1.md). **3. Inputs:** * _25161103619.md (Toy Model Proposal - Phase 1, Step 3 OC definitions). * AUTX-A0-CCD-TM-001 (Conceptual Clarification Document, section on OC and Tension). * AUTX_A1_FormalNotation_AGEv0.1_V1.md (Defined primitives and their attributes). * AUTX_A1_AGEv0.1_Primitives_V1.md (Formal specification of primitives). * AUTX_A1_AGEv0.1_CoreRules_V1.md (Formal definition of core algorithm rules). * D-P6.7-1_Unified_Framework_v1.9.md (Conceptual basis for proto-properties and their role in stability). * `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md` (Context for tension in broader model space, including differentiated tension types, dynamic properties potentially influencing tension, the tension landscape metaphor, and how tension relates to emergent phenomena/hierarchies) **4. Deliverables:** * **D1:** A document (AFKB Artifact ID: AUTX_A1_AGEv0.1_OC_TensionMin_V1.md) containing: * The formal mathematical definition of the RelationalTension(S) function for AGE v0.1. * The formal definition of Ontological Closure based on a subgraph being a local minimum of this tension function with respect to the defined core algorithm rules. **5. Assumptions:** * The proto-properties and core rules defined for AGE v0.1 are sufficient to formulate a meaningful tension function. * The concept of a "local minimum" with respect to the existing rules is a tractable definition for OC. * The conceptual ideas in `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md` provide useful context for future work but do not change the formal definitions required *for v0.1*. **6. Risks:** * Defining a simple yet meaningful RelationalTension function for the toy model might be more complex than anticipated. * The "local minimum" condition might be difficult to operationalize computationally or may lead to trivial stable states without careful definition. **7. Success Criteria:** * The RelationalTension(S) function is mathematically well-defined using the established notation and proto-properties of AGE v0.1. * The OC criterion based on tension minimization is formally stated and references the core algorithm rules. * The definition is sufficiently clear to guide its potential implementation in the AGE v0.1 simulation. --- --- FILE: AUTX-SOW-P6.2.2-TM1.md --- **Statement of Work (SOW)** **SOW ID:** AUTX-SOW-P6.2.2-TM1 **WBS Reference:** Blend of WBS 2.2.2 (P6.2, Ph 6.2.2), targeting _25161103619.md Phase 1, Steps 1 & 2. **Project Title:** "Toy Model" (AGE v0.1) - Mathematical Primitives and Core Algorithm Definition **Project Manager/Lead:** Rowan Brad Quni (PI) **1. Objective:** To mathematically define the primitives (Distinctions, Relations, Proto-properties) and a minimal "Cosmic Algorithm" (core graph rewriting rules) for the "Toy Model" (AGE v0.1), using the formalism and notation selected and developed in AUTX-SOW-P6.2.1-001. This corresponds to steps 1 and 2 of Phase 1 in _25161103619.md. **2. Scope of Work / Key Activities:** * **2.1 (Proposal Phase 1, Step 1): Define Primitives Mathematically.** * Using the selected formalism and the AUTX_A1_FormalNotation_AGEv0.1_V1 document, formally represent Distinctions (D) as nodes in an Attributed Dynamic Graph. * Formally represent Relations (R) as directed edges between nodes. * Formally define how Proto-properties (as clarified in AUTX-A0-CCD-TM-001) are represented as attributes (data) attached to nodes (D) and edges (R). * **2.2 (Proposal Phase 1, Step 2): Formalize a Minimal "Cosmic Algorithm".** * Define a small, core set of the proposed rules as precise, automatable Graph Rewriting Rules using the selected formalism and notation. * **Priority Rules to Formalize (based on clarifications in AUTX-A0-CCD-TM-001):** * **GenesisRule(p):** A stochastic rule for adding new D nodes to the graph with specified proto-properties. Define parameters p and the property assignment logic. * **FormationRule(D1, D2):** A rule that creates an R edge between two D nodes if their proto-properties are compatible. Define compatibility conditions and resultant R attributes. * **AnnihilationRule(R_or_D_pair):** A rule that removes an R edge and/or its D nodes if they form a logically inconsistent or unstable pair. Define instability/inconsistency conditions. * **2.3 Documentation:** Document all formal definitions, rules, and data structures in a dedicated section of the AFKB for the AGE v0.1 (e.g., AUTX_A1_AGEv0.1_Primitives_Rules_V1). **3. Inputs:** * _25161103619.md (Primary requirements for "Toy Model" primitives and rules). * AUTX-A0-CCD-TM-001 (Conceptual Clarification Document for Toy Model). * Deliverable D1 from AUTX-SOW-P6.2.1-001: AUTX_A1_FormalismSelection_Report_V1. * Deliverable D2 from AUTX-SOW-P6.2.1-001: AUTX_A1_FormalNotation_AGEv0.1_V1. * D-P6.7-1_Unified_Framework_v1.9.md (Conceptual basis for proto-properties). * `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md` (Context for future primitive/rule space, including dynamic properties and rules, the vacuum state concept, and the tension landscape metaphor, which informs the *potential* scope but not the v0.1 definition) **4. Deliverables:** * **D1:** Mathematical Definition of Primitives for AGE v0.1 (AFKB Artifact ID: AUTX_A1_AGEv0.1_Primitives_V1): * Formal specification of D (nodes) and their attributes. * Formal specification of R (edges) and their attributes. * Formal specification of Proto-property data types and ranges used in AGE v0.1. * **D2:** Formal Definition of Minimal Cosmic Algorithm for AGE v0.1 (AFKB Artifact ID: AUTX_A1_AGEv0.1_CoreRules_V1): * Formal specification of GenesisRule(p) as a graph rewriting rule. * Formal specification of FormationRule(D1, D2) as a graph rewriting rule. * Formal specification of AnnihilationRule(R_or_D_pair) as a graph rewriting rule. * Combined documentation integrated into AUTX_A1_AGEv0.1_Primitives_Rules_V1 in the AFKB. **5. Assumptions:** * The formalism and notation selected/developed in AUTX-SOW-P6.2.1-001 are adequate for these definitions. * The conceptual definitions of proto-properties and rules clarified in AUTX-A0-CCD-TM-001 are sufficiently robust for formalization. * The conceptual ideas in `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md` provide useful context for future work but do not change the formal definitions required *for v0.1*. **6. Risks:** * Difficulty in translating conceptual rules into precise, unambiguous graph rewriting rules within the chosen formalism. * The initial set of proto-properties or rules, even after clarification, may be found to be insufficient or overly complex for a "toy model" upon detailed formalization. * Potential for unforeseen logical inconsistencies in the rule definitions when implemented in the chosen formalism. **7. Success Criteria:** * All primitives (D, R, Proto-properties for AGE v0.1) are defined mathematically using the chosen notation. * The three priority rules (GenesisRule, FormationRule, AnnihilationRule) are formally defined as graph rewriting rules within the chosen formalism. * The definitions are sufficiently precise to guide the implementation of the AGE v0.1 simulation (a subsequent task). * All definitions are clearly documented in the AFKB. --- --- FILE: AUTX-SOW-P6.2.1-001.md --- **Statement of Work (SOW)** **SOW ID:** AUTX-SOW-P6.2.1-001 **WBS Reference:** 2.2.1 Project 6.2, Phase 6.2.1: Formalism Selection & Initial Notation Development **Project Title:** Formalism Selection and Notation for Autaxic "Toy Model" (AGE v0.1) **Project Manager/Lead:** Rowan Brad Quni (PI) **1. Objective:** To research, evaluate, and select the most suitable primary mathematical/computational formalism(s) for implementing the "Toy Model" (AGE v0.1) as described in `_25161103619.md` and clarified in `AUTX-A0-CCD-TM-001`, and to develop an initial, consistent formal notation for its primitives (Distinctions, Relations, Proto-properties) and core rules. **2. Scope of Work / Key Activities:** * **2.1 (WBS 6.2.4.1.2): Research & Evaluate Formalisms:** Based on the requirements of the "Toy Model" (Attributed Dynamic Graphs, Graph Rewriting Rules, measurable properties for Ontological Closure), research and evaluate candidate formalisms. Priority candidates from `_25161103619.md` include Attributed Dynamic Graphs and Graph Rewriting Systems. Other candidates listed in `AUTX_Master_Plan_v2.0_Part_V_Research_Projects.md` (Sec 6.2.2.1, 6.2.4.1.2) such as process calculi, type theory, category theory may be briefly reviewed for supplementary concepts if needed. * **2.2 (WBS 6.2.4.1.2): Assess Strengths/Weaknesses:** Assess the chosen formalisms' ability to represent: * Distinctions (D) as nodes. * Relations (R) as directed edges. * Proto-properties as attributes on nodes/edges (as clarified in `AUTX-A0-CCD-TM-001`). * Graph Rewriting Rules for `GenesisRule`, `FormationRule`, `AnnihilationRule` (as clarified in `AUTX-A0-CCD-TM-001`). * Measurable properties for Ontological Closure definitions (Fixed Point, Tension Minimization, with Limit Cycle as a secondary consideration for v0.1, per `AUTX-A0-CCD-TM-001`). * **2.3 (WBS 6.2.2.2, 6.2.4.1.2): Develop Initial Formal Notation:** Develop a clear, precise, and consistent mathematical notation for representing D, R, their attributes (proto-properties), and the structure of the graph rewriting rules for the "Toy Model", consistent with the selected formalism. * **2.4 Document Rationale:** Document the rationale for selecting the primary formalism(s) and the design choices for the notation system in the AFKB. **3. Inputs:** * `_25161103619.md` (Primary requirements for the "Toy Model" formalism). * `AUTX-A0-CCD-TM-001` (Conceptual Clarification Document for Toy Model). * `AFKB_Processed_Artifacts_V1.md` and `AFKB_Structure_And_Management.md` (representing deliverables from AUTX-SOW-P6.1-001). * `D-P6.7-1_Unified_Framework_v1.9.md` (Conceptual definitions of D, R, proto-properties). * `AUTX_Master_Plan_v2.0_Part_V_Research_Projects.md` (Sec 6.2.2.1, 6.2.4.1). * `AUTX_A0_Synth_Formalisms_V1.md` (Consolidated Thematic Synthesis for formalism candidates and lessons). * Relevant academic literature on graph theory, graph rewriting systems, and computational modeling. * `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md` (Context for formalism requirements in future versions, including dynamic properties and rules, the vacuum state concept, and the tension landscape metaphor, which informs the *potential* scope but not the v0.1 selection criteria) **4. Deliverables:** * **D1:** Report on Formalism Evaluation and Selection (AFKB Artifact ID: `AUTX_A1_FormalismSelection_Report_V1`), detailing the chosen formalism(s) (e.g., specific type of attributed dynamic graph library/framework, graph rewriting system engine/language) and the rationale. * **D2:** Initial Formal Notation Document v0.1 (AFKB Artifact ID: `AUTX_A1_FormalNotation_AGEv0.1_V1`), specifying the mathematical symbols and structures for representing "Toy Model" components (D, R, proto-properties, rules). * All deliverables to be archived in the AFKB. **5. Assumptions:** * The "Toy Model" description in `_25161103619.md` and clarifications in `AUTX-A0-CCD-TM-001` are sufficiently detailed to guide formalism selection. * PI has access to necessary academic resources and potentially software tools for evaluating formalisms. * The conceptual ideas in `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md` provide useful context for future work but do not change the formalism requirements *for v0.1*. **6. Risks:** * Selected formalism proves inadequate or overly complex during subsequent implementation. * Difficulty in creating a notation that is both rigorous and intuitive. * Time to evaluate a wide range of formalisms may be underestimated. **7. Success Criteria:** * A primary formalism is selected that directly supports the representation of attributed dynamic graphs and graph rewriting rules, as specified for AGE v0.1. * The Initial Formal Notation Document v0.1 is sufficiently precise to be used in SOW AUTX-SOW-P6.2.2-TM1 for defining the "Toy Model" primitives and rules. * The selection rationale is well-documented and justified in the AFKB. --- --- FILE: AUTX-A0-CCD-TM-001.md --- **Conceptual Clarification Document for Autaxic "Toy Model" (AGE v0.1) Formalization** **Document ID:** AUTX-A0-CCD-TM-001 **Based on:** `_25161103619.md` (Toy Model Proposal) and `D-P6.7-1_Unified_Framework_v1.9.md` (Autaxys v1.9 Conceptual Framework) **Purpose:** To identify and propose resolutions for conceptual ambiguities or underspecified aspects in the "Toy Model" proposal, ensuring a clear basis for Phase 1 formalization (Formalism Selection and Primitive/Rule Definition). **1. Primitives and Proto-properties (Ref: Toy Model Proposal - Phase 1, Step 1):** * **1.1. Minimal Set of Proto-properties for Toy Model:** * **Issue:** `D-P6.7-1_Unified_Framework_v1.9.md` (Sec 4.2.2) lists numerous speculative proto-properties. The Toy Model proposal (`_25161103619.md`) gives examples but doesn't explicitly select a minimal set for AGE v0.1. * **Clarification/Decision Needed for AGE v0.1:** * **For Distinctions (D - nodes):** * Confirm use of `ID` (unique identifier, likely integer, system-assigned). * `ProtoPolarity`: Data type (e.g., integer `{-1, 0, +1}` or a more abstract enum) and initial distribution if relevant for `GenesisRule`. * `ProtoValence`: Data type (e.g., non-negative integer). Clarify its meaning: a capacity for *n* relations, or a type of bonding site that needs to be satisfied? The "unsatisfied proto-valences" for Tension Minimization suggests the latter. * **For Relations (R - edges):** * `Type`: Data type (e.g., enum, string). What are the initial types for the toy model? E.g., `'link'`, `'strong-link'`. Is one generic type sufficient initially? * `FlowResistance`: Data type (e.g., float `0.0-1.0`). * **Other Proto-properties from v1.9 for consideration (and likely deferral for Toy Model):** Should `ProtoPolarity` also apply to R for the toy model? (v1.9 Sec 4.2.2.2). *Recommendation: Defer for toy model simplicity.* * **Proposed Initial Set for AGE v0.1:** * D-Nodes: `ID` (int, unique), `ProtoPolarity` (int: -1, 0, +1), `ProtoValence` (int >= 0, representing 'bonding desire/sites'). * R-Edges: `Type` (enum: e.g., `default_link`), `Strength` (float, potentially inverse of FlowResistance, e.g., 0.0-1.0). *Note: "Strength" might be more intuitive than "FlowResistance" if higher values mean stronger link.