CVBA Mathematical Formalization: A Systems-Level Intelligence Framework

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Copyright ©: Coherent Intelligence 2025 Authors: Coherent Intelligence Inc. Research Division
Date: June 5th 2025
Classification: Academic Research Paper
Framework: Universal Coherence Principle Applied Analysis | OM v2.0

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Abstract: This paper presents a mathematical formalization of the Contextual Value-Behavior-Action (CVBA) framework, designed to model systems-level intelligence. The CVBA model structures internal reasoning around Context, Values, Behaviors, and Actions, prioritizing coherence and emergent functionality. This formalization utilizes set theory, function notation, optimization principles, and outlines a roadmap for operationalization through specific function definitions, constraint integration, and sensitivity analysis.

1. Introduction

The pursuit of artificial general intelligence (AGI) necessitates frameworks capable of representing and reasoning about complex systems. The Contextual Value-Behavior-Action (CVBA) framework offers a structured approach to this challenge, emphasizing the interplay between a system’s environment (Context), guiding principles (Values), resulting patterns (Behaviors), and resulting interventions (Actions). This paper provides a rigorous mathematical formalization of the CVBA model, laying the groundwork for its computational implementation and analysis.

2. Core Formalization

2.1 Context (C): System/Domain

The Context represents the system or domain under consideration. It is formally defined as a set of variables and parameters:

• C = {x₁, x₂, ..., xₙ, p₁, p₂, ..., pₘ} where xᵢ are state variables describing the current state of the system, and pᵢ are parameters defining the system’s inherent properties.

2.2 Values (V): Core Principles

Values represent the fundamental principles guiding the system’s operation. They are defined as a vector of weighting functions applied to system states, representing desirability:

• V = [v₁(x), v₂(x), v₃(x)] where each vᵢ(x) maps system state x to a real number indicating the value associated with that state. Normalization ensures 0 ≤ vᵢ(x) ≤ 1, representing a value between undesirable and desirable.

2.3 Behaviors (B): Emergent Properties

Behaviors are emergent properties resulting from the interaction between Context and Values. They are modeled as functions derived from this interplay:

• B = f(C, V) Specifically:
  • _Bᵢ = Σ (wᵢⱼ _ vⱼ(x))* for j = 1 to 3 and i = 1 to k where k is the number of identified behaviors. wᵢⱼ represents the weighting of value vⱼ on behavior *Bᵢ*. This creates a behavioral profile, quantifying the influence of each value on each observed behavior.

2.4 Actions (A): Control Inputs

Actions represent the control inputs applied to the system, optimized to maximize desired behavioral outcomes:

• A = g(B) Formally:
  • _A = argmax_a [Σ (αᵢ _ Bᵢ(a))]* where αᵢ are weighting coefficients representing the prioritization of each behavior Bᵢ, and *a* represents the action space – the set of all possible actions the system can take. This formulation casts action selection as an optimization problem.

2.5 Overall CVBA Function:

The complete CVBA process can be represented as a composition of functions:

• A = g(f(C, V)) – Actions are a function of Behaviors, which are a function of Context and Values.

3. Operationalization & Development Roadmap

The core formalization provides a foundational structure. To move towards a truly operational model, further development is required, specifically in defining the functions f and g, incorporating constraints, and conducting sensitivity analysis.

3.1 Function 'f' Specification (C & V → B)

The function f mapping Context and Values to Behaviors requires concrete implementation. Several approaches are viable:

• Neural Network: f(C, V) = Ψ(C, V, Θ) where Ψ is a feedforward neural network, Θ represents learnable parameters, and the network is trained to predict behavioral outputs based on context and value inputs. This allows for complex, non-linear relationships.
• Fuzzy Logic System: f(C, V) can utilize fuzzy sets and rules to map context and values to behavioral outputs, enabling representation of uncertainty and vagueness.
• Differential Equations: For dynamic systems, f(C, V) could be a set of differential equations where values influence the rate of change of system states, leading to emergent behaviors.

3.2 Function 'g' Specification & Constraint Integration (B → A)

The function g mapping Behaviors to Actions requires an optimization strategy, incorporating realistic constraints:

• _g(B) = argmax_{a ∈ A_feasible} [Σ (αᵢ _ Bᵢ(a))]* where *A_feasible* is the set of feasible actions, defined by constraints.
• Constraint Incorporation: Constraints are formalized as:
  • hᵢ(a) ≤ 0 (inequality constraints)
  • kⱼ(a) = 0 (equality constraints)
• Optimization Techniques: The argmax problem can be solved using:
  • Lagrange Multipliers: For equality constraints.
  • Sequential Quadratic Programming (SQP): For constrained non-linear optimization.
  • Convex Optimization: If the objective function and constraints are convex, efficient global optima can be found.

3.3 Sensitivity Analysis

Understanding the robustness of the model to changes in weighting coefficients is crucial. The following methods can be employed:

• Global Sensitivity Analysis (GSA): Utilizing Sobol indices to quantify the contribution of each weighting coefficient (wᵢⱼ, αᵢ) to the variance in the resulting actions A.
• Monte Carlo Simulation: Randomly sampling weighting coefficients from defined distributions and observing the resulting variations in A.
• Local Sensitivity Analysis: Perturbing each weighting coefficient individually and observing the change in A.

4. Discussion & Conclusion

This paper has presented a mathematical formalization of the CVBA framework, providing a rigorous foundation for modeling systems-level intelligence. The formalization utilizes established mathematical tools and outlines a clear path towards operationalization. The proposed roadmap, encompassing function specification, constraint integration, and sensitivity analysis, will enable the development of computationally implementable and analytically robust CVBA-based systems. Future work will focus on implementing these components and validating the framework through application to real-world problems.