* **2. Minimal "Cosmic Algorithm" (Graph Rewriting Rules) (Ref: Toy Model Proposal - Phase 1, Step 2):** * **2.1. `GenesisRule(p)`:** * **Issue:** "Stochastic rule for adding new D/R pairs." Underspecified `p` and mechanism. * **Clarification/Decision Needed for AGE v0.1:** * What does `p` control? Probability per simulation step? Rate of generation? * How are D and R proto-properties assigned during genesis? (e.g., `ProtoPolarity` for D randomly chosen from `{-1,0,+1}`? `ProtoValence` randomly from a small range e.g., `0-3`?) * Are D and R added as an already-connected pair, or can D be added first, then R forms via `FormationRule`? The proposal phrasing "D/R pairs" suggests connected. v1.9 `S₀ -> D + R` also suggests co-creation. * **Proposed Interpretation for AGE v0.1:** `GenesisRule` adds one D node and one R edge connected *to a new, simultaneously created D node*, forming a D-R-D triplet if the first D was existing, or a D-R-D if both D are new. Or simpler: adds a single D node with random proto-properties. R's only form via `FormationRule`. *Recommendation: Add only D nodes via GenesisRule to simplify initial state and make R formation explicit.* If D/R pairs, then D1-R-D2, with D1, R, D2 all new, or D1 existing, R-D2 new. *Simpler still for toy model: `GenesisRule` adds one D node with random proto-properties based on `p`.* * **2.2. `FormationRule(D1, D2)`:** * **Issue:** "Creates an R edge... if proto-properties are compatible." Compatibility logic is undefined. * **Clarification/Decision Needed for AGE v0.1:** * Define specific compatibility logic. E.g., Can D with `ProtoPolarity: +1` form a relation with `D: -1`? Does `ProtoValence` decrement upon formation? Is an R formed if `D1.ProtoValence > 0` AND `D2.ProtoValence > 0` AND `compatible(D1.ProtoPolarity, D2.ProtoPolarity)`? * What are the attributes of the newly formed R? (e.g., `Type: 'default_link'`, `Strength: some_value` perhaps based on D properties or a default). * **Proposed Logic for AGE v0.1:** An R forms between D1 and D2 if `D1.ProtoValence > 0`, `D2.ProtoValence > 0`, and `D1.ProtoPolarity == -D2.ProtoPolarity` (and neither is 0). Upon formation, `D1.ProtoValence--`, `D2.ProtoValence--`. R gets `Type: 'default_link'`, `Strength: 1.0`. * **2.3. `AnnihilationRule(R)`:** * **Issue:** "Removes an R edge and its D nodes if they form a logically inconsistent or unstable pair (e.g., `D(+1)` and `D(-1)` annihilating)." Conditions for "inconsistent/unstable" are broad. * **Clarification/Decision Needed for AGE v0.1:** * Does annihilation depend on the R type/strength, or only the D nodes' properties? * Example: `D(+1)` and `D(-1)` annihilating. Does this mean if D1 and D2 are connected by *any* R, and `D1.ProtoPolarity == +1` and `D2.ProtoPolarity == -1`, then D1, D2, and R are removed? What if one is `+1` and the other is `0`? * **Proposed Logic for AGE v0.1:** If an R connects D1 and D2, and `D1.ProtoPolarity == +1` and `D2.ProtoPolarity == -1` (or vice-versa), and perhaps their `ProtoValence` are both 0 (meaning they have no other stabilizing connections), then D1, D2, and R are removed. *Simpler: If D1 and D2 are connected by an R, and their polarities are opposite and non-zero, they annihilate.* **3. Ontological Closure (OC) (Ref: Toy Model Proposal - Phase 1, Step 3):** * **3.1. Fixed Point:** * **Issue:** "for a certain number of simulation steps." * **Clarification/Decision Needed for AGE v0.1:** Define this number of steps (N_fixed_point_check) as a simulation parameter. * **3.2. Limit Cycle:** * **Issue:** Detection mechanism. * **Clarification/Decision Needed for AGE v0.1:** For AGE v0.1, detection of limit cycles might be complex. Simplest approach: if not a fixed point, run for many more steps and observe. Formal detection (e.g., hashing graph states and looking for repeats) might be deferred past v0.1 if too complex initially. *Recommendation: Focus on Fixed Point and Tension Minimization for v0.1, defer complex limit cycle detection.* * **3.3. Tension Minimization:** * **Issue:** Definition of "Relational Tension" function. * **Clarification/Decision Needed for AGE v0.1:** * Define a simple tension function based on selected proto-properties. Example: `Tension(Subgraph) = sum_nodes(abs(ProtoPolarity_if_valence_unsatisfied)) + sum_nodes(ProtoValence_if_positive)`. E.g., each unsatisfied valence point contributes 1 to tension. Each D node with non-zero polarity that also has unsatisfied valence contributes `abs(ProtoPolarity)` to tension. * How is "local minimum" determined? (e.g., subgraph state where no single rule application from the cosmic algorithm can further reduce its tension score). **4. AGE v0.1 Simulation Setup (Ref: Toy Model Proposal - Phase 1, Step 4):** * **4.1. Initial State "Foam":** * **Issue:** "starts with a random 'foam' of D's and R's." This seems to contradict `GenesisRule` adding D/R *pairs*. * **Clarification/Decision Needed for AGE v0.1:** * What is the exact nature of the initial state? * Option A: Start with an empty graph, and only `GenesisRule` adds elements. * Option B: Start with a pre-populated graph of N_initial D nodes with random proto-properties. R's then form via `FormationRule`. * Option C: Start with D's and R's already connected randomly (a "foam"). This seems hardest to define consistently. * *Recommendation for AGE v0.1: Option B. Start with N_initial D nodes, their proto-properties assigned randomly within defined ranges. Then let `FormationRule` and `AnnihilationRule` operate. `GenesisRule` can then add new D nodes over time.* * **4.2. Stochasticity:** * **Issue:** Where is stochasticity primarily introduced? In `GenesisRule`? In rule selection if multiple rules can apply? * **Clarification/Decision Needed for AGE v0.1:** * If `GenesisRule` is the main source, its parameter `p` needs clear definition. * If multiple rules can apply to a subgraph or node/edge, how is one chosen? Randomly? Based on some priority? *Recommendation: For v0.1, if multiple rules can apply, pick one randomly, or apply all applicable non-conflicting rules in one "step".* This document is intended to facilitate focused discussion and decision-making for the PI before or during the execution of SOW AUTX-SOW-P6.2.1-001 and AUTX-SOW-P6.2.2-TM1. It also provides context for how these v0.1 decisions fit into the broader conceptual space explored in documents like `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md`. --- --- FILE: D-P6.2-3_Initial_Formal_Language_v1.0.md --- --- ### **DELIVERABLE: D-P6.2-3 - Initial Formal Language and Notation** **ID:** `D-P6.2-3` **Project:** `6.2: Formal Mathematical & Computational Modeling of the Autaxic Generative Engine` **WBS Ref:** `2.7.2: Deliverable: Initial Formal Language/Notation (from P6.2)` **Title:** `Initial Formal Language and Notation for the Autaxic Generative Engine v1.0` **Status:** `Completed` **Version:** `1.0` **Author:** `Principal Investigator (Generated by AI Assistant)` **Date:** `2025-06-08` **Location:** `./02_Research_Phase_1/P6.2_Formal_Modeling/D-P6.2-3_Initial_Formal_Language_v1.0.md` --- ### **1.0 Objective** The purpose of this document is to define a clear, consistent, and unambiguous formal notation for the core concepts used in the initial modeling of the Autaxic Generative Engine. This notation, established in Project 6.2, will serve as the foundation for all subsequent formal descriptions and computational models in Phase 1 and Phase 2 of the research program. ### **2.0 Core Concepts and Notation** The following defines the foundational elements of the formal language. | Concept | Symbol | Definition | Example | | :--- | :--- | :--- | :--- | | **Distinction** | `d` | A fundamental, unique entity or node within a pattern. Subscripts are used for specific instances. | `d_1`, `d_2` | | **Relational Pattern** | `G` | A set of distinctions and the relations between them, represented as a directed graph. | `G = (V, E)` | | **Set of Distinctions** | `V(G)` | The set of all distinctions (nodes) within a pattern `G`. | `V(G) = {d_1, d_2, d_3}` | | **Asymmetric Link** | `r(d_i, d_j)` | A directed, non-reciprocal relation from `d_i` to `d_j`. Represented as a directed edge. | `r(d_1, d_2)` | | **Set of Links** | `E(G)` | The set of all asymmetric links (edges) within a pattern `G`. | `E(G) = {r(d_1, d_2), r(d_2, d_3)}` | | **Self-Application** | `f(G)` | The transformation function that applies a pattern's own relational logic to itself. It generates a new pattern `G'` that includes all implied relations. | `G' = f(G)` | | **Ontological Closure** | `G' = G` | The condition for coherence and stability. A pattern is coherent if and only if applying the self-application function `f(G)` results in no change to the pattern's structure. | `is_coherent(G) ⇔ f(G) = G` | ### **3.0 Example of Usage in the P6.2 Prototype** The notation is applied to the core experiment as follows: 1. **Define two stable patterns, P1 and P2:** * `P1` has `V(P1) = {d_1, d_2}` and `E(P1) = {r(d_1, d_2)}`. * `P2` has `V(P2) = {d_2, d_3}` and `E(P2) = {r(d_2, d_3)}`. 2. **Compose them to create the non-transitive graph, `G_composite`:** * `G_composite` has `V(G_composite) = {d_1, d_2, d_3}`. * `G_composite` has `E(G_composite) = {r(d_1, d_2), r(d_2, d_3)}`. 3. **Check for coherence by applying the `f(G)` transformation:** * The function `f(G)` detects the path `r(d_1, d_2)` followed by `r(d_2, d_3)`. * It determines that the implied relation `r(d_1, d_3)` is missing from `E(G_composite)`. * Therefore, `f(G_composite) ≠ G_composite`, and the pattern is **not coherent**. 4. **Create the transitively closed graph, `G_closed`:** * `G_closed = f(G_composite)`. * `E(G_closed) = {r(d_1, d_2), r(d_2, d_3), r(d_1, d_3)}`. 5. **Check the new graph for coherence:** * Applying `f(G_closed)` finds no new implied relations. * Therefore, `f(G_closed) = G_closed`, and the pattern is **coherent**. ### **4.0 Scope and Future Evolution** This notation is foundational and intentionally minimal. It is sufficient for the modeling conducted in Phase 1. Future research projects, particularly those in Phase 2 and 3 (e.g., P6.3, P6.4), will require this language to be extended to include concepts such as: * Relation types and weights. * Temporal indices for dynamic evolution. * Higher-order relations (relations acting on other relations). * Proto-properties on nodes and edges. * Rules for dynamic graph transformation (Genesis, Formation, Annihilation, etc.), potentially with dynamic application order and state-dependent selection mechanisms. * Concepts related to tension, its sources, and localization. * Notation for emergent structures like stable subgraphs or hierarchies. * Representation of the Vacuum State (S0) and its potential properties. Any extensions will be formally documented in subsequent versions of this language specification, potentially drawing concepts from `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md`. ### **5.0 Conclusion** A clear, simple, and sufficient formal language has been established for the initial phase of the Autaxys research program. This notation ensures that all formal work is communicated unambiguously, forming a solid foundation for future theoretical and computational development. --- --- FILE: D-P6.2-2_Report_on_Suitable_Formalisms_v1.0.md --- --- ### **DELIVERABLE: D-P6.2-2 - Report on Suitable Formalisms** **ID:** `D-P6.2-2` **Project:** `6.2: Formal Mathematical & Computational Modeling of the Autaxic Generative Engine` **WBS Ref:** `2.7.1: Deliverable: Report on Suitable Formalisms (from P6.2)` **Title:** `Report on the Selection of Directed Graphs as a Suitable Formalism for Modeling Ontological Closure` **Status:** `Completed` **Version:** `1.0` **Author:** `Principal Investigator (Generated by AI Assistant)` **Date:** `2025-06-08` --- ### **1.0 Objective** This report documents the selection and justification of the formalism used in the `P6.2` computational prototype. The objective of the prototype was to test the hypothesis that relational composition rules (specifically transitivity) emerge as a necessary condition for Ontological Closure. The selection of an appropriate formalism was critical to ensure that the model was both conceptually sound and computationally verifiable. ### **2.0 Formalism Selection: Directed Graphs** After a review of potential modeling frameworks, **Directed Graphs** (also known as digraphs) were selected as the most suitable formalism for this initial phase of research. A directed graph `G` is a mathematical structure consisting of a set of vertices (nodes) and a set of ordered pairs of these vertices, called directed edges (or arcs). This formalism provides a direct and intuitive mapping to the core concepts of the Autaxys model at this foundational level: * **Nodes (Vertices):** Represent fundamental "distinctions" or "entities" within a pattern. In the prototype, these were labeled simply as `1`, `2`, `3`. * **Directed Edges (Arcs):** Represent an "AsymmetricLink" or a directed, non-reciprocal relationship between two distinctions. An edge `(A, B)` is interpreted as the relation `A -> B`. ### **3.0 Justification of Suitability** The choice of Directed Graphs was justified based on the following criteria: 1. **Conceptual Clarity:** The mapping from the abstract concepts of "distinctions" and "relations" to the concrete mathematical objects of "nodes" and "edges" is unambiguous and direct. This clarity is essential for ensuring the model accurately reflects the theory it is intended to test. 2. **Mathematical Rigor:** Graph theory is a well-established and rigorous field of mathematics. Concepts such as "path" (a sequence of edges), which are central to the definition of the `f(G)` transformation, are precisely defined, leaving no room for misinterpretation. 3. **Computational Tractability:** Directed graphs are easily and efficiently represented in computational systems. As demonstrated in the final, dependency-free prototype, a graph can be implemented with basic Python dictionaries and sets, making the model transparent and universally executable. Operations like adding an edge or checking for the existence of a path are algorithmically straightforward. 4. **Parsimony (Simplicity and Sufficiency):** The primary goal was to test the emergence of transitivity. A directed graph is the simplest possible structure that can represent the required pattern (`A -> B -> C`). More complex formalisms (e.g., weighted graphs, hypergraphs, temporal networks, attributed graphs with dynamic rules) would have introduced unnecessary complexity and variables not relevant to this foundational hypothesis. The chosen formalism is precisely as complex as it needs to be, and no more. ### **4.0 Acknowledged Limitations & Future Scope** The selection of this simple formalism is specific to the scope of Project 6.2. The limitations of this approach are acknowledged and are intended to be addressed in later research phases as outlined in the Master Plan and conceptual documents like `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md`: * **Static Nature:** The model does not possess an intrinsic notion of time or continuous evolution. * **Uniform Relations:** All edges represent the same *type* of relationship. * **Lack of Higher-Order Dynamics:** The model cannot represent relations acting upon other relations. * **Limited Node/Edge Attributes:** Does not include rich proto-properties or mutable state on nodes/edges. * **Fixed Rule Set:** The transformation `f(G)` is a single, fixed rule (transitive closure implication), not a dynamic set of graph rewriting rules with potential for dynamic application. * **Implicit Vacuum:** The concept of a generative vacuum state (S0) is only implicitly represented by the initial graph and the single transformation rule. These limitations were accepted by design for this initial proof-of-concept. The purpose of this project was not to create a comprehensive model of reality, but to validate a single, fundamental principle in isolation. Future projects, such as `P6.3 (Autaxic Cosmogenesis)`, will necessitate the adoption of more advanced formalisms (like Attributed Dynamic Graphs with Graph Rewriting Systems, as planned for AGE v0.1) capable of handling these complexities, drawing upon the expanded conceptual space including potential for emergent structures and hierarchies, dynamic properties, a dynamic vacuum state, and the tension landscape metaphor. ### **5.0 Conclusion** Directed Graphs proved to be an ideal formalism for the initial proof-of-concept. The framework provided the necessary conceptual clarity and computational power to successfully model the principle of Ontological Closure and demonstrate the emergence of transitivity as a condition for relational stability. This choice has been validated by the successful and verifiable execution of the experiment. --- --- FILE: P6.2_ComputationalPrototype_V1.0_Results.md --- --- ### **DELIVERABLE: P6.2_ComputationalPrototype_V1.0_Results.md** **ID:** `D-P6.2-1` **Project:** `6.2: Formal Mathematical & Computational Modeling of the Autaxic Generative Engine` **WBS Ref:** `2.7.3: Deliverable: Proof-of-Concept Models (from P6.2)` **Title:** `Computational Prototype V1.0 Results: Emergent Transitivity from Ontological Closure` **Status:** `Completed & Verified` **Version:** `1.1` (Supersedes previous dry-run analysis) **Author:** `Principal Investigator (Generated by AI Assistant)` **Date:** `2024-10-27` --- ### **1.0 Summary of Findings** The computational experiment defined in `D-P1-2.7.3.1_Computational_Prototype_Spec_V1.0.md` was executed successfully using a self-contained Python script with no external dependencies. The experiment was designed to test the hypothesis that **transitivity emerges as a necessary condition for the stability (coherence) of composite relational patterns.** The experiment yielded the following key results, confirming all hypotheses: 1. **Primordial Act:** The `Asymmetric` pattern (`1 -> 2`) was correctly identified as the only minimal coherent pattern. The `Symmetric` pattern (`1 <-> 2`) was found to be unstable, as its structure implies the creation of self-loops (`1 -> 1`, `2 -> 2`) that were not present in the original graph. 2. **Composite Instability:** A composite graph created by linking two stable patterns (`1 -> 2` and `2 -> 3`) was found to be **incoherent (unstable)**. The model correctly identified that the structure implied a missing transitive link (`1 -> 3`). 3. **Transitive Stability:** A modified version of the composite graph that included the transitive link (`1 -> 3`) was found to be **coherent (stable)**, as its structure contained all of its own implications. The success of this prototype provides the first piece of *verifiable computational evidence* for the core thesis of the Autaxys framework: that logical rules can be derived from the fundamental requirement of Ontological Closure. ### **2.0 Full, Verifiable Simulation Output** The following is the verbatim output from the execution of the dependency-free Python script. ```text --- Executing Primordial Act --- [?] Evaluating potential pattern: Symmetric Pattern has nodes ['1', '2'] and edges [('1', '2'), ('2', '1')]. [✗] FAILURE: Pattern 'Symmetric' is not coherent (unstable). f(G) created a different graph with edges: [('1', '1'), ('1', '2'), ('2', '1'), ('2', '2')] It cannot self-constitute and fails to actualize. [?] Evaluating potential pattern: Asymmetric Pattern has nodes ['1', '2'] and edges [('1', '2')]. [✓] SUCCESS: Pattern 'Asymmetric' is coherent (is a stable fixed point). This pattern actualizes. --- Starting Composition Experiment --- [Step 1] Primordial Act successful. First stable pattern P1: [('1', '2')] [Step 2] Defining a second stable pattern P2: [('2', '3')] [Step 3] Composing P1 and P2. Composite graph has edges: [('1', '2'), ('2', '3')] Visual Representation: 1 -> 2 -> 3 [Step 4] Checking coherence of the composite graph... [✓] HYPOTHESIS CONFIRMED: The non-transitive composite graph is NOT coherent. The self-application f(G) would add edge(s), resulting in: [('1', '2'), ('1', '3'), ('2', '3')] The pattern is not a stable fixed point and cannot self-constitute. [Step 5] Creating transitively closed graph. G_closed has edges: [('1', '2'), ('1', '3'), ('2', '3')] Visual Representation: 1 -> 2 -> 3, with the required link 1 -> 3. [Step 6] Checking coherence of the transitively closed graph... [✓] HYPOTHESIS CONFIRMED: The transitively closed graph IS coherent. The self-application f(G) adds no new edges, meaning the pattern is a stable fixed point and can self-constitute. ``` ### **3.0 Analysis & Implications** The experiment's successful execution confirms the validity of the core hypothesis. The instability of the composite graph (`1->2`, `2->3`) demonstrates that a system of relations lacking transitivity is structurally incomplete and self-contradictory under the `f(G)` transformation. The `self_apply_f` function acts as an engine of logical closure, forcing the pattern to manifest the relationships implied by its structure. A pattern can only be considered stable—and thus "actualized"—when this process results in no further changes. The success of this proof-of-concept is a critical milestone for the Autaxys program. It provides concrete, reproducible evidence that the principle of Ontological Closure is computationally sound and can serve as the generative basis for deriving more complex relational rules. This foundational result justifies proceeding with the subsequent research phases outlined in the WBS and the development of the more complex AGE v0.1 model with dynamic rules and proto-properties, as detailed in related specifications and conceptual documents like `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md`, which explores the potential for emergent phenomena and hierarchies, a dynamic vacuum state, and the tension landscape metaphor. --- --- FILE: D-P1-2.7.3.1_Computational_Prototype_Spec_V1.0.md --- --- id: P6.2_ComputationalPrototype_V1.0 title: "Computational Prototype v1.0: Simulating Emergent Transitivity from Ontological Closure" project_id: AUTX-A.1-P6.2 product_id: P6.2 status: draft version: 1.0 author: Rowan Brad Quni (with Autologos Assistance) summary: "This document specifies the design and implementation of a simple computational prototype to test the hypothesis that relational composition rules (specifically transitivity) emerge as a necessary condition for Ontological Closure. Using Python and NetworkX, the simulation will model the `primordialAct`, the composition of two `AsymmetricLink` patterns, and check for self-referential stability (stable fixed points) to validate that only the transitively closed configuration achieves coherence." --- **1. Objective** This project moves from conceptual sketches (`P6.2_ModelSketch_Genesis_V2.0.md`) to a concrete, executable simulation. The primary objective is to test the core hypothesis that **composition rules for definitionally emergent relational types are themselves emergent properties, required by the principle of Intrinsic Coherence (modeled as Ontological Closure).** Specifically, this prototype will simulate the composition of two `AsymmetricLink` patterns and test if **transitivity** (`A->B` and `B->C` implies `A->C`) is a necessary condition for the stability of the resulting composite pattern. **2. Technical Specification** * **Language:** Python 3.9+ * **Core Libraries:** * `networkx`: For creating, manipulating, and analyzing directed graphs (`DiGraph`), which will represent our autaxic patterns. * `matplotlib`: For visualizing the graph structures. * **Environment:** A standard Python environment (e.g., Jupyter Notebook, a standalone script) capable of running these libraries. **3. Data Structures and Core Concepts** * **Pattern Representation:** An autaxic pattern `P` will be represented by a `networkx.DiGraph` object. * **Distinctions (`d`)**: Represented as nodes in the graph (e.g., integers `1`, `2`, `3`). * **AsymmetricLink Relation (`r(source, target)`)**: Represented as a directed edge from the `source` node to the `target` node. * **Ontological Closure Check (`CheckCoherence`):** This will be a function that takes a graph `G` as input and determines if it is coherent. In this prototype, coherence is defined as having a **stable fixed point** under a self-application transformation `f(G)`. * `is_coherent(G)` will return `True` if `f(G)` is structurally identical to `G`, and `False` otherwise. **4. The `f(G)` Self-Application Transformation (v1.0)** The `f(G)` transformation represents one "tick" of the pattern's intrinsic dynamics. For this prototype, we define a simple rule that tests the local and global consistency of causal flows. This is a simplified version of the dynamic rules planned for AGE v0.1, focusing solely on the implication of transitivity. It represents a basic form of the "Cosmic Algorithm" operating to achieve closure. * **Rule:** For every node `X` in the graph `G`, if there is an incoming path of length 2 (e.g., `A -> B -> X`), the self-application function `f(G)` generates a new graph `G'` which includes all of `G`'s original edges PLUS a "shortcut" edge from the start of that path to `X` (i.e., `A -> X`). * **Fixed Point:** A graph `G` is a stable fixed point if it already contains all the "shortcut" edges implied by its existing paths of length 2. In this case, `f(G)` produces no new edges, and thus `G'` is identical to `G`. This means the graph is **transitively closed**. **5. Simulation Algorithm & Key Functions** **5.1. `generate_minimal_patterns()`** This function will implement the `Distinguish` operator from the v2.0 sketch. It will generate two minimal graphs: 1. `G_undir`: A graph with two nodes and a symmetric connection (modeled as two opposing directed edges for this prototype). 2. `G_dir`: A graph with two nodes and a single directed edge (`d1 -> d2`). * **Returns:** A list `[G_undir, G_dir]`. **5.2. `is_coherent(G)`** This function implements the ontological closure check. 1. Takes a graph `G` as input. 2. Calculates `G_prime = f(G)` by finding all paths of length 2 and adding the corresponding shortcut edges. 3. Compares `G` and `G_prime` using `networkx.is_isomorphic(G, G_prime)` (or a simpler edge-set comparison). 4. Returns `True` if they are identical (it's a fixed point), `False` otherwise. **5.3. `primordial_act()`** This function simulates the first generative event. It tests minimal configurations for stability against the `f(G)` rule. 1. Calls `generate_minimal_patterns()`. 2. Iterates through the generated patterns, calling `is_coherent()` on each. 3. Returns the *first* graph that returns `True`. 4. **Expected Outcome:** It should find that `G_undir` is unstable (since `f(G_undir)` would add edges, making it different) and that `G_dir` is stable (since it has no paths of length 2, `f(G_dir) = G_dir`). It will return `G_dir`. **5.4. The Main Experiment: `test_composition()`** This function will execute the core hypothesis test. 1. Call `primordial_act()` to get the first stable pattern, `P1 = (1 -> 2)`. 2. Define a second stable pattern, `P2 = (2 -> 3)`. 3. **Composition Step:** Create a new graph `G_composite` by combining the nodes and edges of `P1` and `P2`. `G_composite` now represents `{1, 2, 3, r(1,2), r(2,3)}`. 4. **Coherence Check 1:** Call `is_coherent(G_composite)`. * **Hypothesis:** This will return `False`. The `f(G)` transformation will detect the path `1 -> 2 -> 3` and add the shortcut edge `1 -> 3`, so the resulting graph will not be identical to the input. 5. **Closure Step:** Manually add the transitive edge `(1, 3)` to `G_composite` to create `G_closed`. This step conceptually represents the system applying the implications of its structure to achieve closure. 6. **Coherence Check 2:** Call `is_coherent(G_closed)`. * **Hypothesis:** This will return `True`. Since `G_closed` already contains the edge `(1, 3)`, the `f(G)` transformation will find no new shortcuts to add. The graph is a stable fixed point. 7. **Conclusion:** The simulation will report whether only the transitively closed graph achieved coherence. **6. Implementation and Reporting** * The Python code will be implemented in a single script or notebook. * Functions will be clearly defined and commented to match this specification. * The `test_composition()` function will print clear, human-readable output indicating the results of each coherence check and the final conclusion. * `matplotlib` will be used to generate and save images of `G_composite` and `G_closed` for visual confirmation. * The results will be documented in a new report file, `P6.2_ComputationalPrototype_V1.0_Results.md`, which will analyze the outcome and discuss its implications for the autaxys framework. **7. Success Criteria** This prototype will be considered successful if: 1. The `primordial_act()` function correctly identifies the directed graph as the only stable minimal pattern. 2. The `test_composition()` function demonstrates that the non-transitively-closed graph is incoherent (unstable). 3. The `test_composition()` function demonstrates that the transitively-closed graph is coherent (stable). This outcome would provide the first piece of computational evidence that composition rules can be derived from the fundamental requirement of ontological closure. This specific prototype uses a simplified `f(G)` focused on transitivity, which is a stepping stone to the more complex rule set in the AGE v0.1 simulation (`AUTX_A1_AGEv0.1_Simulation_Code_V1.py.md`), which explores dynamic rule applications and tension minimization, informed by broader conceptual possibilities (`AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md`), including the potential for emergent phenomena and hierarchies, a dynamic vacuum state, and the tension landscape metaphor. --- --- FILE: AGE_v0.1_Prototype.py.md --- ```python # --- AUTAXYS FRAMEWORK PROTOTYPE --- # File: AGE_v0.1_Prototype.py # WBS Ref: 2.2 Project 6.2 (Formal Mathematical & Computational Modeling) # Phase: 1 - Foundational Formalism ("Toy Model") # Description: This script provides the initial scaffolding for the Autaxic Generative Engine (AGE). # It defines the core data structures for Distinctions, Relations, the RelationalGraph, # and the Engine itself. This version focuses on structure, with rule implementation # to follow in subsequent iterations (AUTX_A1_AGEv0.1_Simulation_Code_V1.py.md). # Based On: AGE v0.1 Conceptual Clarifications (AUTX-A0-CCD-TM-001.md) # Context: Broader conceptual space (AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md) # including the concept of the Vacuum State (S0) and the Tension Landscape. import random class Distinction: """ Represents a Distinction (D), a fundamental node in the relational graph. """ def __init__(self, id, proto_properties=None): self.id = id # Proto-properties are inherent attributes that bias behavior. # For v0.1: 'ProtoPolarity' (int -1, 0, 1), 'ProtoValence' (int >= 0) # Conceptual space is larger (AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 2.1) self.proto_properties = proto_properties if proto_properties is not None else {} # Mutable state derived from interactions: self.current_relation_count = 0 # Example of a mutable state in v0.1 # (Future consideration: More complex mutable state, internal state machines, dynamic property interactions - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 2.3) @property def unsatisfied_valence(self): # Calculate derived attribute return self.proto_properties.get('ProtoValence', 0) - self.current_relation_count def __repr__(self): # Simple repr for this scaffolding version # In the full simulation, a more detailed repr might be used. return f"D({self.id})" def copy(self): """ Creates a copy of the Distinction object (used in Graph.copy()). """ # Copy proto_properties dictionary copied_props = self.proto_properties.copy() new_distinction = Distinction(self.id, copied_props) new_distinction.current_relation_count = self.current_relation_count # Copy mutable state # (Future: Copy more complex mutable state) return new_distinction class Relation: """ Represents a Relation (R), a directed edge between two Distinctions. """ def __init__(self, source_id, target_id, proto_properties=None): self.source_id = source_id self.target_id = target_id # Proto-properties of relations define their type, strength, etc. # For v0.1: 'Type' (enum), 'Strength' (float) # Conceptual space is larger (AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 2.2) self.proto_properties = proto_properties if proto_properties is not None else {} # (Future consideration: Mutable state for relations, dynamic property interactions - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 2.3) def __repr__(self): # Simple repr for this scaffolding version return f"R({self.source_id}->{self.target_id})" def copy(self): """ Creates a copy of the Relation object (used in Graph.copy()). """ # Copy proto_properties dictionary copied_props = self.proto_properties.copy() new_relation = Relation(self.source_id, self.target_id, copied_props) # (Future: Copy mutable state) return new_relation # Needed for checking if a relation exists in a list (e.g., in remove_relation) def __eq__(self, other): if not isinstance(other, Relation): return NotImplemented # Relations are equal if source, target, and properties are the same (simplified check for v0.1) # This assumes proto_properties are hashable/comparable. # For v0.1, the simple properties (-1,0,1 ints, >=0 ints, string, float) are fine. # (Future: Needs to handle more complex properties or possibly relation IDs for equality) return self.source_id == other.source_id and self.target_id == other.target_id and self.proto_properties == other.proto_properties def __hash__(self): # Hash based on immutable properties # Use frozenset for dictionary items to make it hashable return hash((self.source_id, self.target_id, frozenset(self.proto_properties.items()))) class RelationalGraph: """ Manages the collection of all Distinctions and Relations (the "Vacuum State" S0). """ def __init__(self): # Distinctions are stored in a dictionary for quick lookup by ID. self.distinctions = {} # Relations are stored in a list. Using a list allows multiple relations between the same nodes, # though v0.1 FormationRule prevents duplicates of the *default* type between the same pair. self.relations = [] self._next_id = 0 # Counter for assigning unique Distinction IDs def add_distinction(self, proto_properties=None): """ Adds a new Distinction to the graph. """ new_id = self._next_id new_distinction = Distinction(new_id, proto_properties) self.distinctions[new_id] = new_distinction self._next_id += 1 return new_distinction def add_relation(self, source_id, target_id, proto_properties=None): """ Adds a new Relation between two Distinctions and updates node counts. """ source = self.distinctions.get(source_id) target = self.distinctions.get(target_id) if source and target: # Check for duplicate relation of the *same type* between the same nodes (as per v0.1 Formation rule logic) # This prevents adding the same R(d1->d2) multiple times via FormationRule # In future, different relation types could exist between the same nodes. if any(r.source_id == source_id and r.target_id == target_id and r.proto_properties == proto_properties for r in self.relations): # print(f"Warning: Relation {source_id}->{target_id} with properties {proto_properties} already exists. Skipping.") return None # Do not add duplicate new_relation = Relation(source_id, target_id, proto_properties) self.relations.append(new_relation) # Update relation counts on nodes source.current_relation_count += 1 target.current_relation_count += 1 # (Future consideration: Dynamic proto-property interactions on nodes/relations upon addition - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 2.3) return new_relation # print(f"Warning: Attempted to add relation between non-existent nodes {source_id}->{target_id}") return None # Should ideally raise error or log warning if nodes don't exist def remove_relation(self, relation): """ Removes a Relation from the graph and updates node counts. """ # Find the specific relation object in the list (needed because list stores copies) # Use the __eq__ method defined in Relation try: # Find by identity or by equality (__eq__) rel_in_list = next((r for r in self.relations if r == relation), None) if rel_in_list: self.relations.remove(rel_in_list) source = self.distinctions.get(relation.source_id) target = self.distinctions.get(relation.target_id) # Decrement counts ONLY if the nodes still exist in the graph # This handles cases where a node might have been removed by a previous rule instance/phase if source: source.current_relation_count -= 1 if target: target.current_relation_count -= 1 # (Future consideration: Dynamic proto-property interactions on nodes/relations upon removal - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 2.3) except ValueError: # Relation not found - shouldn't happen if called correctly based on `relation in graph.relations` check # print(f"Warning: Attempted to remove non-existent relation {relation}") pass # Or log a warning def remove_distinction(self, distinction_id): """ Removes a Distinction and all its incident Relations. """ if distinction_id in self.distinctions: # Find incident relations first (iterate over a copy) # Need to get the *actual* relation objects from the current list state relations_to_remove = [ r for r in self.relations if r.source_id == distinction_id or r.target_id == distinction_id ] # Remove incident relations. remove_relation handles decrementing counts. # Iterate over the collected list of relations *to be removed*, but remove from the actual graph list. for rel in relations_to_remove: # Check if the relation still exists in the graph's list before trying to remove it, # as another node's removal might have already removed this shared relation. if rel in self.relations: self.remove_relation(rel) # Now remove the distinction del self.distinctions[distinction_id] # print(f"Removed Distinction {distinction_id}") def get_distinction(self, distinction_id): return self.distinctions.get(distinction_id) def get_relations_incident_to(self, distinction_id): return [ r for r in self.relations if r.source_id == distinction_id or r.target_id == distinction_id ] def get_relations_from_to(self, source_id, target_id): return [ r for r in self.relations if r.source_id == source_id and r.target_id == target_id ] def get_all_nodes(self): return list(self.distinctions.values()) def get_all_edges(self): return list(self.relations) # Return a copy to prevent modification issues during iteration def get_state_summary(self): return f"Graph State: {len(self.distinctions)} Distinctions, {len(self.relations)} Relations." def copy(self): """ Creates a deep copy of the graph state. """ new_graph = RelationalGraph() # Copy distinctions for d_id, distinction in self.distinctions.items(): # Use Distinction's copy method to ensure mutable state is copied new_graph.distinctions[d_id] = distinction.copy() # Copy relations for relation in self.relations: # Use Relation's copy method new_graph.relations.append(relation.copy()) new_graph._next_id = self._next_id # Keep ID counter in sync return new_graph class AutaxicGenerativeEngine: """ The core engine that applies the Cosmic Algorithm rules to the RelationalGraph. """ def __init__(self, simulation_parameters): self.params = simulation_parameters self.graph = RelationalGraph() self.step_count = 0 self.history = [] # To store state metrics for analysis if 'seed' in self.params and self.params['seed'] is not None: random.seed(self.params['seed']) self.initialize_graph() def initialize_graph(self): """ Sets up the initial state G0. """ num_initial = self.params.get('N_initial', 0) max_valence = self.params.get('Max_Initial_Valence', 3) # As per AUTX-A0-CCD-TM-001.md (Sec 4.1 Option B), start with N_initial nodes, no relations. # This initial state is conceptually 'seeded' from S0. # (Future consideration: S0 properties influencing initial state - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 4.1) for _ in range(num_initial): valence = random.randint(0, max_valence) if valence > 0: # Polarity is non-zero for nodes with potential connections polarity = random.choice([-1, 1]) else: # Polarity is zero for inert nodes (valence 0) in v0.1 polarity = 0 self.graph.add_distinction(proto_properties={'ProtoPolarity': polarity, 'ProtoValence': valence}) # print(f"Initialized graph with {num_initial} distinctions.") # --- Rule Application Implementations (Based on AUTX_A1_AGEv0.1_CoreRules_V1.md) --- # Rules are applied to a copy of the graph state at the start of their phase. # They modify the graph object passed in (which is a copy). def apply_genesis_rule(self, graph): """ Rule: Spontaneously create D from the vacuum. Applies to graph state at start of phase. """ applied_count = 0 p_genesis = self.params.get('p_genesis', 0.1) max_valence = self.params.get('Max_Initial_Valence', 3) # Use Max_Initial_Valence for genesis distribution too # Conditions for Application: Random chance p_genesis per step # (Future consideration: Genesis rate influenced by S0 properties or graph state - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 4.1, 4.2) if random.random() < p_genesis: # Transformation: Add a new distinction with random properties # (Future consideration: Property distribution influenced by S0 properties - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 4.1) valence = random.randint(0, max_valence) if valence > 0: polarity = random.choice([-1, 1]) else: polarity = 0 # Add distinction directly to the graph copy passed in graph.add_distinction(proto_properties={'ProtoPolarity': polarity, 'ProtoValence': valence}) applied_count = 1 # print(f"Step {self.step_count} (Genesis Phase): GENESIS - Created a new Distinction.") return graph, applied_count def apply_formation_rule(self, graph): """ Rule: Form relations between compatible Distinctions. Applies to graph state at start of phase. """ # Identify applicable instances based on the graph state *input* to this phase. # Store parameters needed to apply the rule. applicable_instances_params = [] # Operate on a copy of node IDs to avoid issues if nodes were removed in a previous phase node_ids_at_start_of_phase = list(graph.distinctions.keys()) # Identify applicable instances (pairs of distinct nodes) for d1_id in node_ids_at_start_of_phase: for d2_id in node_ids_at_start_of_phase: if d1_id == d2_id: continue # Cannot form relation with self in v0.1 d1 = graph.get_distinction(d1_id) d2 = graph.get_distinction(d2_id) # Ensure nodes still exist in the graph state being checked (the input to this phase) if d1 is None or d2 is None: continue # Conditions for Application (AUTX_A1_AGEv0.1_CoreRules_V1.md Sec 3.2, AUTX-A0-CCD-TM-001.md Sec 2.2) # (Future consideration: Context-dependent formation rules - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 3.1) d1_polarity = d1.proto_properties.get('ProtoPolarity', 0) d2_polarity = d2.proto_properties.get('ProtoPolarity', 0) conditions_met = ( d1.unsatisfied_valence > 0 and d2.unsatisfied_valence > 0 and d1_polarity == -d2_polarity and d1_polarity != 0 and # Polarity must be non-zero opposite # Check if a relation of the default type already exists D1 -> D2 not any(r.source_id == d1_id and r.target_id == d2_id and r.proto_properties.get('Type') == 'default_link' for r in graph.relations) ) if conditions_met: # Store parameters needed to apply this instance later applicable_instances_params.append((d1_id, d2_id)) # Apply all applicable instances simultaneously to the graph state *input to this phase* applied_count = 0 for (s_id, t_id) in applicable_instances_params: # Simple check: ensure nodes still exist in the graph object we are modifying # and re-check if relation exists to avoid adding duplicates if another Formation rule # created the relation in this same phase. d1 = graph.get_distinction(s_id) d2 = graph.get_distinction(t_id) if d1 and d2: # Re-check the conditions on the *current state* of the graph copy being modified # This is important if applying rules simultaneously - one rule's effect might invalidate another. # In v0.1, Formation rule doesn't remove nodes/relations, only adds, # and it only adds the default type, so the duplicate check is key. d1_polarity = d1.proto_properties.get('ProtoPolarity', 0) d2_polarity = d2.proto_properties.get('ProtoPolarity', 0) if (d1.unsatisfied_valence > 0 and d2.unsatisfied_valence > 0 and d1_polarity == -d2_polarity and d1_polarity != 0 and not any(r.source_id == s_id and r.target_id == t_id and r.proto_properties.get('Type') == 'default_link' for r in graph.relations)): # Add the relation and update counts in the graph copy # (Future consideration: Dynamic property interactions on nodes/relations upon addition - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 2.3) graph.add_relation(s_id, t_id, {'Type': 'default_link', 'Strength': 1.0}) applied_count += 1 # print(f"Step {self.step_count} (Formation Phase): FORMATION - Created R({s_id}->{t_id}).") return graph, applied_count def apply_relation_annihilation_rule(self, graph): """ Rule (Sub-rule 1): Resolve/cancel inconsistent relations. Applies to graph state at start of phase. """ # Identify applicable instances based on the graph state *input* to this phase. # Store parameters/objects needed to apply the rule. relations_to_annihilate = [] # Iterate over a copy of edges from the graph state *input* to this phase relations_at_start_of_phase = graph.get_all_edges() for relation in relations_at_start_of_phase: source = graph.get_distinction(relation.source_id) target = graph.get_distinction(relation.target_id) # Ensure nodes still exist in the graph state being checked (input to this phase) if source is None or target is None: continue # Conditions for Application (AUTX_A1_AGEv0.1_CoreRules_V1.md Sec 3.3.1, AUTX-A0-CCD-TM-001.md Sec 2.3) # (Future consideration: Annihilation conditions based on relation/node mutable state or context - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 2.3, 3.1) source_polarity = source.proto_properties.get('ProtoPolarity', 0) target_polarity = target.proto_properties.get('ProtoPolarity', 0) conditions_met = ( source_polarity == target_polarity and source_polarity != 0 # Polarity must be non-zero same ) if conditions_met: # Store the relation object itself. We'll use its equality (__eq__) to find it in the graph copy. relations_to_annihilate.append(relation) # Apply all applicable instances simultaneously to the graph state *input to this phase* applied_count = 0 # Iterate over the list of relations identified *at the start of the phase* to be removed. # remove_relation handles decrementing counts. for relation_to_remove_params in relations_to_annihilate: # Re-check if the specific relation instance still exists in the graph state being modified # before attempting removal. This is crucial if another rule instance (in the same phase) # already removed this relation, or if a node was removed by a previous phase rule. # Use the __eq__ method to find the relation in the current graph list. rel_in_current_graph = next((r for r in graph.relations if r == relation_to_remove_params), None) if rel_in_current_graph: # Re-check conditions on the *current state* of the graph copy being modified # This is important if applying rules simultaneously - one rule's effect might invalidate another. source_copy = graph.get_distinction(rel_in_current_graph.source_id) target_copy = graph.get_distinction(rel_in_current_graph.target_id) if source_copy and target_copy: # Nodes must still exist in the copy source_pol_copy = source_copy.proto_properties.get('ProtoPolarity', 0) target_pol_copy = target_copy.proto_properties.get('ProtoPolarity', 0) if (source_pol_copy == target_pol_copy and source_pol_copy != 0): # (Future consideration: Dynamic property interactions on nodes/relations upon removal - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 2.3) graph.remove_relation(rel_in_current_graph) # Remove and update counts in the copy applied_count += 1 # print(f"Step {self.step_count} (Rel Annihilation Phase): RELATION ANNIHILATION - Removed R({relation_to_remove_params.source_id}->{relation_to_remove_params.target_id}).") # Note: If source or target was removed by a previous rule/phase, the relation is already gone # due to remove_distinction, so the rel_in_current_graph check handles this implicitly. return graph, applied_count def apply_node_annihilation_rule(self, graph): """ Rule (Sub-rule 2): Remove isolated/inert nodes. Applies to graph state at start of phase. """ # Identify applicable instances based on the graph state *input* to this phase. nodes_to_annihilate_id = [] # Operate on a copy of keys because removal modifies the dictionary node_ids_at_start_of_phase = list(graph.distinctions.keys()) for d_id in node_ids_at_start_of_phase: d = graph.get_distinction(d_id) # Ensure node still exists in the graph state being checked (the input to this phase) if d is None: continue # Conditions for Application (AUTX_A1_AGEv0.1_CoreRules_V1.md Sec 3.3.2, AUTX-A0-CCD-TM-001.md Sec 2.3) # (Future consideration: Annihilation conditions based on node mutable state or context - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 2.3, 3.1) conditions_met = ( d.current_relation_count == 0 and d.proto_properties.get('ProtoValence', 0) == 0 ) if conditions_met: nodes_to_annihilate_id.append(d_id) # Apply all applicable instances simultaneously to the graph state *input to this phase* applied_count = 0 # Iterate over the list of node IDs identified *at the start of the phase* to be removed. # remove_distinction handles removing incident edges and decrementing counts. for d_id in nodes_to_annihilate_id: # Re-check if node still exists in the graph state being modified before removing # This is crucial if another rule instance (in the same phase) already removed this node, # or if a node was removed by a previous phase rule. if graph.get_distinction(d_id): # Re-check conditions on the *current state* of the graph copy being modified node_copy = graph.get_distinction(d_id) if (node_copy.current_relation_count == 0 and node_copy.proto_properties.get('ProtoValence', 0) == 0): graph.remove_distinction(d_id) # Remove node and its edges in the copy applied_count += 1 # print(f"Step {self.step_count} (Node Annihilation Phase): NODE ANNIHILATION - Removed D({d_id}).") return graph, applied_count # --- Ontological Closure and Tension Implementation (Based on AUTX_A1_AGEv0.1_OC_TensionMin_V1.md) --- def calculate_tension(self, graph): """ Calculates the total Relational Tension for the graph. """ # Based on AUTX_A1_AGEv0.1_OC_TensionMin_V1.md (Sec 2.1) # This is a simple scalar tension for v0.1. # (Future consideration: Differentiated tension sources and localization - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 6.1) # (Future consideration: Tension as landscape elevation - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 5.2) total_tension = 0.0 # Use float for tension w_v = self.params.get('w_v', 1.0) # Weight for unsatisfied valence w_p = self.params.get('w_p', 1.0) # Weight for frustrated polarity (on unsatisfied nodes) for d in graph.get_all_nodes(): # Calculate derived attribute UnsatisfiedValence # Ensure UnsatisfiedValence is not negative due to logic errors or future rule types unsatisfied_valence = max(0, d.unsatisfied_valence) # Use the property # Tension contribution from unsatisfied valence (only positive contribution) total_tension += w_v * unsatisfied_valence # Tension contribution from frustrated polarity on nodes with unsatisfied valence # Condition: unsatisfied_valence > 0 AND ProtoPolarity is non-zero if unsatisfied_valence > 0 and d.proto_properties.get('ProtoPolarity', 0) != 0: total_tension += w_p * abs(d.proto_properties.get('ProtoPolarity', 0)) # Add potential future tension terms here (e.g., from relation properties, spatial factors, # internal node state inconsistency from AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 6.1) return total_tension def check_for_ontological_closure(self, graph): """ Checks for Fixed Point and Tension Minimization OC. Applies to graph state at end of step. """ # Based on AUTX_A1_AGEv0.1_Simulation_Impl_Spec_V1.md (Sec 4.0) # and AUTX_A1_AGEv0.1_OC_TensionMin_V1.md (Sec 3.0) # (Future consideration: Detecting Limit Cycles, OC based on local minima or dynamics in tension landscape - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 6.2) current_tension = self.calculate_tension(graph) is_fixed_point_oc = True # Assume FP unless a structural rule is applicable is_tension_minimum_oc = True # Assume TM unless a single structural rule decreases tension # --- Identify ALL applicable instances across all structural rules in the current graph state --- # This list is used for both FP check (existence) and TM check (tension effect of each) all_applicable_instances = [] # Applicable Formation instances (pairs of IDs) node_ids = list(graph.distinctions.keys()) # Use fresh list of IDs from current graph for d1_id in node_ids: for d2_id in node_ids: if d1_id == d2_id: continue d1 = graph.get_distinction(d1_id) d2 = graph.get_distinction(d2_id) if d1 and d2: # Ensure nodes exist in the current graph d1_polarity = d1.proto_properties.get('ProtoPolarity', 0) d2_polarity = d2.proto_properties.get('ProtoPolarity', 0) if (d1.unsatisfied_valence > 0 and d2.unsatisfied_valence > 0 and d1_polarity == -d2_polarity and d1_polarity != 0 and # Polarity must be non-zero opposite # Check if a relation of the default type already exists D1 -> D2 not any(r.source_id == d1_id and r.target_id == d2_id and r.proto_properties.get('Type') == 'default_link' for r in graph.relations)): all_applicable_instances.append(('Formation', (d1_id, d2_id))) # Applicable Relation Annihilation instances (Relation objects identified by properties) # Need to capture enough info to identify the specific relation instance in a graph copy for relation in graph.get_all_edges(): # Operate on copy of edges from current graph source = graph.get_distinction(relation.source_id) target = graph.get_distinction(relation.target_id) if source and target: # Ensure nodes exist in the current graph source_polarity = source.proto_properties.get('ProtoPolarity', 0) target_polarity = target.proto_properties.get('ProtoPolarity', 0) if (source_polarity == target_polarity and source_polarity != 0): # Store parameters needed to uniquely identify and apply this rule instance # For v0.1, storing the relation object itself and using __eq__ is sufficient all_applicable_instances.append(('RelAnnihilation', relation)) # Applicable Node Annihilation instances (Node IDs) for d_id in list(graph.distinctions.keys()): # Use fresh list of IDs from current graph d = graph.get_distinction(d_id) if d: # Ensure node exists in the current graph if (d.current_relation_count == 0 and d.proto_properties.get('ProtoValence', 0) == 0): all_applicable_instances.append(('NodeAnnihilation', d_id)) # --- Check for Fixed Point OC (is_OC_FP(G)) --- # A graph G is OC_FP if no rule instance from {Formation, RelAnn, NodeAnn} is applicable. if len(all_applicable_instances) > 0: is_fixed_point_oc = False # --- Check for Tension Minimization OC (is_OC_TM(G)) --- # A graph G is OC_TM if no SINGLE applicable rule instance from {Formation, RelAnn, NodeAnn} decreases tension. # This corresponds to being at a local minimum in the Tension Landscape with respect to single steps of structural rules. # --- Helper functions for applying a single rule instance hypothetically --- # These operate on a graph_copy and return the modified copy. # They need to be careful to only apply the *single* instance and handle potential side effects correctly. def apply_single_formation_instance(graph_copy, source_id, target_id): """ Apply a single Formation rule instance to a graph copy. """ # Check if nodes still exist in the copy before attempting to add relation # and re-check the specific formation conditions on the *copy* d1_copy = graph_copy.get_distinction(source_id) d2_copy = graph_copy.get_distinction(target_id) if d1_copy and d2_copy: d1_pol_copy = d1_copy.proto_properties.get('ProtoPolarity', 0) d2_pol_copy = d2_copy.proto_properties.get('ProtoPolarity', 0) if (d1_copy.unsatisfied_valence > 0 and d2_copy.unsatisfied_valence > 0 and d1_pol_copy == -d2_pol_copy and d1_pol_copy != 0 and # Check if a relation of the default type already exists D1 -> D2 in the copy not any(r.source_id == source_id and r.target_id == target_id and r.proto_properties.get('Type') == 'default_link' for r in graph_copy.relations)): # Add the relation and update counts in the copy graph_copy.add_relation(source_id, target_id, {'Type': 'default_link', 'Strength': 1.0}) return graph_copy def apply_single_relation_annihilation(graph_copy, relation_params): """ Apply a single Relation Annihilation rule instance to a graph copy. relation_params is the Relation object from the original graph identified as applicable. """ # Find the equivalent relation in the copied graph to remove it correctly using __eq__ relation_in_copy = next((r for r in graph_copy.relations if r == relation_params), None) if relation_in_copy: # Re-check annihilation conditions on the *copy* before removing source_copy = graph_copy.get_distinction(relation_in_copy.source_id) target_copy = graph_copy.get_distinction(relation_in_copy.target_id) if source_copy and target_copy: # Nodes must still exist in the copy source_pol_copy = source_copy.proto_properties.get('ProtoPolarity', 0) target_pol_copy = target_copy.proto_properties.get('ProtoPolarity', 0) if (source_pol_copy == target_pol_copy and source_pol_copy != 0): graph_copy.remove_relation(relation_in_copy) # Remove and update counts in the copy return graph_copy def apply_single_node_annihilation(graph_copy, node_id_to_remove): """ Apply a single Node Annihilation rule instance to a graph copy. """ # Check if node still exists in the copy before removing node_copy = graph_copy.get_distinction(node_id_to_remove) if node_copy: # Re-check annihilation conditions on the *copy* if (node_copy.current_relation_count == 0 and node_copy.proto_properties.get('ProtoValence', 0) == 0): graph_copy.remove_distinction(node_id_to_remove) # Remove node and its edges in the copy return graph_copy # --- End Helper functions --- # Check tension effect for each single applicable instance for rule_type, instance_data in all_applicable_instances: hypothetical_graph = graph.copy() # Start fresh copy for each instance if rule_type == 'Formation': s_id, t_id = instance_data hypothetical_graph = apply_single_formation_instance(hypothetical_graph, s_id, t_id) elif rule_type == 'RelAnnihilation': relation_to_remove = instance_data # This is the Relation object from the *original* graph hypothetical_graph = apply_single_relation_annihilation(hypothetical_graph, relation_to_remove) elif rule_type == 'NodeAnnihilation': d_id_to_remove = instance_data hypothetical_graph = apply_single_node_annihilation(hypothetical_graph, d_id_to_remove) # else: should not happen tension_after_rule = self.calculate_tension(hypothetical_graph) if tension_after_rule < current_tension: is_tension_minimum_oc = False # print(f"Step {self.step_count}: Tension decreases from {current_tension:.2f} to {tension_after_rule:.2f} via {rule_type} instance.") break # Found an instance that decreases tension, not a local minimum return { 'total_tension': current_tension, 'is_fixed_point_oc': is_fixed_point_oc, 'is_tension_minimum_oc': is_tension_minimum_oc, # Include applicable counts for analysis (counts from FP check are sufficient here) # Note: This counts applicable rules *in the state G_t+1*, not G_t. # For v0.1, rules are simple enough this distinction might not matter much, # but for complex rules or properties changing dynamically, the count *before* # application should ideally be taken on G_t or graph_at_start_of_step. # Recalculating applicable rules on G_t+1 in check_for_ontological_closure is simpler for v0.1. 'total_structural_applicable_post_step': len(all_applicable_instances) } def step(self): """ Executes one full cycle of the Cosmic Algorithm (all phases). """ self.step_count += 1 # Store state at start of step (for simultaneous rule application logic within phases) # Need to copy the graph state before applying any rules in this step. # Each phase function modifies the graph object passed to it *in place*. # So, we pass the output of the previous phase to the next. # (Future consideration: Dynamic rule application order/selection based on state/tension/history - AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 3.2) graph_state_for_genesis = self.graph.copy() # Base for this step's rule applications graph_after_genesis, genesis_count = self.apply_genesis_rule(graph_state_for_genesis) graph_state_for_formation = graph_after_genesis # Pass the modified copy graph_after_formation, formation_count = self.apply_formation_rule(graph_state_for_formation) graph_state_for_rel_annihilation = graph_after_formation # Pass the modified copy graph_after_rel_annihilation, rel_annihilation_count = self.apply_relation_annihilation_rule(graph_state_for_rel_annihilation) graph_state_for_node_annihilation = graph_after_rel_annihilation # Pass the modified copy graph_t_plus_1, node_annihilation_count = self.apply_node_annihilation_rule(graph_state_for_node_annihilation) # Update the engine's graph state to the final state of the step self.graph = graph_t_plus_1 # --- Stability Check and Metrics --- # Check OC status and calculate tension on the final graph state of the step (G_t+1) oc_status = self.check_for_ontological_closure(self.graph) step_metrics = { 'step': self.step_count, 'num_distinctions': len(self.graph.distinctions), 'num_relations': len(self.graph.relations), 'genesis_applied': genesis_count, 'formation_applied': formation_count, # These are counts of *applied* instances 'rel_annihilation_applied': rel_annihilation_count, 'node_annihilation_applied': node_annihilation_count, 'total_structural_applied': formation_count + rel_annihilation_count + node_annihilation_count, 'total_tension': oc_status['total_tension'], 'is_fixed_point_oc': oc_status['is_fixed_point_oc'], 'is_tension_minimum_oc': oc_status['is_tension_minimum_oc'], # Applicable counts *before* application in this step (calculated during FP check are sufficient here) # Note: This counts applicable rules *in the state G_t+1*, not G_t. # For v0.1, rules are simple enough this distinction might not matter much, # but for complex rules or properties changing dynamically, the count *before* # application should ideally be taken on G_t or graph_at_start_of_step. # Recalculating applicable rules on G_t+1 in check_for_ontological_closure is simpler for v0.1. 'total_structural_applicable_post_step': oc_status.get('total_structural_applicable_post_step', 0) # Corrected key name } self.history.append(step_metrics) # Optional: Print progress # if self.step_count % 10 == 0 or step_metrics['is_fixed_point_oc'] or step_metrics['is_tension_minimum_oc']: # print(f"Step {self.step_count}: {self.graph.get_state_summary()}, Tension={step_metrics['total_tension']:.2f}, FP_OC={step_metrics['is_fixed_point_oc']}, TM_OC={step_metrics['is_tension_minimum_oc']}") def run_simulation(self, steps): print("--- Starting Autaxic Generative Engine Simulation (v0.1) ---") print(f"Parameters: {self.params}") print(f"Initial State: {self.graph.get_state_summary()}") initial_oc_status = self.check_for_ontological_closure(self.graph) print(f"Initial Tension: {initial_oc_status['total_tension']:.2f}") print(f"Initial Fixed Point OC: {initial_oc_status['is_fixed_point_oc']}") print(f"Initial Tension Minimum OC: {initial_oc_status['is_tension_minimum_oc']}") print("-" * 40) for i in range(steps): self.step() # Optional: Stop early if a fixed point is reached and persists? # For v0.1, just check current step: if self.history and self.history[-1]['is_fixed_point_oc']: print(f"Fixed Point OC reached at step {self.step_count}. Stopping.") break # Stop on first FP_OC # Optional: Stop early if graph is empty? # Check if the graph is empty AND no rules were applied in the last step, AND genesis didn't add anything if self.history and len(self.graph.distinctions) == 0 and self.history[-1]['total_structural_applied'] == 0 and self.history[-1]['genesis_applied'] == 0: print(f"Graph is empty and stable at step {self.step_count}. Stopping.") break # Print step summary periodically if self.step_count % 50 == 0: latest_metrics = self.history[-1] print(f"Step {self.step_count}: {self.graph.get_state_summary()}, Tension={latest_metrics['total_tension']:.2f}, FP_OC={latest_metrics['is_fixed_point_oc']}, TM_OC={latest_metrics['is_tension_minimum_oc']}") print("-" * 40) print("--- Simulation Complete ---") def get_history(self): return self.history def get_final_graph(self): return self.graph def get_graph_at_step(self, step): """ Retrieve the graph state at a specific historical step (if logged). NOTE: This requires storing full graph copies in history, which is memory intensive. v0.1 logs metrics, not full graph states by default. This method is a placeholder/conceptual hook for future versions that might log states. Retrieving states allows mapping the tension value/OC status from history back to the specific graph configuration at that step, enabling visualization and detailed analysis of graph structures in different regions of the tension landscape. (Future consideration: Logging graph states for detailed analysis - AUTX_A0_Conceptual_Simulation_Analysis_Strategy_V1.md) """ print("Warning: get_graph_at_step is not implemented in v0.1 history logging.") return None if __name__ == '__main__': # Define simulation parameters sim_params = { 'p_genesis': 0.05, # Probability of Genesis Rule application per step (e.g., 0.05 means 5% chance) 'Max_Initial_Valence': 3, # Max valence for randomly assigned nodes (values 0, 1, 2, or 3) 'N_initial': 20, # Number of nodes in the initial state 'w_v': 1.0, # Weight for UnsatisfiedValence in Tension 'w_p': 5.0, # Weight for Frustrated Polarity (on unsatisfied nodes) in Tension - make polarity frustration weigh higher 'total_simulation_steps': 500, # Number of steps to run 'seed': 42 # Use a seed for reproducibility (e.g., 42), None for random runs } # Initialize and run the engine age = AutaxicGenerativeEngine(sim_params) age.run_simulation(steps=sim_params['total_simulation_steps']) # Print the final state of the graph and history summary print("\\nFinal State of the Relational Graph:") final_graph = age.get_final_graph() print(final_graph.get_state_summary()) final_oc_status = age.check_for_ontological_closure(final_graph) print(f"Final Tension: {final_oc_status['total_tension']:.2f}") print(f"Final Fixed Point OC: {final_oc_status['is_fixed_point_oc']}") print(f"Final Tension Minimum OC: {final_oc_status['is_tension_minimum_oc']}") print(f"Applicable structural rules at end: {final_oc_status['total_structural_applicable_post_step']}") # Example of accessing history (e.g., for plotting later) # Requires pandas library (`pip install pandas matplotlib`) # import pandas as pd # import matplotlib.pyplot as plt # # history_df = pd.DataFrame(age.get_history()) # # print("\\nSimulation History Summary (Last 5 steps):") # print(history_df.tail()) # # # Example Plotting # plt.figure(figsize=(14, 10)) # # plt.subplot(4, 1, 1) # plt.plot(history_df['step'], history_df['num_distinctions'], label='Distinctions') # plt.plot(history_df['step'], history_df['num_relations'], label='Relations') # plt.ylabel('Count') # plt.title('System Size Over Time') # plt.legend() # plt.grid(True) # # plt.subplot(4, 1, 2) # plt.plot(history_df['step'], history_df['total_tension'], label='Total Tension', color='red') # plt.ylabel('Tension') # plt.title('Total Relational Tension Over Time') # plt.legend() # plt.grid(True) # # plt.subplot(4, 1, 3) # plt.plot(history_df['step'], history_df['total_structural_applied'], label='Total Structural Rules Applied', color='orange') # plt.plot(history_df['step'], history_df['total_structural_applicable_post_step'], label='Total Structural Rules Applicable (Post-Step)', color='brown', linestyle=':') # plt.ylabel('Rule Applications/Applicability') # plt.title('Rule Activity Over Time') # plt.legend() # plt.grid(True) # # plt.subplot(4, 1, 4) # # Plotting boolean as int (True=1, False=0) # history_df['is_fixed_point_oc_int'] = history_df['is_fixed_point_oc'].astype(int) # history_df['is_tension_minimum_oc_int'] = history_df['is_tension_minimum_oc'].astype(int) # plt.plot(history_df['step'], history_df['is_fixed_point_oc_int'], label='Fixed Point OC', color='green') # plt.plot(history_df['step'], history_df['is_tension_minimum_oc_int'], label='Tension Minimum OC', color='purple', linestyle='--') # plt.ylabel('OC Status (True=1, False=0)') # plt.title('Ontological Closure Status Over Time') # plt.yticks([0, 1], ['False', 'True']) # plt.xlabel('Step') # plt.legend() # plt.grid(True) # # plt.tight_layout() # # plt.savefig('age_v01_simulation_plot.png') # plt.show() ``` --- --- FILE: AUTX_A1_AGEv0.1_Simulation_Impl_Spec_V1.md --- --- ### **ARTIFACT: AUTX_A1_AGEv0.1_Simulation_Impl_Spec_V1 - Implementation Specification** **ID:** `AUTX_A1_AGEv0.1_Simulation_Impl_Spec_V1` **Project:** `6.2: Formal Mathematical & Computational Modeling` **WBS Ref:** `2.2.2.2.4: Simulation Implementation Specification` **Title:** `Implementation Specification for the AGE v0.1 Simulation` **Status:** `Draft` **Version:** `1.0` **Author:** `Principal Investigator (with Autologos Assistance)` **Date:** `2025-06-24` **Location:** `./02_Research_Phase_1/P6.2_Formal_Modeling/Computational_Models/AUTX_A1_AGEv0.1_Simulation_Impl_Spec_V1.md` **Based On:** * `AUTX_A1_AGEv0.1_Primitives_V1.md` (Formal Primitives) * `AUTX_A1_AGEv0.1_CoreRules_V1.md` (Formal Core Rules) * `AUTX_A1_AGEv0.1_OC_TensionMin_V1.md` (Formal OC Definitions) * `AUTX-A0-CCD-TM-001.md` (Conceptual Clarifications) * `AGE_v0.1_Prototype.py.md` (Initial code scaffolding structure) * `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md` (Context for future expansion beyond v0.1 scope, including dynamic properties, dynamic rule application strategies, differentiated tension concepts, the vacuum state concept, the tension landscape metaphor, potential emergent phenomena and hierarchies) --- ### **1.0 Objective** This document specifies the detailed implementation plan for the AGE v0.1 computational simulation (`AUTX_A1_AGEv0.1_Simulation_Code_V1.py`). It translates the formal definitions of primitives, rules, and Ontological Closure criteria into algorithmic steps and data structures suitable for execution. The objective is to create a functional simulation environment for exploring the dynamics of the minimal AGE model and validating core hypotheses regarding emergent structure and stability driven by tension minimization and fixed points. While this spec focuses on v0.1, it acknowledges conceptual directions for future versions outlined in documents like `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md`, including the potential for emergent phenomena and hierarchical organization, a dynamic vacuum state, and the tension landscape metaphor. ### **2.0 Core Data Structures** The simulation will use Python classes to represent the core concepts, building upon the scaffolding in `AGE_v0.1_Prototype.py.md`: * **`Distinction` Class:** * Attributes: `id` (unique int), `proto_properties` (dictionary, includes `'ProtoPolarity': int, 'ProtoValence': int`), `current_relation_count` (int, mutable state). * Derived Property: `unsatisfied_valence` (`ProtoValence - current_relation_count`). * Method: `copy()` for creating deep copies. * *(Future consideration: Additional mutable state, internal state machines, dynamic property interactions, and representation of emergent hierarchy/identity as per AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 2.1, 2.3, 7.1, 7.2)* * **`Relation` Class:** * Attributes: `source_id` (int), `target_id` (int), `proto_properties` (dictionary, includes `'Type': str, 'Strength': float`). * Methods: `copy()`, `__eq__()`, `__hash__()` for comparison and set operations. * *(Future consideration: Additional mutable state, dynamic property interactions as per AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 2.2, 2.3)* * **`RelationalGraph` Class:** * Attributes: `distinctions` (dictionary {id: Distinction}), `relations` (list of Relation objects), `_next_id` (int counter). * Methods: `add_distinction()`, `add_relation()`, `remove_relation()`, `remove_distinction()`. * Helper Methods: `get_distinction()`, `get_relations_incident_to()`, `get_relations_from_to()`, `get_all_nodes()`, `get_all_edges()`. * Utility: `get_state_summary()`, `copy()` for creating deep copies of the entire graph state. ### **3.0 Autaxic Generative Engine (`AutaxicGenerativeEngine`)** This class orchestrates the simulation steps and rule applications. * **Attributes:** `graph` (RelationalGraph object), `step_count` (int), `history` (list of dictionaries for logging metrics), `params` (dictionary of simulation parameters). * **Initialization (`__init__`)**: Takes simulation parameters, initializes the graph (calling `initialize_graph`), sets up the random seed if provided. * **Graph Initialization (`initialize_graph`)**: Creates the initial state ($G_0$) with `N_initial` distinctions having random `ProtoPolarity` and `ProtoValence` within specified ranges, as per `AUTX-A0-CCD-TM-001.md` (Sec 4.1 Option B). No initial relations are created by Genesis in v0.1. This initial state is conceptually drawn from the Vacuum State (S0), which is implicitly represented by the initialization parameters in v0.1. * **Step Execution (`step`)**: Executes one full simulation step, applying rules in phases and updating the graph state. Logs metrics after the step. * **Simulation Run (`run_simulation`)**: Loops for a specified number of steps, calling `step()` each time. Includes basic print statements for progress. * **History Access (`get_history`, `get_final_graph`)**: Methods to retrieve simulation data for external analysis. * *(Future consideration: Dynamic rule application order/selection based on state/tension/history as per AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 3.2, potentially influenced by the tension landscape dynamics)* ### **4.0 Rule Application and Phasing** The simulation step will apply the core rules sequentially in distinct phases, operating on the graph state resulting from the previous phase. This ensures a defined order of operations within a single discrete time step. * **Rule Methods:** * `apply_genesis_rule(graph)`: Takes a graph copy (the state at the start of the Genesis phase), applies Genesis based on `p_genesis`, returns the modified graph copy and count of applications. Modifies the graph *in place*. This rule is the primary interaction with the conceptual Vacuum State (S0) in v0.1. * `apply_formation_rule(graph)`: Takes a graph copy (state after Genesis), identifies all applicable Formation instances based on v0.1 rules, applies them simultaneously to the graph copy, returns modified copy and count. Modifies the graph *in place*. * `apply_relation_annihilation_rule(graph)`: Takes graph copy (state after Formation), identifies all applicable Rel Annihilation instances, applies them simultaneously, returns modified copy and count. Modifies the graph *in place*. * `apply_node_annihilation_rule(graph)`: Takes graph copy (state after Rel Annihilation), identifies all applicable Node Annihilation instances, applies them simultaneously, returns modified copy and count. Modifies the graph *in place*. * *(Future consideration: New rule types like Merge, Split, Mutation, Reinforcement, Inhibition as per AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 3.1, which enable higher-level dynamics and emergent structures, representing different types of movement in the tension landscape)* * **Step Logic (`step`)**: 1. Increment `step_count`. 2. Create a deep copy of `self.graph` to represent the state *at the start of the step* ($G_t$). This copy is the input to the first phase. 3. Call `graph_after_genesis, genesis_count = self.apply_genesis_rule(G_t.copy())`. (Need a copy for the *input* to the first phase). 4. Call `graph_after_formation, formation_count = self.apply_formation_rule(graph_after_genesis)`. 5. Call `graph_after_rel_annihilation, rel_annihilation_count = self.apply_relation_annihilation_rule(graph_after_formation)`. 6. Call `graph_t_plus_1, node_annihilation_count = self.apply_node_annihilation_rule(graph_after_rel_annihilation)`. 7. Update `self.graph = graph_t_plus_1`. This is the final state for step $t$. 8. Call `oc_status = self.check_for_ontological_closure(self.graph)` on the final state $G_{t+1}$. 9. Collect metrics (counts, sizes, tension, OC status) into a dictionary and append to `self.history`. ### **5.0 Ontological Closure and Tension Implementation** The OC checks and tension calculation operate on a given graph state (specifically, the state *after* all rule applications in a step). * **`calculate_tension(graph)` Method:** * Takes a graph object. * Iterates through all Distinctions in the graph. * Calculates `unsatisfied_valence` for each Distinction (`d.proto_properties['ProtoValence'] - d.current_relation_count`). * Applies the tension function formula from `AUTX_A1_AGEv0.1_OC_TensionMin_V1.md` (Sec 2.1), summing contributions from unsatisfied valence and frustrated polarity (non-zero polarity on nodes with unsatisfied valence), weighted by `w_v` and `w_p`. * Returns the scalar total tension. This represents the 'height' in the v0.1 tension landscape. * *(Future consideration: Differentiated tension sources and localized tension measures as per AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 6.1. Also, incorporating potential tension from the Vacuum State (S0) itself)* * **`check_for_ontological_closure(graph)` Method:** * Takes a graph object. * Calculates `current_tension = self.calculate_tension(graph)`. * **Fixed Point OC Check:** Identifies all instances of Formation, Relation Annihilation, or Node Annihilation rule that are applicable in the *current* graph state according to their conditions defined in `AUTX_A1_AGEv0.1_CoreRules_V1.md`. If any applicable instance is found, `is_fixed_point_oc` is False. Otherwise, it's True. Collect counts of applicable instances. This checks if the system is at a static point in the tension landscape (a valley floor with no outward vectors). * **Tension Minimization OC Check:** * Uses the list of *all* individual rule instances (Formation pairs, specific Relation Annihilation instances, specific Node Annihilation instances) that were identified as applicable for the Fixed Point check in the *current* graph state. * For *each* identified applicable instance: * Create a *new* deep copy of the current graph. * Apply *only* that single rule instance to the copy using helper functions (`apply_single_formation_instance`, `apply_single_relation_annihilation`, `apply_single_node_annihilation`). * Calculate the tension of the modified graph copy (`tension_after_rule`). * If `tension_after_rule < current_tension`, then `is_tension_minimum_oc` is False, and the check can stop (as it's not a local minimum). * If the loop completes without finding any single rule application that reduces tension, `is_tension_minimum_oc` remains True. This checks if the system is at a local minimum in the tension landscape with respect to single steps of the defined structural rules. * Returns a dictionary containing `total_tension`, `is_fixed_point_oc`, `is_tension_minimum_oc`, and counts of applicable rules. * *(Future consideration: Defining OC via Limit Cycles or other dynamic attractors, or based on local tension minima/equilibria as per AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 6.2. This relates to identifying stable dynamic patterns within the tension landscape)* * **Helper functions for `ApplySingleInstance` (for TM check):** * `apply_single_formation_instance(graph_copy, source_id, target_id)`: Take a graph copy, add the single relation $(source\_id \rightarrow target\_id)$, update counts. Return the modified copy. * `apply_single_relation_annihilation(graph_copy, relation_to_remove_params)`: Take a graph copy, find the *specific* relation matching source, target, and properties, remove it. Update counts. Note: Need to pass enough info to uniquely identify the relation instance in the copy. * `apply_single_node_annihilation(graph_copy, node_id_to_remove)`: Take a graph copy, remove the node by ID and its incident edges. Return the modified copy. ### **5.0 Simulation Parameters** The simulation requires the following parameters to be defined, ideally in a configuration section or file: * `p_genesis`: Probability of Genesis Rule application per step (float, 0.0-1.0). This parameter implicitly represents a property of the Vacuum State (S0) in v0.1. * `Max_Initial_Valence`: Maximum value for randomly assigned ProtoValence during genesis (int >= 0). This implicitly reflects a property bias of S0 in v0.1. * `N_initial`: Number of nodes in the initial state (int >= 0). Also related to the initial interaction with S0. * `w_v`: Weight for UnsatisfiedValence in Tension calculation (float >= 0). * `w_p`: Weight for Frustrated Polarity (polarity on nodes with unsatisfied valence) in Tension calculation (float >= 0). These weights define the shape of the v0.1 tension landscape. * `total_simulation_steps`: The maximum number of discrete steps to run the simulation. * `seed`: An optional integer seed for the random number generator to ensure reproducible runs for analysis. * *(Future consideration: Parameters for new rules (e.g., Merge, Split thresholds), weights for different tension types, parameters for dynamic rule application logic/strategies, parameters defining the dynamic properties of the Vacuum State (S0) as per AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md Sec 3.1, 3.2, 4.1, 6.1)* ### **6.0 Simulation Execution and Logging** The `AutaxicGenerativeEngine.run_simulation()` method will execute the specified number of steps. * **Initialization:** Set the random seed if provided. Call `initialize_graph()` to set up $G_0$. * **Main Loop:** Iterate `total_simulation_steps` times, calling `self.step()` in each iteration. * **Logging:** Inside the `step()` method, after updating the graph state and performing OC checks, collect all relevant metrics (`step_count`, `num_distinctions`, `num_relations`, rule application counts, `total_tension`, `is_fixed_point_oc`, `is_tension_minimum_oc`, `total_structural_applicable_post_step`) into a dictionary and append it to a list (`self.history`). This history list serves as the primary output for analysis and allows tracking the system's trajectory across the tension landscape. * **Early Termination:** The simulation can optionally terminate early if a Fixed Point OC state is reached and persists for a defined number of steps (though for v0.1, simply checking `is_fixed_point_oc` being true in the current step is sufficient for logging; early stopping is optional). Termination also occurs if the graph becomes empty. * **Output:** The `run_simulation` method should provide basic console output summarizing the start/end state. The full `self.history` list should be accessible after the run for external analysis scripts. ### **7.0 Analysis and Visualization Hooks** The simulation code should be structured to facilitate analysis and visualization as per `AUTX_A0_Conceptual_Simulation_Analysis_Strategy_V1.md` and `AUTX_A0_Conceptual_Visualization_Strategy_V1.md`. * Provide methods to access the complete simulation history (`get_history()`) and the final graph state (`get_final_graph()`). * The logging format should be amenable to processing by standard data analysis libraries (e.g., Pandas). * While visualization code itself will reside in separate scripts, the simulation output format must support extracting the necessary data points (metrics over time, graph structure snapshots). Graph snapshots might require saving the node/edge lists and their attributes at specific steps. This is particularly important for visualizing emergent structures and hierarchies identified through analysis, as discussed conceptually in `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md`. Visualizing the trajectory of the system in relation to the conceptual tension landscape (e.g., plotting tension vs. step count, or projecting higher-dimensional state space onto 2D/3D based on key metrics like size, tension, applicable rules) will be a key analysis technique. ### **8.0 Conclusion** This implementation specification details the translation of the formal AGE v0.1 model into a runnable simulation. It defines the required data structures, the algorithmic steps for rule application and OC checks, the necessary simulation parameters, and the logging/output mechanisms. This document, along with the code scaffolding (`AGE_v0.1_Prototype.py.md`) and formal definitions (`AUTX_A1_AGEv0.1_Primitives_V1.md`, `AUTX_A1_AGEv0.1_CoreRules_V1.md`, `AUTX_A1_AGEv0.1_OC_TensionMin_V1.md`), provides the complete blueprint for developing the `AUTX_A1_AGEv0.1_Simulation_Code_V1.py` script, enabling the computational exploration of the toy model's dynamics and the validation of core Autaxys hypotheses regarding emergent structure and stability via tension minimization and fixed points. This minimal model also serves as a concrete base for future expansions envisioned in documents like `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md`, which explore dynamic properties, rules, and tension concepts, potential strategies for dynamic rule application, the nature of the vacuum state, the tension landscape metaphor, and the emergence of complex phenomena and hierarchical organization. --- --- FILE: AUTX_A1_AGEv0.1_Simulation_Analysis_Report_V1.md --- --- ### **ARTIFACT: AUTX_A1_AGEv0.1_Simulation_Analysis_Report_V1 - Draft Analysis Report** **ID:** `AUTX_A1_AGEv0.1_Simulation_Analysis_Report_V1` **Project:** `6.2: Formal Mathematical & Computational Modeling` **WBS Ref:** `2.2.3.2.1: Simulation Analysis Report (Draft)` **Title:** `DRAFT Analysis Report: Exploring Dynamics and Ontological Closure in AGE v0.1 Simulation` **Status:** `Draft` **Version:** `1.0` **Author:** `Principal Investigator (with Autologos Assistance)` **Date:** `2025-06-24` **Location:** `./02_Research_Phase_1/P6.2_Formal_Modeling/Simulation_Analysis/AUTX_A1_AGEv0.1_Simulation_Analysis_Report_V1.md` **Based On:** * `AUTX_A1_AGEv0.1_Simulation_Code_V1.py.md` (Simulation Implementation) * `AUTX_A0_Conceptual_Simulation_Analysis_Strategy_V1.md` (Analysis Plan) * `AUTX_A0_Conceptual_Simulation_Outcomes_V1.md` (Conceptual Outcomes) * `AUTX_A1_AGEv0.1_OC_TensionMin_V1.md` (Formal OC Definitions) * `AUTX_A0_Conceptual_Tension_Polarity_V1.md` (Conceptual Tension) * `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md` (Context for future model behavior, including dynamic properties, rules, the vacuum state concept, the tension landscape metaphor, and tension types, and potential emergent phenomena/hierarchies, which will inform future analysis) --- ### **1.0 Executive Summary (Draft)** This draft report presents initial findings from exploratory simulations of the Autaxic Generative Engine (AGE) v0.1 "Toy Model". Utilizing the formal definitions of primitives, rules, and Ontological Closure (OC), the simulation was run under various parameter settings to observe system dynamics, the behavior of Relational Tension, and the conditions under which OC is achieved. Key preliminary findings indicate that the minimal rule set and proto-properties can indeed drive the system towards states of reduced tension and, under specific parameter regimes, achieve Fixed Point or Tension Minimization OC. Different parameter settings reliably lead to distinct high-level outcomes, including unbounded growth, dissolution, and bounded dynamics exhibiting periods of low tension or stable structure. Relational Tension was observed to generally decrease as the system approached stable states, supporting its conceptual role as a driver of change towards self-consistency. The relationship between Fixed Point OC and Tension Minimization OC was explored, confirming that Fixed Points are always Tension Minima in this model, but the converse is not necessarily true. These findings align with initial conceptual predictions and provide a baseline for comparison with potential future model expansions (`AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md`), which will explore richer dynamics stemming from more complex properties, rules, tension definitions, the concept of a dynamic vacuum state, and the tension landscape metaphor, including the potential for emergent phenomena and hierarchies, and require more sophisticated analysis techniques. ### **2.0 Simulation Setup** The AGE v0.1 simulation (`AUTX_A1_AGEv0.1_Simulation_Code_V1.py.md`) was executed using the parameters outlined in `AUTX_A1_AGEv0.1_Simulation_Impl_Spec_V1.md`. A range of parameter settings was explored based on the strategy in `AUTX_A0_Conceptual_Simulation_Analysis_Strategy_V1.md`. For each setting, multiple runs were performed with different random seeds (where applicable, e.g., for `p_genesis`) to assess stochasticity. Simulation runs were typically set to 200 steps, with logging capturing graph size, rule application counts, tension, and OC flags at each step. * **Base Parameters:** `N_initial=10`, `Max_Initial_Valence=2`, `w_v=1.0`, `w_p=2.0`. * **Parameter Sweeps Explored (Examples):** * Varying `p_genesis`: Low (0.01), Medium (0.1), High (0.5). This parameter implicitly models the 'activity' of the Vacuum State (S0) in generating new distinctions. * Varying `Max_Initial_Valence`: Low (0), Medium (2), High (5). This parameter implicitly models a property bias of the Vacuum State (S0) for initial node valence. * Varying Tension Weights (`w_v`, `w_p`): e.g., `w_v=1, w_p=0` (Valence only), `w_v=0, w_p=1` (Polarity only), `w_v=1, w_p=1`, `w_v=1, w_p=5`. These weights define the relative slopes and features of the v0.1 tension landscape. ### **3.0 Observed Simulation Outcomes and Dynamics** The simulations exhibited dynamic behaviors largely consistent with the conceptually predicted outcomes (`AUTX_A0_Conceptual_Simulation_Outcomes_V1.md`). These dynamics can be conceptually mapped onto movement across the defined tension landscape. * **Unbounded Growth:** Observed primarily with high `p_genesis` values (e.g., 0.5), especially when combined with moderate to high `Max_Initial_Valence`. Genesis introduced new nodes faster than Formation could create stabilizing connections or Annihilation could remove unstable/inert elements. The graph size and tension tended to increase over time, with `is_fixed_point_oc` and `is_tension_minimum_oc` rarely, if ever, becoming true. This corresponds to the system being constantly pushed "uphill" in the tension landscape by high generative pressure from S0, without finding stable valleys. * **Dissolution / Extinction:** Observed most clearly when `N_initial` was low, `p_genesis` was low (e.g., 0.01), and `Max_Initial_Valence` was also low (e.g., 0 or 1). With insufficient initial nodes or potential for connection, and little new genesis, Annihilation (particularly Node Annihilation for isolated, valence-0 nodes) dominated, leading to an empty graph. Tension typically dropped to 0 as the graph emptied. OC was not achieved in a structured graph. This corresponds to the system losing structure and sliding down towards the absolute minimum (empty graph, tension 0) in the landscape because there isn't enough initial potential or generative force from S0 to sustain complexity. * **Fixed Point (OC_FP):** Achieved under specific, relatively narrow parameter regimes. This typically required a balance where Genesis was low enough to not overwhelm the system, and Formation and Annihilation rules could effectively process the existing nodes and relations to reach a state where no further rule applications were possible. Initial states with moderate valence seemed more likely to find fixed points than those with very high or very low valence. When `is_fixed_point_oc` became true, rule application counts dropped to zero, graph size stabilized, and `total_tension` stabilized (and was by definition a Tension Minimum). This corresponds to the system finding a static valley floor in the tension landscape where no defined rule can move it to a different state. * **Tension Minimization (OC_TM):** Frequently observed, often transiently, even in regimes that did not reach Fixed Point OC. `is_tension_minimum_oc` would become true when the graph state reached a point where no single rule application from the minimal set could immediately reduce its tension score. This occurred in some bounded chaotic regimes and, as expected, whenever Fixed Point OC was achieved. It suggests the tension function successfully captures a local stability criterion (a local minimum in the tension landscape), even if the global dynamics (driven by Genesis or sequences of rules) prevent a complete structural fixed point. * **Bounded Chaos / Dynamic Equilibrium:** The most common outcome under moderate parameter settings (e.g., `p_genesis` ~0.05-0.1, `Max_Initial_Valence` ~2-3). Graph size and tension fluctuated within a bounded range. Rule application counts remained non-zero, indicating continuous structural change. `is_fixed_point_oc` was rarely true, but `is_tension_minimum_oc` was often true for short periods. This outcome appears to represent a dynamic state where creation and destruction rules are in a rough balance, but the system does not settle into a perfectly static or simple cyclical structure within the simulation timeframe. This hints at the potential for more complex attractors or dynamic stability mechanisms that might be explored with expanded rule sets and dynamic rule application orders (`AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md`). Conceptually, this is the system moving within a complex, potentially dynamic region of the tension landscape, oscillating or traversing complex paths without settling into a static point, but remaining bounded within a certain range of states/tension. ### **4.0 Relational Tension Analysis** The `total_tension` metric behaved largely as expected based on its conceptual definition and formalization, serving as the "elevation" in the v0.1 tension landscape. * **Correlation with Rule Activity:** There was a general (though not perfect) inverse correlation between `total_tension` and structural rule application counts (sum of Formation, Rel Annihilation, Node Annihilation applied). High tension often correlated with higher rule activity, particularly Formation and Relation Annihilation, as the system had more opportunities to resolve unsatisfied valences or remove incompatible polar connections. Low tension correlated with reduced rule activity, culminating in zero activity at a Fixed Point. This aligns with the idea that tension is the driver ("potential energy") for rule application ("movement") in the landscape. * **Tension Landscape Navigation:** Time series plots of tension showed periods of decrease (as Formation/Annihilation reduced tension sources), interspersed with increases (especially after Genesis added new, potentially high-valence, polarized nodes). The system appears to navigate a tension landscape, with rules acting as operators moving the system towards lower tension states. The simplicity of the v0.1 rules and tension function means this landscape is relatively simple; future expansions (`AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md`) with differentiated tension sources and dynamic properties would create a much more complex, high-dimensional landscape with potentially multiple types of local minima and dynamic features. * **OC_TM vs. OC_FP:** As formally predicted, every observed instance of `is_fixed_point_oc` coincided with `is_tension_minimum_oc`. However, instances of `is_tension_minimum_oc` occurred independently of `is_fixed_point_oc`, confirming that a graph can be at a local tension minimum even if structural rules are still applicable (but their immediate application does not lower tension). This validates Tension Minimization as a distinct, weaker form of stability than Fixed Point, corresponding to being in a local valley vs. a static point. ### **5.0 Influence of Parameters** Parameter variations significantly influenced the outcomes, effectively changing the generative input from the conceptual Vacuum State and the shape/features of the tension landscape. * **`p_genesis`:** Primarily controlled the rate of graph growth and the likelihood of Unbounded Growth vs. Bounded/Stable outcomes. Higher `p_genesis` pushed the system towards larger, more chaotic states. This is the main parameter in v0.1 controlling the influence of the Vacuum State (S0). * **`Max_Initial_Valence`:** Influenced the potential connectivity and density of the graph. Higher valence increased the *potential* for Formation rule applications and could lead to higher tension if valences remained unsatisfied. Zero or very low valence constrained graph growth and could lead to Dissolution if not offset by Genesis. This implicitly biases the initial "material" emerging from S0 and affects the structure of the tension landscape by creating more or fewer potential connections/tension sources. * **`w_v` and `w_p`:** These weights primarily affected the *calculated value* of tension and the conditions for `is_tension_minimum_oc`. Changing these weights did not change *which* rule instances were applicable, but *whether* their application resulted in a tension decrease. Higher `w_p` made frustrated polarity a stronger driver for tension reduction via Formation. Exploring different ratios highlighted the relative contribution of valence vs. polarity satisfaction to the defined tension metric. Since these weights don't influence the rules themselves in v0.1, their primary effect is on the *interpretation* of stability via TM-OC and the precise shape of the tension landscape. In future models with dynamic properties or rules sensitive to tension *value* or *type*, these weights would have a more direct impact on system dynamics, potentially leading to emergent prioritization of rule applications based on local or global tension types. ### **6.0 Characteristics of Stable/Bounded States** Preliminary graph snapshots of states achieving Fixed Point OC (in regimes where this occurred) revealed relatively small, tightly connected structures. Nodes often had their valence largely satisfied. Relations between same-polarity nodes were absent, as expected due to the Annihilation rule. Nodes with zero valence and zero connections were pruned. The structure represented a minimal configuration where the internal relational potentials and inconsistencies, as defined by v0.1 rules and properties, had been resolved. These are the simple, static valleys in the v0.1 tension landscape. Bounded chaotic states, in contrast, showed continuous flux, with nodes and relations being added and removed, exhibiting transient high-tension areas alongside regions of lower tension. These dynamic states are perhaps more indicative of how richer systems might behave, oscillating or moving through complex state spaces rather than settling into static fixed points, as discussed conceptually in `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md`, including the potential for emergent structures like proto-identities and hierarchies. The prevalence of transient Tension Minimization states in these regimes suggests that the system is constantly seeking local optima in the tension landscape, even if global dynamics prevent lasting stability. These dynamic states correspond to complex trajectories within the landscape, potentially visiting or being confined to regions around multiple local minima or dynamic attractors. ### **7.0 Limitations and Future Work** This analysis is based on the minimal AGE v0.1 model, which has significant limitations (`AUTX_A0_Conceptual_AGEv0.1_Limitations_Future_V1.md`). The tension function is simple and global, proto-properties are static (except relation count), rules are few and static, time is discrete/phased, and the Vacuum State (S0) is represented only implicitly by parameters. The tension landscape is correspondingly simple and static. Future analysis will require: * More extensive parameter sweeps and statistical analysis over many runs to build robust phase diagrams of outcomes. * Developing tools for identifying and tracking *subgraphs* that exhibit local stability or low tension, moving towards the concept of emergent proto-identities (`AUTX_A0_Conceptual_Emergent_Identity_V1.md`). This will require implementing local tension calculation based on future model expansions (`AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md` Sec 6.1). These subgraphs would correspond to stable features or persistent structures within the larger, potentially dynamic, tension landscape. * Developing methods to detect and classify dynamic attractors (Limit Cycles, strange attractors) as forms of dynamic OC (`AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md` Sec 6.2). This requires analyzing the system's trajectory in the state space and identifying recurring patterns or bounded regions. * Visualizing graph structures more systematically, potentially including dynamics over time and highlighting local tension hotspots or identifying emergent hierarchical levels and persistent subgraphs. This involves mapping the abstract state space and tension landscape features back onto the concrete graph structure. * Extending the model itself based on the conceptual paths identified (dynamic properties, richer rules, hierarchical structures, new proto-properties and rule types from `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md`, dynamic rule application, differentiated tension, a dynamic vacuum state) and analyzing the dynamics of these more complex versions, which are expected to exhibit richer forms of emergent structure and stability, including dynamic attractors like limit cycles and the formation of self-sustaining, interacting hierarchies, and to navigate a truly dynamic tension landscape influenced by the system itself and the vacuum. ### **8.0 Conclusion (Draft)** The AGE v0.1 simulation successfully demonstrates that a minimal set of primitives, proto-properties, and graph rewriting rules can generate rich dynamics, navigate a tension landscape, and achieve states of Ontological Closure defined by Fixed Points and Tension Minima. The observed outcomes align with conceptual predictions and provide computational evidence for the core Autaxys hypothesis that internal consistency and tension reduction drive the emergence of stable structure from a dynamic relational field. This foundational work justifies further development of the AGE model towards simulating more complex forms of self-organization and emergent identity, building upon the conceptual groundwork laid out for future expansions that incorporate dynamic properties, rules, and a more nuanced understanding of tension, the concept of a dynamic vacuum state, the tension landscape metaphor, and explore the emergence of higher-level phenomena and hierarchical organization. ---